Proceedings of the International Group for the Psychology of Mathematics Education 2790-3648 49 2026 07 49th Conference of the International Group for the Psychology of Mathematics Education Helsinki, Finland

3 10 Shira Abad Michal Ayalon Nadav Marco University of Haifa Israel The David Yellin Academic College of Education Israel This study explores how the level of detail in written critical events shapes mathematics teachers’ noticing. Using the MOST framework and gap-filling theory, the study compares teachers’ interpretations of the same events across condensed and expanded versions. Seven experienced secondary mathematics teachers analyzed three events through written reports and semi-structured interviews. The analysis focused on teachers’ expressions of certainty and the range of interpretations generated in response to each version. Condensed versions elicited more tentative statements and greater interpretive diversity than expanded versions, suggesting that reduced detail promotes gap filling, interpretative flexibility, and openness in teachers’ noticing. Chen, C., & Herbst, P. (2012). Teachers’ conceptions of instructional explanations: A comparison of videos and comics. International Journal of Science and Mathematics Education, 10(3), 673–699. https://doi.org/10.1007/s10763-011-9296-8 Herbst, P., & Kosko, K. W. (2013). Using representations of practice to elicit mathematics teachers’ tacit knowledge of practice: A comparison of responses to animations and videos. Journal of Mathematics Teacher Education, 17(6), 515–537. https://doi.org/10.1007/s10857-013-9267-y Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Marco, N., Palatnik, A., & Schwarz, B. B. (2021). Mind the gaps: Gap-filling in proving activities. For the Learning of Mathematics, 41(2), 21–25. Marco, N., Palatnik, A., & Schwarz, B. B. (2022). When is less more? Investigating gapfilling in proofs without words activities. Educational Studies in Mathematics, 110(2), 297–315. https://doi.org/10.1007/s10649-022-10159-z Perry, M., & Sternberg, M. (1986). The king through ironic eyes: Biblical narrative and the literary reading process. Poetics Today, 7(2), 275–322. https://doi.org/10.2307/1772124 Rotem, S. H., & Ayalon, M. (2022). Using critical events during a field-based teaching preparation program: Characterizing changes in professional noticing skills among preservice mathematics teachers. Journal of Mathematics Teacher Education, 26(3), 281– 309. Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2019). Relationships among prospective secondary mathematics teachers’ skills of attending, interpreting and responding to students’ understanding. Educational Studies in Mathematics, 100(4), 83– 99. https://doi.org/10.1007/s10649-018-9855-y Stockero, S. L., Leatham, K. R., Ochieng, M. A., Van Zoest, L. R., & Peterson, B. E. (2020). Teachers’ orientations toward using student mathematical thinking as a resource during whole-class discussion. Journal of Mathematics Teacher Education, 23, 237–267.

11 18 Jill Adler Shemunyenge Hamukwaya Anthony Essien University of the Witwatersrand South Africa From 2025, South Africa introduced mother tongue-based bilingual education to replace the early exit language transition. The study reported in this paper uses a case of a teacher in a Grade 4 classroom to illustrate the complexity of implementing mother tongue-based bilingual education (MTBBE) as envisioned by the curriculum. Using a translanguaging framework, merged with language responsive mathematics teaching, this qualitative study provides empirical evidence of the complexity of simultaneously working across languages, representations and registers where MTBBE is mandated. Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht: Kluwer Academic Publishers. Essien, A. A. (2025). Harnessing the Epistemic Potential of Multiple Languages in Transitional Multilingual Mathematics Classrooms in South Africa. Norrag Special Issue 11, 80-84. Essien, A. A., Sapire, I., & Moleko, M. M. (2024). The concept of language-asresource/sources of meaning from an African perspective: Challenges and opportunities in mathematics education. In A. Essien (Ed.), Multilingualism in mathematics education in Africa (pp. 1–20). Bloomsbury Publisher. https://doi.org/10.5040/9781350369238.ch-001. García, O., & Li, W. (2014). Translanguaging: Language, bilingualism and education. Palgrave Macmillan. García, O., Johnson, S., & Seltzer, K. (2017). The translanguaging classroom: Leveraging student bilingualism for learning. Philadelphia, PA: Caslon. Li, W. (2018). Translanguaging as a practical theory of language. Applied Linguistics, 39(1), 9–30. https://doi.org/10.1093/applin/amx039 Planas, N., & Setati-Phakeng, M. (2014). On the process of gaining language as a resource in mathematics education. ZDM–Mathematics Education, 46, 883–893. Prediger, S. (2019). Investigating and promoting teachers’ expertise for language responsive mathematics teaching. Mathematics Education Research Journal, 31, 367–392. Probyn, M. (2009). ‘Smuggling the vernacular into the classroom’: conflicts and tensions in classroom codeswitching in township/rural schools in South Africa. International Journal of Bilingual Education and Bilingualism, 12(2), 123-136. Trudell, B. (2023), Early-exit language transitioning programming: The rationale, the benefits and the limitations. Paper commissioned for International Mother Language Day 2023. © UNESCO 2023 Zacharos, K. (2006). Prevailing educational practices for area and perimeter. Journal of Mathematical Behavior, 25(3), 224–239. https://doi.org/10.1016/j.jmathb.2006.09.003

19 26 Veysel Akcakin Julia L. Hill Hanife Unlutuk Akcakin Usak University Turkey RMIT University Australia Science and Art Center Turkey Despite growing research on mathematical wellbeing (MWB), no studies have examined MWB among gifted students. We report on 261 gifted primary students' (Mage = 9.13) MWB and mathematical thinking styles during mathematics learning moments. Latent class analysis identified three thinking styles: integrated (n = 118), visual (n= 94), and analytic (n = 49). ANOVA revealed significant effects for thinking styles (η² = .162), with integrated thinkers generally reporting higher MWB than visual thinkers. No gender or grade-level differences emerged. Findings suggest flexible thinking supports gifted students' MWB, while visual thinkers experience lower MWB when instruction emphasises symbolic approaches, highlighting the need for differentiated instruction accommodating diverse thinking styles. Akcakin, V., & Kaya, G. (2020). Determining high school students’ mathematical thinking styles: Latent class analysis. Education and Science, 45(201), 39-54. Akcakin, Hill, Unluturk-Akcakin Borromeo Ferri, R. (2015). Mathematical thinking styles in school and across cultures. In S. J. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 153-174). Springer. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum. Grootenboer, P., & Marshman, M. (2015). Mathematics, affect and learning: Middle school students' beliefs and attitudes about mathematics education. Springer. Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. (2022). A primer on partial least squares structural equation modeling (PLS-SEM) (3rd ed.). Sage. Hill, J. L., & Hunter, J. (2024). Diverse students' mathematical wellbeing. New Zealand Journal of Educational Studies, 59(1), 211-228. Hill, J. L., Kern, M. L., Seah, W. T., & van Driel, J. (2021). Feeling good and functioning well in mathematics education: Exploring students' conceptions of mathematical wellbeing and values. Journal for Research in Mathematics Education, 52(4), 463-484. Hill, J.L., Kern, M.L., Seah, W.T., van Driel, J. (2024). To what extent are students fulfilling their values and thriving in mathematics education?—The case for Victoria, Australia. In Y. Dede, G. Marschall, & P. Clarkson. (Eds.) Values and valuing in mathematics education. (pp. 259–283). Springer. Hill, J. L. & Seah, W.T. (2023). Student values and wellbeing in mathematics education: perspectives of Chinese primary students. ZDM Mathematics Education, 55(2), 385-398 Jöreskog, K. G., Olsson, U. H., & Wallentin, F. Y. (2016). Multivariate analysis with LISREL. Springer. Leikin, R. (2021). When practice needs more research: The nature and nurture of mathematical giftedness. ZDM Mathematics Education, 53(7), 1579-1589. Little, C. A. (2012). Curriculum as motivation for gifted students. Psychology in the Schools, 49(7), 695-705. Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 205-235). Sense Publishers. Smedsrud, J. H., Nordahl-Hansen, A., & Idsøe, E. (2022). Mathematically gifted students' experience with their teachers' mathematical competence and boredom in school: A qualitative interview study. Frontiers in Psychology, 13, 876350.

27 34 Hatice Akkoc Marmara University Turkey This study examines how pre-service mathematics teachers (PMTs) integrate Scratch into their micro-teaching lessons based on the levels of computational thinking (CT) integration. The data of a case study with three PMTs include their micro-teaching lesson videos, lesson plans, and teaching notes. The data analysis is based on the SAMR-based CT integration matrix with an emphasis on a disciplinary approach. Findings showed that participants integrated CT into their lessons at the substitution level only, but also demonstrated some characteristics of the augmentation level. They prioritized Scratch use over fostering CT and approached mathematics and CT as separate domains. The paper discusses the theoretical implications for operationalizing the SAMR-based CT integration. Blundell, C. N., Mukherjee, M., & Nykvist, S. (2022). A scoping review of the application of the SAMR model in research. Computers and Education Open, 3, 100093. Brennan, K., & Resnick, M. (2012, April). New frameworks for studying and assessing the development of computational thinking. In Proceedings of the 2012 Annual Meeting of the American Educational Research Association (pp. 1–25). Vancouver, Canada. Cabrera, L., Ketelhut, D. J., Mills, K., Killen, H., Coenraad, M., Byrne, V. L., & Plane, J. D. (2023). Designing a framework for teachers’ integration of computational thinking into elementary science. Journal of Research in Science Teaching, 61(6), 1326–1361. Grover, S., & Pea, R. (2013). Computational thinking in K–12: A review of the state of the field. Educational Researcher, 42(1), 38–43. Hamilton, E. R., Rosenberg, J. M., & Akcaoglu, M. (2016). The Substitution Augmentation Modification Redefinition (SAMR) model: A critical review and suggestions for its use. TechTrends, 60(5), 433–441. Kite, V., & Park, S. (2023). What’s computational thinking?: Secondary science teachers’ conceptualizations of computational thinking (CT) and perceived barriers to CT integration. Journal of Science Teacher Education, 34(4), 391–414. National Research Council. (2011). Report of a workshop on the pedagogical aspects of computational thinking. The National Academies Press. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books. Puentedura, R. R. (2006). Transformation, technology, and education. Hippasus. Rich, K. M., Yadav, A., & Fessler, C. J. (2024). Computational thinking practices as tools for creating high cognitive demand mathematics instruction. Journal of Mathematics Teacher Education, 27(2), 235–255. Waterman, K. P., Goldsmith, L., & Pasquale, M. (2019). Integrating computational thinking into elementary science curriculum. J. of Science Education and Technology, 29, 53–64. Weintrop, D., Beheshti, E., Horn, M., Orton, K., Jona, K., Trouille, L., & Wilensky, U. (2016). Defining computational thinking for mathematics and science classrooms. Journal of Science Education and Technology, 25, 127–147. Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35. Yeni, S., Grgurina, N., Saeli, M., Hermans, F., Tolboom, J., & Barendsen, E. (2024). Interdisciplinary integration of computational thinking in K–12 education: A systematic review. Informatics in Education, 23(1), 223–278. Yin, R. K. (2014). Case study research: Design and methods (5th ed.). SAGE. Yun, M., & Crippen, K. J. (2024). Computational Thinking Integration into Pre-Service Science Teacher Education: A Systematic Review. Journal of Science Teacher Education, 36(2), 225–254. Zhang, L., & Nouri, J. (2019). A systematic review of learning computational thinking through Scratch in K–9. Computers & Education, 141, 103607.

35 42 Ismael Almahdi Osama Swidan The Ben-Gurion University of the Negev, Beer-Sheva Israel This paper proposes conceptualising the mathematics e-textbook as a platform for instrumental orchestration. Building on the instrumental approach, we treat the e-textbook as the locus where didactical configurations and intended exploitation are constructed through design and enacted in classroom performance. We illustrate that through a case study of a secondary teacher who designed and taught an e-textbook chapter on parallelograms. Data include observations of the design process, an interview, and a lesson recording. The analysis shows how the chapter structured a progression from a video-based introduction to GeoGebra-supported verification. The paper contributes a conceptual extension of instrumental orchestration that links e-textbook design and classroom enactment within a single analytical framework. Clinton-Lisell, V. et al. (2023) Interactive features of e-texts’ effects on learning: A systematic review and meta-analysis. Interactive Learning Environments, 31(6), 3728– 3743. https://doi.org/10.1080/10494820.2021.1943453 Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75(2), 213–234. https://doi.org/10.1007/s10649-0109254-5 Greefrath, G; Li, S; Radmehr, F; Swidan, O; & Winslow, C. (2025). Theorizing digital curriculum resources: Recent cases from research in mathematics education. In B. Pepin, I. Kohanova, & M. B. Langfeldt (Eds.), Proceedings of the Fifth International Conference on Mathematics Textbook Research and Development (pp. 52–71). Norwegian University of Science and Technology. Pepin, B., Gueudet, G., Yerushalmy, M., Trouche, L., & Chazan, D. I. (2016). E-textbooks in/for teaching and learning mathematics. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education. (3rd ed., pp. 636– 661). Routledge. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307. https://doi.org/10.1007/s10758-004-3468-5 Wijaya, T. T., Cao, Y., Weinhandl, R., & Tamur, M. (2022). A meta-analysis of the effects of E-books on students' mathematics achievement. Heliyon, 8(6), 09432. https://doi.org/10.1016/j.heliyon.2022.e09432

43 50 Gabriel Amancio Rúbia Amaral Beatriz Litoldo Danyal Farsani UNESP - São Paulo State University Brazil Contextualization is often linked to deeper cognitive engagement in mathematics, but evidence is scarce. Against this background, this study addresses the research question: How do different contextual references in geometry tasks relate to the levels of cognitive demand mobilized by these tasks in mathematics textbooks from Brazil and Norway? Using a qualitative Theorem System framework, tasks were classified by context and demand and compared. Findings reveal the dominance of Purely Mathematical contexts and low-demand tasks in both countries. Semi-reality contexts show relatively higher demand but are not predominant. Overall patterns remain consistent across the two collections. The study points out that cognitive demand is more associated with task planning intentions than with the nature of the context itself. Alves, G. A., Amaral, R. B., & Litoldo, B. F. (2024). Teorizando contextos: Fundamentos para a análise de livros didáticos de matemática do Novo Ensino Médio. In Anais do 2º Colóquio de Livros Didáticos de Matemática. UFMS. Amaral, R. B., Mazzi, L. C., Vieira, L., & Perovano, A. P. (2023). Livro didático de matemática: Compreensões e reflexões no âmbito da educação matemática. Mercado de Letras. Dekker, T., & Querelle, N. (2002). Chapter 6 – Context. In Great assessment problems. Freudenthal Institute. Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: Development status and directions. ZDM, 45(5), 633-646. Jackson, K., Garrison, A., Wilson, J., Gibbons, L. & Shahan, E. (2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class discussions in middle-grades mathematics instruction. Journal for Research in Mathematics Education, 44(4), 646-682. Lana, M. A., & Carrião, A. (2015). A contextualização das atividades no livro didático de matemática do ensino médio. In Anais do VII Encontro Mineiro de Educação Matemática. Litoldo, B. F. (2021). A contextualização e os níveis de demanda cognitiva de tarefas de geometria presentes em livros didáticos de matemática sob a perspectiva do opportunityto-learn (Tese de Doutorado). UNICAMP. Özgeldi, M.; Esen, Y. Analysis of mathematical tasks in Turkish elementary school mathematics textbooks. Procedia-Social and Behavioral Sciences, v. 2(2), n. Elsevier Ltd, p. 2277-2281, 2010. Sierra, C. D. (2025). High cognitive demand didactic experience for teaching inferential statistics. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proceedings of the 48th PME: General Contributions. Skovsmose, O. (2001). Landscapes of Investigation. ZDM 33, 123-132. Stein, M. K., Henningsen, M. A.; Smith, M. S. & Silver, E. A. (2000). Implementing standards-based mathematics instruction: a Casebook for Professional Development. New York: Teachers College Press. Teixeira, L. A. (2020a). Diálogos matemática e suas tecnologias: Geometria plana. Moderna. Teixeira, L. A. (2020b). Diálogos matemática e suas tecnologias: Geometria espacial. Moderna. Van den Heuvel-Panhuizen, M. (2005). The role of contexts in assessment problems in mathematics. For the Learning of Mathematics, 25(2), 2-9. Wallace, A. K., & Kongsnes, A. L. (2020a). Matemagisk 8. Aschehoug Undervisning. Wallace, A. K., & Kongsnes, A. L. (2020b). Matemagisk 9. Aschehoug Undervisning. Wallace, A. K., & Kongsnes, A. L. (2020c). Matemagisk 10. Aschehoug Undervisning.

51 58 Chiara Andrà Domenico Brunetto Umberto Dello Iacono Università del Piemonte Orientale Italy Politecnico di Milano Italy Università della Campania “L. Vanvitelli” In a research conducted within the NATION National project “Scientific degrees”, we attempt to deal with the challenge of transforming the preparation for a university entrance test into an opportunity to learn and reflect on mathematics learning, rather than obsessing students with performance. More specifically, in a series of five lessons, the students engage in tasks by taking on roles that are the cognitive functions that come into play when tackling a math problem. The students work in groups and the roles/cognitive functions are gradually internalized through reflection questions asked at the end of tasks. The results show that the students still focus on performance, but a number of them shift their attention to other dimensions, even considering how they constructed the answer together with their mates. Albano, G., Coppola, C., & Dello Iacono, U. (2021). What does ‘Inside Out’ mean in problem solving?. For the learning of mathematics, 41(2), 32-36. De Lange, J. (2007). Large-scale assessment and mathematics education. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1111– 1142). Charlotte, NC: Information Age Publishing. Klieme, E., Avenarius, H., Blum, W., Döbrich, P., Gruber, H., Prenzel, M., et al. (2004). The development of national educational standards: An expertise. Berlin, Germany: Federal Ministry of Education and Research. Kollar, I., Fischer, F. & Hesse, F. W. (2006). Collaboration scripts – a conceptual analysis. Educational Psychology Review, 18(2), 159-185. Kopparlo, M. (2024). Role of mathematics in retention of undergraduate STEM majors: A meta analysis. Journal of Mathematics Education, 12(1). Newton, P. E. (2007). Clarifying the purposes of educational assessment. Assessment in Education: Principles, Policy and Practice, 14(2), 149–170. Olsher, S., Chazan, D., Drijvers, P., Sangwin, C. J. & Yerushalmy, M. (2023). Digital assessment and the machine. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), Handbook of digital resources in mathematics education. Springer. https://doi.org/10.1007/978-3-03095060-6_44-1 Suurtamm, C., Thompson, D. R., Kim, R. Y., Moreno, L. D., Sayac, N., Schukajlow, S., ... & Vos, P. (2016). Assessment in mathematics education: Large-scale assessment and classroom assessment. Springer Nature. Weigand, H. G., Trgalova, J., & Tabach, M. (2024). Mathematics teaching, learning, and assessment in the digital age. ZDM–Mathematics Education, 56(4), 525-541. Weinberger, A., & Fischer, F. (2006). A framework to analyze argumentative knowledge construction in computer-supported collaborative learning. Computers & education, 46(1), 71-95.

59 66 Fannie Skoland Andreassen Guri A. Nortvedt Alexander Jonas Viktor Selling University of Oslo Norway Researchers have found that desirable difficulties contribute to student learning. Therefore, we carried out a study that examines impasses in a teaching approach that mathematics teachers increasingly implement, thinking classrooms. In this study, we use video data from an upper secondary classroom to examine the impasses a group of three students encounter when solving a problem-solving task on vertical whiteboards and how they overcome the impasses. Impasses and navigation out of them occurred in various ways. We also found that the students depended on external assistance to overcome most of the impasses, drawing help from neighbouring groups and from the teacher. When the students navigated out of impasses by collaboration, the study illustrates that students established a joint problem space (JPS). Blikstad-Balas, M. (2017). Key challenges of using video when investigating social practices in education: Contextualization, magnification, and representation. International Journal of Research and Method in Education, 40(5), 511–523. https://doiorg.ezproxy.uio.no/10.1080/1743727X.2016.1181162 Derry, S. J., Pea, R. D., Barron, B., Engle, R. A., Erickson, F., Goldman, R., Hall, R., Koschmann, T., Lemke, J. L., Sherin, M. G. & Sherin, B. L. (2010). Conducting Video Research in the Learning Sciences: Guidance on Selection, Analysis, Technology, and Ethics. The Journal of the Learning Sciences, 19(1), 3–53. https://doi.org/10.1080/10508400903452884 Jonsson, B., Norqvist, M., Liljekvist, Y. & Lithner, J. (2014). Learning mathematics through algorithmic and creative reasoning. The Journal of Mathematical Behavior, 36, 20–32. https://doi.org/10.1016/j.jmathb.2014.08.003 Liljedahl, P. (2016). Building Thinking Classrooms: Conditions for Problem-Solving. In P. Felmer, E. Pehkonen & J. Kilpatrick (Ed.), Posing and Solving Mathematical Problems: Advances and New Perspectives (pp. 361–386). Springer. https://doi.org/10.1007/978-3319-28023-3_21 Ohlsson, S. (2011). Deep Learning: How the mind overrides experience. Cambridge University Press. https://doi.org/10.1017/CBO9780511780295 Pólya, G. (1945). How to solve it. Princeton University Press. Roschelle, J., & Teasley, S. (1994). The construction of shared knowledge in collaborative problem solving. Computer Systems Sciences, (128), 69–97. https://doi.org/10.1007/978-3-642-85098-1_5 Rott, B., Specht, B. & Knipping, C. (2021). A descriptive phase model of problem-solving processes. ZDM - Mathematics Education, 53, 737–752. https://doi.org/10.1007/s11858021-01244-3 Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press. Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (1st ed., pp. 189–215). Lawrence Erlbaum Associates. https://doi.org/10.4324/9780203062685-8 VanLehn, K., Siler, S., Murray, C., Yamauchi, T., & Baggett, W. (2003). Why do only some events cause learning during human tutoring? Cognition & Instruction, 21(3), 209–249 Warshauer, H. K. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18(4), 375–400. https://doi.org/10.1007/s10857-014-9286-3

67 74 Ratih Ayu Apsari Dor Abrahamson University of California, Berkeley United States A group of four 5th-grade participants in a design-based research study that evaluated body-scale geometry activities (n=76) formed a circle in response to a dance task prompt. Drawing on coordination dynamics and ethnomethodological conversation analysis, we examine microgenetically how, without a shared plan, students self-organized via micro-adjustments. The analysis exposes the group’s heteroperception (i.e. multiple divergent perspectives) evidenced as conflicting attentional anchors (i.e. perceptual structures for action). Initially implicit, heteroperception became explicit through discussion and resolved as perspectival synergy that promoted geometry discourse. Visualizing heteroperception enables researchers to implicate the emergent coalescence of collaborative enactment. Abrahamson, D., Lomos, C., & Palatnik, A. (2025). The icosahedron in the room: Revisiting pedagogical dilemmas of individual learning in collaborative activities. In M. Bosch & S. Carreira (Eds.), Proceedings of CERME 14. ERME. Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The future of embodied design for mathematics teaching and learning. Frontiers in Education, 5, 147. Abrahamson, D., & Shulman, A. (2019). Co-constructing movement in mathematics and dance: An interdisciplinary pedagogical dialogue on subjectivity and awareness. Feldenkrais Research Journal, 6, 1–24. Alberto, R., Shvarts, A., Drijvers, P., & Bakker, A. (2022). Action-based embodied design for mathematics learning: A decade of variations on a theme. International Journal of Child-Computer Interaction, 32, 100419. Apsari, R. A., & Abrahamson, D. (2024). Dancing geometry: Imagining auxiliary lines by reflecting on physical movement. International Journal of Mathematical Education in Science and Technology. Apsari, R. A., & Abrahamson, D. (2025). Culturally situated embodied design: A cognitiveanthropological investigation of Balinese spatial practice toward grounding geometry. In R. M. Zbiek, X. Yao, A. McCloskey, & F. Arbaugh (Eds.), Proceedings of PME-NA 47 (pp. 1473–1482). PSU. Benally, J., Palatnik, A., Ryokai, K., & Abrahamson, D. (2022). Learning through negotiating conceptually generative perspectival complementarities: The case of geometry. For the Learning of Mathematics, 42(3), 34–41. Chorney, S. (2017). Circles, materiality and movement. For the Learning of Mathematics, 37(3), 45–48. Clark, J. O., & Ando, T. (2014). Geometry, embodied cognition and choreographic praxis. International Journal of Performance Arts and Digital Media, 10(2), 179–192. Herbst, P., Gonzales, G., & Macke, M. (2005). How can geometry students understand what it means to define in mathematics? The Mathematics Educator, 15(2), 17–24. Hutto, D. D., & Sánchez-García, R. (2015). Choking RECtified: Embodied expertise beyond Dreyfus. Phenomenology and the Cognitive Sciences, 14, 309–331. Kirsh, D. (2010). Thinking with external representations. AI & Society, 25(4), 441–454. Lajos, J., Soto, H., & De Jesus Pagan, F. (2025). Imagination as an embodied space for engaging with abstract algebra concepts. The Journal of Mathematical Behavior, 80, 101275. Lockhart, P. (2009). A mathematician’s lament. Bellevue Literary Press. Mondada, L. (2019). Contemporary issues in conversation analysis: Embodiment and materiality, multimodality and multisensoriality in social interaction. Journal of Pragmatics, 145, 47–62. Palatnik, A., & Abrahamson, D. (in press). Building knowledge: Studying geometry through collaborative construction of body-scale structures. Journal of the Learning Sciences. Radford, L. (2021). The theory of objectification: A Vygotskian perspective on knowing and becoming in mathematics teaching and learning. Brill Sense. Shvarts, A., & Abrahamson, D. (2023). Coordination dynamics of semiotic mediation: A functional dynamic systems perspective on mathematics teaching/learning. Constructivist Foundations, 18(2), 220–234. Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn & R. Siegler (Eds.), Handbook of child psychology. Wiley. Sinclair, N., Bartolini Bussi, M. G., De Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: An ICME-13 survey team report. ZDM, 48(5), 691–719. Smith, C., & Walkington, C. (2019). Four principles for designing embodied mathematics activities. Australian Mathematics Education Journal, 1(4), 16–20. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Erlbaum. Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. ZDM, 51(2), 331–346. Wilson, M. (2002). Six views of embodied cognition. Psychonomic Bulletin & Review, 9(4), 625–636.

75 82 Miglena Asenova Agnese Del Zozzo Marzia Garzetti University of Milan Italy University of Trento SUPSI Locarno This theoretical and methodological paper introduces the construct of researcher’s semiotic noticing, defined as the operational use of Semiotic Interpretative Knowledge to analyse students’ written productions. Semiotic noticing is grounded in Duval’s semio-cognitive theory and focuses on three analytical functions: the identification of distinctive features, treatments, and conversions. Drawing on Rabardel’s theory of instrumental genesis, these functions are transformed from a theoretical artefact into an analytical tool through the abstraction of a scheme of use that emerges from the detailed analysis of three students’ written solutions. Asenova, M., Del Zozzo, A., & Santi, G. (2023). Unfolding Teachers’ Interpretative Knowledge into Semiotic Interpretative Knowledge to Understand and Improve Mathematical Learning in an Inclusive Perspective. Education Sciences, 13(1), 65. Asenova, M., Del Zozzo, A., & Garzetti, M. (2025). Investigating the effectiveness of a training model for developing Semiotic Interpretative Knowledge in mathematics education. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.). Proceedings of the 48th Conference of PME (Vol. 1, pp. 67–74). PME. D’Amore, B. (2003). La complexité de la noétique en mathématiques ou les raisons de la dévolution manquée [The Complexity of Noetics in Mathematics, or the Reasons for a Failed Devolution]. For the Learning of Mathematics, 23(1), 47–51. Duval, R. (1993). Registres de représentations sémiotique et fonctionnement cognitif de la pensée [Registers of semiotic representations and cognitive functioning of thought]. Annales de Didactique et de Sciences Cognitives, 5(1), 37– 65. Duval, R. (2017). Understanding the Mathematical Way of Thinking: The Registers of Semiotic Representations. Springer. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge Falmer. Mason, J. (2021). Learning about noticing, by, and through, noticing. ZDM Mathematics Education 53(1), 231–243. https://doi.org/10.1007/s11858-020-01192-4 Presmeg, N., Radford, L., Roth, W. M., & Kadunz, G. (2016). Semiotics in Mathematics Education. ICME-13 Topical Surveys. Springer Open. Rabardel, P., & Béguin, P. (2005). Instrument mediated activity: from subject development to anthropocentric design. Theoretical Issues in Ergonomics Science, 6(5), 429–461. Ribeiro, M., Mellone, M., & Jakobsen, A. (2016). Interpreting students’ nonstandard reasoning: insights for mathematics teacher education. For The learning of mathematics, 36(2), 8–13.

83 90 Parviz Asghari Maike Schindler Franziska Schulte Achim J. Lilienthal Technical University of Munich Germany University of Cologne Germany Örebro University Eye tracking (ET) is a valuable tool for identifying students’ learning processes, also in number line activities. While prior research has shown that AI can successfully identify student number line processes, the approaches operate as a “black box” and require extensive data collection and manual labelling. Vision-Language Models (VLMs) offer a promising, generally applicable, alternative. They can analyze ET data without prior labelling by leveraging domain knowledge. The aim of this study is to investigate how effectively VLMs can identify student number line processes. Based on an ET study with 136 fifth graders, we found that our VLM framework VETA (VLMs for ET Analysis) achieved very high agreement with the researchers’ manual coding (Cohen’s κ ≥ 0.80 – 1.0), demonstrating reliability and potential for ET analyses. Andrá, C., Lindström, P., Arzarello, F., Holmqvist, K., Robutti, O., & Sabena, C. (2015). Reading mathematics representations: An eye tracking study. International Journal of Science and Mathematics Education, 13, 237–259. https://doi.org/10.1007/s10763-0139484-y Boels, L., Moreno-Esteva, E.G., Bakker, A., & Drijvers, P. (2024). Automated gaze-based identification of students’ strategies in histogram tasks through an interpretable mathematical model and a machine learning algorithm. International Journal of Artificial Intelligence in Education, 34, 931–973. https://doi.org/10.1007/s40593-023-00368-9 Bordes, F., Pang, R. Y., Ajay, A., Li, A. C., Bardes, A., Petryk, S., ... & Chandra, V. (2024). An introduction to vision-language modeling. arXiv preprint arXiv:2405.17247. Duchowski, A. (2017). Eye tracking methodology: Theory and practice. Springer. https://doi.org/10.1007/978-3-319-57883-5 Hao, S., Pan, H., & Zhang, D. A. (2025). Process-oriented approach to assessing high school students’ mathematical problem-solving competence: Insights from multidimensional eyetracking analysis. Education Sciences, 15(6), 761. https://doi.org/10.3390/educsci15060761 Kennel, K., Ishimaru, S., Küchemann, S., Steinert, S., Kuhn, J., & Ruzika, S. (2025). GazeBased Prediction of Students’ Math Difficulties-A Time Dynamic Machine Learning Approach to Enable Early Individual Assistance. International Journal of Artificial Intelligence in Education, 1-25. https://doi.org/10.1007/s40593-024-00447-5 Klein, C., & Ettinger, U. (2019). Eye movement research: An introduction to its scientific foundations and applications. Springer. https://doi.org/10.1007/978-3-030-20085-5 Murphy, K. P. (2012). Machine learning: A probabilistic perspective. MIT press. Simon, A. L., Asghari, P., Lilienthal, A. J., & Schindler, M. (2023). Strategy use in number line tasks of students with and without mathematical difficulties: A study using eye tracking and AI. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel., & M. Tabach (Eds.), Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 211-218). PME. Schindler, M., Schaffernicht, E., & Lilienthal, A. J. (2019). Differences in Quantity Recognition Between Students with and without Mathematical Difficulties Analyzed Through Eye: Analysis Through Eye-Tracking and AI. In 43rd Conference of the International Group for the Psychology of Mathematics Education, Pretoria, South Africa, 7-12 July, 2019 (Vol. 3, pp. 281-288). PME. Schindler, M., Simon, A. L., Baumanns, L., & Lilienthal, A. J. (2025). Eye-tracking research in mathematics and statistics education: Recent developments and future trends. A systematic literature review. ZDM Mathematics Education, 57, 727–743. https://doi.org/10.1007/s11858-025-01699-8

91 98 Daniela Assmus Nina Bohlmann Kerstin Bräuning Caren Feskorn Torsten Fritzlar Heike Hahn Simone Reinhold Denise Resche University of Halle-Wittenberg Germany Leipzig University Germany University of Erfurt Freie Universität Berlin (Germany) Germany Visual representations are essential for teaching fundamental properties of multiplication such as commutativity. However, little is known about how beginning preservice teachers (PSTs) assess such representations. This study investigates how first-year PSTs at three German universities (N = 863) evaluate visual representations of the commutative law of multiplication and which criteria underlie their judgments. Results indicate that many students relied primarily on surface-related features (e.g., clarity or familiarity) rather than content-related aspects for evaluation. Even conceptually problematic representations were often rated positively. The findings point to a general surface orientation of some PSTs and highlight the need for teacher education to promote content-focused and comparative analyses of representations. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. ESM, 52(3), 215–241. https://doi.org/10.1023/A:1024312321077 Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554 Battey, D., Kafai, Y., & Franke, M. (2005). Evaluation of mathematical inquiry in commercial rational number software. In C. Vrasidas & G. Glass (Eds.), Preparing teachers to teach with technology (pp. 241–256). Information Age. Barmby, P., & Milinković, J. (2011). Pre-service teachers' use of visual representations of multiplication. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, 2-105 - 2-110). PME. Ding, M., Li, X., & Capraro, M. M. (2013). Preservice elementary teachers’ knowledge for teaching the associative property of multiplication: A preliminary analysis. The Journal of Mathematical Behavior, 32(1), 36–52. https://doi.org/10.1016/j.jmathb.2012.09.002 Dreher, A., Wang, T.‑Y., Feltes, P., Hsieh, F.‑J., & Lindmeier, A. (2024). High-quality use of representations in the mathematics classroom – a matter of the cultural perspective? ZDM – Mathematics Education, 56(5), 965–980. https://doi.org/10.1007/s11858-02401597-5 Harries, T., & Barmby, P. (2007). Representing and understanding multiplication. Research in Mathematics Education, 9(1), 33–45. https://doi.org/10.1080/14794800008520169 Hurst, C. (2015). The multiplicative situation. APMC, 20(3), 10–16. Kosko, K. W. (2019). Third‐grade teachers’ self‐reported use of multiplication and division models. School Sci Math, 119(5), 262–274. https://doi.org/10.1111/ssm.12337 Lindmeier, A. (2011). Modeling and measuring knowledge and competencies of teachers. Waxmann. Mayring, P. (2015). Qualitative content analysis: theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to Qualitative Research in Mathematics Education (pp. 365–380). Springer Netherlands. https://doi.org/10.1007/978-94-017-9181-6_13 Milinković, J. (2012). Pre-service teachers’ representational preferences. In T.-Y. Tso (Ed.), Opportunities to Learn in Mathematics Education. Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, Vol. 3, 209–215). PME. Schoenherr, J., Strohmaier, A. R., & Schukajlow, S. (2024). Learning with visualizations helps: A meta-analysis of visualization interventions in mathematics education. Educational Research Review, 45, 100639. https://doi.org/10.1016/j.edurev.2024.100639 Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures. Visible Language, 16(3), 281–288.

99 106 Theodora Avgeri Maria Bebeni Konstantinos Tatsis Xenia Vamvakoussi university of Ioannina Greece This paper reports findings from an ongoing design-based research study aimed at enhancing primary teachers’ conceptual understanding of fraction multiplication and division, focusing on the latter. An instructional sequence was designed to support the development of a measurement interpretation of division, transferable from natural to rational numbers. Data from 36 prospective teachers were collected through a pretest, a group task, and individual assignments involving interpretation of fraction division and task design. Analysis indicated that although many participants employed a measurement interpretation after the intervention, most did so inconsistently, often exhibiting misinterpretations of what constitutes a measurement situation. The results highlight specific challenges with division to be considered in teacher education. Copur-Gencturk, Y. (2021). Teachers’ conceptual understanding of fraction operations: results from a national sample of elementary school teachers. Educational Studies in Mathematics, 107(3), 525–545. https://doi.org/10.1007/s10649-021-10033-4 Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17. https://doi.org/10.5951/jresematheduc.16.1.0003 Gravemeijer, K., & Prediger, S. (2019). Topic-specific design research: An introduction. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 33–57). Springer. https://doi.org/10.1007/978-3-030-15636-7_2 Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). National Council of Teachers of Mathematics. Ma, L. (1999). Knowing and teaching elementary mathematics. Lawrence Erlbaum Associates. Nunes, T., & Bryant, P. (2022). Number systems as models of quantitative relations. In G. Karagöz Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative reasoning in mathematics and science education (pp. 71–105). Springer. https://doi.org/10.1007/9783-031-14553-7_4 Olanoff, D., Lo, J.-J., & Tobias, J. M. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267– 310. https://doi.org/10.54870/1551-3440.1304 Tchoshanov, M., Quinones, M. C., Shakirova, K. B., Ibragimova, E. N., & Shakirova, L. R. (2017). Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics. The Journal of Mathematical Behavior, 47, 54–69. https://doi.org/10.1016/j.jmathb.2017.06.005 Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25. https://doi.org/10.2307/749817 Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). The framework theory approach to the problem of conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 3–34). Routledge. https://doi.org/10.4324/9780203874813

107 114 Mela Aziza Chris Sangwin Universitas Islam Negeri Fatmawati Sukarno The University of Edinburgh United Kingdom Robust algebraic reasoning depends on recognising when expressions or equations are equivalent and justifying that judgement. This study investigates students' definitions and their actual justification strategies, and the alignment between them. Open written responses from 13 first-year undergraduate mathematics students were analysed thematically, both in a traditional manner and using an LLM. The tasks required definitions, examples of equivalent or non-equivalent pairs, and justification for equivalence in expressions and equations. For equivalence of expressions, students primarily provided normative definitions, the same value for every x, whereas in practice, students often relied on transformations or numerical-value checks. Meanwhile, students showed greater alignment in equivalent equations. Alcock, L., & Simpson, A. (2017). Interactions between defining, explaining and classifying: The case of increasing and decreasing sequences. Educational Studies in Mathematics, 94(1), 5–19. https://doi.org/10.1007/s10649-016-9709-4 Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). Research commentary: An alternative reconceptualization of procedural and conceptual knowledge. Journal for research in mathematics education, 38(2), 115–131. https://doi.org/10.2307/30034952 Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa Cook, J. P., Reed, Z., & Lockwood, E. (2022). An initial framework for analyzing students’ reasoning with equivalence across mathematical domains. The Journal of Mathematical Behavior, 66, 100935. https://doi.org/10.1016/j.jmathb.2022.100935 Daro, P., & Burkhard, H. (2012). A population of assessment tasks. Journal of Mathematics Education at Teachers College, 3(1), 19–25. https://doi.org/10.7916/jmetc.v3i1.732 Godfrey, D., & Thomas, M. O. J. (2008). Student perspectives on equation: The transition from school to university. Mathematics Education Research Journal, 20(2), 71–92. https://doi.org/10.1007/BF03217478 Jones, I., Inglis, M., Gilmore, C., & Dowens, M. (2012). Substitution and sameness: Two components of a relational conception of the equals sign. Journal of experimental child psychology, 113 (1), 166–176. https://doi.org/10.1016/j.jecp.2012.05.003. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? evidence from solving equations. Journal for research in Mathematics Education, 37(4), 297–312. https://doi.org/10.2307/30034852 Melhuish, K. (2018). Three conceptual replication studies in group theory. Journal for Research in Mathematics Education, 49(1), 9–38. https://doi.org/10.5951/jresematheduc.49.1.00095 Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715 Steinberg, R. M., Sleeman, D., & Ktorza, D. (1991). Algebra students’ knowledge of equivalence of equations. Journal for research in mathematics education, 22(2), 112–121. https://doi.org/10.2307/749588 Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151– 169. https://doi.org/10.1007/BF00305619 Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In Advanced mathematical thinking (pp. 65–81). Dordrecht: Springer. Wladis, C., Sencindiver, B., Offenholley, K., Jaffe, E., & Taton, J. (2022). Student definitions of equivalence: Structural vs. operational conceptions, and extracted vs. stipulated definition construction. In Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) (No. 26). Zhang, H., Wu, C., Xie, J., Lyu, Y., Cai, J., & Carroll, J. M. (2025). Harnessing the power of AI in qualitative research: Exploring, using and redesigning ChatGPT. Computers in Human Behavior: Artificial Humans, 4, 100144.

115 122 Anna Baccaglini-Frank Bernardo Nannini Federica Poli University of Pisa Italy University of Florence Italy The change side–change sign procedure (CS procedure) for equation solving is often used by students as a memorized rule without justification. This paper reports on findings from the DynaMat project that sees remediation not as repetition of procedures, but as access to mathematical meaning through Digital Exploratory Objects (DEOs). Grounded in Commognition, the study analyzes a case of two grade-10 students to investigate whether and how the remedial activities supported the substantiation of the CS procedure. The findings show that the intervention supported students through a guided process of discourse expansion with the DEOs, followed by a compression in which the CS procedure came to be seen as a substantiated shortcut. Antonini, A., Bonadiman, C., & Nannini, B. (2023). Dynamic interactive mediators in discourse on intermediate quantities: a case study. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel & M. Tabach (Eds.), Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 43–50). PME 46. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 390–419). Macmillan Publishing Co, Inc. Kristinsdóttir, B. (2022) Opportunities and challenges that silent video tasks bring to the mathematics classroom. In A. Clark-Wilson, N. Sinclair, & O. Robutti (Eds.), The Mathematics Teacher in the Digital Era. Mathematics Education in the Digital Era, vol 16 (pp. 119–149). Springer. Kurz, A., Elliott, S. N., Lemons, C. J., Zigmond, N., Kloo, A., & Kettler, R. J. (2014). Assessing opportunity- to-learn for students with disabilities in general and special education classes. Assessment for Effective Intervention, 40(1), 24–39. Lambert, R. (2018). “Indefensible, illogical, and unsupported"; Countering deficit mythologies about the potential of students with learning disabilities. Ed. Sciences, 8(2). Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. Educational Studies in Mathematics, 101, 153–176. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press. Thurston, W.P. (1990). Mathematics Education. Notices of the AMS, 37, 844–850. Weingarden, M., Lisarelli, G., & Baccaglini-Frank, A. (2024). Exploring mathematics learning opportunities afforded by a balanced scale digital activity. Digital Experiences in Mathematics Education, 10, 382–394.

123 130 Julia Bagdadi Michal Yerushalmy University of Haifa Israel This paper reports on the design of an automated assessment for exploring students’ conceptions of equivalence of linear equations in a multiple-linked representation (MLR) environment. Students’ approaches to equivalence are examined in a function-based context, where an equation is represented as a comparison of two functions. Common solutions are visually represented by a shared X value of intersection points, derived by representation of a sequence of solution-preserving transformations. Implemented in the STEP platform, answers to Example Eliciting Tasks (EETs) –including triangulation of choice of examples from lists of equations, learner-generated examples and choice items are automatically analysed. Cook, J. P., Richardson, A., Reed, Z., & Lockwood, E. (2023). Using conceptual analyses to resolve the tension between advanced and secondary mathematics: the cases of equivalence and inverse. ZDM–Mathematics Education, 55(4), 753-766. Gelfand, I. M., & Shen, A. (1993). Algebra. Birkhäuser. Kieran, C., & Martínez-Hernández, C. (2022). Coordinating visible and invisible sameness in equivalence transformations. ZDM–Mathematics Education, 54, 1215–1227. https://doi.org/10.1007/s11858-022-01374-9 Larkin, J. H. (1989). Robust performance in algebra: The role of the problem representation. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra, Vol. 4, pp. 120–134). Lawrence Erlbaum Associates, Inc; National Council of Teachers of Mathematics. Noster, N., & Siller, H.-S. (2025). Transforming equations into equivalent equations: An empirical study of equation transformation capability as a multidimensional construct. Educational Studies in Mathematics, 120, 249–267. https://doi.org/10.1007/s10649-02410287-3 Olsher, S. (2019). Making good practice common using computer aided formative assessment. In G. Aldon & J. Trgalová (Eds.), Technology in Mathematics Teaching (pp. 31–47). Springer. https://doi.org/10.1007/978-3-030-19741-4_2 Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Lawrence Erlbaum. Yerushalmy, M. & Chazan D. (2002). Flux in school algebra: Curricular change, graphing technology, and research on student learning and teacher knowledge. In L. English et al. (Eds.) Handbook of International Research in Mathematics Education. pp. 725-756. Hillsdale, NJ: Erlbaum. Yerushalmy, M., Olsher, S. (2020). Online assessment of students’ reasoning when solving example-eliciting tasks: using conjunction and disjunction to increase the power of examples. ZDM–Mathematics Education, 52, 1033–1049. https://doi.org/10.1007/s11858020-01134-0

131 138 Oliver Baumann Johanna Schoenherr Pia Gödecke Jascha Quarder Stanislaw Schukajlow Maike Schindler Osnabrueck University Germany Different types of drawings influence their effectiveness in solving modelling problems, however, the underlying processes are still not fully known. This study investigates how a drawing’s level of abstraction relates to students’ drawing construction, drawing use, and modelling performance in geometry. Sixty-six ninth- and tenth-graders solved six geometry modelling problems with instructions to make a drawing while their gaze movements were recorded. Path analysis showed that students whose gaze movements indicated more effective drawing construction created more abstract drawings. More abstract drawings were indirectly related to better modelling performance via more effective drawing use. Thus, instructional support should focus on fostering both the construction of abstract drawings and their effective use in modelling. Bassi, C., & Brunetto, D. (2025). Shared drawings in a mathematical modelling activity: An exploratory study. Journal of Mathematical Behavior, 78, 101234. https://doi.org/10.1016/j.jmathb.2024.101234 Baumann, O., Goedecke, P., Quarder, J., Schukajlow, S., Schindler, M., & Schoenherr, J. (under review). Examining the role of drawing construction and use in mathematical modelling – An eye tracking study. ZDM – Mathematics Education. Boonen, A. J., van Wesel, F., Jolles, J., & van der Schoot, M. (2014). The role of visual representation type, spatial ability, and reading comprehension in word problem solving: An item-level analysis in elementary school children. International Journal of Educational Research, 68, 15-26. https://doi.org/10.1016/j.ijer.2014.08.001 Cox, R. (1999). Representation construction, externalised cognition and individual differences. Learning and Instruction, 9(4), 343-363. https://doi.org/10.1016/S09594752(98)00051-6 de Bock, D., Verschaffel, L., Janssens, D., van Dooren, W., & Claes, K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modelling non-linear geometry problems. Learning and Instruction, 13(4), 441-463. Galbraith, P. L., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM, 38(2), 143-162. https://doi.org/10.1007/BF02655886 Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684-689. https://doi.org/10.1037/0022-0663.91.4.684 Baumann, Schoenherr, Goedecke, Quarder, Schukajlow, Schindler Hellenbrand, J., Mayer, R. E., Opfermann, M., Schmeck, A., & Leutner, D. (2019). How generative drawing affects the learning process: An eye‐tracking analysis. Applied Cognitive Psychology, 33(6), 1147-1164. https://doi.org/10.1002/acp.3559 Holzäpfel, L., & Leiss, D. (2014). Modellieren in der Sekundarstufe [Mathematical modelling in secondary education]. In H. Linneweber-Lammerskitten (Ed.), Fachdidaktik Mathematik: Grundbildung und Kompetenzaufbau im Unterricht der Sekundarstufe I und II (pp. 159-178). Kallmeyer. Jian, Y.‑C. (2018). Reading instructions influence cognitive processes of illustrated text reading not subject perception: An eye-tracking study. Frontiers in Psychology, 9, 2263. https://doi.org/10.3389/fpsyg.2018.02263 Mason, L., Pluchino, P., Tornatora, M. C., & Ariasi, N. (2013). An eye-tracking study of learning from science text with concrete and abstract illustrations. The Journal of Experimental Education, 81(3), 356-384. https://doi.org/10.1080/00220973.2012.727885 Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. https://doi.org/10.4324/9781315189314 Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. L. Galbraith, H.W. Henn, & M. Niss (Eds.), Modelling and Applications in Mathematics Education: The 14th ICMI Study (pp. 3-32). Springer US. https://doi.org/10.1007/978-0-387-29822-1_1 Rellensmann, J., Schukajlow, S., Blomberg, J., & Leopold, C. (2022). Effects of drawing instructions and strategic knowledge on mathematical modeling performance: Mediated by the use of the drawing strategy. Applied Cognitive Psychology, 36(2), 402-417. Rellensmann, J., Schukajlow, S., & Leopold, C. (2017). Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53-78. https://doi.org/10.1007/s10649-016-9736-1 Schoenherr, J., Baumann, O., Goedecke, P., Quarder, J., Schukajlow, S., & Schindler, M. (under review). Strategy-based motivation and learner-generated drawing: Linking motivation and drawing processes in geometry modelling. ZDM – Mathematics Education. Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: towards an organizing scheme. Educational Studies in Mathematics, 76(3), 265-280. van Garderen, D., Scheuermann, A., & Jackson, C. (2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Quarterly, 36(3), 145-160. https://doi.org/10.1177/0731948712438558 van Meter, P., & Firetto, C. M. (2013). Cognitive model of drawing construction: Learning through the construction of drawings. In G. Schraw, M. T. McCrudden, & D. H. Robinson (Eds.), Current perspectives on cognition, learning, and instruction. Learning through visual displays (pp. 247-280). Information Age Publishing. Zimmermann, W., & Cunningham, S. (1991). Editor’s introduction: What is mathematical visualization? In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 1-8). Mathematical Association of America.

139 146 Lukas Baumanns University of Duisburg-Essen Germany Patterning skills in first grade are a robust predictor of later mathematics achievement, underscoring the need to foster them. This study examines not only whether a support program for first-graders at risk of developing mathematical difficulties (RMD) reduces errors on repeating-pattern tasks, but also what eye tracking can reveal about pattern-recognition processes. Eighteen RMD first graders were randomly assigned to an RMD support group that received one-to-one instruction on repeating patterns and an RMD control group without additional support. Pre–post test measures showed significant reductions in error rates for both groups. However, only the RMD support group showed gaze shifts towards relational processes. These findings highlight the value of combining process with performance measures when evaluating interventions. Baumanns, L., Pitta-Pantazi, D., Demosthenous, E., Lilienthal, A. J., Christou, C., & Schindler, M. (2024). Pattern-recognition processes of first-grade students: an explorative eye-tracking study. Int. Journal of Science and Mathematics Education, 22, 1663–1682. Clements, D. H., & Sarama, J. (2020). Learning and Teaching Early Math. The Learning Trajectories Approach (Vol. 3). Routledge. Hendricks, C., Trueblood, L., & Pasnak, R. (2006). Effects of teaching patterning to 1stgraders. Journal of Research in Childhood Education, 21(1), 79–89. Junker, A., Nortvedt, G. A., & Farsani, D. (2024). Patterning strategies in grade 1 students with low and high number sense proficiency. Ed. Studies in Mathematics, 118, 29–51. Kidd, J. K., Pasnak, R., Gadzichowski, K. M., Gallington, D. A., McKnight, P., Boyer, C. E., & Carlson, A. (2013). Instructing first-grade children on patterning improves reading and mathematics. Early Education and Development, 25(1), 134–151. Krajewski, K. (2018). MBK 0. Test mathematischer Basiskompetenzen im Kindergartenalter [MBK 0. Test of basic mathematical skills in kindergarten age children]. Hogrefe. Lüken, M., Peter-Koop, A., & Kollhoff, S. (2014). Influence of early repeating patterning ability on school mathematics learning. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the PME 38 and PME-NA 36 (Vol. 4, pp. 137–144), PME. Lüken, M. M., & Sauzet, O. (2021). Patterning strategies in early childhood: a mixed methods study examining 3- to 5-year-old children’s patterning competencies. Mathematical Thinking and Learning, 23(1), 28–48. Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers’ mathematical patterning. J. for Research in Math. Ed., 42(3), 237–269. Rittle-Johnson, B., Fyfe, E. R., Loehr, A. M., & Miller, M. R. (2015). Beyond numeracy in preschool: Adding patterns to the equation. Early Childhood R. Quarterly, 31, 101–112. Rittle-Johnson, B., Zippert, E. L., & Boice, K. L. (2019). The roles of patterning and spatial skills in early mathematics development. Early Childhood R. Quarterly, 46, 166–178. Schindler, M., Simon, A. L., Baumanns, L., & Lilienthal, A. (2025). Eye-tracking research in mathematics and statistics education: recent developments and future trends. A systematic literature review. ZDM – Mathematics Education, 57(4), 727–743. Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing young children’s mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19(1), 99–120. Wijns, N., Verschaffel, L., De Smedt, B., De Keyser, L., & Torbeyns, J. (2021). Stimulating preschoolers’ focus on structure in repeating and growing patterns. Learning and Instruction, 74, 101444.

147 154 Federica Becker Julia Bruns Institute of Mathematics, Paderborn University Germany Research on teacher noticing highlights differences between novice and expert teachers, yet little is known about early career and intermediate expertise levels and how these expertise-related differences emerge in central teaching tasks and content areas. Facilitating whole-class discussions represents a central teaching task, such as when fostering flexible strategy use. This study examines expertise-related differences in mathematics teachers’ noticing in observed video scenes of whole-class discussions on flexible strategy use in Grade 1 using Epistemic Network Analysis. The preliminary results show no substantial differences between intermediate and expert teachers but between beginning teachers and the other two expertise groups regarding their noticing focus and the cognitive structures underlying their noticing processes. Berliner, D. C. (2001). Learning about and learning from expert teachers. International Journal of Educational Research, 35(5), 463–482. Blömeke, S., Gustafsson, J.-E., & Shavelson, R. J. (2015). Beyond dichotomies: Competence viewed as a continuum. Zeitschrift für Psychologie, 223(1), 3–13. https://doi.org/10.1027/2151-2604/a000194 Blömeke, S., Jentsch, A., Ross, N., Kaiser, G., & König, J. (2022). Opening up the black box: Teacher competence, instructional quality, and students’ learning progress. Learning and Instruction, 79, Article 101600. https://doi.org/10.1016/j.learninstruc.2022.101600 Ericsson, K. A., Hoffman, R. R., Kozbelt, A., &Williams, A. M. (Eds.) (2018). The Cambridge handbook of expertise and expert performance (2nd ed.). Cambridge University Press. https://doi.org/10.1017/9781316480748 Kaiser, G., König, J., Krauss, S., Buchholtz, N., & Gruber, H. (2025). Are teacher expertise and teacher competence complementary or contradictory research paradigms? Reflections on a possible integration and first conceptualizations. ZDM – Mathematics Education, 57(6), 1077–1092. https://doi.org/10.1007/s11858-025-01743-7 Kuckartz, U., & Rädiker, S. (2023). Qualitative content analysis: Methods, practice and software. SAGE Publications. Marquart, C. L., Hinojosa, C., Swiecki, Z., Eagan, B., & Shaffer, D. W. (2018). Epistemic network analysis tool (Version 1.7.0) [Computer software]. Available from http://app.epistemicnetwork.org (accessed January 14, 2026). Prediger, S., Götze, D., Holzäpfel, L., Rösken-Winter, B., & Selter, C. (2022). Five principles for high-quality mathematics teaching: Combining normative, epistemological, empirical, and pragmatic perspectives for specifying the content of professional development. Frontiers in Education, 7, Article 969212, 1–15. https://doi.org/10.3389/feduc.2022.969212 Rechtsteiner, C., & Rathgeb-Schnierer, E. (2017). “Zahlenblickschulung” as approach to develop flexibility in mental calculation in all students. Journal of Mathematics Education, 10(1), 1–16. https://doi.org/10.1590/2175-623687078 Seidel, T., Kosel, C., Böheim, R., Gegenfurtner, A., & Stürmer, K. (2024). A cognitive model of professional vision and acquisition of visual expertise using video excerpts in the teaching profession. In A. Gegenfurtner & R. Stahnke (Eds.), Teacher professional vision: Theoretical methodological https://doi.org/10.4324/9781003370901-7 advances (pp. 43–56). Routledge. Shaffer, D. W., & Ruis, A. R. (2017). Epistemic network analysis: A worked example of theory-based learning analytics. In C. Lang, Siemens, G., A. Wise, & D. Gašević (Eds.), The handbook of learning analytics (pp. 175-187). Society for Learning Analytics Research. https://doi.org/10.18608/hla17.015 Shaughnessy, M., Garcia, N. M., O'Neill, M. K., Selling, S. K., Willis, A. T., Wilkes, C. E., Salazar, S. B., Ball, D. L. (2021). Formatively assessing prospective teachers' skills in leading mathematics discussions. Educational Studies in Mathematics, 108(3), 451–472. https://doi.org/10.1007/s10649-021-10070-z Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Situating the study of teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Studies in mathematical thinking and learning. Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 3–13). Routledge. https://doi.org/10.4324/9780203832714 Stahnke, R. (2024). Research on teacher professional vision: Mapping the landscape of theoretical, methodological, and empirical contributions. In A. Gegenfurtner & R. Stahnke (Eds.), Teacher professional vision: Theoretical and methodological advances. Volume 1 (pp. 11–26). Routledge. https://doi.org/10.4324/9781003370901-3 Verschaffel, L. (2024). Strategy flexibility in mathematics. ZDM – Mathematics Education, 56(1), 115–126. https://doi.org/10.1007/s11858-023-01491-6 Webb, N. M., Franke, M. L., Ing, M., Turrou, A. C., Johnson, N. C., & Zimmerman, J. (2019). Teacher practices that promote productive dialogue and learning in mathematics classrooms. International Journal of Educational Research, 97, 176–186. https://doi.org/10.1016/j.ijer.2017.07.009

155 162 Catharina Beckschulte Jascha Quarder Annika Rosendahl Stanislaw Schukajlow University of Münster Germany Digital portfolios have been proposed as a promising assessment format for mathematical modelling. However, empirical evidence on whether such portfolios can be used to reliably and validly assess analytically defined modelling sub-competencies is limited. We investigated the development and validation of an assessment instrument that combines digital portfolios based on holistic modelling tasks with an analytic assessment rubric. Using data from 116 ninth-grade students, we examined the internal structure, interrater reliability, and criterion-related validity. Results indicate that modelling sub-competencies can be empirically distinguished, assessed with good reliability, and meaningfully related to an established modelling test, suggesting that digital portfolios can bridge holistic task design and analytic assessment. Beckschulte, C., & Quarder, J. (2025). Alternative forms of assessment in mathematical modelling: The AlPMo project in practice and research. University of Münster (MIAMI – Institutional Repository). https://doi.org/10.17879/02958629587 Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (pp. 222– 231). Woodhead Publishing. https://doi.org/10.1533/9780857099419.5.221 Borromeo Ferri, R., & Greefrath, G. (2026). Pre-service teachers’ assessment competence for grading students’ solutions of mathematical modelling problems. Teaching and Teacher Education, 169, 105281. https://doi.org/10.1016/j.tate.2025.105281 Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). Guilford Press. Cevikbas, M., Kaiser, G., & Schukajlow, S. (2022). A systematic literature review of the current discussion on mathematical modelling competencies: State-of-the-art developments in conceptualizing, measuring, and fostering. Educational Studies in Mathematics, 109(2), 205–236. https://doi.org/10.1007/s10649-021-10104-6 Frejd, P. (2013). Modes of modelling assessment—A literature review. Educational Studies in Mathematics, 84(3), 413–438. https://doi.org/10.1007/s10649-013-9491-5 Hankeln, C., Adamek, C., & Greefrath, G. (2019). Assessing sub-competencies of mathematical modelling—Development of a new test instrument. In G. Stillman & J. Brown (Eds.), Lines of inquiry in mathematical modelling research in education (pp. 143– 160). Springer. https://doi.org/10.1007/978-3-030-14931-4_8 Koo, T. K., & Li, M. Y. (2016). A guideline of selecting and reporting intraclass correlation coefficients for reliability research. Journal of Chiropractic Medicine, 15(2), 155–163. https://doi.org/10.1016/j.jcm.2016.02.012 Krall, G. (2023). Teacher-designed mathematical portfolio assessments: Motivations, potential benefits, and lessons learned. Canadian Journal of Science, Mathematics and Technology Education, 23(2), 303–321. https://doi.org/10.1007/s42330-023-00280-3 Krawitz, J., Schukajlow, S., Yang, X., & Geiger, V. (2025). A systematic review of international perspectives on mathematical modelling: Modelling goals and task characteristics. ZDM – Mathematics Education, 57, 193–212. https://doi.org/10.1007/s11858-025-01683-2 Lu, X., Kaiser, G., Zhu, Y., Ma, H., & Yan, Y. (2025). Mathematical creativity in modelling: Further development of the construct, its measurement, and its empirical implementation. ZDM – Mathematics Education, 57(2), 365–379. https://doi.org/10.1007/s11858-02501652-9 Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. https://doi.org/10.4324/9781315189314 Stillman, G. (2011). Applying metacognitive knowledge and strategies in applications and modelling tasks at secondary school. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 165–180). Springer. https://doi.org/10.1007/978-94-007-0910-2_18 Wilson, M. (2008). Constructing measures—An item response modeling approach. Routledge. https://doi.org/10.4324/9781410611697 Yang, M., Wang, T., & Lim, C. P. (2017). E-portfolios as digital assessment tools in higher education. In M. J. Spector, B. B. Lockee, & M. D. Childress (Eds.), Learning, design, and technology: An international compendium of theory, research, practice, and policy (pp. 1– 23). Springer. https://doi.org/10.1007/978-3-319-17727-4_83-1

163 170 Mervenur Belin Işıl İşler-Baykal Gülseren Karagöz Akar Boğaziçi University, Türkiye Middle East Technical University Turkey As part of an intensive year-long multi-case study, in this paper, we investigated a mathematics teacher’s functional approaches and levels of decentering while teaching quadratic functions. We specifically report from the data of three hours of teaching on the introduction to quadratic functions. Results showed that the teacher used correspondence and covariation approaches at four levels of decentering. Importantly, at decentering levels 2 and 3, the covariation approach was the dominant approach in the discourse. In this regard, a relationship was observed between the teacher’s emphasis on functional approaches and the quality of interaction with students. Importantly, this study provided empirical evidence that covariational reasoning is a significant component of teachers’ quality of teaching quadratic functions. Baş-Ader, S., & Carlson, M. P. (2022). Decentering framework: A characterization of graduate student instructors’ actions to understand and act on student thinking. Mathematical Thinking and Learning, 24(2), 99– 122. Carlson, M. P., & Moore, K. C. (2015). The role of covariational reasoning in understanding and using the function concept. In E. A. Silver & P. A. Kenney (Eds.), Lessons learned from research: Useful and useable research related to core mathematical practices (Vol. 1) (pp. 279– 291). Reston, VA: National Council of Teachers of Mathematics. Ellis, A. B. (2011). Algebra in the middle school: Developing functional relationships through quantitative reasoning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 215– 238). New York: Springer. Ellis, A.B., Özgür, Z., Doğan, M.F. (2023). A Conceptual Analysis of Early Function Through Quantitative Reasoning. In G. Karagöz Akar, İ.Ö. Zembat, S. Arslan, P.W. Thompson, (Eds.) Quantitative reasoning in mathematics and science education. Mathematics education in the digital era, (Vol 21., pp. 169 – 197). Springer, Cham. Hohensee, C., Gartland, S., Willoughby, L., & Melville, M. (2021). Backward transfer influences from quadratic functions instruction on students’ prior ways of covariational reasoning about linear functions. The Journal of Mathematical Behavior, 61, 100834. Moore, K. C. (2016). Graphing as figurative and operative thought. In Csíkos, C., Rausch, A., & Szitányi, J. (Eds.). Proc. of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 323– 330). Szeged, Hungary: PME. Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics. (Vol. 73, pp. 27–42). Mathematical Association of America. Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics. Yoon, H. (2019). The relationships between meanings teachers hold and meanings their students construct [Unpublished Doctoral Dissertation]. Arizona State University.

171 178 Valentina Bernabè University of Modena and Reggio Emilia Italy This research report examines instrument-mediated activity in linear algebra learning in a non-visual context. Using the instrumental approach, we analyse a task-based interview with a student with visual impairment solving matrix-related tasks. The analysis focuses on processes of instrumentation and instrumentalisation in accessing mathematical structure through sequential representations. Results show how digital artefacts are transformed into cognitive instruments supporting control and coherence of mathematical meaning. The study suggests that non-visual contexts pose explicit challenges for linear algebra that are often implicit in visually mediated learning. Abrahamson, D., Flood, V. J., Miele, J. A., & Siu, Y. T. (2019). Enactivism and ethnomethodological conversation analysis as instruments for expanding universal design for learning: The case of visually impaired mathematics students. ZDM – Mathematics Education, 51(2), 291–303. https://doi.org/10.1007/s11858-018-0997-z Archambault, D., Stöger, B., Fitzpatrick, D., Miesenberger, K., & Pontelli, E. (2007). Access to scientific content by visually impaired people. Upgrade, 8(2), 14–23. Cohen, L., Manion, L., & Morrison, K. (2002). Research methods in education (5th ed.). Routledge. De Freitas, E., & Sinclair, N. (2014). Mathematics and the body: Material entanglements in the classroom. Cambridge University Press. Dorier, J.-L. (2003). Teaching https://arxiv.org/abs/math/0305018 linear algebra at university. arXiv. Goldin, G. A. (2012). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Routledge. Hodges, J. S., & Keller, M. J. (1999). Visually impaired students’ perceptions of their social integration in college. Journal of Visual Impairment & Blindness, 93(3), 153–165. Maffia, A., Manolino, C., & Miragliotta, E. (2025). There is more to algebra than meets the eye: the case of blindness. Educational Studies in Mathematics, 119(1), 63–77. https://doi.org/10.1007/s10649-025-10394-0 Mason, H., & McCall, S. (2013). Visual impairment: Access to education for children and young people. Routledge. Pedersen, T. V. (2007). Design of a didactic situation-mathematical experiments in linear algebra. Nordic Studies in Mathematics Education, 12(4), 27–52. https://doi.org/10.7146/nomad.v12i4.148045 Piaget, J. (1977). La naissance de l’intelligence chez l’enfant. FeniXX. (Original work published 1936). Radford, L. (2013). Perceiving with the eyes and with the hands. REPIME, 3(1), 56-77. Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting with Computers, 15(5), 665–691. Rabardel, P., & Samurçay, R. (2001). From artifact to instrument-mediated learning. In Symposium on New Challenges to Research on Learning (pp. 21–23). University of Helsinki. Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101. White, T. (2008). Debugging an artifact, instrumenting a bug: Dialectics of instrumentation and design in technology-rich learning environments. International Journal of Computers for Mathematical Learning, 13(1), 1–26.

179 186 Meeri-Liisa Beste Boris Girnat Henning Sievert University of Hildesheim Germany Graph theory is considered to have great educational and motivational potential for school teaching. However, so far, it has only been used peripherally in this context. This study investigates the impact of a graph theory lecture on student teachers’ motivation and their intention to integrate this topic in future teaching. We examined 42 student teachers, divided into a group that attended a graph theory lecture and a control group. While no significant differences were found in motivation, attendees of the graph theory lecture showed a higher tendency to use graph-theoretical ideas in future teaching. Regression analyses show that students’ intentions to use graph-theoretical ideas in future teaching are associated with graph theory lecture attendance, intrinsic motivation, and secondary school specialisation. Anthony, G. (2000). Factors influencing first-year students’ success in mathematics. International Journal of Mathematical Education in Science and Technology, 31(1), 3–14. https://doi.org/10.1080/002073900287336 Coufal, P., Hubálovský, S., & Hubálovská, M. (2021). Application of Basic Graph Theory in Autonomous Motion of Robots. Mathematics, 9(9). https://doi.org/10.3390/math9090919 Girnat, B. (2019). Selbstkonzept, Selbstwirksamkeitserwartung und Emotionen von Studienanfänger im Fach Mathematik [Self-concept, self-efficacy expectations and emotions of first-year maths students]. https://doi.org/10.17877/DE290R-20828 Greefrath, G., Siller, H., Vorholter, K., & Kaiser, G. (2022). Mathematical modelling and discrete mathematics: Opportunities for modern mathematics teaching. ZDM – Mathematics Education, 54(4), 865–879. https://doi.org/10.1007/s11858-022-01339-5 Hart, E. W., & Martin, W. G. (2018). Discrete Mathematics Is Essential Mathematics in a 21st Century School Curriculum. In E. W. Hart & J. Sandefur (Eds.), Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research (pp. 3–19). Springer International Publishing. https://doi.org/10.1007/978-3-319-70308-4_1 Hazzan, O., & Hadar, I. (2005). Reducing Abstraction When Learning Graph Theory. Journal of Computers in Mathematics and Science Teaching, 24(3), 255–272. Hibi, W. (2025). Developing Mathematics Education By Integrating Graph Theory Into College Curricula. Multicultural Education, 11(1). https://doi.org/10.5281/ZENODO.14639723 Klimm, F., & Maier, B. F. (2020). Commentary: A network science summer course for highschool students. Network Science, 8(4), 596–608. Scopus. https://doi.org/10.1017/nws.2020.12 Krus, D. J., & Wilkinson, S. M. (1986). Demonstration of properties of a suppressor variable. Behavior Research Methods, Instruments, & Computers, 18(1), 21–24. https://doi.org/10.3758/BF03200988 Lens, W., Paixão, M. P., & Herrera, D. (2009). Instrumental Motivation is Extrinsic Motivation: So What??? Psychologica, 50, 21–40. https://doi.org/10.14195/16478606_50_2 Medová, J., Milicic, G., & Ludwig, M. (2022). Graph problems as a means for accessing the abstraction skills. Twelfth Congress of the European Society for Research in Mathematics Education (CERME12), TWG11. https://hal.science/hal-03748478 Milková, E. (2011). Puzzles—A creative way of development of logical thinking. Journal on Efficiency and Responsibility in Education and Science, 4(4), 207–218. Scopus. Prayitno, A. T., Nusantara, T., Hidayanto, E., & Rahardjo, S. (2022). Identification of Graph Thinking in Solving Mathematical Problems Naturally. Participatory Educational Research, 9(2), Article 2. https://doi.org/10.17275/per.22.32.9.2 Radmehr, F., Tohidinasab, K., & Tavakoli, M. (2023). Teaching and Learning of Vertex Coloring Using an Inquiry-Based Approach. PRIMUS, 33(7), Article 7. https://doi.org/10.1080/10511970.2023.2170505 Rencher, A. C., & Schaalje, G. B. (2007). Linear Models in Statistics (1st ed.). Wiley. https://doi.org/10.1002/9780470192610 Ryan, R. M., & Deci, E. L. (2000). Intrinsic and Extrinsic Motivations: Classic Definitions and New Directions. Contemporary Educational Psychology, 25(1), 54–67. https://doi.org/10.1006/ceps.1999.1020 Sawilowsky, S. S. (2009). New Effect Size Rules of Thumb. Journal of Modern Applied Statistical Methods, 8(2), 597–599. https://doi.org/10.22237/jmasm/1257035100 Uyangör, S. M. (2019). Investigation of the mathematical thinking processes of students in mathematics education supported with graph theory. Universal Journal of Educational Research, 7(1), 1–9. Scopus. https://doi.org/10.13189/ujer.2019.070101 Wahyuningsih, S., Satyananda, D., Qohar, A., & Atan, N. A. (2020). An Integration of ‘online interactive apps’ for learning application of graph theory to enhance creative problem solving of mathematics students. International Journal of Interactive Mobile Technologies, 14(12), 97–109. Scopus. https://doi.org/10.3991/IJIM.V14I12.15583 Windler, M. (2018). Der Einfluss graphentheoretischer Konzepte im Mathematikunterricht der Grundschule auf psychologische Schülerinnen- und Schülermerkmale [The influence of graph-theoretical concepts in primary school maths lessons on pupils’ psychological characteristics] [Universität Hildesheim]. https://doi.org/10.25528/858

187 194 Alistair Bissell Mathematics in Education and Industry, UK United Kingdom This paper examines how facilitation is enacted and negotiated in interaction during an online mathematics teacher professional development session. Drawing on an enactivist perspective and van Es’ framework of facilitation moves, the analysis focuses on transcript excerpts from a synchronous session involving two facilitators and two participants. The findings show that facilitative work is distributed across interaction, with participants frequently taking on facilitation roles such as pressing for clarification and directing attention to mathematical detail. The paper argues that facilitation in online professional development is negotiated rather than fixed. Amador, J., Carson, C., Gillespie, R., & Choppin, J. (2019). Online Video Coaching: An analysis of teachers and coaches' noticing. In S. Otten, A. Candela, Z. de Araujo, C. Haines, & C. Munter (Ed.), Proc. 41st Conf. of the International Group for the Psychology of Mathematics Education (pp. 509-513). St. Louis, MO: University of Missouri. Borba, M. C., Askar, P., Engelbrecht, J., Gadanidis, G., Llinares, S., & Aguilar, M. S. (2016). Blended learning, e-learning and mobile learning in mathematics education. ZDM, 48(5), 589-610. doi:10.1007/s11858-016-0798-4 Brown, L. (2020). Bringing Forth Worlds. For the Learning of Mathematics, 55-58. Eaton, S. E. (2018). The Impact of Technology on How Instructors Teach and How Students Learn. In R. J. Harnish, K. R. Bridges, D. N. Sattler, M. L. Signorella, & M. Munson (Eds.), The Use of Technology in Teaching and Learning (pp. 74-80). Society for the Teaching of Psychology. Retrieved January 18, 2024, from http://teachpsych.org/ebooks/ Enser, M. (2020, May 3). A Remote Chance of Learning. Retrieved from Teaching It Real: https://teachreal.wordpress.com/2020/05/03/a-remote-chance-of-learning/ Korthagen, F. (2016). Inconvenient truths about teacher learning: towards professional development 3.0. Teachers and Teaching, 1-19. doi:10.1080/13540602.2016.1211523 Lee, S. (2014). From the physical classroom to the online classroom - providing tuition, revision and professional development in 16-19 education. Pope, S. (Ed.) Proceedings of the 8th British Congress of Mathematics Education 2014, (pp. 239-246). van Es, E. A., Tunney, J., Goldsmith, L. T., & Seago, N. (2014). A Framework for the Facilitation of Teachers' Analysis of Video. Journal of Teacher Education, 65(4), 340-356. Varela, F., Thompson, E., & Rosch, E. (1991). The embodied mind: cognitive science and human experience. Cambridge: MIT Press.

195 202 Camilla Björklund Hanna Palmér University of Gothenburg Sweden Linnaeus University Sweden In this paper, we present the ways in which fingers are used as representations in teaching the basic meaning of numbers to young learners, and what learning opportunities the different ways entail. This is done through an analysis of 38 video recordings from authentic Swedish preschool teaching situations. Three ways of using fingers as representations were identified in the observed interactions: Fingers as countables, as symbols, and as bridging tool. Different finger-use opened for different meaning to be discerned; cardinality as a novel notion was made discernible when fingers were used as bridging tool, connecting different quantities but also different modes of representation. The results provide empirical evidence for how fingers are used as primary to if fingers are used in early number teaching. Baroody, A., & Purpura, D. (2017). Early number and operations: Whole numbers. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 308–354). National Council of Teachers of Mathematics. Berteletti, I., & Booth, J. R. (2016). Finger representation and finger-based strategies in the acquisition of number meaning and arithmetic. In D. B. Berch, D. C. Geary, & K. Mann Koepke (Eds.), Development of mathematical cognition: Neural substrates and genetic influences (pp. 109–139). Elsevier. Björklund, C. (2025). Finger use mirroring young children’s ways of experiencing numbers. Journal of Experimental Child Psychology, 249, 106076. https://doi.org/10.1016/j.jecp.2024.106076 Björklund, C., Kullberg, A., & Runesson Kempe, U. (2019). Structuring versus counting critical ways of using fingers in subtraction. ZDM Mathematics Education, 51(1), 13–24. https://doi.org/10.1007/s11858-018-0962-0 Björklund, C., Marton, F., & Kullberg, A. (2021). What is to be learnt? Critical aspects of elementary arithmetic skills. Educational Studies in Mathematics, 107(2), 261–284. https://doi.org/10.1007/s10649-021-10045-0 Björklund, C., & Palmér, H. (2022). Teaching toddlers the meaning of numbers - Connecting modes of mathematical representations in book reading. Educational Studies in Mathematics, 110(3), 525–544. https://doi.org/10.1007/s10649-022-10147-3. Björklund, C., & Palmér, H. (2025). Enhancing Swedish toddlers’ learning opportunities through interactions with pictures and narrative designed for numerical learning purposes. Early Childhood Education Journal, 53, 37–48. https://doi.org/10.1007/s10643-02301556-x Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131. https://doi.org.10.1007/s10649-006-0400-z Elgavi, O., & Hamo, P. (2024). Math on the brain: Seven principles from neuroscience for early childhood educators. Early Childhood Education Journal. https://doi.org/10.1007/s10643-024-01656-2 Frey, M., Gashaj, V., Nuerk, H. C., & Moeller, K. (2024). You can count on your fingers: Finger-based intervention improves first-graders’ arithmetic learning. Journal of Experimental Child Psychology, 244, 105934. https://doi.org/10.1016/j.jecp.2024.105934 Jay, T., & Betenson, J. (2017, June). Mathematics at your fingertips: Testing a finger training intervention to improve quantitative skills. In Frontiers in Education (Vol. 2, p. 22). Frontiers Media SA. https://doi.org.10.3389/feduc.2017.00022 Krenger, M., & Thevenot, C. (2025). Do children need counting principle knowledge to count on their fingers?. Journal of Experimental Child Psychology, 249, 106073. https://doi.org/10.1016/j.jecp.2024.106073 Marton, F. (2015). Necessary conditions of learning. Routledge. Neuman, D. (1987). The origin of arithmetic skills. Acta Universitatis Gothoburgensis. Ollivier, F., Noël, Y., Legrand, A., & Bonneton-Botté, N. (2020). A teacher-implemented intervention program to promote finger use in numerical tasks. European Journal of Psychology of Education, 35(3), 589-606. https://doi.org/10.1007/s10212-019-00441-9 Paliwal, V., & Baroody, A. J. (2020). Cardinality principle understanding: The role of focusing on the subitizing ability. ZDM Mathematics Education, 52(4) 649–661. https://doi.org/10.1007/s11858-020-01150-0 Palmér, H., & Björklund, C. (2023). The teaching of numbers in common preschool activities: A delicate balancing act. Early Childhood Education Journal, 51(5), 971–980. https://doi.org/10.1007/s10643-022-01354-x Roesch, S., Moeller, K., & Bahnmueller, J. (2024). Finger counting, finger number gesturing, and basic numerical skills: A cross-sectional study in 3- to 5-year-olds. Journal of Experimental Child Psychology, 242, 105892. https://doi.org/10.1016/j.jecp.2024.105892

203 210 Sarah Hopkins Janette Bobis James Russo Jane Hubbard Anne Roche Monash University, Melbourne, Australia The University of Sydney, Sydney Australia Deakin University, Geelong Australia Three Grade 4 students (aged 9-10) were selected for an intervention because of their problematic reliance on counting-based strategies for single-digit addition. Using a multiple baseline design, we examined their daily use of retrieval and derived-fact strategies before and after an intervention over a period of 20 days. Although one participant benefited greatly from the intervention, the other two showed only limited improvement. Using microgenetic methods to analyse data, we examined features of participants’ prior performances that appeared to enable or hamper the development of derived-fact strategies. Findings provide evidence of the critical knowledge students need to be able to bridge through 10. Canobi, K. H. (2009). Concept–procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102(2), 131–149. http://dx.doi.org/10.1016/j.jecp.2008.07.008 Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552. https://doi.org/10.1037/a0025510 Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552. Graven, M., & Venkat, H. (2021). Piloting national diagnostic assessment for strategic calculation. Mathematics Education Research Journal, 33(1), 23–42. https://doi.org/10.1007/s13394-019-00291-0 Hopkins, S., & Bayliss, D. (2017). The prevalence and disadvantage of min-counting in seventh grade: Problems with confidence as well as accuracy? Mathematical Thinking and Learning, 19(1), 19–32. https://doi.org/10.1080/10986065.2017.1258613 Hopkins, S., & Lawson, M. (2002). Explaining the acquisition of a complex skill: Methodological and theoretical considerations uncovered in the study of simple addition and the moving-on process. Educational Psychology Review, 14(2), 121–154. https://doi.org/10.1023/A:1014629604663 Hopkins, S., Russo, J., & Bobis, J. (2023). Why many children persist with counting. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel & M. Tabach (Eds.). Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 83-90). PME. Hopkins, S., Russo, J., & Siegler, R. (2022). Is counting hindering learning? An investigation into children’s proficiency with simple addition and their flexibility with mental computation strategies. Mathematical Thinking and Learning, 24(1), 52-69. https://doi.org/10.1080/10986065.2020.1842968. Kullberg, A. & Nord, M. (2025). Experiencing part-whole relations of numbers in a partitioning task. Nordic Studies in Mathematics Education, 30(2), 101-121. Kullberg, A., & Björklund, C. (2020). Preschoolers’ different ways of structuring part-partwhole relations with finger patterns when solving an arithmetic task. ZDM, 52(4), 767– 778. https://doi.org/10.1007/s11858-019-01119-8 Murata, A. (2004). Paths to learning ten-structured understandings of teen sums: addition solution methods of Japanese grade 1 students. Cognition and Instruction, 22(2), 185-218. Russo, J., Hopkins, S., Hubbard, J., & Roche, A. (2025). Affordances of a Novel Mathematical Manipulative: Teacher Perspectives on the Keyboard. In S. M. Patahuddin, L. Gaunt, D. Harris & K. Tripet (Eds.), Proc. 47th Annual Conf. of the Mathematics Education Research Group of Australasia (pp. 397–404). MERGA. Shrager, J., & Siegler, R. S. (1998). SCADS: A model of children's strategy choices and strategy discoveries. Psychological Science, 9(5), 405–410. https://doi.org/10.1111/14679280.00076 Siegler, R. (2005). Children’s learning, American Psychologist, 60, 769-778. Siegler, R. & Cowley, K. (1991). The microgenetic method: A direct means for studying cognitive development. American psychologist, 46(6), 606.

211 218 Alessandra Boscolo Silvia Funghi Matteo Casale LUMSA University of Rome Italy University of Genoa I.I.S. “Norberto Bobbio” We report a design-based case study on grade 11 students’ sense-making of sinusoidal parameters in a GeoGebra environment that dynamically couples the unit circle and the Cartesian graph. Drawing on Ainsworth’s DeFT framework, the coupling was designed to serve a constraining function while students predicted and justified graphs associated with transformed motions (e.g., phase shift, reversed rotation). Analyses of small-group discourse and interviews reveal that the constraint can serve as a resource, supporting students’ monitoring and self-correction; however, it can also be interpreted in locally plausible ways that hinder the intended coordination across representations. Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learn. Instr., 16(3), 183–198. Bornstein, N. (2020). Teaching transformations of trigonometric functions with technology. J. Interact. Media Educ., 2020(1), 1–9. https://doi.org/10.5334/jime.503 Carlsen, M. (2018). Upper secondary students’ mathematical reasoning on a sinusoidal function. Educ. Stud. Math., 99(3), 277–291. https://doi.org/10.1007/s10649-018-9844-1 Kidron, I. (2020). Calculus teaching and learning. In S. Lerman (Ed.), Encycl. Math. Educ. (2nd ed., pp. 87–94). Springer. https://doi.org/10.1007/978-3-030-15789-0_18 Kristinsdóttir, B., Hreinsdóttir, F., & Lavicza, Z. (2020). Using silent video tasks for formative assessment. In B. Barzel, et al. (Eds.), Proc. ICTMT 14 (pp. 189–196). Duisburg-Essen. Palatnik, A., Abrahamson, D., Baccaglini-Frank, A., Ng, O.-L., Shvarts, A., & Swidan, O. (2023). Theory and practice of designing embodied mathematics learning. In M. Ayalon et al. (Eds.), Proc. PME 46 (Vol. 1, pp. 159–189). PME. Robutti, O. (2009). Teacher’s semiotic games in mathematics laboratory. Jornal Internacional de Estudos em Educação Matemática, 1(1), 1–25. Presmeg, N. (2008). Trigonometric connections through a semiotic lens. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 103–119). Sense Publishers. Shvarts, A., & Alberto, R. (2021). Melting cultural artifacts back to personal actions: Embodied design for a sine graph. In M. Inprasitha et al. (Eds.), Proc. PME 44 (Vol. 4, pp. 49–56). PME. Shvarts, A., & van Helden, G. (2023). Embodied learning at a distance: From sensory-motor experience to constructing and understanding a sine graph. Math. T. L.., 25(4), 409–437. Weber, K. (2005). Students’ understanding of trigonometric functions. Math. Educ. Res. J., 17(3), 91–112.

219 226 Nicola Bretscher Jill Adler University College London United Kingdom University of Witwatersrand South Africa We investigate how university-based mathematics teacher educators (MTEs) use an ideational resource when providing feedback to pre-service teachers after observing them teach a lesson in secondary school. The specific ideational resource was a framework for teaching mathematics introduced into university-taught sessions of our pre-service teacher education programme to support lesson-planning. We selected two MTE-pre-service-teacher pairs as telling cases to illuminate when and how MTEs use elements of the framework in providing lesson observation feedback. For each pair, we analysed two lesson observations, drawing on notions of situated abstraction and transparency. We discuss our initial findings and implications. Adler, J. (2021). Levering change: the contributory role of a mathematics teaching framework. ZDM Mathematics Education, 53(6), 1207–1220. https://doi.org/10.1007/s11858-021-01273-y Barbosa, J.C. (2025). Bridge-building as a dimension of mathematics teacher educators’ expertise. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.). Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 83–90). PME. Bretscher, N., Adler, J., Saunders, P., & Ghosh, S. (2025). Exploring university tutors’ use of a mathematics teaching framework when providing lesson observation feedback in initial teacher education. In T. Fujita (Ed.), Proceedings of the British Society for Research into Learning Mathematics, 45(3). Lave, J., & Wenger, E. (1991). Situated learning: legitimate peripheral participation. Cambridge University Press. https://doi.org/10.1017/CBO9780511815355 Miles, M.B., Huberman, A. M., & Saldaña, J. (2020). Qualitative data analysis: a methods sourcebook. Fourth edition. SAGE. Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Kluwer Academic Publishers. Pepin, B., & Gueudet, G. (2020). Curriculum resources and textbooks in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 172–176). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_40 Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255–281. doi:10.1007/s10857-005-0853-5 Wenger, E. (1998). Communities of practice: learning, meaning, and identity. Cambridge University Press. https://doi.org/10.1017/CBO9780511803932

227 234 Beatrice Bruno Franco Marzia Garzetti Ornella Robutti Università di Torino, University of Applied Sciences and Arts of Southern Switzerland (SUPSI) Switzerland AR/VR research in mathematics education has highlighted new possibilities for embodied interaction with mathematical objects. Less is known about how collaboration is organised in asymmetric classroom settings where only one student has immersive access to AR/VR and peers rely on a mirrored display. This qualitative case study examines mirroring, the real-time projection of the headset user’s view onto a shared screen, in a Grade 8 task on translations in the prototype Transforming Math (TM). Data were analysed through multimodal semiotic analysis grounded in the semiotic bundle to trace joint attention and joint action. Findings shows that mirroring stabilises shared referents, enables request, execution – monitoring - coordination, supports conjecture checking, and fosters shared semiotic resources. Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM, 34, 66–72. https://doi.org/10.1007/BF02655708 Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70, 97–109. https://doi.org/10.1007/s10649-008-9163-z Arzarello, F., Robutti, O., & Thomas, M. (2015). Growth point and gestures: Looking inside mathematical meanings. Educational Studies in Mathematics, 90, 19–37. https://doi.org/10.1007/s10649-015-9611-5 Böckler, A., & Sebanz, N. (2013). Linking joint attention and joint action. In J. Metcalfe & H. Terrace (Eds.), Agency and joint attention (pp. 206–215). Oxford University Press. Bulut, M., & Borromeo Ferri, R. (2023). A systematic literature review on augmented reality in mathematics education. European Journal of Science and Mathematics Education, 11(3), 556–572. https://doi.org/10.30935/scimath/13124 Shvarts, A., & Abrahamson, D. (2019). Dual-eye-tracking Vygotsky: A microgenetic account of a teaching/learning collaboration in an embodied-interaction technological tutorial for mathematics. Learning, Culture Social Interaction, 22. https://doi.org/10.1016/j.lcsi.2019.05.003 Walkington, C., Nathan, M. J., Huang, W., et al. (2024). Multimodal analysis of interaction data from embodied education technologies. Educational Technology Research and Development, 72, 2565–2584. https://doi.org/10.1007/s11423-023-10254-9

235 242 Neslihan Bulut Rita Borromeo Ferri University of Kassel, Institute of Mathematics Germany This qualitative case study investigates how two pre-service primary mathematics teachers developed their understanding of Education for Sustainable Development through an ESD-modelling seminar. Analysis of interviews, a designed ESD-mathematical modelling task, and written reflections revealed a shift from limited, behaviour-focused views of sustainability toward a broader understanding of ESD and the role of mathematics. The findings show that mathematical modelling supported engagement with sustainability issues, while also posing challenges related to scaffolding and task complexity. The study underscores the value of integrating ESD-modelling tasks into primary mathematics teacher education. Alsina, Á., & Silva-Hormazábal, M. (2023). Promoting mathematics teacher education for sustainability through a STEAM approach. Avances de Investigación en Educación Matemática, 23. Alsina, Á., & Vásquez, C. (2024). Professional development and teacher agency in mathematics teacher education for sustainability. Mathematics Education Research Journal, 36, 1–24. https://doi.org/10.1007/s13394-024-00488-y Borromeo Ferri, R. (2025). Mathematical modelling as a key practice for promoting learners’ competencies targeting the transformation of global sustainability. In Le Roux et al. (Eds.), Proceedings of the 27th ICMI Study Conference (Mathematics Education and the SocioEcological), Quezon City (Philippines) (pp. 284-291). MATHTED and ICMI. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. Bulut, N., & Borromeo Ferri, R. (2025). Bridging mathematical modelling and education for sustainable development in pre-service primary teacher education. Education Sciences, 15(2), 248. https://doi.org/10.3390/educsci15020248 English, L. D., & Watters, J. J. (2005). Mathematical modelling in the early school years. Mathematics Education Research Journal, 16(3), 58–79. García-Alonso, I., Sosa-Martín, D., & Trujillo-González, R. (2023). Assessing sustainability competencies present in class proposals developed by prospective mathematics teachers. Avances de Investigación en Educación Matemática, 23. Just, J., Siller, H. S., & Vorhölter, K. (2025). Directions of intentionalities in education for sustainable development in mathematics education: a systematic literature review. Educational Studies in Mathematics, 1-25. https://doi.org/10.1007/s10649-025-10449-2 Makramalla, M., Coles, A., le Roux, K., & Wagner, D. (2025). Mathematics education for sustainable futures: a strengths-based survey of the field to invite further research action. Educational Studies in Mathematics, 119(3), 535-556. Merriam, S. B. (2009). Qualitative research: a guide to design and implementation. Steffensen, L. (2023). Sustainability and mathematical modelling in 5th grade. Prometeica Revista De Filosofía Y Ciencias, 27, 241–251. UNESCO. (2017). Education for sustainable development goals: Learning objectives. UNESCO. Available online: https://www.unesco.org/en/articles/education-sustainabledevelopment-goals-learning-objectives (accessed on 20 November 2025). Vesterinen, M., & Ratinen, I. (2024). Sustainability competences in primary school education-A systematic literature review. Environmental Education Research, 30, 56-67. Vorhölter, K., Siller, H.S., Just, J., Orschulik, A. B., & Zieriacks, C. (2025) Education for sustainable development in mathematics classrooms. Environmental Education Research, 1-30. https://doi.org/10.1080/13504622.2025.2565461 Wiegand, S., & Borromeo Ferri, R. (2023). Promoting pre-service teachers’ professionalism in steam education and education for sustainable development through mathematical modelling activities. ZDM–Mathematics Education, 55, 1269–1282. Wiegand, S. & Borromeo Ferri, R. (in press). Die Baumwollproduktion in Kasachstan zur Jeansherstellung- eine BNE-Modellierungsaufgabe. In Siller, H.S. & Vorhölter, K. (Eds.) ISTRON-Series, Vol. 11. Springer

243 250 Ernesto Daniel Calleros California State University Fresno United States This paper examines the community cultural wealth (CCW) mobilized by Johan, an international, multilingual undergraduate student from Malaysia, as he participated in an inquiry-based mathematics course at a U.S. university. Drawing on Yosso’s (2005) CCW framework, this paper presents an empirical counterstory that makes visible the linguistic, spiritual, social, navigational, aspirational, and resistance capital Johan leveraged in response to English-dominant, religiously blind, and verbally oriented participation norms. This study extends CCW research into undergraduate mathematics education by illustrating the labor required to persist in inquiry-based classrooms that are often assumed to be equitable. Cortes, A. (2017). The experiences and studying practices of community college Latin@ STEM majors taking Calculus I (Unpublished doctoral dissertation). University of California, Davis. Dika, S. L., Pando, M. A., Tempest, B. Q., & Allen, M. E. (2017). Examining the cultural wealth of underrepresented minority engineering persisters. Journal of Professional Issues in Engineering Education and Practice, 144(2), 1–9. Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences of the United States of America, 111(23), 8410–8415. Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Johnson, E., Andrews-Larson, C., Keene, K., Melhuish, K., Keller, R., & Fortune, N. (2020). Inquiry and gender inequity in the undergraduate mathematics classroom. Journal for Research in Mathematics Education, 51(4), 504–516. Ladson-Billings, G., & Tate, W. F. (1995). Toward a critical race theory of education. Teachers College Record, 97(1), 47–68. Laursen, S. L., & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 5(2), 129–146. Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Qualitative data analysis: A methods sourcebook (3rd ed.). Sage. Rios, J. (2023). The role of language-as-resource and language-as-political in college mathematics courses. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 91–98). PME. Solórzano, D. G., & Delgado Bernal, D. (2001). Examining transformational resistance through a critical race and LatCrit theory framework: Chicana and Chicano students in an urban context. Urban Education, 36(3), 308–342. Solórzano, D. G., & Yosso, T. J. (2002). Critical race methodology: Counter-storytelling as an analytical framework for education research. Qualitative Inquiry, 8(1), 23–44. Yosso, T. J. (2005). Whose culture has capital? A critical race theory discussion of community cultural wealth. Race, Ethnicity and Education, 8(1), 69–91.

251 258 Marijke Ann Campbell Simon Fraser University Canada This paper argues that Bakhtin’s concept of addressivity (the orientation of the speaker toward an addressee) can enrich interactionist gesture models to explain why gesture scale changes during mathematical communication. The analysis shows that during moments of conflicting perspectives of mathematics (in asymmetric engagement), recapitulations occur alongside larger, whole-body gestures. Conversely, when children share the same perspective on mathematics (in symmetric engagement), they produce small-scale gestures with recapitulations. Shared perspectives are seen to coincide with children's adoption of the other’s gestures (hereafter, a gestural relation of agreement). The analysis argues that, in a Bakhtinian sense, speaker gestures are anticipatory, responsive, and multi-voiced. Bakhtin, M. M. (2010). The dialogic imagination: Four essays. University of Texas Press. Gerofsky, S. (2011). Seeing the graph vs. being the graph. Integrating gestures, 4, 254–265. Hostetter, A.B., & Alibali, M.W. (2019). Gesture as simulated action: Revisiting the framework. Psychonomic Bulletin and Review, 26, 721 – 752. Pimm, D. & Sinclair, N. (2009). Audience, style and criticism. For the Learning of Mathematics, 29(2), 23–27. Spronck, S. (2020). Grammar and levels of Addressivity: Exploring Ungarinyin engagement. Open Linguistics, 6(1), 1–18. https://doi.org/10.1515/opli-2020-0001 Streeck, J. (2009). Gesturecraft. John Benjamins Publishing. https://doi.org/10.1075/gs.2

259 266 Maria Giuseppina Adesso Roberto Capone Eleonora Faggiano Oriana Fiore University of Bari Aldo Moro Italy This paper proposes a theoretical framework for understanding interdisciplinarity in mathematics education as a comparison of rationalities. To this aim we refer to Habermas’s construct of rationality and conceptualise disciplinary borders as semiotic spaces of negotiation, according to the the lens offered by De Luca Picione and Valsiner. The framework is operationalised by analysing a case study, based on the Fagnano’s problem, illustrating how students navigate mathematical and physical rationalities. The findings show how communicative acts mediate epistemological negotiations at the border between the disciplines, supporting the co-construction of meaning and a synthesised interdisciplinary understanding Adesso, M. G., Capone, R., & Fiore, O. (2025). Teaching on the Border Between Mathematics and Physics. Through the History of Mathematics: Fagnano’s Problem. In History and Epistemology in Mathematics Education: Trends, Practices, Future Developments (pp. 181-194). Cham: Springer Nature. Bakker, A., & Akkerman, S. F. (2014). A boundary-crossing approach to support students’ integration of statistical and work-related knowledge. Educational Studies in Mathematics, 86(2), 223–237. https://doi.org/10.1007/s10649-013-9517-z Boero, P. (2006). Habermas’ theory of rationality as a comprehensive frame for conjecturing and proving in school. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 185-192). Capone, R. (2022). Interdisciplinarity in Mathematics Education: From Semiotic to Educational Processes. EURASIA Journal of Mathematics, Science and Techn. Ed., 18(2). Capone, R. & Faggiano, E. (2024) The border as a semiotic cross-cultural space: Semiotics and Pragmatics aspects of Interdisciplinarity in Mathematics Education. For the Learning in Mathematics, 44(3), 22–28. De Luca Picione, R. & Valsiner, J. (2017). Psychological functions of semiotic borders in sense-making: Liminality of narrative processes. Europe’s journal of psychology, 13(3), 532–547. https://doi.org/10.5964/ejop.v13i3.1136 Goos, M., Carreira, S., & Namukasa, I. K. (2023). Mathematics and interdisciplinary STEM education: recent developments and future directions. ZDM, 55(7), 1199–1217. https://doi.org/10.1007/s11858-023-01533-z Habermas, J. (1981). Theorie des kommunikativen. Handelns, Suhrkamp. Klein, J. T. (1990). Interdisciplinarity: History, theory, and practice. Wayne Univ. Press. Star, S. L. (1989). The structure of ill-structured solutions: Boundary objects and heterogeneous distributed problem solving. In L. Gasser & M. Huhns (Eds.), Distributed artificial intelligence (pp. 37–54). Morgan Kaufmann. Vergnaud, G. (2013). Conceptual development and learning. Revista Qurriculum, 26, 39–59.

267 274 Busra Caylan-Ergene Ozkan Ergene Igor’ Kontorovich Sakarya University Turkey University of Hong Kong This paper examines how pre-service teachers interact with AI-generated responses, with attention to perceived credibility of AI outputs and their impact on teachers’ solutions. Twelve participants completed a task on representing eπ on the number line and subsequently interacted with a specially designed GPT that deliberately produced erroneous responses. GPT responses included hallucinations primarily involving fabricated concepts, principles, and academic sources. Only one participant noted and questioned the validity of AI-generated responses, whereas others endorsed them as shared. Delayed recognition, identified in the post-interaction written reflections when participants revisited their earlier GPT dialogues, was linked to limited subject-matter confidence, the professional tone of AI explanations, and trust in AI. Caylan Ergene, B., & Ergene, Ö. (2020). Repeating decimals and irrational numbers on the number line: Through the lens of pre-service and in-service mathematics teachers. Acta Didactica Napocensia, 13(2), 215–232. https://doi.org/10.24193/adn.13.2.15 Cui, Y., & Zhang, H. (2025). Can student accurately identify artificial intelligence generated content? An exploration of AIGC credibility from user perspective in education. Education and Information Technologies, 30, 16321–16346. https://doi.org/10.1007/s10639-02513448-1 Elsayed, H. (2024). The impact of hallucinated information in large language models on student learning outcomes: A critical examination of misinformation risks in AI-assisted education. Northern Reviews, 11. Geiger, V., Scheiner, T., Beswick, K., Fielding, J., Kaiser, G., Goos, M., Fernandez, K., & Arwan, S. (2025). Critical mathematical thinking: a framework and illustrative examples. In K. Le Roux, A. Coles, A. SolaresRojas, A. Bose, C. Vistro-Yu, P. Valero, N. Sinclair, M. Makramalla, R. Gutiérrez, V. Geiger, M. Borba. (Eds.), Proceedings of the 27th ICMI Study Conference (Mathematics Education and the Socio-Ecological) (pp. 307-314). ICMI and MATHTED Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997). Culturally relevant mathematics teaching in a Mexican American context. Journal of Research in Mathematics Education, 28, 709–737. https://doi.org/10.2307/749639 Leal Filho, W., Kim, E., Borsatto, J. M. L. S., & Marcolin, C. B. (2025). Using artificial intelligence in sustainability teaching and learning. Environmental Sciences Europe, 37(124). https://doi.org/10.1186/s12302-025-01159-w Mustaffa, N. E., Lai, K. E., Preece, C. N., & Wong, F. Y. (2025). A bibliometric review of large language model hallucination. International Journal of Research and Innovation in Social Science (IJRISS), 9(9), 5025–5037. https://dx.doi.org/10.47772/IJRISS.2025.909000409 Torres, R. C. (2024). AI hallucination in the context of education: Exploring college students’ use of generative AI for academic tasks. Mapúa Malayan Colleges Laguna. Ye, H., Xu, J., Huang, D., Xie, M., Guo, J., Yang, J., Bao, H., Zhang, M., & Zheng, C. (2025). Assessment of large language models’ performances and hallucinations for Chinese postgraduate medical entrance examination. Discover Education, 4(59). https://doi.org/10.1007/s44217-025-00446-1

275 282 Maria Chimoni Constantinos Christou Demetra Pitta-Pantazi University of Cyprus This study compares four classroom-based interventions designed to support Grade 6 students’ understanding of equation solving. The first intervention employed a digital environment that emphasized working with non-letter-symbolic and letter-symbolic equations via a balance-scale model. The second intervention employed a digital environment focused exclusively on transformational activities with numerical equalities and letter-symbolic-equations. The third intervention combined these two approaches, while the fourth relied solely on the mathematics textbook. Using a pre-test–post-test design, learning outcomes were analyzed. Results indicated that the combined intervention was the most effective, indicating that coordinating modeling and transformational activities can support algebraic thinking. Ellis, A. B., & Özgür, Z. (2024). Trends, insights, and developments in research on the teaching and learning of algebra. ZDM–Mathematics Education, 56, 199–210. https://doi.org/10.1007/s11858-023-01545-9 Kapon, S., Halloun, A., & Tabach, M. (2019). Incorporating a digital game into the formal instruction of algebra. Journal for Research in Mathematics Education, 50(5), 555–591. Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). Routledge. Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8(1), 139–151. Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM–Mathematics Education, 54, 1131– 1150. https://doi.org/10.1007/s11858-022-01435-6 Kieran, C., & Martinez-Hernandez, C. (2022). Coordinating invisible and visible sameness within equivalence transformations of numerical equalities by 10-to 12-year-olds in their movement from computational to structural approaches. ZDM–Mathematics Education, 54, 1215–1227. https://doi.org/10.1007/ s11858-022-01355-5 Lenz, D. (2022). The role of variables in relational thinking: an interview study with kindergarten and primary school children. ZDM–Mathematics Education. https://doi.org/10.1007/s11858-022-01419-6 Naftaliev, E., & Yerushalmy, M. (2022). Guiding students’ learning with an interactive diagram: The case of equations. Journal of Mathematical Behavior, 68, 101008. https://doi.org/10.1016/j. jmathb.2022.101008 Otten, M., Van den Heuvel-Panhuizen, M., Veldhuis, M., Boom, J., & Heinze, A. (2020). Are physical experiences with the balance model beneficial for students’ algebraic reasoning? An evaluation of two learning environments for linear equations. Education Sciences, 10(6), 163. https://doi.org/10.3390/educsci10060163 Pitta-Pantazi, D., Chimoni, M., & Christou, C. (2024). Revisiting the relationship of arithmetical thinking and letter-symbolic algebra. International Journal of Science and Mathematics Education, 23(3), 689–711. https://doi.org/10.1007/s10763-024-10493-z Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26, 257–277. https://doi.org/10.1007/s13394013-0087-2 Radford, L. (2022). Early algebra: Simplifying equations. In J. Hodgen, E. Geraniou, G. Bolondi, & F. Ferretti (Eds.), Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) (pp. 588–595). ERME. Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49(3), 341–359. https://doi.org/10.1023/A:1020229023965 Xie, S., & Cai, J. (2022). Fifth graders’ learning to solve equations: the impact of early arithmetic strategies. ZDM–Mathematics Education. https://doi.org/ 10.1007/s11858-02201417-8.

283 290 Maria Chiara Cibien University of Ferrara Italy This study investigates how secondary school mathematics teachers identify and interpret students’ errors in a geometry item, analysed through the theoretical lens of spatial reasoning. The item was first administered to a sample of 10th-grade students to examine whether students’ errors were framed by spatial reasoning. Subsequently, a questionnaire was administered to 19 in-service secondary school teachers to collect information on their interpretations. Results indicate that while some teachers correctly identify spatial errors, those who do not adopt a spatial reasoning perspective tend to have greater difficulty in recognizing and interpreting students’ errors. These findings highlight the need to strengthen teachers’ hermeneutical competence through targeted professional development programs. Arzarello, F., & Ferretti, F. (2022). Links between the INVALSI Mathematics test and teaching practices: an exploratory study. In P. Falzetti (Ed.), INVALSI data to investigate the characteristics of students, schools and society (pp. 96–109). FrancoAngeli. Bishop, A. J. (1980). Spatial abilities and mathematics education—A review. Educational Studies in Mathematics, 11(3), 257–269. https://doi.org/10.1007/BF00697739 Bolondi, G., & Ferretti, F. (2021). Quantifying solid findings in mathematics education: Loss of meaning for algebraic symbols. International Journal of Innovation in Science and Mathematics Education, 29(1), 1–15. https://dx.doi.org/10.30722/IJISME.29.01.001 Burte, H., Gardony, A. L., Hutton, A., & Taylor, H. A. (2020). Elementary teachers’ attitudes and beliefs about spatial thinking and mathematics. Cognitive Research: Principles and Implications, 5(1), 17. https://doi.org/10.1186/s41235-020-00221-w Cibien, M. C. (2025). Pre-service teachers’ professional development and mathematics Large-Scale Assessment. In M. Bosh, S. Carreira, G. Bolondi, M. Gaidoschik & C. Spagnolo (Eds.), Proceedings of CERME14. Free University of Bozen-Bolzano; ERME. Cibien, M. C. (2026). Teachers’ engagement with students’ errors in mathematics: Introducing Hermeneutical Competence. In J. Fauskanger, F. Ferretti, R. Mosvold, & N. Vondrová (Eds.), Proceedings of the Nineteenth ERME Topic Conference: Connecting the Learning of Mathematics Teaching to Practice. Charles University in Prague, Faculty of Education; ERME. Cohen, L., Manion, L., & Morrison, K. (2018). Research Methods in Education (8th ed.). Routledge. Hawes, Z., Tepylo, D., & Moss, J. (2015). Developing spatial reasoning. In B. David & The Spatial Reasoning Study Group (Eds.), Spatial reasoning in the early years (pp. 29–44). Routledge. Heinrichs, H., & Kaiser, G. (2018). Diagnostic competence for dealing with students’ errors: Fostering diagnostic competence in error situations. In T. Leuders, K. Philipp, & J. Leuders (Eds.), Diagnostic competence of mathematics teachers: Unpacking a complex construct in teacher education and teacher practice (Vol. 11, pp. 79–94). Springer. Hidayat, R., Saad, M. R. M., Ismail, N., & Noor, M. A. M. (2023). Systematic review of spatial thinking in mathematics education. In Educational Studies: Conference Series, 3(1), 47–60). https://doi.org/10.30872/escs.v3i1.2571 Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. British Journal of Educational Psychology, 87(2), 170–186. https://doi.org/10.1111/bjep.12142 Lowrie, T., Resnick, I., Harris, D., & Logan, T. (2020). In search of the mechanisms that enable transfer from spatial reasoning to mathematics understanding. Mathematics Education Research Journal, 32(2), 175–188. https://doi.org/10.1007/s13394-020-003369 Patton, M. Q. (1980). Qualitative Evaluation Methods. Sage. Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM – The International Journal on Mathematics Education, 40(2), 165–178. https://doi.org/10.1007/s11858-008-0086-z Ribeiro, M., Mellone, M., & Jakobsen, A. (2013). Characterizing prospective teachers’ knowledge in/for interpreting students’ solutions. In Proc. 37th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 89–96). PME. Sinclair, N., & Bruce, C. D. (2014). Spatial reasoning for young learners. In Proc. 38th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 5, pp. 15–20). PME. Viola, G., Ferretti, F., Gambini, A., Martignone, F., Soldano, C., & Spagnolo, C. (2024). Teachers’ professional development and mathematics LSA: first results of National Project. In Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 223–224). PME.

291 298 Francesco Contel Annalisa Cusi Sapienza University of Rome Italy In this paper, we explore secondary students’ interaction with GPT-4o in the context of activities of proof construction through algebraic language with the aim of investigating how the interplay between internal and external factors affects the represent-manipulate-interpret essential algebraic cycle. A qualitative micro-analysis of chats, students’ interviews and their written work was conducted. The findings show that bias and hallucinations in the GenAI tool interact with students’ epistemic beliefs, producing different forms of short circuits of the algebraic cycle. Implications are raised for the design of GenAI systems that support algebraic reasoning as meaning-making rather than procedural execution. Arzarello, F., Bazzini, L., & Chiappini, G. (2001). A model for analyzing algebraic thinking. In R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds.), Perspectives on school algebra (pp 61–81). Kluwer. Bell, A. (1996). Problem-Solving Approaches to Algebra: Two Aspects. In N. Bernarz, C. Kieran & L. Lee (Eds.), Approaches to Algebra. Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds.), Perspectives on school algebra (pp 99–119). Kluwer. Contel, F., & Cusi, A. (2025). Investigating the Role of ChatGPT in Supporting Metacognitive Processes During Problem-Solving Activities. Digital Experiences in Mathematics Education, 11, 167–191. https://doi.org/10.1007/s40751-024-00164-7 Cusi, A., & Contel, F. (2026). A reconceptualisation of the instrumental genesis process in the context of human-LLM interactions: exploring students' perspective on their use of GPT-4o as a tool to support self-assessment. In E. Geraniou, C. Crisan & M. Mavrikis (Eds.), Digital Technology and Artificial Intelligence in Mathematics Education Assessment. Routledge. https://doi.org/10.4324/9781003533764-8 Frieder, S., Bayer, J., Collins, K. M., Berner, J., Loader, J., Juhász, A., Ruehle, F., Welleck, S., Poesia, G., Griffiths, R., Weller, A., Goyal, A., Lukasiewicz, T., Freer, C., Buzzard, K. & Gowers, T. (2025). Data for mathematical copilots: Better ways of presenting proofs for machine learning. arXiv preprint arXiv:2412.15184 Harel, G., & Sowder, L. (1998). Students’ proof schemes: results from exploratory studies. In E. Dubinsky, A. Shoenfeld e J. Kaput (Eds.), Research Issues in collegiate Mathematics Education III (pp. 234–283). American Mathematical Society. Huang, L., Yu, W., Ma, W., Zhong, W., Feng, Z., Wang, H., Chen, Q., Peng, W., Feng, X., Qin, B. & Liu, T. (2025). A survey on hallucination in large language models: Principles, taxonomy, challenges, and open questions. ACM Transactions on Information Systems, 43(2), 1–55. https://doi.org/10.1145/3703155 Rabardel, P. (2002). People and technology – a cognitive approach to contemporary instruments. https://hal.archives-ouvertes.fr/hal-01020705 Yoon, H., Hwang, J., Lee, K., Roh, K. H., & Kwon, O. N. (2024). Students’ use of generative artificial intelligence for proving mathematical statements. ZDM – Mathematics Education, 56(7), 1531–1551. https://doi.org/10.1007/s11858-024-01629-0 Zhuang, Y. (2025). Lessons from Using ChatGPT in Calculus: Insights from Two Contrasting Cases. Journal of Formative Design in Learning, 1–11.

299 306 Annalisa Cusi Despina Potari Sapienza University of Rome Italy National and Kapodistrian University of Athens Greece This paper investigates how tasks designed by mathematics teacher educators can support prospective mathematics teachers in using research-based theoretical ideas within teaching-related practices. Drawing on the Meta-Didactical Transposition model, seven meta-tasks were identified and characterised in relation to the research and teaching practices they mobilise. An a priori analysis of meta-tasks was developed to highlight the boundary objects involved and the potential actions at syntactic, semantic and pragmatic levels that may foster internalisation processes. Arzarello, F., Robutti, O., Sabena, C., Cusi, A., Garuti, R., Malara, N.A., & Martignone, F. (2014). Meta-Didactical Transposition: A Theoretical Model for Teacher Education Programs. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era, Mathematics Education in the Digital Era (pp. 347–372). Springer. https://doi.org/10.1007/978-94-007-4638-1_15 Carlile, P. R. (2004). Transferring, translating, and transforming: An integrative framework for managing knowledge across boundaries. Organization Science, 15(5), 555–568. https://doi.org/10.1287/orsc.1040.0094 Cusi, A., Robutti, O., Panero, M., Taranto, E., & Aldon, G. (2022). Meta-Didactical Transposition.2: the evolution of a framework to analyse teachers' collaborative work with researchers in technological settings. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), Mathematics Teacher in the Digital Era – 2nd Edition (pp. 365–389). Springer. https://doi.org/10.1007/978-3-031-05254-5_14 Jackson, A., Serbin, K. S., & Bae, Y. (2024). Prospective and in-service teachers’ use of pedagogical mathematical practices in approximations of teaching practice. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-024-09669-9 Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of mathematics teacher education, 9(2), 187–211. https://doi.org/10.1007/s10857-005-1223-z Lendínez Muñoz, E. M., García García, F. J., Lerma Fernández, A. M., & Abril Gallego, A. M. (2024). Increase in self-efficacy in prospective teachers through theory-based lesson study. Journal of Mathematics Teacher Education, 27(4), 717–742. https://doi.org/10.1007/s10857-023-09597-0 Robutti, O., Aldon, G., Cusi, A., Olsher, S., Panero, M., Cooper, J., Carante, P., & Prodromou, T. (2019). Boundary objects in mathematics education and their role across communities of teachers and researchers in interaction. In G.M. Lloyd & O. Chapman (Eds.), International Handbook of Mathematics Teacher Education (second edition). (Volume 3, pp. 211–240). Brill - Sense Publisher. https://doi.org/10.1163/9789004419230_009 Rotem, S. H., Potari, D., & Psycharis, G. (2024). Using critical incidents as a tool for promoting prospective teachers’ noticing during reflective discussions in a fieldworkbased university course. Educational Studies in Mathematics, 117(1), 67–95. https://doi.org/10.1007/s10649-024-10336-2 Ruthven, K., & Goodchild, S. (2015). Knowledge creation through dialogic interaction between the practices of teaching and researching. In L.D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education – 3rd edition (pp. 523–540). Routledge. Star, S. L., & Griesemer, J. (1989). Institutional ecology, ‘translations’, and boundary objects: Amateurs and professionals on Berkeley’s museum of vertebrate zoology. Social Studies of Science, 19(3), 387–420. https://doi.org/10.1177/030631289019003001

315 322 Milena Damrau LMU Munich Germany This experimental study investigates how first-year university students validate universal statements and how their success is influenced by statement characteristics and argument types. A total of 430 participants validated five statements that varied in familiarity and truth value and were randomly assigned to receive no arguments, empirical arguments, generic proofs, or ordinary proofs. Logistic regression analyses show that both familiarity and truth value significantly affect validation success. Reading empirical arguments positively affects validation success, whereas ordinary proofs do not. These findings suggest that students need support in using proofs to validate statements, and that example-based reasoning can help students develop an initial understanding of universal statements. Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111–129. https://doi.org/10.1007/s10649-008-9149-x Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2002). Proving or refuting arithmetic claims: The case of elementary school teachers. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Meeting of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 57–64). PME. Buchbinder, O., & Zaslavsky, O. (2007). How to decide? Student’s ways of determining the validity of mathematical statements. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 561–570). Christensen, R. H. B. (2022). ordinal: Regression Models for Ordinal Data (Version R package version 2022.11-16) [Computer software]. https://cran.rproject.org/web/packages/ordinal/ordinal.pdf Conner, K. A. (2022). A multi-faceted framework for identifying students’ understanding of the generality requirement of proof. International Electronic Journal of Mathematics Education, 17(4), em0702. https://doi.org/10.29333/iejme/12270 Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42(3), 225–235. https://doi.org/10.1023/A:1017502919000 Damrau, M. (2023). Understanding the generality of mathematical statements: An experimental study at the transition from school to university. Springer. https://doi.org/10.1007/978-3-658-43763-3 de Villiers, M. (2010). Experimentation and proof in mathematics. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 205–221). Springer. https://doi.org/10.1007/978-1-4419-0576-5_14 Frederick, S. (2005). Cognitive reflection and decision making. Journal of Economic Perspectives, 19(4), 25–42. https://doi.org/10.1257/089533005775196732 Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20–23. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education III (Vol. 7, pp. 234–283). American Mathematical Society. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428. https://doi.org/10.2307/749651 Hoyles, C., & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51(3), 193–223. https://doi.org/10.1023/A:1023629608614 Kempen, L. (2019). Begründen und Beweisen im Übergang von der Schule zur Hochschule: Theoretische Begründung, Weiterentwicklung und Evaluation einer universitären Erstsemesterveranstaltung unter der Perspektive der doppelten Diskontinuität. Springer. https://doi.org/10.1007/978-3-658-24415-6 Ko, Y.-Y. (2011). True or false? Pre-service secondary mathematics teachers’ strategies for evaluating statements. In L. R. Wiest & T. Lamberg (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 478–486). University of Nevada, Reno. Ko, Y.-Y., & Knuth, E. (2009). Problems manifested in prospective secondary mathematics teachers’ proofs and counterexamples in differentiation. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the ICMI study 19 conference: Proof and proving in mathematics education (Vol. 1, pp. 262–267). The Department of Mathematics, National Taiwan Normal University. Kultusministerkonferenz (Ed.). (2012). Bildungsstandards im Fach Mathematik für die Allgemeine Hochschulreife: Beschluss der Kultusministerkonferenz vom 18.10.2012. Mejía Ramos, J. P., & Inglis, M. (2009). What are the argumentative activities associated with proof? Research in Mathematics Education, 11, 77–78. https://doi.org/10.1080/14794800902732258 National Council of Teachers of Mathematics (Ed.). (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics. Riley, K. J. (2003). An investigation of prospective secondary mathematics teachers’ conceptions of proof and refutations [Doctoral dissertation, Montana State University]. https://scholarworks.montana.edu/xmlui/bitstream/handle/1/8385/31762103856330.pdf;s equence=1 Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321. Weber, K., & Mejía Ramos, J. P. (2015). On relative and absolute conviction in mathematics. For the Learning of Mathematics, 35(2), 15–21. Zeybek Simsek, Z. (2021). “Is it valid or not?”: Pre-service teachers judge the validity of mathematical statements and student arguments. European Journal of Science and Mathematics Education, 9(2), 26–42. https://doi.org/10.30935/scimath/10772

323 330 Jon Dewayne Davis Western Michigan University United States This study investigates the visibility and conceptualization of sense-making in PME proceedings (1978–2025). Proceedings were converted to searchable text as needed and searched for sense making and sense-making; papers were included when the term appeared outside references and paratext, yielding 277 papers (first occurrence: 1990). Visibility was coded by section (Title–Conclusion) using binary presence/absence indicators, and conceptual specificity was coded by type (e.g., implied definition). Most papers used the term in one section (mean = 1.43). Implied definitions predominated (N=158), sense-making appeared without a definition in 97 papers, while explicit definitions (N=15) and definitions with operationalization (N=8) were rare. Implied and explicit definitions were examined for their thematic content. Aquilina, G., Di Martino, P., & Lisarelli, G. (2025). The construct of attitude in mathematics education research: current trends and new research challenges from a systematic literature review. ZDM, 57, 569-581. Diego-Mantecón, J. M., Blanco, T. F., Chamoso, J. M., & José Cáceres, M. (2019). An attempt to identify the issues underlying the lack of consistent conceptualisations in the field of student mathematics-related beliefs. PLoS ONE, 14(11), 1-19. Glass, B., & Maher, C. A. (2004). Students problem solving and justification. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 463-470). Bergen, Norway. Hollebrands, K., Cayton, C., & Boehm, E. (2013). Pivotal teaching moments in a technologyintensive secondary geometry classroom. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 73-80). Kiel, Germany: PME. Inglis, M., & Foster, C. (2018). Five decades of mathematics education research. Journal for Research in Mathematics Education, 49(4), 462-500. Rios, J. (2023). The role of language-as-resource and language-as-political in college mathematics courses. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 91-106). Haifa, Israel. Scheiner, T. (2021). Towards a more comprehensive model of teacher noticing. ZDM, 53(1), 85-94. https://doi.org/10.1007/s11858-020-01202-5 Scheiner, T., & Pinto, M. M. F. (2022). Sense-making in mathematics: An act of comprehension or creation? For the Learning of Mathematics, 42(3), 16-20. Simon, M. A. (2017). Explicating mathematical concept and mathematical conception as theoretical constructs for mathematics education research. Educational Studies in Mathematics, 94, 117-137. Urquhart, C., Cheuk, B., Lam, L., & Snowden, D. (2024). Sense-making, sensemaking, and sense making – A systematic review and meta-synthesis of literature in information science and education. Annual Review of Information Science and Technology (ARIST), 76(1), 3-97.

339 346 Charlotte Krog Skott Liping Ding Svein Arne Sikko University College Absalon Norwegian University of Science and Technology Norway Facilitators seem important for the effect of lesson study, yet research on their role is scarce and there is a lack of theoretical approaches to examine this role. In this paper, we adapt a theoretical framework, the knowledge quartet, that focuses on how teachers’ knowledge of mathematics and its teaching and learning play out at the classroom level. We use a lifting strategy to adapt the quartet to the facilitator level and apply it to analyse the facilitators’ role in a Norwegian lesson study that targeted primary teachers’ development of inquiry-based teaching. We conclude that the adapted framework appears as a promising theoretical approach that can provide deep insight into the facilitator role and potentially serve as a means to develop it. However, more research beyond this single case is needed to further develop the framework. Clivaz, S., Batteau, V., Pellet, J. P., Bünzli, L. O., Daina, A., & Presutti, S. (2023). Teachers’ mathematical problem-solving knowledge: In what way is it constructed during teachers’ collaborative work? Journal of Mathematical Behavior, 69. Karsenty, R. (2020). The role of frameworks in researching knowledge and practices of mathematics teachers and teacher educators. In S. Zehetmeier (Ed.), Professional development and knowledge of mathematics teachers. Routledge. Lewis, C., Perry, R., & Murata, A. (2006). How should research contribute to instructional improvement? The case of lesson study. Educational Researcher, 35 (3). 3–14. Maaß, K. & Reitz-Koncebovski, K. (Eds.) (2013). Inquriy-based learning in maths and science classes. PRIMAS project Pädagogische Hochschule Freiburg. https://primasproject.eu/wp-content/uploads/sites/323/2017/11/primas_final_publication.pdf Seino, T., & Foster, C. (2021). Analysis of the final comments provided by a knowledgeable other in lesson study. Journal of Mathematics Teacher Education, 24(5), 507–528. https://doi.org/10.1007/s10857-020-09468-y Skott, C. K. (2024). Lesson study beyond its initial adaptation: a case without external support. Journal of Mathematics Teacher Education, 27(2), 163–183. https://doi.org/10.1007/s10857-022-09556-1 Skott, C. K., & Ding, L. (2022). A comparative study of the role of the external facilitator in lesson studies in Denmark and China. In J. Hodgen, E. Geraniou, G. Bolondi, & F. Feretti (Eds.), Proc. 12th Congress of the European Society for Research in Mathematics Education (pp. 3294–3301). Free University of Bozen-Bolzano and ERME. Skott, C. K., & Psycharis, G. (in press). Facilitators’ role in supporting core practices in a lesson-study context. Takahashi, A. (2014). The role of the knowledgeable other in lesson study: examining the final comments of experienced lesson study practitioners. Mathematics Teacher Education and Development, 16(1), 1–17. Turner, F., & Rowland, T. (2011). The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers´ Mathematical Knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical Knowledge in Teaching (pp. 195–212). Springer. Utdanningsdirektoratet (2006). Læreplanverket for kunnskapsløftet. https://www.nb.no/items/a9713c6622111b63c53947c7f01ee6f0?page=0&searchText=lær eplanen for kunnskapsløftet

347 354 Rui Ding Boyuan Xue Xixi Deng Jiajia Li Northeast Normal University China, People's Republic of Same-Unit Coordination (SUC) has been proposed as a necessary requisite for Mixed-Unit Coordination (MUC) and Place-Value, Base-10 reasoning (PV-B10), yet longitudinal evidence is limited. We followed 225 Grade 3-4 students in Northeast China across three waves and employed Latent Class Growth Modeling to identify the role of SUC on the advanced multiplicative schemes. Two latent classes (high-decreasing vs. low-increasing) were identified based on their developmental pattern on SUC. Mixed-design repeated-measures ANOVA showed significant Class × Time interactions for both MUC and PV-B10: the high-decreasing class declined on both schemes, whereas the low-increasing class improved. Results suggest SUC as an adaptor in the development of advanced multiplicative reasoning schemes. Bond, T. G., Yan, Z., & Moritz, H. (2020). Applying the Rasch model: Fundamental measurement in the human sciences (4th ed.). Routledge. Boone, W. J., Staver, J. R., & Yale, M. S. (2014). Rasch analysis in the human sciences. Springer. Ding, R., Tzur, R., Wei, B., Jin, X., Jin, X., Davis, A. (2019). Conceptual Analysis of an Error Pattern in Chinese Elementary Students. In M. Graven, H. Venkat, A. Essien & P. Vale (Eds.), Proc 43rd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp.93-100). PME. Fisher, W. P. (2007). Rating scale instrument quality criteria. Rasch Measurement Transactions, 21(1), 1095. Geiser, C. (2020). Longitudinal structural equation modeling with Mplus: A latent state-trait perspective. Guilford Press. Lin, C. D. (2018). Developmental psychology. Beijing: People’s Education Press. [In Chinese]. Linacre, J. M. (2002). What do infit and outfit, mean-square and standardized mean? Rasch Measurement Transactions, 16(2), 878. Linacre, J. M. (2025). A user’s guide to WINSTEPS: Rasch-model computer programs. Winsteps. Lubke, G., & Muthén, B. O. (2007). Performance of factor mixture models as a function of model size, covariate effects, and class-specific parameters. Structural Equation Modeling: A Multidisciplinary Journal, 14(1), 26-47. Moss, J., & McNab, S. L. (2011). An approach to geometric and numeric patterning that fosters second grade students’ reasoning and generalizing about functions and covariation. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 277-302). Berlin: Springer-Verlag. Norton, A., Boyce, S., Phillips, N., Anwyll, T., Ulrich, C., & Wilkins, J. (2015). A written instrument for assessing students’ units coordination structures. Journal on Mathematics Education, 10(2), 111-136. Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intellectual development (T. Brown & K. J. Thampy, Trans.). The University of Chicago Press. Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259-309. Tzur, R., Wei, B., Smith, A., Norton, A., Davis, A., & Johnson, H. L. (2018). Same unit coordination: A conceptual screener for mixed unit coordination and base-10, place value reasoning. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proc. 42nd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 323-330). PME. Tzur, R., Johnson, H. L., McClintock, E., Kenney, R. H., Xin, Y. P., Si, L., . . . Jin, X. (2013). Distinguishing schemes and tasks in children’s development of multiplicative reasoning. PNA-Pensamiento Numérico Avanzado, 7(3), 85-101. Tzur, R., & Xin, Y. P. (2022). Nurturing multiplicative reasoning with whole numbers. In Y. P. Xin, R. Tzur, & H. Thouless (Eds.), Enabling mathematics learning of struggling students (1st ed., pp. 291-314). Springer - Nature. Ulrich, C. (2016). Stages in constructing and coordinating units additively and multiplicatively (part 2). For the Learning of Mathematics, 36(1), 34-39. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Falmer.

355 362 Giovannina Albano Cristina Coppola Angela Donatiello University of Salerno Italy This study frames theorem learning as a process of scientific “enculturation” fostering rational behaviour. We present a case study of a teaching setting that supports 10th graders’ development of theoretical validation within Euclidean geometry. The analysis of the evolution of shared geometrical arguments along the didactical path shows a movement from naïve justification to theoretical validation within a geometrical system aligned with the standards of the mathematical community. Albano, G., Coppola, C. & Donatiello, A. (2024). Cognitive roles’ characterisation of the Inside- Out Model in Habermas’ rationality perspective: the cases of pest and blogger. In M. Ní Ríordáin, and K. Erath (Eds.), Proc. of the 16th ETC on Language and Social Interaction in Mathematics Classrooms (pp. 4 – 11). https://hal.science/hal-04833321 Albano, G., Coppola, C., & Dello Iacono, U. (2021a). What Does 'Inside Out' Mean in Problem Solving?. For the learning of mathematics, 41(2), 32-36. Albano, G., Dello Iacono, U., & Mariotti, M. A. (2021b). An e-learning innovative approach for mathematical argumentative thinking. International Journal for Technology in Mathematics Education, 28(1), 3-14. Balacheff N. (2010) Bridging knowing and proving in mathematics. An essay from a didactical perspective. In Hanna G., Jahnke H. N., Pulte H. (Eds.) Explanation and Proof in Mathematics (pp. 115-135). Springer. Boero, P., & Azrou, N. (2023) Analysis of proving in the Habermas’ rationality perspective: The meta-knowledge – knowledge interplay. In Drijvers, P., Csapodi, C., Palmér, H., Gosztonyi, K., & Kónya, E. (Eds.), Proc. CERME13, (pp. 72-79). Alfréd Rényi Institute of Mathematics and ERME. Cohen, L., Manion, L., & Morrison, K. (2002). Research methods in education. Routledge. de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24. Ferrari. P.L. (2017). Language in argumentation and solution of problems with graphs. In T. Dooley & G. Gueudet (Eds.): Proc. of CERME10 (pp. 1276-1283). DCU Institute of Education and ERME. hal-01937159 Habermas, J. (1998). On the pragmatics of communication (M. Cook, Ed.). MIT Press. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds): Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173-204). Sense publishers. Morselli, F., & Boero, P. (2009). Habermas' construct of rational behaviour as a comprehensive frame for research on the teaching and learning of proof. In F. L. Lin et al (Eds.), Proc. of the ICMI Study 19 Conference: Proof and proving in mathematics education (Vol. 2, pp. 100-105). National Taiwan Normal University. Stylianides, A. L. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. Zaslavsky, O., Nicherson, S. D., Stylianides, A. J., Kidron, I., &Winicki-Landman, G. (2021). The Need for Proof and Proving: Mathematical and Pedagogoical Perspectives. In G. Hanna and M. de Villiers (Eds.), Proof and Proving in Mathematics Education, New ICMI Study Series, Vol. 15 (pp. 215 - 230), Springer Zhuang, Y., & Conner, A. (2022). Teachers’ use of rational questioning strategies to promote student participation in collective argumentation. Educational Studies in Mathematics, 111(2), 345-365

363 370 Marie Dreier Tjorven Seibold Bettina Roesken-Winter University of Münster Germany The quality of professional development (PD) essentially depends on facilitators and their knowledge. However, research shows that facilitators primarily draw on general pedagogical knowledge (GPK) and less on content-specific aspects for teacher learning. To further investigate this phenomenon, we examined how 54 facilitators reacted to a fictional PD-situation, examining the knowledge categories they used and the focus they pursued within these categories. The findings indicate a strong focus on classroom content without reference to the PD-level, indicating a struggle between the roles of facilitator and teacher. We also found that many facilitators used GPK, especially with a focus on general moderation. Therefore, content-specific support for teacher learning should be an integral part of facilitators’ qualification. Bialy, V., Shure, V., Lehmann, M. Roesken-Winter, B. (2024). Facilitators’ categories when noticing a fictional PD-situation: Pedagogical content vs. general pedagogic focus. In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp.80–87). PME. Borko, H., Koellner, K., & Jacobs, J. (2014). Examining novice teacher leaders’ facilitation of mathematics professional development. Journal of Mathematical Behavior, 33, 149– 167. https://doi.org/10.1016/j.jmathb.2013.11.003 Cheeseman, J., McDonough, A., & Golemac, D. (2017). Investigating children’s thinking about suspended balances. New Zealand Journal of Educational Studies, 52(1), 143–158. https://doi.org/10.1007/s40841-016-0073-9 Jacobs, J., Seago, N., & Koellner, K. (2017). Preparing facilitators to use and adapt mathematics professional development materials productively. International Journal of STEM Education, 4(1), 30. https://doi.org/10.1186/s40594-017-0089-9 Kilpatrick, J., Swafford, J., & Findel, B. (2001). Adding it up. Helping children learn mathematics. National Academy Press. Pöhler-Friedrich, B., Prediger, S., Schwarts, G., & Karsenty, R. (2024). Learning to leverage teachers’ ideas: Facilitators’ growth in moderating activities with simultaneously strong goal focus and participant focus. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-024-09660-4 Peters, S., & Prediger, S. (2025). Enhancing student’s understanding of unit conversion by explicating underlying structures: design research on schematization in middle schools. Educational Studies in Mathematics. Advance online publication. https://doi.org/10.1007/s10649-025-10437-6 Prediger, S., & Selter, C. (2024). Establish shared visions and support productive adaptations on all levels: Aims, strategies, and architecture of a nationwide implementation program. Implementation and Replication Studies in Mathematics Education, 4(1), 15– 49. https://doi.org/10.1163/26670127-bja10020 Prediger, S., Roesken-Winter, B., Stahnke, R., & Pöhler, B. (2022). Conceptualizing contentrelated PD facilitator expertise. Journal of Mathematics Teacher Education, 25(4), 403– 428. https://doi.org/10.1007/s10857-021-09497-1 Schwarts, G., Karsenty, R., & Arcavi, A. (2025, online first). To share your stance, or not to share? Positioning, identity, and agency of mathematics teachers becoming facilitators. School Science and Mathematics. https://doi.org/10.1111/ssm.18364 Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.3102/0013189X01500200

371 378 Anni E Julia Hill Jodie Hunter The University of Melbourne, RMIT University, Massey University Australia Student wellbeing encompasses both hedonic (feeling good) and eudaimonic (functioning well) aspects, yet how students articulate these dimensions in mathematics remains underexplored. Drawing on survey responses from 1,192 students, this study examines the extent to which students mention hedonic, eudaimonic, or both aspects when describing their mathematical wellbeing. Findings showed 61% mentioned eudaimonic-only, 36% mentioned both, and 3% mentioned hedonic-only. Among those describing both, 542 described functioning before feeling, whereas only 41 described the reverse. This suggests positive emotions are experienced alongside successful functioning—particularly through relationships and accomplishment—highlighting the importance of supporting students to function well in mathematics. Boaler, J. (2015). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. John Wiley & Sons. Campbell, T. G., & Bean, B. (2025). Factors influencing young children’s mathematical wellbeing in the United States. Social Indicators Research, 178(1), 255-278. Guo, J., Parker, P. D., Marsh, H. W., & Morin, A. J. S. (2015). Achievement, motivation, and educational choices: A longitudinal study of expectancy and value using a multiplicative perspective. Developmental Psychology, 51(8), 1163–1176. Hill, J. L. (2024). Uncovering student values and wellbeing across mathematics and science education. [PhD thesis, University of Melbourne]. http://hdl.handle.net/11343/344720 Hill J. L., Kern M. L., Seah WT, & van Driel J. (2021). Feeling good and functioning well in mathematics education: Exploring students’ conceptions of mathematical well-being and values. ECNU Review of Education. 4(2), 349-375. Hill, J. L., Kern, Ml. L. Seah, W. T., & van Driel, J. (2022). Developing a model of mathematical wellbeing through a thematic analysis of the literature. In M. Vollstedt, I. Kontorovich, K. Jones, & D. Sommerhoff (Eds.), Proceedings of the 45th Conference of the International Group Psychology of Mathematics Education. PME. Hill, J.L., Kern, M.L., Seah, W.T., van Driel, J. (2024). To what extent are students fulfilling their values and thriving in mathematics education?—The case for Victoria, Australia. In Y. Dede, G. Marschall, & P. Clarkson. (Eds.) Values and valuing in mathematics education. (pp. 259–283). Springer. Hunter, J., & Hill, J. L. (2025). Exploring mathematical wellbeing across cultures: Insights from diverse students. Mathematics Education Research Journal, 37(3), 529–556. Huppert, F. A., & So, T. T. C. (2013). Flourishing across Europe: Application of a new conceptual framework for defining well-being. Social Indicators Research, 110(3), 837– 861. Huta, V., & Ryan, R. M. (2010). Pursuing pleasure or virtue: The differential and overlapping well-being benefits of hedonic and eudaimonic motives. Journal of Happiness Studies, 11(6), 735–762. Mullis, I. V. S., Martin, M. O., Foy, P., Kelly, D. L., & Fishbein, B. (2020). TIMSS 2019 international results in mathematics science. https://timssandpirls.bc.edu/timss2019/international-results/ OECD. (2024). PISA 2022 Results (Volume V): Learning strategies and attitudes for life. PISA. https://doi.org/10.1787/c2e44201-en Ryan, R. M., & Deci, E. L. (2001). On happiness and human potentials: A review of research on hedonic and eudaimonic well-being. Annual Review of Psychology, 52(1), 141–166. Ryff, C. D., & Singer, B. H. (2008). Know thyself and become what you are: A eudaimonic approach to psychological well-being. Journal of Happiness Studies, 9(1), 13–39. Wigfield, A., & Eccles, J. S. (2000). Expectancy–value theory of achievement motivation. Contemporary Educational Psychology, 25(1), 68–81.

379 386 Andreas Eckert Andreas Ebbelind Linnaeus University Sweden This study investigates how language expressing uncertainty can support learning in probability. Using Systemic Functional Linguistics, we analysed classroom discourse during a five-lesson sequence with sixth-grade students. Video transcripts were coded for modality to examine how teachers and students articulated certainty and conjecture. Early lessons featured low modality linked to personal beliefs, while later lessons connected modal expressions to data-based reasoning. Teacher strategies shifting from mirroring to challenging and grounding statements in evidence proved crucial in fostering probabilistic thinking. Findings underscore the role of language as a resource for teaching probability and call for further research on linguistic approaches to statistical uncertainty. Ben-Zvi, D., Aridor, K., Makar, K., & Bakker, A. (2012). Students’ emergent articulations of uncertainty while making informal statistical inferences. ZDM - the International Journal on Mathematics Education, 44(7), 913–925. https://doi.org/10.1007/s11858-012-0420-3 Braham, H.M., & Ben-Zvi, D. (2019). Design for reasoning with uncertainty. In G. Burrill & D. Ben-Zvi (Eds.), Topics and trends in current statistics education research. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-030-03472-6_5 Dolor, J., & Noll, J. (2015). Using guided reinvention to develop teachers’ understanding of hypothesis testing concepts. Statistics Education Research Journal, 14(1), 60–89. https://doi.org/10.52041/serj.v14i1.269 Eckert, A., & Nilsson, P. (2015). Introducing a symbolic interactionist approach on teaching mathematics: The case of revoicing as an interactional strategy in the teaching of probability. Journal of Mathematics Teacher Education, 20(1) 31–48. https://doi.org/10.1007/s10857-015-9313-z Groth, R. E. (2013). Characterizing key developmental understandings and pedagogically powerful ideas within a statistical knowledge for teaching framework. Mathematical Thinking and Learning, 15(2), 121–145. https://doi.org/10.1080/10986065.2013.770718 Halliday, M., & Matthiessen, C. (2004). An introduction to functional grammar (4th edition). Arnold. Langrall, C., Makar, K., Nilsson, P., & Shaughnessy, J. M. (2017). Teaching and learning probability and statistics: An integrated perspective. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 490–525). NCTM. Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical inference. Mathematical Thinking Learning, 13(1–2), 152–173. https://doi.org/10.1080/10986065.2011.538301 Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105. Morgan, C., Craig, T., Schuette, M., & Wagner, D. (2014). Language and communication in mathematics education: an overview of research in the field. ZDM, 46(6), 843–853. https://doi.org/10.1007/s11858-014-0624-9 Steinbring, H. (1991). The concept of chance in everyday teaching: Aspects of a social epistemology of mathematical knowledge. Educational Studies in Mathematics, 22(6), 503–522. https://doi.org/10.1007/bf00312713 Steinbring, H. (1993). The context for the concept of chance: Everyday experiences in classroom interactions. Acta didactica Universitatis Comenianae Mathematics, (2), 20–32. Voigt, J. (1996). Negotiation of mathematical meaning in classroom processes: Social interaction and learning mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of Mathematical Learning. Lawrence Erlbaum Associates. Wagner, D., Dicks, J., & Kristmanson, P. (2015). Students’ language repertoires for prediction. The Mathematics Enthusiast, 12(1-3), 246–261.https://doi.org/10.54870/15513440.1346

395 402 Dafna Efron Nadav Marco Alik Palatnik The Hebrew University of Jerusalem Israel David Yellin Academic College of Education Israel This multiple-case exploratory study examines what mathematical ideas pre-service mathematics teachers (PSTs) discern and represent when interpreting a non-verbal, rule-governed movement system. Twenty-nine PSTs viewed Samuel Beckett’s Quad and produced textual and diagrammatic accounts of the choreography. Analysed through the lens of professional vision, the accounts reveal a variety of mathematical ideas alongside systematic gaps and representational tensions, and in some cases show PSTs extending beyond Beckett’s movement score by drawing on prior mathematical knowledge. Taken together, the findings suggest that observing embodied movement can serve as a diagnostic context for examining emerging professional vision and for informing pedagogical support in embodied mathematics teacher education. Abrahamson, D., & Lindgren, R. (2014). Embodiment and embodied design. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (2nd ed., pp. 358–376). Cambridge University Press. https://doi.org/10.1017/CBO9781139519526.022 Apsari, R. A., & Abrahamson, D. (2024). Dancing geometry: Imagining auxiliary lines by reflecting on physical movement. International Journal of Mathematical Education in Science and Technology, 1–28. https://doi.org/10.1080/0020739X.2024.2427099 Beckett, S. (1984). Quad. Faber & Faber. Boonstra, K., Kool, M., Shvarts, A., & Drijvers, P. (2025). Primary school teachers’ practices for fostering embodied abstraction in geometry. Journal of Mathematics Teacher Education, 1–25. https://doi.org/10.1007/s10857-025-09702-5 Efron, D., & Palatnik, A. (2025). “Can I use my leg too?” Dancing with uncertainty: Exploring probabilistic thinking through embodied learning in a Jerusalem art high school classroom. Education Sciences, 15(9), Article 1248. https://doi.org/10.3390/educsci15091248 Eshkol, N., & Wachman, A. (1958). Movement notation. Movement Notation Society. Flood, V. J., Shvarts, A., & Abrahamson, D. (2022). Responsive teaching for embodied learning with technology. In S. Macrine & J. M. B. Fugate (Eds.), Movement matters: How embodied cognition informs teaching and learning (pp. 179–196). MIT Press. https://doi.org/10.7551/mitpress/13593.003.0020 Gerofsky, S. (2025). Experiencing mathematical relationships at a variety of scales through body movement, voice, and touch. In L. D. Edwards & C. M. Krause (Eds.), The body in mathematics: Theoretical and methodological lenses (pp. 198–228). Brill. https://doi.org/10.1163/9789004717701_008 Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606–633. Hegna, H. M., & Ørbæk, T. (2021). Traces of embodied teaching and learning: A review of empirical studies in higher education. Teaching in Higher Education. https://doi.org/10.1080/13562517.2021.1989582 Palatnik, A., Lomos, C., & Abrahamson, D. (2026). In their hands: Multimodal learning analytics as reflective-practice resources empowering mathematics teachers’ professional development. Learning Instruction, 102, 102272. https://doi.org/10.1016/j.learninstruc.2025.102272 Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books. Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM—Mathematics Education, 46(3), 349–361. https://doi.org/10.1007/s11858-014-0591-1 Stake, R. E. (2013). Multiple case study analysis. Guilford Press. Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sépulveda (Eds.), Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 45–64). PME.

403 410 Samantha Anne Ehrlich Department of Teaching and Learning, Stockholm University Sweden Large-scale multi-format assessments include short-response, constructed-response, oral, and investigative mathematics tasks, offering different ways for students to demonstrate their competencies. However, given their complex design, it is non-trivial to assume that items designed to operationalise the same competency work together to produce sum scores that can be inferred as measures of that competency. I found that although there was consistent evidence that items from a Swedish large-scale multi-format assessment measured some curriculum-defined competencies, evidence for the measurement of others was mixed, underscoring the opportunities and challenges of using such assessments to measure complex educational constructs at scale. Andrich, D., Lyne, A., Sheridan, B., & Luo, G. (2010). RUMM2030 [Computer software]. RUMM Laboratory. Bennett, J., Tognolini, J., & Pickering, S. (2012). Establishing and applying performance standards for curriculum-based examinations. Assessment in Education: Principles, Policy & Practice, 19(3), 321–339. https://doi.org/10.1080/0969594X.2011.614219 Braswell, J., & Kupin, J. (1993). Item formats for assessment in mathematics. In W. Ward C. & R. E. Bennett (Eds.), Construction Versus Choice in Cognitive Measurement: Issues in Constructed Response, Performance Testing, and Portfolio Assessment. Routledge. Christensen, K. B., Makransky, G., & Horton, M. (2017). Critical Values for Yen’s Q3: Identification of Local Dependence in the Rasch Model Using Residual Correlations. Applied Psychological Measurement, 41(3), 178–194. https://doi.org/10.1177/0146621616677520 de Lange, J. (2007). Large-scale assessment and mathematics education. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. Information Age Publishers. Mari, L., Wilson, M., & Maul, A. (2023). Measurement Across the Sciences: Developing a Shared Concept System for Measurement. Springer International Publishing. https://doi.org/10.1007/978-3-031-22448-5 Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests (Expanded ed.). University of Chicago Press. (Original work published 1960). Smith, E. V. (2002). Detecting and evaluating the impact of multidimensionality using item fit statistics and principal component analysis of residuals. Journal of Applied Measurement, 3(2), 205–231. The Swedish National Agency for Education. (2011). Curriculum for the compulsory school, preschool class school-age educare revised 2018. https://www.skolverket.se/publikationer?id=3984 Acknowledgments This work was supported by the ASSESS research school funded by the Swedish Research Council under Grant [2021–00540]. Special thanks to the test developers and content experts at PRIM, Stockholm University, Sweden.

411 418 Anna-Lena Ekdahl Camilla Björklund Jönköping University Sweden University of Gothenburg Sweden Critical for developing multiplicative reasoning skills is recognizing the multiplicative structure in the problem. In this study, we investigate which discerned and undiscerned aspects explain differences in 6 yearolds’ ways of solving a task with a multiplicative structure. Drawing on variation theory of learning, qualitative analyses were conducted of interview data from 199 students solving an arithmetic task with a multiplicative structure. Four critical aspects were identified: the size of the unit, its iteration, the number of iterations, and the relation between iterations and the whole. Results show that learning to handle multiplicative structures requires discernment and coordination of all four aspects simultaneously. Askew, M., Venkat, H., Mathews, C., Ramsingh, V., Takane, T. & Roberts, N. (2019). Multiplicative reasoning: An intervention’s impact on Foundation Phase learners’ understanding. South African Journal of Childhood Education, 9(1), a622. https://doi.org/10.4102/sajce.v9i1.622 Björklund, C., Elofsson, J., Kullberg, A., Ekdahl, A-L., Runesson Kempe, U., & Alkhede, M. (2024). Preschool class students’ discernment of number structure in a spatial pattern. In J. Häggström et al. (Eds.), Proceedings of MADIF14 (pp. 85–96). SMDF. Cheeseman, J., Downton, A., Roche, A., & Ferguson, S. (2020). Investigating young students’ multiplicative thinking: The 12 little ducks problem. Journal of Mathematical Behavior, 60, 100817. Marton, F. (2015). Necessary conditions of learning. Routledge. Nuñes, T., & Bryant, P. (1996). Children doing mathematics. Wiley. Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel, & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–39). State University of New York Press. Tzur, R., Johnson, H. L., Norton, A., Davis, A., Wang, X., Ferrara, M., Harrington, C., & Hodkowski, N. M. (2021). Children’s spontaneous additive strategy relates to multiplicative reasoning. Cognition Instruction, 39(4), 451–476, http://doi.org/10.1080/07370008.2021.1896521 Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (part 1). For the Learning of Mathematics, 35(3), 2–7. Van den Heuvel-Panhuizen, M., & Elia, I. (2020). Mapping kindergartners’ quantitative competence. ZDM Mathematics Education (4), 805–819. https://doi.org/10.1007/s11858-020-01138-w Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication … and back: The development of students’ additive and multiplicative reasoning skills. Cognition Instruction, 28(3), 360–381. https://doi.org/10.1080/07370008.2010.488306 Zwanch, K., & Wilkins, J. L. M. (2021). Releasing the conceptual spring to construct multiplicative reasoning. Educational Studies in Mathematics, 106, 151–170. https://doi.org/10.1007/s10649-020-09999-4

419 426 Eleonora Faggiano Roberto Capone University of Bari Aldo Moro Italy To address the persistent gap between the drawing (perceptual) and the figure (theoretical) in geometry learning, we examine how a four‑phase sequence, implemented with 20 undergraduates, based on the synergy between manipulative and digital artefacts, supports robust construction in Dynamic Geometry Environments. We integrate Duval’s semiotic registers and the Instrumental Approach to investigate how an intermediate writing instructions task triggers semiotic conversions from perceptual to verbal register and impacted on instrumental genesis: students who identified functional dependencies developed robust schemes of use that withstand the drag-test, while perception-based approaches led to “soft” constructions, confirming the connection and underscoring the teacher’s role. Bartolini Bussi M. G. & Mariotti M. A. (2008). Semiotic Mediation in the Mathematics Classroom: Artefacts and Signs after a Vygotskian Perspective, In L. English et al. (eds.), Handbook of International Research in Mathematics Education (pp. 746–783). LEA. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(2), 103–131. https://doi.org/10.1007/s10649-006-0400-z Faggiano, E. (2022). The Semiotic Bundle as a reflective tool in pre-service mathematics teachers’ education. In Frontiers in Education, 7, (p. 1043710), 1–12. https://doi.org/10.3389/feduc.2022.1043710 Faggiano, E., Montone, A. and Mariotti, M. A., (2016). Creating a synergy between manipulatives and virtual artefacts to conceptualize axial symmetry at primary school, In Csíkos, C., Rausch, A., & Szitányi, J. (Eds.). Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 235–242). Szeged, Hungary: PME Faggiano, E., Montone, A., & Mariotti, M. A. (2018). Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school. International Journal of Mathematical Education in Science and Technology, 49(8), 1165–1180. Guin, D., & Trouche, L. (1998). The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning, 3(3), 195–227. https://doi.org/10.1023/A:1009892720043 Healy, L., & Hoyles, C. (2002). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6(3), 235– 256. https://doi.org/10.1023/A:1013305627916 Hölzl, R. (1996). How does ‘dragging’ affect the learning of geometry. International Journal of Computers for Mathematical Learning 1, 169–187 https://doi.org/10.1007/BF00571077 Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1-2), 55–85. Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational studies in Mathematics, 44(1), 151–161. Laborde, C. (2005). Robust and soft constructions: Two sides of the use of dynamic geometry environments. In Chu et al. (Eds) Proceedings of the 10th Asian technology conference in mathematics (pp. 22–35). Cheong‐Ju: Korea National University of Education. Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. International Journal of Mathematical Education in Science and Technology, 37(6), 665–679. https://doi.org/10.1080/00207390600712539 Mariotti, M. A. (2002). Justifying and proving in the Cabri environment. International Journal of Computers for Mathematical Learning, 6(3), 257–281. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Sense Publishers. Santos-Trigo, M., & Espinosa-Perez, H. (2002). Searching and exploring properties of geometric configurations using dynamic software. International Journal of Mathematical Education in Science and Technology, 33(1), 37–50. Strässer, R. (2002). Cabri-Geometre: Does dynamic geometry software (DGS) change geometry and its teaching and learning? International Journal of Computers for Mathematical Learning, 6(3), 319–333. https://doi.org/10.1023/A:1013361712895 Trouche, L. (2004). Managing the complexity of human-machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. 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427 434 Sarah Falke Bettina Roesken-Winter Lars Meyer-Jenßen University of Münster Germany HMU Health and Medical University Erfurt Emotions play a significant role in professional teacher competence. Pride is one of the most common emotions experienced by primary pre-service teachers (PSTs) during their university mathematics education. However, this emotion has only been marginally investigated. This study examined the pride experiences of 67 primary PSTs, classifying situations that evoke either authentic or hubristic pride. Overall, the participants reported moderately high levels of authentic pride and low levels of hubristic pride; the two facets showed no significant correlation. Situations that evoke authentic pride fall into two categories: evaluative situations and demonstrating content knowledge situations. Hubristic pride situations fall into to the second category only, focusing on social comparison. Anttila, H., Pyhältö, K., Soini, T., & Pietarinen, J. (2016). How does it feel to become a teacher? Emotions in teacher education. Social Psychology of Education: An International Journal, 19(3), 451–473. https://doi.org/10.1007/s11218-016-9335-0 Bekdemir, M. (2010). The pre-service teachers’ mathematics anxiety related to depth of negative experiences in mathematics classroom while they were students. Educational Studies in Mathematics, 75(3), 311–328. https://doi.org/10.1007/s10649-010-9260-7 Blömeke, S., Gustafsson, J.-E., & Shavelson, R. J. (2015). Competence viewed as a continuum. Zeitschrift für Psychologie, 223(1), 3–13. https://doi.org/10.1027/21512604/a000194 Brígido, M., Bermejo, M. L., Conde, M. C., & Mellado, V. (2010). The emotions in teaching and learning nature sciences and physics/chemistry in pre-service primary teachers. US China Education Review, 7(12), 25–32. Frenzel, A. C., Daniels, L., & Burić, I. (2021). Teacher emotions in the classroom and their implications for students. Educational Psychologist, 56(4), 250–264. Hannula, M. S. (2015). Emotions in problem solving. In S. J. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 269–288). Springer International Publishing. https://doi.org/10.1007/978-3-319-17187-6_16 Jenßen, L. (2021). What affects the arrogant, proud or ashamed pre-service teacher in mathematics? Effects of social comparison, gender and self-concept on self-conscious emotions. Social Psychology of Education, 24(5), 1105–1123. https://doi.org/10.1007/s11218-021-09655-6 Jenßen, L., Möller, R., Eilerts, K., & Roesken-Winter, B. (2022). Pre-service primary teachers’ shame experiences during their schooling time: Characteristics and effects on their subject-choices at university. Educational Studies in Mathematics, 110(3), 435–455. https://doi.org/10.1007/s10649-021-10141-1 Liu, C., Yuan, L., Chen, C., & Yu, G. (2016). Authentic and hubristic pride as assessed by self, friends, and strangers. Social Psychological and Personality Science, 7(7), 690–696. https://doi.org/10.1177/1948550616649240 Oades-Sese, G., Matthews, T. A., & Lewis, M. (2014). Shame and pride and their effects on student achievement. In R. Pekrun, & L. Linnenbrink-Garcia (Eds), International Handbook of Emotions in Education (pp. 246–264). Routledge. Pekrun, R. (2024). Control-value theory: From achievement emotion to a general theory of human emotions. Educational Psychology Review, 36(3), 83. https://doi.org/10.1007/s10648-024-09909-7 Timoštšuk, I., & Ugaste, A. (2012). The role of emotions in student teachers’ professional identity. European Journal of Teacher Education, 35(4), 421–433. Tracy, J. L., & Robins, R. W. (2007). The psychological structure of pride: A tale of two facets. Journal of Personality and Social Psychology, 92(3), 506–525. https://doi.org/10.1037/0022-3514.92.3.506 Tracy, J. L., & Robins, R. W. (2014). Conceptual and empirical strengths of the authentic/hubristic model of pride. Emotion, 14(1), 33–37. https://doi.org/10.1037/a0034490 Williams, L. A., & DeSteno, D. (2008). Pride and perseverance: The motivational role of pride. Journal of Personality and Social Psychology, 94(6), 1007–1017.

435 442 Janne Fauskanger Reidar Mosvold University of Stavanger Norway This study examines how student engagement is observed in an inexperienced Building Thinking Classroom (BTC) teacher’s practice. The analysis draws on four dimensions of engagement—behavioral, cognitive, emotional, and social—and indicates that the BTC may encourage each of these in different ways. The BTC structure was found to foster active and social participation, but it did not reliably produce deeper mathematical thinking. Students often stayed in procedural routines and appeared behaviourally engaged, showing that cognitive engagement requires deliberate teacher actions such as clear goals and responses that prompt further exploration. Christenson, S. L., Reschly, A. L., & Wylie, C. (Eds.). (2012). Handbook of Research on Student Engagement. Springer US. https://doi.org/10.1007/978-1-4614-2018-7 Fall, A. M., & Roberts, G. (2012). High school dropouts: Interactions between social context, self-perceptions, school engagement, and student dropout. Journal of Adolescence, 35(4), 787–798. https://doi.org/10.1016/j.adolescence.2011.11.004 Fredricks, J. A., Wang, M. T., Schall Linn, J., Hofkens, T. L., Sung, H., Parr, A., & Allerton, J. (2016). Using qualitative methods to develop a survey measure of math and science engagement. Learning Instruction, 43, 5–15. https://doi.org/10.1016/j.learninstruc.2016.01.009 Fung, F., Tan, C. Y., & Chen, G. (2018). Student engagement and mathematics achievement: Unraveling main and interactive effects. Psychology in the Schools, 55(7), 815–831. https://doi.org/10.1002/pits.22139¨ Hsieh, H.-F. & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288. Liljedahl, P. (2021). Building Thinking Classrooms in mathematics, Grades K-12. 14 teaching practices for enhancing learning. Corvin Press Inc. Liljedahl, P., & Cai, J. (2021). Empirical research on problem solving and problem posing: a look at the state of the art. ZDM - Mathematics Education, 53, 723–735. https://doi.org/10.1007/s11858-021-01291-w Liljedahl, P., & Girox, M. (2024). Mathematics Tasks for the Thinking Classroom, Grades K-5. Corvin Press Inc. Maamin, M., Maat, S. M., & H. Iksan, Z. (2022). The influence of student engagement on mathematical achievement among secondary school students. Mathematics, 10(1), Article 1. https://doi.org/10.3390/math10010041 Reschly, A. L., & Christenson, S. L. (Eds.). (2022). Handbook of Research on Student Engagement. Springer International Publishing.

443 450 Marianela Alpízar-Vargas Ceneida Fernández Salvador Llinares Universidad Nacional Universidad de Alicante Spain To date, the research has focused on how mathematics teachers acquire isolated core professional practices. Hitherto unexplored, however, is how these practices develop jointly. This case study focused on how a primary school teacher developed different core practices during a professional development course on the teaching and learning of magnitudes. The study participant was selected based on the specific context of a single-teacher school. The teacher noticed students’ mathematical thinking and anticipated students’ answers integrating the lenses provided in the course (Hypothetical Learning Trajectories and curricular guidelines). The lesson planning, however, remained closely aligned with institutional requirements (curricular guidelines) and was therefore work context-sensitive. Amador, J. M., Males, L. M., Earnest, D., & Dietiker, L. (2017). Curricular noticing: Theory on and practice of teachers’ curricular use. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 427–443). Springer. https://doi.org/10.1007/978-3-319-46753-5_25 Callejo, M.L., Pérez-Tyteca, P., Moreno, M., & Sánchez-Matamoros, G. (2022). The use of a length and measurement HLT by pre-Service kindergarten teachers to notice children’s mathematical thinking. International Journal of Science and Mathematics Education, 20, 597–617. https://doi.org/10.1007/s10763-021-10163-4 Fernández, C., & Choy, B. H. (2020). Theoretical lenses to develop mathematics teacher noticing. Learning, teaching, psychological, and social perspectives. In S. Llinares, & O. Chapman (Eds.), International Handbook of Mathematics Teacher Education: Volume 2. (Second Edition) (pp. 337–360). Brill. https://doi.org/10.1163/9789004418967_013 Breda, A., Sala-Sebastià, G., & Pino-Fan, L. (2024). Future teachers’ reflection on mathematical errors made in their teaching practice. ZDM – Mathematics Education, 56, 1169–1181. https://doi.org/10.1007/s11858-024-01574-y Grossman P. (Ed.) (2018). Teaching core practices in teacher education. Harvard Education Press. Ivars, P., & Fernández, C. (2016). Narratives and the development of the skill of noticing. In Csíkos, C., Rausch, A. & Szitányi, J. (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 19–26). Szeged, Hungary: PME. Jacobs, V. R., Lamb, L. C., & Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Ministerio de Educación Pública (MEP). (2012). Programas de Estudio en Matemáticas para la Educación General Básica y el Ciclo Diversificado. San José, Costa Rica: autor. Moreno, M., Santonja, P., & Llinares, S. (2025). Levels of articulation in prospective mathematics teacher noticing. Educational Studies in Mathematics. Sarama, J., & Clements, C. (2009). Early Childhood Mathematics Education Research. Learning Trajectories for Young Children. Routledge. Scheiner, T. (2023). Shifting the ways prospective teachers frame and notice student mathematical thinking: From deficits to strengths. Educational Studies in Mathematics, 114(1), 35–61. https://doi.org/10.1007/s10649-023-10235-y. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 114-145. Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing a high-level task. Mathematics Teaching in the Middle School, 14(3), 132–138. Weyers, J., König, J., Scheiner, T., Santagata, R., & Kaiser, G. (2024). Teacher noticing in mathematics education: a review of recent developments. ZDM–Mathematics Education, 56, 249–264. https://doi.org/10.1007/s11858-023-01527-x

451 458 Michael Fischer Johanna Hackl Jana Sitte Robin Göller Universität Graz Austria Universität Wien PH Steiermark Universität Klagenfurt This research report examines how 41 pre-service teachers design sustainability-related tasks for lower secondary school and reflect on them after a short, theory-informed intervention. We analyse which topics are addressed, how far tasks meet key mathematics-educational quality criteria, and the depth of participants’ theory-informed reflections. The results inform teacher education by indicating which criteria are salient in participants’ reflections and where explicit scaffolding is needed. Abels, S. (2011). LehrerInnen als „Reflective Practitioner“ [Teachers as ’reflective practitioners’]. VS Verlag für Sozialwissenschaften. Barwell, R., & Hauge, K. H. (2021). A Critical Mathematics Education for Climate Change: A Post-Normal Approach. In Applying Critical Math. Education (pp. 166–184). Brill. Coles, A., & Helliwell, T. (2023). The role of mathematics teacher educators in preparing teachers of mathematics to respond to global challenges within their classrooms. London Review of Education, 21(1). https://doi.org/10.14324/LRE.21.1.02 Gildehaus, L., Göller, R., Vajen, B., & Liebendoerfer, M. (2023). CiviMatics – Challenges and potentials of an interdisciplinary approach on normative modelling. Proceedings of the 12th International Conference of Mathematics Education and Society. International Conference of Mathematics Education and Society. Hackl, J., Schlüter, D., & Fischer, M. (2026). Textaufgaben als pragmatischer erster Schritt? Eine Studie zur Wirkung und Erstellung von BNE-Textaufgaben im Mathematikunterricht [Word problems as a pragmatic first step? A study on the effect and design of ESD textbased tasks in mathematics lessons]. Mathematik im Unterricht, 16. Hatton, N., & Smith, D. (1995). Reflection in teacher education: Towards definition and implementation. Teaching and Teacher Education, 11(1), 33–49. Hwang, S., & Han, S. (2025). Pre-service mathematics teachers’ competencies in designing mathematical modeling tasks. Journal of Mathematics Teacher Ed., 28(3), 713–743. Kaiser, G. (2020). Mathematical Modelling and Applications in Education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 553–561). Springer Int. Publishing. Schlüter, D., & Besser, M. (2024). Assessing authenticity in modelling test items: Deriving a theoretical model. Frontiers in Education, 9. https://doi.org/10.3389/feduc.2024.1343510 Schukajlow, S., Krawitz, J., Kanefke, J., Blum, W., & Rakoczy, K. (2023). Open modelling problems: Cognitive barriers and instructional prompts. Educational Studies in Mathematics, 114(3), 417–438. https://doi.org/10.1007/s10649-023-10265-6 Siller, H.-S., & Vorhölter, K. (2025). Bildung für nachhaltige Entwicklung im Mathematikunterricht: Potentiale und Herausforderungen für die Mathematikdidaktik [ESD in mathematics teaching: potential and challenges for mathematics education]. Mitteilungen der Gesellschaft für Didaktik der Mathematik, 118, Article 118. Skovsmose, O. (2021). Mathematics and crises. Educational Studies in Mathematics, 108(1– 2), Article 1–2. https://doi.org/10.1007/s10649-021-10037-0 Smith, D. (2021). Evaluating evidence of reflection and reflective practice. In R. Ewing, F. Waugh, & D. L. Smith, Reflective Practice in Education and Social Work (1st ed, pp. 150– 172). Routledge. https://doi.org/10.4324/9781003155010-11 UNESCO. (2017). Education for Sustainable Development Goals: Learning objectives. UNESCO. https://doi.org/10.54675/CGBA9153 Vorhölter, K., Siller, H.-S., Just, J., Orschulik, A., & Zieriacks, C. (2025). Education for sustainable development in mathematics classrooms. Environmental Education Research.

459 466 Arya Fu Shanghai Experimental Foreign Language School China, People's Republic of Mathematical modelling is widely regarded as a promising approach in mathematics education for exploring Sustainable Development Goals (SDGs). However, opportunities for mathematics teachers to design and teach modelling tasks are unevenly distributed. Generative artificial intelligence (GenAI) offers new possibilities to support such teaching activities. This study is a classroom-based exploratory research project. Starting with the design of mathematical modelling tasks on sustainable development themes, the researchers explored the potential of integrating AI into mathematical modelling instruction and cultivating students' sustainable development thinking by analyzing teacher-AI dialogue transcripts, mathematical modelling lesson plans, and classroom reflections. Barwell, R., Boylan, M., & Coles, A. (2022). Mathematics education and the living world: A dialogic response to a global crisis. The Journal of Mathematical Behavior, 68, 101013. Blum, W., & Leiss, D. (2007). How do students and teachers deal with modelling problems. Mathematical modelling (ICTMA12): Ed., engineering and economics, 222-231. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education (pp. 73-96). Cham: Springer. Consortium for Mathematics and Its Applications. (2019). High School Mathematical Contest in modelling (HiMCM): Problem B – Bottle Battles. COMAP. Consortium for Mathematics and Its Applications. (2019). Mathematical Contest in modelling (MCM): Problem A – Game of Ecology. COMAP. Giordano, F. R., Fox, W. P., Horton, S. B., & Weir, M. D. (2014). A first course in mathematical modelling (5th ed.). Brooks/Cole, Cengage Learning. Makramalla, M., Coles, A., le Roux, K., & Wagner, D. (2025). Mathematics education for sustainable futures: a strengths-based survey of the field to invite further research action: A. Coles et al. Educational Studies in Mathematics, 119(3), 535-556. Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American psychologist, 55(1), 68. Stillman, G. A., Kaiser, G., Blum, W., & Brown, J. P. (Eds.). (2013). Teaching mathematical modelling: Connecting to research and practice. Springer Science & Business Media. Helliwell, T., & Ng, O. L. (2022). Imagining possibilities: Innovating mathematics (teacher) education for sustainable futures. Research in Mathematics Education, 24(2), 128-149. Werth, J. C., Taylor, P. C. S., & Taylor, E. (2024). Integrating ethics for sustainability into mathematics instruction using generative AI. Australian Mathematics Education Journal, 6(1), 17-24. Wiegand, S., & Borromeo Ferri, R. (2023). Promoting pre-service teachers’ professionalism in steam education and education for sustainable development through mathematical modelling activities. ZDM–Mathematics Education, 55(7), 1269-1282. Yang, X., Li, X., Deng, Z., & Kaiser, G. (2025). Obstacles to in-service Chinese high school mathematics teachers’ implementation of mathematical modelling in classrooms: An empirical investigation of teachers’ perspectives. ZDM – Mathematics Education, 57(3), 535–551.

467 474 Selen Galiç Selin Urhan Zsolt Lavicza Johannes Kepler University Austria Hacettepe University This paper argues that students’ mathematical thinking can be reframed as rational behaviour manifested in their participation in mathematical activity. Drawing on Habermas’s theory of rationality, we conceptualise students’ engagement through epistemic, teleological, and communicative components, which provide observable and theoretically grounded indicators of understanding. To address the long-standing challenge of specifying what teachers should notice, we integrate this perspective with the expanded teacher noticing framework and propose the Teacher Noticing of Rationality Framework. The paper contributes to the networking of theories in mathematics education by articulating connections, and offers implications for research on noticing and the design of teacher educations Blömeke, S., Gustafsson, J., & Shavelson, R. (2015). Beyond dichotomies: Competence viewed as a continuum. Zeitschrift für Psychologie, 223(1), 3–13. Boero, P. (2006). Habermas’ theory of rationality as a comprehensive frame for conjecturing and proving in school. Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 2, 185–192. Boero, P., & Morselli, F. (2009). The use of algebraic language in mathematical modelling and proving in the perspective of Habermas’ theory of rationality. In Proceedings of CERME 6, January 28th-February 1st. Cai, J., & Ding, M. (2017). On mathematical understanding: perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education, 20, 5-29. https://doi.org/10.1007/s10857-015-9325-8 Erickson, F. (2011). On noticing teacher noticing. In Mathematics teacher noticing (pp. 4764). Routledge. Galiç, S., Urhan, S., Dost, Ş., & Lavicza, Z. (2025). Examining Mathematics Teachers Noticing the Rationality: Scenario-Based Training with AI Chatbot. Science & Education, 34(2), 2752-2790. https://doi.org/10.1007/s11191-025-00618-3 García-García, J., & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing Calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227-252. Habermas, J. (1999). Warheit und Rechtfertigung. Frankfurt: Suhrkamp Verlag (it. translation: 2001. Verità e giustificazione. Bari, Editori Laterza). Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41, 169–202. Jazby, D. (2023). Conceptualising mathematics teacher noticing as a perception/action cycle. Mathematics education research journal, 35(Suppl 1), 133-155. https://doi.org/10.1007/s13394-021-00392-9 Kastberg, S. E. (2002). Understanding mathematical concepts: The case of the logarithmic function (Doctoral dissertation, University of Georgia). König, J., Santagata, R., Scheiner, T., Adleff, A.-K., Yang, X., & Kaiser, G. (2022). Teachers’ noticing: A systematic literature review of conceptualizations, research designs, and findings on learning to notice. Educational Research Review, 36. https://doi.org/10.1016/j.edurev.2022.100453 Morselli, F., & Boero, P. (2009). Proving as a rational behaviour: Habermas’ construct of rationality as a comprehensive frame for research on the teaching and learning of proof. In Proceedings of CERME 6, January 28th-February 1st. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics. Mason, J. (2025). Critical Noticing and its Consequences. Journal of Mathematics Teacher Education, 28(5), 1267-1285. https://doi.org/10.1007/s10857-025-09718-x Santagata, R., & Yeh, C. (2014). Learning to teach mathematics and to analyze teaching effectiveness: Evidence from a video- and practice-based approach. Journal of Mathematics Teacher Education, 16, 1–24. Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (Eds.). (2011). Mathematics teacher noticing: Seeing through teachers’ eyes. New York: Routledge. Sherin, M. G., & van Es, E. A. (2009). Effects of video club participation on teachers' professional vision. Journal Of Teacher Education, 60(1), 20-37. https://doi.org/10.1177/0022487108328155 Urhan, S., & Bülbül, A. (2023a). Analysis of mathematical proving in geometry based on Habermas’ construct of rationality. Mathematics Education Research Journal, 35, 929– 959. https://doi.org/10.1007/s13394-022-00420-2 Urhan, S., & Bülbül, A. (2023b). Habermas’ construct of rationality in the analysis of the mathematical problem-solving process. Education Studies in Mathematics, 112, 175–197. https://doi.org/10.1007/s10649-022-10188-8 Urhan, S., & Zengin, Y. (2025). Creativity in Computational Problem-solving: The Perspective of Rationality. International Journal for Technology in Mathematics Education, 32(2), 51-72. https://doi.org/10.1564/tme_v32.2.01 van Es, E.A., & Sherin, M.G. (2021) Expanding on prior conceptualizations of teacher noticing. ZDM Mathematics Education 53, 17–27. https://doi.org/10.1007/s11858-02001211-4 van Es, E. A., & Sherin, M. G. (2025). Exploring teacher noticing in practice: the role of shaping. ZDM–Mathematics Education, Advanced online publication. https://doi.org/10.1007/s11858-025-01740-w Zhuang, Y., & Conner, A. M. (2022). Teachers’ use of rational questioning strategies to promote student participation in collective argumentation. Educational Studies in Mathematics, 111(2), 345–365. https://doi.org/10.1007/s10649-022-10160-6

475 482 Clara García-Cerdá Irene Ferrando Carlos Segura Universitat de València Spain This study examines whether performance and confidence gaps between students and pre-service teachers depend on the amount of information in modelling problems, and whether the confidence–performance correlation varies by type of problem and solver's profile. Using modelling problems with missing data (MD) and excess data (ED), we compared performance and confidence patterns across groups. Pre-service teachers outperformed students on ED-problems, while students exhibited lower confidence in MD-problems despite higher performance. The confidence–performance correlation remained stable for pre-service teachers but weakened for students on ED-problems. The findings highlight the importance of developing pre-service teachers' diagnostic competence to recognise metacognitive miscalibration in students. Böswald, V. & Schukajlow, S. (2023). I value the problem, but I don’t think my students will: Preservice teachers’ judgments of task value and self-efficacy for modelling, word, and intramathematical problems. ZDM - Mathematics Education, 55(2), 331-344. https://doi.org/10.1007/s11858-022-01412-z Ferrando, I., Barquero, B., & Segura, C. (2025). Does training matter? Effect of training strategies on how pre-service teachers pose and assess modelling problems. ZDM– Mathematics Education, 57(2), 275-288. https://doi.org/10.1007/s11858-025-01658-3 Ferrando, I., Segura, C. & Pla-Castells, M. (2020). Relations entre contexte, situation et schéma de résolution dans les problèmes d’estimation. Canadian Journal of Science, Mathematics and Technology Education, 20(3), 557-573. https://doi.org/10.1007/s42330020-00102-w Foster, C. (2016). Confidence and competence with mathematical procedures. Educational Studies in Mathematics, 91(2), 271-288. https://doi.org/10.1007/s10649-015-9660-9 García-Cerdá, C., Segura, C. & Ferrando, I. (in press). Effect of reading literacy on performance and confidence solving modelling problems with missing or excess data. In The 22nd International Conference on the Teaching of Mathematical Modelling and Applications (ICTMA 22). Linköping, Sweden, August 10-15, 2025. Krawitz, J. & Schukajlow, S. (2018). Do students value modelling problems, and are they confident they can solve such problems? Value and self-efficacy for modelling, word, and intra-mathematical problems. ZDM - Mathematics Education, 50(1-2), 143-157. https://doi.org/10.1007/s11858-017-0893-1 Krawitz, J., Schukajlow, S. & Van Dooren, W. (2018). Unrealistic responses to realistic problems with missing information: what are important barriers? Educational Psychology, 38(10), 1221-1238. https://doi.org/10.1080/01443410.2018.1502413 Klock, S. & Siller, H.-S. (2020). A time-based measurement of the intensity of difficulties in the modelling process. In G. A. Stillman, G. Kaiser, & C. E. Lampen, (Eds.), Mathematical modelling education sense-making (pp. 163-173). Springer. https://doi.org/10.1007/978-3-030-37673-4_15 Leiß, D., Plath, J. & Schwippert, K. (2019). Language and mathematics - key factors influencing the comprehension process in reality-based tasks. Mathematical Thinking and Learning, 21(2), 131-153. https://doi.org/10.1080/10986065.2019.1570835 Leuders, T., Dörfler, T., Leuders, J. & Philipp, K. (2018). Diagnostic competence of mathematics teachers: Unpacking a complex construct. In T. Leuders, K. Philipp & J. Leuders (Eds.), Diagnostic Competence of Mathematics Teachers (pp. 3-31). Springer. https://doi.org/10.1007/978-3-319-66327-2_1 Nederhand, M. L., Tabbers, H. K., De Bruin, A. B. & Rikers, R. M. (2021). Metacognitive awareness as measured by second-order judgements among university and secondary school students. Metacognition and Learning, 16(1), 1-14. https://doi.org/10.1007/s11409020-09228-6 Niss, M. & Blum, W. (2020). The Learning and Teaching of Mathematical Modelling. Routledge. https://doi.org/10.4324/9781315189314 Segura, C., Gallart, C. & Ferrando, I. (2025). Influence of pre-service primary school teachers’ prior knowledge of measurement and measurement estimation in solving modelling problems. Journal of Mathematics Teacher Education, 1-26. https://doi.org/10.1007/s10857-025-09685-3 Stankov, L., Lee, J., Luo, W. & Hogan, D. J. (2012). Confidence: A better predictor of academic achievement than self-efficacy, self-concept and anxiety? Learning and individual differences, 22(6), 747-758. https://doi.org/10.1016/j.lindif.2012.05.013 Vicente, S., Manchado, E., & Verschaffel, L. (2018). Solving arithmetic word problems. An analysis of Spanish textbooks. Culture and Education, 30, 71–104. https://doi.org/10.1080/11356405.2017.1421606

483 490 Jesus Garrido-Rojas Antonio Moreno Juan F. Ruiz-Hidalgo University of Granada Spain This study examines the professional identity of secondary mathematics teachers as a dynamic construct related to training, professional experience, and professional perceptions. The aim of the study is to analyse the relationship between teachers’ professional identity profiles and their evaluations of initial and continuing teacher education. Questionnaire data from 401 teachers of Compulsory and Upper Secondary Education were analysed using descriptive statistics, chi-square tests, and latent class analysis. The latent class analysis identified four differentiated teacher profiles combining academic background, professional trajectories, and perceptions of teacher education. The findings highlight the heterogeneity of secondary mathematics teachers and reinforce the need for flexible and differentiated models of teacher education. Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education. Routledge. Davydovskaia, O., De Coster, I., Birch, P., Vasiliou, N., & Motiejunaite-Schulmeister, A. (2021). Teachers in Europe: Careers, Development and Well-Being. Eurydice Report. Education, Audiovisual and Culture Executive Agency, European Commission. García-Longoria, M., y Blanco, L. J. (2005). La formación inicial de profesores de Matemáticas: Una encuesta entre los profesores de secundaria. Campo Ab., (28), 111-128. Garrido-Rojas, J., Moreno, A., y Ruiz-Hidalgo, J.F. (2024). Tendencias en educación matemática: explorando la relación entre el perfil del profesorado y la percepción de la formación en Enseñanza Secundaria. En N. Adamuz-Povedano, E. Fernández-Ahumada, N. Climent y C. Jiménez-Gestal (Eds.), Investigación en Educación Matemática XXVII (pp. 273-280). SEIEM. Hartl, A., y Holzberger, D. (2022). Identifying teachers’ motivational profiles and their changes from teacher education into practice: A longitudinal study. Z Erziehungswiss, 25, 427–451. https://doi.org/10.1007/s11618-022-01093-0 Holzberger, D., Praetorius, A. K., Seidel, T., & Kunter, M. (2019). Identifying effective teachers: The relation between teaching profiles and students’ development in achievement and enjoyment. European Journal of Psychology of Education, 34(4), 801-823. Howard, M. C., & Hoffman, M. E. (2017). Variable-centered, person-centered, and personspecific approaches: Where theory meets the method. Organizational Research Methods, 21(4), 846–876. https://doi.org/10.1177/1094428117744021 Losano, L., Fiorentini, D., & Villarreal, M. (2018). The development of a mathematics teacher’s professional identity during her first year teaching. Journal of Mathematics Teacher Education, 21, 287-315. https://doi.org/10.1007/s10857-017-9364-4 Martín de Diego, D. (Coord.) (2020). Libro Blanco de las Matemáticas. Real Sociedad Matemática Española y Fundación Ramón Areces. https://bit.ly/4bnvqZe Madeleine M. Müller & Bernadette Gold (2026): Exploring profiles of pre-service teachers’ situation-specific skills regarding classroom management, European J. of Teacher Ed. Muñiz-Rodríguez, L., Alonso Velázquez, P., Blanco, T. F., Rodríguez-Muñiz, L. J., y Valcke, M. (2015). ¿Cuál es el perfil de los futuros profesores de matemáticas en Educación Secundaria? En C. Fernández, M. Molina y N. Planas (Eds.), Investigación en Educación Matemática XIX (p. 575). Sociedad Española de Investigación en Educación Matemática (SEIEM). https://bit.ly/3S2XrOm Muñiz-Rodríguez, L., Aguilar-González, Á., y Rodríguez-Muñiz, L.J. (2020). Perfiles del futuro profesorado de matemáticas a partir de sus competencias profesionales. Enseñanza de las Ciencias, 38(2), 141-161. https://doi.org/10.5565/rev/ensciencias.3161 OECD. (2019). TALIS 2018 Results (Vol. 1): Teachers and School Leaders as Lifelong Learners. OECD Publishing. https://doi.org/10.1787/1d0bc92a-en

491 498 Sebastian Geisler University of Potsdam Germany High dropout rates illustrate students’ difficulties with university mathematics. Previous studies showed that interest and self-concept for university mathematics as well as the development thereof predict dropout. Therefore, this study examined the effects of students’ beliefs on interest, self-concept and the development of both variables. Using latent change score models, data from 286 first year students was analysed. The results show that students’ beliefs are related to interest and self-concept regarding university mathematics at the beginning of the first term, but beliefs do not predict the development of interest and self-concept. Moreover, beliefs regarding university mathematics were more relevant than beliefs regarding school mathematics. Implications for teaching and further research are discussed. Bong, M., & Skaalvik, E. M. (2003). Academic self-concept and self-efficacy: How different are they really? Educational Psychology Review,15(1), 1–40. Daskalogianni, K., & Simpson, A. (2001). Beliefs overhang: The transition from school to university. Proc. of the British Congress of Mathematics Education (vol. 21, pp. 97–108). Daskalogianni, K., & Simpson, A. (2002). “Cooling-off”: The phenomenon of a problematic transition from school to university. In I. Vakalis (Chair), Proceedings of the 2nd International Conference on the Teaching of Mathematics (at the Undergraduate Level) (pp. 103–110). Symposium conducted at the meeting of University of Crete. Engelbrecht, J. (2010). Adding structure to the transition process to advanced mathematical activity. IJMEST, 41(2), 143–154. Geiser, C. (2021). Longitudinal structural equation modeling with Mplus: A latent state-trait perspective. The Guilford Press. Geisler, S. (2024). Students’ beliefs concerning the nature of mathematics – Are they different with regard to school and University Mathematics? In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.). Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 224-231). PME. Geisler, S., & Rach, S. (2022). Development of Attitudes during the Transition to University Mathematics - different for Students who drop out? In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proc. 45th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2 S. 283-290). PME. Grigutsch, S., & Törner, G. (1998). World views of mathematics held by university teachers of mathematics science (Preprint Series No. 420). University of Duisburg. Gueudet, G. (2008). Investigating the secondary–tertiary transition. ESM, 67(3), 237–254. Heublein, U., Hutzsch, C., & Schmelzer, R. (2022). Die Entwicklung der Studienabbruchquoten in Deutschland. https://doi.org/10.34878/2022.05.dzhw_brief Kaya, S. (2007). The influences of student views related to mathematics and self-regulated learning on achievement of algebra I students. Ohio State University. Kirsten, K., Greefrath, G. & Geisler, S. (online first). Development of interest in mathematics and mathematical self-concept at the transition from school to university and its relation to study success. ESM. https://doi.org/10.1007/s10649-025-10469-y Krapp, A. (2007). An educational-psychological conceptualisation of interest. International Journal for Educational and Vocational Guidance, 7(1), 5–21. Liebendörfer, M., & Schukajlow, S. (2017). Interest development during the first year at university: do mathematical beliefs predict interest in mathematics? ZDM, 49(3), 355–366. Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 257–315). Information Age Publication. Rach, S., Ufer, S., & Kosiol, T. (2021). Die Rolle des Selbstkonzepts im Mathematikstudium – Wie fit fühlen sich Studierende in Mathematik? [The role of self-concept in mathematics study programs - How fit do students feel in mathematics?]. ZfE, 24(6), 1549–1571. Shavelson, R. J., Hubner, J. J., & Stanton, G. C. (1976). Self-concept: Validation of construct interpretations. Review of Educational Research, 46(3), 407–441 Sonnert, G., Sadler, P. M., Sadler, S. M., & Bressoud, D. M. (2015). The impact of instructor pedagogy on college calculus students’ attitude toward mathematics. IJMEST, 46(3), 370– 387. Ufer, S., Rach, S., & Kosiol, T. (2017). Interest in mathematics = interest in mathematics? What general measures of interest reflect when the object of interest changes. ZDM, 49(3), 397–409.

499 506 Chiara Giberti Consuelo Mameli Valentina Grazia University of Modena and Reggio Emilia, Alma Mater Studiorum University of Bologna Italy University of Parma Student agency has become a central construct in contemporary educational research, yet its meaning and operationalization remain fragmented, particularly in mathematics. This theoretical essay analyses how student agency in mathematics is conceptualized and investigated within educational psychology and mathematics education. Through a dialogic analysis, we highlight convergences and tensions between psychosocial and disciplinary perspectives, with particular attention to proactivity, resistance, authority, and epistemic norms. We propose an integration of the two perspectives, arguing that this can guide a more comprehensive definition of student agency in mathematics and inform the development of methodological tools sensitive to the specificities of mathematical teaching and learning. Bae, C. L., Matewos, A. M., & Fife, J. (2025). Supporting student voice in science classrooms: The limits of psychosocial approaches and the importance of sociocultural and critical perspectives on student agency. Educational Psychologist, 60(1), 48–68. Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22, 42–47. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171– 200). Ablex. Brown, R. (2020). Re-conceptualizing the development of agency in the school mathematics classroom. Theory Into Practice, 59(2), 139–149. Etelapelto, A., Vahasantanen, K., Hokka, P., & Paloniemi, S. (2013). What is agency? Conceptualizing professional agency at work. Educational Research Review, 10, 45–65. Grazia, V., Giberti, C., & Mameli, C. (2025). The trajectory of student agentic engagement in mathematics: The role of individual variables. Learning and Individual Differences, 123, 102766. Gresalfi, M. S., Martin, T., Hand, V., & Greeno, J. G. (2009). Constructing competence: An analysis of student participation in the activity systems of mathematics classrooms. Educational Studies in Mathematics, 70(1), 49–70. Leijen, A., Baucal, A., Pikk, K., Uibu, K., Pajula, L., & Sormus, M. (2024). Opportunities to develop student’s math-related agency in primary education: The role of teacher beliefs. European Journal of Psychology of Education, 39, 1637–1659. Mameli, C., & Passini, S. (2019). Development and validation of an enlarged version of the student agentic engagement scale. Journal of Psychoeducational Assessment, 37(4), 450– 463. Mameli, C., Grazia, V., & Molinari, L. (2023). Student agency: Theoretical elaborations and implications for research and practice. International Journal of Educational Research, 122, 102258. Mueller, M., Yankelewitz, D., & Maher, C. (2012). A framework for analyzing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics, 80(3), 369–387. Nieminen, J. H., Chan, M. C. E., & Clarke, D. (2022). What affordances do open-ended reallife tasks offer for sharing student agency in collaborative problem-solving? Educational Studies in Mathematics, 109(1), 115–136. Nieminen, J., & Hilppo, J. (2020). Methodological and conceptual suggestions for researching the interplay of assessment and student agency. In P. J. White, R. Tytler, J. Ferguson, & J. Cripps Clark (Eds.), Methodological approaches to STEM education research (Vol. 1, pp. 87–107). Cambridge Scholars. Pickering, A. (1995). The Mangle of Practice: Time, Agency, and Science. University of Chicago Press. Reeve, J., & Jang, H. (2022). Agentic engagement. In A. L. Reschly, & S. L. Christenson (Eds.), Handbook of research on student engagement (pp. 95–107). Springer International. Taguma, M., Makowiecki, K., & Gabriel, F. (2023). OECD learning compass 2030: implications for mathematics curricula. In Y. Shimizu, & R. Vithal, (Eds.), Mathematics curriculum reforms around the world: The 24th ICMI Study (pp. 479-509). Springer International Publishing. Wagner, D. (2004). Critical awareness of voice in mathematics classroom discourse: Learning the steps in the “dance of agency”. In M. Johnsen Høines, & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 401–408). IGPME. Woods, D., & Salem, W. (2024). Cultivating students’ agency by infusing number talks with notice and wonder. Investigations in Mathematics Learning, 16(22), 120–136. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 27(4), 458-477.

507 514 Lara Gildehaus Robin Göller University of Klagenfurt Austria This study examined the effect of positive narrative reflection on primary preservice teachers’ (PSTs) mathematics-related identity. Using data from 51 PSTs and a mixed-methods approach, reflection essays were qualitatively coded and integrated into linear mixed models. Results showed significant increases in mathematics-related self-efficacy, identification, and attainment value, while professional teaching identity remained stable. Reflections focusing on social relations and positive emotions acted as key moderators, significantly driving these gains. High intraclass correlation coefficients revealed the enduring resilience of mathematics teacher identities despite the intervention’s success. The findings suggest that the development of mathematics teacher identities relies on the type of narrative reflections which are discussed. Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5), 1860–1863. https://doi.org/10.1073/pnas.0910967107 Bekdemir, M. (2010). The pre-service teachers’ mathematics anxiety related to depth of negative experiences in mathematics classroom while they were students. Educational Studies in Mathematics, 75(3), 311–328. https://doi.org/10.1007/s10649-010-9260-7 Gildehaus, L., & Liebendörfer, M. (2024). Math for teaching or university? - Preservice teachers’ motivation in their first study year. In T. Evans, O. Marmur, J. Hunter, & G. Leach (Eds), Proceedings of the 47th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 232–240). PME. Gildehaus, L., Liebendörfer, M., & Heyd-Metzuyanim, E. (2024). “I’m a bit out of place here.”—Preservice teachers’ positioning in the figured world of university mathematics. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-024-10315-7 Jenßen, L. (2024). “Name Three Good Things About Yourself in Mathematics” – An Intervention to Reduce Pre-Service Teachers’ Shame in Mathematics. International Journal of Applied Positive Psychology, 9(1), 189–208. https://doi.org/10.1007/s41042023-00120-7 Kempen, L., & Liebendörfer, M. (2021). University students’ fully digital study of mathematics: An identification of student-groups via their resources usage and a characterization by personal and affective characteristics. Teaching Mathematics and Its Applications: An International Journal of the IMA, hrab020. https://doi.org/10.1093/teamat/hrab020 Lutovac, S., & Kaasila, R. (2014). Pre-service teachers’ future-oriented mathematical identity work. Educational Studies in Mathematics, 85(1), 129–142. https://doi.org/10.1007/s10649-013-9500-8 Maloney, E. A., Schaeffer, M. W., & Beilock, S. L. (2013). Mathematics anxiety and stereotype threat: Shared mechanisms, negative consequences and promising interventions. Research in Mathematics Education, 15(2), 115–128. https://doi.org/10.1080/14794802.2013.797744 Ramm, G., Adamsen, C., Neubrand, M., & Deutsches PISA-Konsortium (Eds). (2006). PISA 2003: Dokumentation der Erhebungsinstrumente [PISA 2003: Documentation of the survey instruments]. Waxmann. Solomon, Y., Eriksen, E., Smestad, B., Rodal, C., & Bjerke, A. H. (2017). Prospective teachers navigating intersecting communities of practice: Early school placement. Journal of Mathematics Teacher Education, 20(2), 141–158. https://doi.org/10.1007/s10857-0159327-6

515 522 Alejandro S. González-Martín Mathilde Hitier Jonathan Morcos Département de Didactique, Université de Montréal Canada The mathematical training of engineers and their practical application of mathematics have received increased attention from researchers over the past 15 years. In this paper, we present results from the pilot phase of a survey that will eventually be distributed to a large sample of engineers. Drawing on elements of the Anthropological Theory of the Didactic (ATD), we focus on tasks in which mathematics is explicitly or implicitly involved, and on the ways participants complete these tasks. Drawing on responses from ten engineers, we find that three report not needing any university mathematics in their professional practice. Seven carry out tasks that implicitly involve mathematics and seven carry out tasks (albeit infrequently) in which the use of mathematics is explicit. All participants use software to perform these tasks. Artigue, M. (2016). Mathematics education research at university level: Achievements and challenges. In E. Nardi, C. Winsløw, & T. Hausberger (Eds.), Proceedings of the First Conference of the International Network for Didactic Research in University Mathematics (INDRUM2016) (pp.11–27). University of Montpellier and INDRUM. Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at postsecondary level. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 1011–1049). Information Age Publishers. Biza, I., González-Martín, A. S., & Pinto, A. (2022). Scaffolding’ or ‘Filtering’: A review of studies on the diverse roles of calculus courses for students, professionals and teachers. Int. Journal of Research in Undergraduate Mathematics Education, 8(2), 389–418. Bosch, M., Chevallard, Y., García, F. J., & Monaghan, J. (Eds.) (2020). Working with the anthropological theory of the didactic in mathematics education: a comprehensive casebook. Routledge. https://doi.org/10.4324/9780429198168 Faulkner, B., González-Martín, A. S., Gueudet, G., Ralston, P., & Hieb, J. (2025). Calculus in engineering programmes: Issues related to content, student persistence, and the use of applications. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-025-00280-8 𝐵 González-Martín, A. S. (2021). V(B) – V(A) = ∫𝐴 𝑓 (𝑥)𝑑𝑥. The use of integrals in engineering programmes: a praxeological analysis of textbooks and teaching practices in Strength of Materials and Electricity and Magnetism courses. International Journal of Research in Undergraduate Mathematics Education, 7(2), 211–234. https://doi.org/qmz6 Howson, A. G., Kahane, J.-P., Kelly, P. J., Lauginie, P., Nemetz, T., Simons, F. H.,…de Turckheim, E. (1986). Mathematics as a service subject. L’enseignement Mathématique, 32, 159–172. Kent, P., & Noss, R. (2003). Mathematics in the university education of engineers (a report to the Ove Arup Foundation). Noss, R. (2002). Mathematical epistemologies at work. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 47–63). PME. Quéré, P.-V. (2017). French engineers’ training and their mathematical needs in the workplace: Interlinking tools and reasoning. In T. Dooley & G. Gueudet (Eds.), Proceedings of the 10th Congress of the European Mathematical Society for Research in Mathematics Education (pp. 2233–2240). DCU Institute of Education and ERME. Quéré, P.-V. (2019). Les mathématiques dans la formation des ingénieurs et sur leur lieu de travail: études et propositions (cas de la France). [PhD thesis, Université de Bretagne Occidentale]. https://www.theses.fr/2019BRES0041 Saldaña, J. (2013). The coding manual for qualitative researchers (2nd ed.). Sage. Snyder, V. (1912). The fifth International Congress of Mathematicians, Cambridge, 1912. Bulletin of the American Mathematical Society, 19(3), 107–130.

523 530 Myriam Goor Weizmann Institute of Science Israel Discussions between mathematicians and secondary mathematics teachers are important for addressing shared concerns about mathematics teaching, yet productive dialogue across these professional boundaries is challenging. This study examines how mathematicians and secondary mathematics teachers engage in dialogical learning during a joint discussion of how students experience a university preparatory mathematics course. Drawing on the literature on boundary crossing, the analysis focuses on dialogical learning mechanisms to explore how differences between perspectives are articulated and engaged with in interaction. The findings show how processes of identification and reflection can support the re-framing of the problem under discussion, contributing to the emergence of a shared problem space. Akkerman, S. F., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research, 81(2), 132-169. https://doi.org/10.3102/0034654311404435 Corriveau, C. (2023). Collaboration between secondary and post-secondary teachers about their ways of doing mathematics using contexts. In Practice-Oriented Research in Tertiary Mathematics Education (pp. 67-89). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-031-14175-1_4 Fried, M. N. (2014). Mathematics & mathematics education: Searching for common ground. In M. N. Fried & T. Dreyfus (Eds.), Mathematics & mathematics education: Searching for common ground (Advances in Mathematics Education, pp. 3–22). Springer. https://doi.org/10.1007/978-94-007-7473-5_1 Goos, M., & Bennison, A. (2018). Boundary crossing and brokering between disciplines in pre-service mathematics teacher education. Mathematics Education Research Journal, 30(3), 255–275. https://doi.org/10.1007/s13394-017-0232-4 Gueudet, G. (2023). New insights about the secondary–tertiary transition in mathematics. Educational Studies in Mathematics, 113(1), 165–179. https://doi.org/10.1007/s10649023-10223-2 Goor, M., Pinto, A., & Karsenty, R. (2023). Mathematicians and experienced teachers: Crossing the boundary. In M. Trigueros, B. Barquero, R. Hochmuth, & J. Peters (Eds.), Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19–22 October 2022) (pp. 508–517). University of Hannover and INDRUM. Lai, Y., & Baldinger, E. E. (2021). Authority relations and the tertiary-to-secondary (Dis) continuity. For the Learning of Mathematics, 41(2), 26–31. https://www.jstor.org/stable/27091201 Pinto, A., & Cooper J. (2022). The road not taken—Investigating affordances of infinitesimal calculus for enriching the repertoire of secondary mathematics teachers. International Journal of Research in Undergraduate Mathematics Education, 8, 318-338. https://doi.org/10.1007/s40753-021-00161-w Wasserman, N. H., Buchbinder, O., & Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. ZDM Mathematics Education, 55, 719–736. https://doi.org/10.1007/s11858-023-01484-5

531 538 Gilbert Greefrath Emilia Bommert Katharina Kirsten University of Münster Germany Functional thinking is considered a fundamental prerequisite for understanding key mathematical concepts. This study examines the structure of functional thinking based on the facets of input-output assignment, covariation and objects. To this end, an economic testing instrument was developed and evaluated with N = 153 Year 11 students. Confirmatory factor analyses confirm that the theoretically postulated three-dimensional model represents the data significantly better than a one-dimensional model. The instrument enables a differentiated diagnosis of individual thinking patterns at the transition to upper secondary school and provides a valid basis for research and practice in mathematics education. Acevedo Nistal, A., Van Dooren, W., & Verschaffel, L. (2012). What counts as a flexible representational choice? An evaluation of students’ representational choices to solve linear function problems. Instructional Science, 40(6), 999–1019. Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds), Early Algebraization (pp. 5–23). Springer. DeMarois, P., & Tall, D. (1999). Function: Organizing principle or cognitive root? In O. Zaslavsky (Ed.), Proc. 23rd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 257–264). PME. Doorman, M., Drijvers, P., Gravemeijer, K., Boon, P., & Reed, H. (2012). Tool use and the development of the function concept: From repeated calculations to functional thinking. International Journal of Science and Mathematics Education, 10(6), 1243–1267. Dubinsky, E., & Wilson, R. T. (2013). High school students’ understanding of the function concept. The Journal of Mathematical Behavior, 32(1), 83–101. Greefrath, G., Oldenburg, R., Siller, H.-S., Ulm, V., & Weigand, H.-G. (2023). Mathematics students’ characteristics of basic mental models of the derivative. Journal Für MathematikDidaktik, 44(1), 143–169. https://doi.org/10.1007/s13138-022-00207-9 Günster, S. M., & Weigand, H.-G. (2020). Designing digital technology tasks for the development of functional thinking. ZDM – Mathematics Education, 52(7), 1259–1274. Harel, G., Selden, A., & Selden, J. (2006). Advanced mathematical thinking. In Á. Gutiérrez & P. Boero (Eds), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 147–172). Brill. https://doi.org/10.1163/9789087901127 Kaput, J., & Schorr, R. (2008). Changing representational infrastructures changes most everything: The case of SimCalc, algebra and calculus. In G. W. Blume & M. K. Heid (Eds), Research on technology and the learning and teaching of mathematics: Vol. 2 cases and perspectives (Vol. 2, pp. 211–253). Information Age Publishing. Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2020). Young students’ functional thinking modes: The relation between recursive patterning, covariational thinking, and correspondence relations. Journal for Research in Mathematics Education, 51(5), 631–674. Pittalis, M., Sproesser, U., Demosthenous, E., & Odysseos, E. (2025). Enhancing functional thinking in grade 5–6 students through a dynamic mathematics intervention program. Education and Information Technologies, 30(2), 1329–1361. Rolfes, T., Roth, J., & Schnotz, W. (2022). Mono- and multi-representational learning of the covariational aspect of functional thinking. Journal for STEM Education Research, 5(1), 1–27. https://doi.org/10.1007/s41979-021-00060-4 Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. J. Kaput, D. V. Carraher, & M. L. Blanton (Eds), Algebra in the Early Grades (pp. 133–160). Lawrence Erlbaum. Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics. Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305. Wei, H., Bos, R., & Drijvers, P. (2025). A digital-embodied design for functional thinking in the classroom. Mathematical Thinking and Learning, 1–31.

539 546 Pin-Chen Guo Kai-Lin Yang Kristin Litteck Aiso Heinze National Taiwan Normal University IPN – Leibniz Institute for Science and Mathematics Education Kiel Taiwan Using Sfard’s (1991) description of operational (process-oriented) and structural (object-oriented) knowledge, Litteck et al. (2024) empirically validated a derivative knowledge model with German students’ data. This study replicated their work with a Taiwanese sample (N = 373) to examine whether the model holds across educational systems. Operational items were found to be consistently more difficult than structural ones; however, the relative difficulty among structural sub-concepts differed from the German results. Model-comparisons showed that a three-factor model (ratio, limit, function) best represented the Taiwanese data, whereas Litteck et al. identified a two-dimensional structure (operational, structural). These results suggest that while the process-object distinction is robust, its internal organization may vary across contexts. Aguilar, M. S. (2020). Replication studies in mathematics education: What kind of questions would be productive to explore? International Journal of Science and Mathematics Education, 18(Suppl 1), 37–50. Burkhardt, H. (2013). Methodological issues in research and development. In Y. Li & J. N. Moschkovich (Eds.), Proficiency and beliefs in learning and teaching mathematics: Learning from Alan Schoenfeld and Günter Törner (pp. 203–236). Sense. Dreher, A., Lindmeier, A., Feltes, P., Wang, T.-Y., & Hsieh, F.-J. (2021). Do cultural norms influence how teacher noticing is studied in different cultural contexts? A focus on expert norms of responding to students’ mathematical thinking. ZDM, 53(1), 165–179. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 95–126). Springer. Litteck, K., Rolfes, T., & Heinze, A. (2024). The structure of knowledge about the concept of derivative – a study investigating a process-object framework. International Journal of Mathematical Education in Science and Technology, 56(11), 2164–2184. Ministry of Education (2018). Curriculum Guidelines for the 12-Year Basic Education Elementary School, Junior High School, and Upper Secondary School: The Domain of Mathematics. OECD. (2023). PISA 2022 results (Volume I): The state of learning and equity in education. OECD-Publishing. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Ed. Studies in Mathematics, 22(1), 1–36. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In E. Dubinsky & G. Harel (Eds.), MAA notes: Vol. 25. The concept of function: Aspects of epistemology and pedagogy (pp. 59–84). Mathematical Association of America. Skemp, R. R. (2012). The psychology of learning mathematics: Expanded American edition. Routledge. Star, J. R. (2018). When and why replication studies should be published: Guidelines for mathematics education research journals. Journal for Research in Mathematics Education, 49(1), 98–103. Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in Collegiate Mathematics Education (Vol. IV, pp. 103–127). Am. Mathematical Society.

547 554 Robin Göller University of Klagenfurt Austria High dropout rates in mathematics-intensive study programmes remain a persistent challenge in higher education. This study investigates how students’ dropout intentions are shaped by the interplay between performance feedback and engagement in exercise tutorials. Using a longitudinal design, data of 125 students in mathematics-intensive study programmes were analysed using linear mixed models. Findings reveal that performance feedback from a midterm exam acts as a critical calibration point, triggering a divergence in students’ dropout intentions. Especially for students with lower exercise engagement, midterm performance feedback is critical for dropout intention. In contrast, students with high exercise engagement tend to be less reactive to exam results. However, this immunity comes at a cost that is discussed. Bäulke, L., Grunschel, C., & Dresel, M. (2022). Student dropout at university: A phaseorientated view on quitting studies and changing majors. European Journal of Psychology of Education, 37(3), 853–876. https://doi.org/10.1007/s10212-021-00557-x Büchele, S. (2023). Navigating success in higher education: Engagement as a mediator between learning strategies and performance in mathematics. Assessment & Evaluation in Higher Education, 1–15. https://doi.org/10.1080/02602938.2023.2230387 Burazin, A., & Lovric, M. (2018). Transition from Secondary to Tertiary Mathematics: Culture Shock – Mathematical Symbols, Language, and Reasoning. In A. Kajander, J. Holm, & E. J. Chernoff (Eds), Teaching and Learning Secondary School Mathematics (pp. 601–611). Springer International Publishing. https://doi.org/10.1007/978-3-319-923901_55 Di Martino, P., & Gregorio, F. (2019). The mathematical crisis in secondary–tertiary transition. International Journal of Science and Mathematics Education, 17(4), 825–843. https://doi.org/10.1007/s10763-018-9894-y Di Martino, P., Gregorio, F., & Iannone, P. (2023). The transition from school to university in mathematics education research: New trends and ideas from a systematic literature review. Educational Studies in Mathematics, 113(1), 7–34. Dibbs, R. (2019). Forged in failure: Engagement patterns for successful students repeating calculus. Educational Studies in Mathematics, 101(1), 35–50. Findeisen, S., Brodsky, A., Michaelis, C., Schimmelpenningh, B., & Seifried, J. (2024). Dropout intention: A valid predictor of actual dropout? Empirical Research in Vocational Education and Training, 16(1), 10. https://doi.org/10.1186/s40461-024-00165-1 Geisler, S., Rach, S., & Rolka, K. (2023). The relation between attitudes towards mathematics and dropout from university mathematics—The mediating role of satisfaction and achievement. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-02210198-6 Gildehaus, L., Haak, I., & Liebendörfer, M. (2025). The role of learning groups in first-year university mathematics: An explorative interview study. International Journal of Mathematical Education in Science Technology, 1–16. https://doi.org/10.1080/0020739X.2025.2572375 Gueudet, G. (2008). Investigating the secondary–tertiary transition. Educational Studies in Mathematics, 67(3), 237–254. https://doi.org/10.1007/s10649-007-9100-6 Kwiek, M., & Szymula, L. (2025). Quantifying attrition in science: A cohort-based, longitudinal study of scientists in 38 OECD countries. Higher Education, 89(6), 1465– 1493. https://doi.org/10.1007/s10734-024-01284-0 Oliveira, A. W., Musah, R. A., Zacharia, Z. C., Avlou, Y., Sgro, C. M., Lathrop, R., Kim, Y., Espinosa, K., Japashov, N., & Park, J. (2025). The Persistent Problem of Undergraduate Student Attrition in STEM. International Educational Review, 3(1), 85–111. Pleskac, T. J., Keeney, J., Merritt, S. M., Schmitt, N., & Oswald, F. L. (2011). A detection model of college withdrawal. Organizational Behavior and Human Decision Processes, 115(1), 85–98. https://doi.org/10.1016/j.obhdp.2010.12.001

555 562 Nadav Marco Raz Harel Gil Schwarts David Yellin Academic College of Education Israel Weizmann Institute of Science Israel This exploratory study proposes a novel approach for integrating communication with large language models (LLMs) into teacher education, and a theoretical framework for analyzing the resulting interactions between prospective teachers (PTs) and artificial intelligence (AI) agents. The study involved 20 PTs who were tasked with identifying a mathematical problem from the curriculum where the LLM generates an erroneous solution, and subsequently guiding it to identify and correct its error. To investigate PT–AI interactions, particularly teachers’ mathematical responsiveness (or lack thereof), we employed methods from Systemic Functional Linguistics (SFL). We found substantial variation in responsiveness, shaped by whether PTs framed the interaction as a discussion with AI or adopted a teaching–learning stance. Bishop, J. P., Hardison, H. L., & Przybyla-Kuchek, J. (2022). Responsiveness to students’ mathematical thinking in middle-grades classrooms. Journal for Research in Mathematics Education, 53(1), 10–40. https://doi.org/10.5951/jresematheduc-2020-0188 Buber, M. (1958). I and Thou. Scribner. Ergene, Ö., & Çaylan-Ergene, B. (2025). Pre-service mathematics teachers’ and engineering students’ perceptions of ChatGPT in mathematics: Development, validation and implementation study. Digital Experiences in Mathematics Education. 1–27 [online first] https://doi.org/10.1007/s40751-025-00176-x Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. https://doi.org/10.5951/jresematheduc.41.2.0169 Halliday, M., & Matthiessen, C. M. (2004). Introduction to Functional Grammar (3rd Edition edition). Hodder Arnold. Marco, N., & Stylianides, A. J. (2025). ChatGPT’s mathematical knowledge and the nature of human–AI collaboration in mathematical inquiry. International Journal of Mathematical Education in Science and Technology. 56(11), 2279–2297. https://doi.org/10.1080/0020739X.2025.2543817 Rizos, I., & Gkrekas, N. (2025). The impact of LLMs on mathematics education and research at the university. Social Sciences & Humanities Open, 12, 101969. https://doi.org/10.1016/j.ssaho.2025.101969 Robertson, A. D., Scherr, R. E., & Hammer, D. (Eds.). (2016). Responsive teaching in science and mathematics. Taylor & Francis. https://doi.org/10.4324/9781315689302 Schwarts, G. (2025). Moving beyond moves: Redefining mathematically responsive teaching as a humanistic stance. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.). Proc. 48th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 267–274). PME. Schwarts, G., & Czaczkes, S. (Submitted). Implicit Formative Feedback in AI-Based Simulations for Assessing Mathematically Responsive Teaching. Submitted to the second Feedback and Assessment in Mathematics Education conference (FAME2). Taani, O., & Alabidi, S. (2024). ChatGPT in education: Benefits and challenges of ChatGPT for mathematics and science teaching practices. International Journal of Mathematical Education in Science Technology, 1–30. https://doi.org/10.1080/0020739X.2024.2357341 Zhuang, Y., & Zhang, S. (2025). Integrating ChatGPT in Mathematics Teacher Education: AI-Based Simulation Role-Playing to Support Practice-based Teaching. International Journal of Artificial Intelligence in Education, 1–23.

571 578 Çiğdem Haser Ion Petre Kübra Nur Savaş University of Turku Finland Preservice secondary mathematics teachers’ AI literacy is important because they can help secondary school students develop effective AI literacy. This study investigates how preservice secondary mathematics teachers have developed their own AI literacy for their future career throughout a Master’s level course in Finland that focuses on AI-related mathematics, coding, ethics, teaching implementations and attitudes. Interviews with 4 preservice secondary mathematics teachers revealed that participants benefitted from the course experiences and enhanced their AI literacy, and built connections between different components of the course. Findings indicate that training preservice mathematics teachers for AI should address their AI literacy and mathematics-specific elements that could be brought to the mathematics classrooms. Ayanwale, M. A., Adelana, O. P., Molefi, R. R., Adeeko, O., & Ishola, A. M. (2024). Examining artificial intelligence literacy among pre-service teachers for future classrooms. Computers and Education Open,100179. Carrillo, C., Flores, M. A., & Mena, J. (2025). Artificial intelligence in the context of teacher education: emerging themes and critical issues. European Journal of Teacher Education, 48(5), 1143-1162. Gamlem, S. M., McGrane, J., Brandmo, C., Moltudal, S., Sun, S. Z., & Hopfenbeck, T. N. (2025). Exploring pre-service teachers’ attitudes and experiences with generative AI: a mixed methods study in Norwegian teacher education. Educational Psychology, 1-25. Hur, J. W. (2024). Fostering AI literacy: overcoming concerns and nurturing confidence among preservice teachers. Information and Learning Sciences, 126(1/2), 56-74. Laru, J., Celik, I., Jokela, I., & Mäkitalo, K. (2025). The antecedents of pre-service teachers’ AI literacy: perceptions about own AI driven applications, attitude towards AI and knowledge in machine learning. European Journal of Teacher Education, 48(5), 964-986. Ng, D. T. K., Lee, M., Tan, R. J. Y., Hu, X., Downie, J. S., & Chu, S. K. W. (2023a). A review of AI teaching and learning from 2000 to 2020. Education and Information Technologies, 28, 8445-8501. Ng, D. T. K., Su, J., Leung, J. K. L., & Chu, S. K. W. (2024). Artificial intelligence (AI) literacy education in secondary schools: A review. Interactive Learning Environments, 32(10), 6204-6224. Ng, D. T. K., Leung, J. K. L., Su, J., Ng, R. C. W., & Chu, S. K. W. (2023b). Teachers’ AI digital competencies and twenty-first century skills in the post-pandemic world. Educational Technology Research and Development, 71(1), 137-161. Stenberg, C. J., Åkerfeldt, A., Stenliden, L., & McGrath, C. (2025). Teaching with the algorithm: anticipations of AI in teacher education. European Journal of Teacher Education, 48(5), 1056-1076. Yi, L., Liu, D., Jiang, T., & Xian, Y. (2025). The effectiveness of AI on K-12 students’ mathematics learning: A systematic review and meta-analysis. International Journal of Science and Mathematics Education, 23, 1105-1126.

579 586 Vesife Hatisaru Steven Richardson Edith Cowan University Australia University of Tasmania This paper raises questions about interactions between mathematics content and mathematics pedagogy knowledge. We draw on data from a larger study on examining secondary preservice mathematics teachers’ proficiency in algebra and its teaching. These data were collected in an Australian initial teacher education course, where a set of algebraic word problems were implemented, and were analysed by using sample indicators of school algebra (SA) and algebra teaching (AT) knowledge, drawn from the Teachers’ Knowledge of Algebra for Teaching framework (Reckase et al., 2015). The findings highlight complex interactions between the two knowledge categories, where SA knowledge provides a foundation for AT yet does not guarantee it. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal for Teacher Education, 59(5), 389-407. Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., . . . Tsai, Y. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133-180. Blömeke, S., Busse, A., Kaiser, G., König, J., & Suhl, U. (2016). The relation between contentspecific and general teacher knowledge and skills. Teaching and T Ed., 56, 35-46. Hatisaru, V., Stacey, K., & Star, J.R. (2024). Mathematical connections in preservice secondary mathematics teachers’ solution strategies to algebra problems. AIEM – Avances de Investigación en Educación Matemática, 25, 33-55. Lowrie, T., & Jorgensen, R. (2016). Pre-service teachers’ mathematics content knowledge: implications for how mathematics is taught in higher education. Teaching Mathematics and its Applications: An International Journal of the IMA, 35(4), 202-215. McCrory, R., Floden, R., Ferrini-Mundy, J., Reckase, M. D. & Senk, S. L. (2012). Knowledge of algebra for teaching: A framework of knowledge and practices. Journal of Research in Mathematics Education, 43(5), 584-615. https://doi.org/10.5951/jresematheduc.43.5.0584 Reckase, M. D., McCrory, R., Floden, R. E., Ferrini-Mundy, J., & Senk, S. L. (2015). A multidimensional assessment of teachers’ knowledge of algebra for teaching: Developing an instrument and supporting valid inferences. Educational Assessment, 20(4), 249–267. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Ed.ucational Researcher, 15(2), 4-14. Osei, W., & Agyei, D. D. (2023). Teaching knowledge and difficulties of in-field and out-of-filed junior high school mathematics teachers in algebra. Cogent Education, 10(2).

587 594 Marion Hechler Victoria Shure Malte Lehmann Bettina Rösken-Winter Humboldt Universität zu Berlin Germany Studies on professional development (PD) emphasize the important role of facilitators in supporting teacher learning. Since moderating discussions is an essential part of PD, this study examines how facilitators justify the productivity of PD discussions for teacher learning. Two fictional PD discussions were designed to show a fictional facilitator enhancing teacher learning in a more or less productive way. Fifty-one facilitators were asked to select one of the discussions and explain their choice. A content analysis yielded four categories of justifications. The most important aspect of moderating a productive discussion, according to facilitators, was cognitive activation of teachers. The most important aspect of moderating an unproductive discussion was lack of appreciative feedback. Borko, H., Jacobs, J., Seago, N., & Mangram, C. (2014). Facilitating video-based professional development: Planning and orchestrating productive discussions. In Y. Li, E. A. Silver, & S. Li (Eds.), Transforming mathematics instruction (pp. 259–281). Springer International Publishing. https://doi.org/10.1007/978-3-319-04993-9_16 Cobb, P., Yackel, E., & Wood, T. (1992). Interaction and learning in mathematics classroom situations. Educational Studies in Mathematics, 23(1), 99–122. https://doi.org/10.1007/BF00302315 Desimone, L. M. (2009). Improving impact studies of teachers’ professional development: toward better conceptualizations and measures. Educational Researcher, 38(3), 181–199. https://doi.org/10.3102/0013189X08331140 Elliott, R., Kazemi, E., Lesseig, K., Mumme, J., Carroll, C., & Kelley-Petersen, M. (2009). Conceptualizing the work of leading mathematical tasks in professional development. Journal of Teacher Education, 60(4), 364–379. https://doi.org/10.1177/0022487109341150 Hechler, Shure, Lehmann, Roesken-Winter Jacobs, J., Seago, N., & Koellner, K. (2017). Preparing facilitators to use and adapt mathematics professional development materials productively. International Journal of STEM Education, 4(1), 30. https://doi.org/10.1186/s40594-017-0089-9 Koellner, K., Jacobs, J., & Borko, H. (2011). Mathematics professional development: Critical features for developing leadership skills and building teachers’ capacity. Prediger, S., Roesken-Winter, B., & Leuders, T. (2019). Which research can support PD facilitators? Strategies for content-related PD research in the Three-Tetrahedron Model. Journal of Mathematics Teacher Education, 22(4), 407–425. https://doi.org/10.1007/s10857-019-09434-3 Prediger, S., Roesken-Winter, B., Stahnke, R., & Pöhler, B. (2022). Conceptualizing contentrelated PD facilitator expertise. Journal of Mathematics Teacher Education, 25(4), 403– 428. https://doi.org/10.1007/s10857-021-09497-1 Prediger, S., & Selter, C. (2024). Establish shared visions and support productive adaptations on all levels: Aims, strategies, and architecture of a nationwide implementation program. Implementation and Replication Studies in Mathematics Education, 1–35. https://doi.org/10.1163/26670127-bja10020 Roesken-Winter, B., Stahnke, R., Prediger, S., & Gasteiger, H. (2021). Towards a research base for implementation strategies addressing mathematics teachers and facilitators. ZDM – Mathematics Education, 53(5), 1007–1019. https://doi.org/10.1007/s11858-021-01220x Schifter, D., & Lester, J. B. (2005). Active facilitation: What do Specialists need to know and how might they learn it? Shaughnessy, M., Garcia, N. M., O’Neill, M. K., Selling, S. K., Willis, A. T., Wilkes, C. E., Salazar, S. B., & Ball, D. L. (2021). Formatively assessing prospective teachers’ skills in leading mathematics discussions. Educational Studies in Mathematics, 108(3), 451–472. https://doi.org/10.1007/s10649-021-10070-z Shure, V., Lehmann, M., Friesen, M., Roesken-Winter, B., & Prediger, S. (2025). Professional development research on promoting productive practices for teachers’ handling of typical tasks of mathematics teaching: A systematic review. ZDM – Mathematics Education. https://doi.org/10.1007/s11858-025-01659-2 Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking Learning, 10(4), 313–340. https://doi.org/10.1080/10986060802229675 Zaslavsky, O., & Leikin, R. (2004). Professional development of mathematics teacher educators: Growth through practice. Journal of Mathematics Teacher Education, 7(1), 5– 32. https://doi.org/10.1023/B:JMTE.0000009971.13834.e1

595 602 Laura Heibeyn Carina Büscher University of Cologne Germany As a core competency of the 21st century, promoting computational thinking (CT) is necessary, and should begin at a young age. However, since this is an abstract concept, it is often difficult for children to grasp. Embodiment can help children acquire CT skills, especially in the early stages. As part of a qualitative study, 24 elementary school students were asked to complete a task involving decoding an arrow card algorithm for the Blue-Bot learning robot. This paper presents a particularly interesting case that covers many different approaches. It found that embodiment seems to support the CT skill of algorithmicising. In particular, visualising the position and orientation of the robot after each arrow card in the grid with the index finger appears to be promising. Büscher, C. (2025). Arrow cards as representation when programming educational robots – Algorithmitizing as an interplay between encoding and decoding. Proceedings of CERME14. https://hal.science/CERME14/hal-05228555v1 Ferrara, F. (2025). Computational and mathematical thinking at kindergarten. Moving with the grid. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri & V. Randolph (Eds.). Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education. (Vol. 1, pp. 235–242). PME 48. Hong, L. (2023). The impact of educational robots on students’ computational thinking: A meta-analysis of K-12. Education And Information Technologies, 29(11), 13813–13838. https://doi.org/10.1007/s10639-023-12415-y Kopcha, T. J. & Ocak, C. (2019). Embodiment of Computational Thinking During Collaborative Robotics Activity. Computer Supported Collaborative Learning. https://dblp.uni-trier.de/db/conf/cscl/cscl2019.html#KopchaO19 Kuckartz, U. (2019). Qualitative Text Analysis. A Systematic Approach. In G. Kaiser & N. Presmeg (Eds.), Compendium for Early Career Researchers in Mathematics Education (pp. 181–197). Springer International Publishing. https://doi.org/10.1007/978-3-03015636-7_8 Kwon, K., Jeon, M., Zhou, C., Kim, K. & Brush, T. A. (2022). Embodied learning for computational thinking in early primary education. Journal Of Research On Technology in Education, 56(4), 410–430. https://doi.org/10.1080/15391523.2022.2158146 Radford, L. (2008). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126. https://doi.org/10.1007/s10649-008-9127-3 Skulmowski, A. & Rey, G. D. (2018). Embodied learning: introducing a taxonomy based on bodily engagement and task integration. Cognitive Research Principles And Implications, 3(1), 6. https://doi.org/10.1186/s41235-018-0092-9 Statter, D. & Armoni, M. (2020). Teaching Abstraction in Computer Science to 7th Grade Students. ACM Transactions On Computing Education, 20(1), 1–37. https://doi.org/10.1145/3372143 Wing, J. M. (2017). Computational thinking’s influence on research and education for all. Italian Journal of Educational Technology, 25(2), 7–14. https://doi.org/10.17471/24994324/922

603 610 Tracy Helliwell Alf Coles University of Bristol, UK United Kingdom We situate this theoretical report within the growing swell of interest in socio-ecological and “ecological practices” of teaching mathematics. We take ecological practices to imply a faithfulness to ethics and relationality, and we draw insights from biological ecologies for our work in mathematics education. We argue that ecological practices of teaching mathematics imply an attention to ontology, or the nature of knowing, and, in particular, an ethical stance of “not knowing together”. What we mean by this is, in part, a teacher holding open a space of communication in the classroom which allows for the emergence of the new and a supporting of ambiguity. We illustrate what “not knowing together” might look like through a re-looking of some historical classroom data and draw out implications for mathematics education. Bateson, G. (2000). Steps to an ecology of mind. University of Chicago Press. Bion, W. (1970). Attention and interpretation. Tavistock Publications. Boylan, M. (2016). Ethical dimensions of mathematics education. Educational Studies in Mathematics, 92(3), 395–409. https://doi.org/10.1007/s10649-015-9678-z Coles, A. (2023). Teaching in the new climatic regime: Steps to a socio-ecology of mathematics education. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. T. (Eds.), Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 17-33). PME. de Freitas, E., & Sinclair, N. (2014). Mathematics and the Body: Material Entanglements in the Classroom. Cambridge University Press. Grosz, S. (2025). Love’s labour. Chatto & Windus. Helliwell, T., & Schwarts, G. (2023). Mathematics teacher educators in an unknowable world: Teaching mathematics for climate justice. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. T. (Eds.), Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 35-42). PME. le Roux, K., Coles, A., Solares-Rojas, A., Bose, A., Vistro-Yu, C. P., Valero, P., Sinclair, N., Makramalla, M., Gutiérrez, R., Geiger, V., & Borba, M. (Eds.). (2025). Proc. 27th ICMI Study Conf: Mathematics Education and the Socio-Ecological. MATHTED & ICMI. Makramalla, M., Coles, A., le Roux, K., Wagner, D. (2025). Mathematics education for sustainable futures: a strengths-based survey of the field to invite further research action. Educational Studies in Mathematics, 119, 535–556. https://doi.org/10.1007/s10649-025-10389-x Radford, L. (2025). What does culture have to do with knowledge? A contribution to solving the riddle of the ontological constitution of knowledge from a sociocultural perspective. Hiroshima Journal of Mathematics Education, 18, 71–88. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press. Simon, M. A. (1995). Reconstructing Mathematics Pedagogy from a Constructivist Perspective. Journal for Research in Mathematics Education, 26(2), 114–145. Stengers, I. (2020). The earth won’t let itself be watched. In B. Latour & P. Weibel (Eds.), Critical zones: The science and politics of landing on earth (pp. 228–235). The MIT Press. Stengers, I. (2005). Introductory notes on an ecology of practices. Cultural Studies Review, 11(1), 183–196. https://doi.org/10.5130/csr.v11i1.3459 Valero, P. (2023). Mathematical subjectivation: Death sentence or chance for a terrestrial life? In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. T. (Eds.), Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 53-68). PME.

611 618 Rachel Elizabeth Helme University of Bristol United Kingdom Dominant, deficit discourses can be prevalent in the culture that surrounds students labelled as low attaining in mathematics. Drawing on a model of deficit discourses in mathematics education, this report reflects on how the echoes of dominant discourses pervade the students’ own stories about their lived experiences of the teaching and learning of mathematics. Using the poetic inquiry method of the Listening Guide to lean in and listen carefully, I examined how students talked about themselves in relation to mathematics and found that there was a clear influence of these cultural discourses on the stories told by most students, However, defying these discourses allowed one student to re-narrate themselves from a more positive viewpoint. Adiredja, A., & Louie, N. (2020). Untangling the web of deficit discourses in mathematics education. For the Learning of Mathematics, 40(1), 42-43. Coles, A., & Brown, L. (2021). Differentiation from an advanced standpoint: Outcomes of mathematics teachers' action research studies aimed at raising attainment. Mathematics Teacher Education and Development, 23(3), 166-181. Faulkner, S. L. (2019). Poetic inquiry: Craft, method and practice (2nd ed.). Routledge. Francome, T., & Hewitt, D. (2020). “My math lessons are all about learning from your mistakes”: How mixed-attainment mathematics grouping affects the way students experience mathematics. Educational Review, 72(4), 475-494. Gilligan, C., Spencer, R., Weinberg, M. K., & Bertsch, T. (2006). In the Listening Guide: A voice-centered relational method. In S. N. Hesse-Biber, & P. Leavy (Eds.), Emergent Methods in Social Research (pp. 253-271). SAGE Publications, Inc. Hargreaves, E., Quick, L., & Buchanan, D. (2019). "I got rejected": Investigating the status of ‘low-attaining’ children in primary-schooling. Pedagogy, Culture & Society, 29(1), 7997. Helme, R. (2023). Foregrounding the voices of students: Extending the Listening Guide in a feminist poetic inquiry into identity work in the context of low attainment in mathematics [Doctoral dissertation, University of Bristol]. Explore Bristol Research. http://researchinformation.bristol.ac.uk/. Macaulay, L., & Kaukko, M. (2025). From 'I-Poems' to 'Pronoun Poems': Listening for the collective voice through a voice-centred relational methodology. International Journal of Qualitative Methods, 24, 1-10. Mendick, H., & Moreau, M.-P. (2014). Using popular culture. In D. Leslie, & H. Mendick (Eds.), Debates in mathematics education (pp. 17-26). Routledge. Mkhize, D. R. (2017). Forming positive identities to enhance mathematics learning among adolescents. Universal Journal of Educational Research, 5(2), 175-180. Norris, J. (2022). The positioning of GCSE and Functional Skills in Further Education: How do vocational students make sense of post-16 mathematics pathways? Research in Mathematics Education, 24(2), 1-19. Noyes, A., & Dalby, D. (2020). Students' perspectives on mathematics in Further Education colleges. The mathematics in Further Education colleges project: Interim report 3. https://www.nottingham.ac.uk/research/groups/crme/documents/mifec/interim-report3.pdf Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 1422. Staats, S. (2021). Mathematical poetic structures: The sound shape of collaboration. Journal of Mathematical Behavior, 62. Thompson, G. (2014). Labeling in interactional practice: Applying labeling theory to interactions and interactional analysis to labeling. Symbolic Interaction, 37(4), 458-482. Zavala, M. d., & Hand, V. (2019). Conflicting narratives of success in mathematics and science education: Challenging the achievement-motivation master narrative. Race Ethnicity and Education, 22(6), 802-820.

619 626 Marcel Hier Johanna Schoenherr Osnabrück University Although promoting creativity in mathematics education has been widely recognized as an important educational goal, creativity still plays a subordinate role in classroom practice. This suggests the presence of barriers to promoting creativity in mathematics classrooms. The present study investigates such barriers using a qualitative content analysis of 20 semi-structured interviews with in-practice mathematics teachers. The results reveal a wide range of barriers operating across multiple contexts, including the educational system, institutional, instructional, teacher, and student contexts. Within each context, specific barriers were identified that teachers perceived as hindering the promotion of creativity in mathematics classrooms. Bolden, D. S., Harries, T. V., & Newton, D. P. (2010). Pre-service primary teachers’ conceptions of creativity in mathematics. Educational Studies in Mathematics, 73(2), 143– 157. https://doi.org/10.1007/s10649-009-9207-z Chamberlin, S. A., & Moon, S. M. (2005). Model-eliciting activities as a tool to develop and identify creatively gifted mathematicians. Journal of Secondary Gifted Education, 17(1), 37–47. https://doi.org/10.4219/jsge-2005-393 Cropley, A. J. (1995). Fostering creativity in the classroom: General principles. In M. Runco (Ed.), The creativity research handbook (Vol. 1, pp. 1–46). Hampton Press. Donovan, L., Green, T. D., & Mason, C. (2014). Examining the 21st century classroom: Developing an innovation configuration map. Journal of Educational Computing Research, 50(2), 161–178. https://doi.org/10.2190/EC.50.2.a Elgrably, H., & Leikin, R. (2021). Creativity as a function of problem-solving expertise: Posing new problems through investigations. ZDM, 53(4), 891–904. https://doi.org/10.1007/s11858-021-01228-3 Leikin, R. (2013). Evaluating mathematical creativity: The interplay between multiplicity and insight. Psychological Test and Assessment Modeling, 55(4), 385–400. Leikin, R., & Elgrably, H. (2020). Problem posing through investigations for the development and evaluation of proof-related skills and creativity skills of prospective high school mathematics teachers. International Journal of Educational Research, 102, 101424. https://doi.org/10.1016/j.ijer.2019.04.002 Leikin, R., & Guberman, R. (2023). Creativity and challenge: Task complexity as a function of insight and multiplicity of solutions. In R. Leikin (Ed.), Mathematical Challenges For All (pp. 325–342). Springer International Publishing. https://doi.org/10.1007/978-3-03118868-8_17 Lev-Zamir, H., & Leikin, R. (2011). Creative mathematics teaching in the eye of the beholder: focusing on teachers' conceptions. Research in Mathematics Education, 13(1), 17–32. https://doi.org/10.1080/14794802.2011.550715 Li, G., & Ma, Y. (2025). Exploring the influencing factors of teacher beliefs and their impact on teacher behaviors. BMC Psychology, 13(1), 993. https://doi.org/10.1186/s40359-02503095-z Lu, X., & Kaiser, G. (2022). Creativity in students’ modelling competencies: Conceptualisation and measurement. Educational Studies in Mathematics, 109(2), 287– 311. https://doi.org/10.1007/s10649-021-10055-y OECD. (2024). PISA 2022 results (volume III): Creative minds, creative schools. OECD Publishing. https://doi.org/10.1787/765ee8c2-en Okamoto, H. (2022). Measuring creativity in the fermi problem, a type of mathematical modelling, applying information theory. Journal of Educational Technology Development and Exchange, 15(1). https://doi.org/10.18785/jetde.1501.02 Schoenherr, J., & Schukajlow, S. (2023). Does drawing help or hinder creativity when solving a mathematical modelling problem? Findings from an eye-tracking study. Rivista Di Matematica Della Universita Di Parma, 14(2), 233–248. Shriki, A. (2010). Working like real mathematicians: developing prospective teachers’ awareness of mathematical creativity through generating new concepts. Educational Studies in Mathematics, 73(2), 159–179. https://doi.org/10.1007/s10649-009-9212-2 Sriraman, B. (2005). Are giftedness and creativity synonyms in mathematics? Journal of Secondary Gifted Education, 17(1), 20–36. https://doi.org/10.4219/jsge-2005-389

627 634 Kjellrun Hiis Hauge Suela Kacerja Yasmine Abtahi Western Norway University of Applied Sciences Norway University of South-Eastern Norway Three of the core values in the Norway curriculum are ethical awareness, human dignity and democratic participation. In this report, we highlight how mathematics educators in teacher education include these values in their teaching. Results show that they put different emphasis on the three values, and their approach to teaching varied. Some chose to work explicitly with the values during a session, focusing on learning about the values, for example how they are presented in the curriculum. Others chose to have the values as a red thread through the whole course, putting emphasis on learning through the values, enacting them in class. Statistics and mathematical modelling were seen as easier to connect to the values. Both challenges and opportunities were addressed. We discuss implications of the results. Aguilar, M.S., & Zavaleta, J.G.M. (2012). On the links between mathematics education and democracy: A literature review. Pythagoras, 33(2), Art. #164, 15 pages. http://dx.doi.org/10.4102/pythagoras.v33i2.164 Hauge, K. H., Kacerja, S., Lilland, I. E. & Steffensen, L. (2024). Layers of ethical awareness: Mathematics teachers’ concerns on including socio-political issues. In P. Ernest (Ed.) Ethics and mathematics education - The Good, the Bad and the Ugly, p. 415–439. Springer.Ministry of Education and Research (2019). Mathematics subject curriculum. Retrieved from https://www.udir.no/lk20/mat01-05 Robinson, C., Phillips, L., & Quennerstedt, A. (2020) Human rights education: developing a theoretical understanding of teachers’ responsibilities. Educational Review, 72(2), 220– 241. https://doi.org/10.1080/00131911.2018.1495182 Scott, M., & Philip, T. M. (2023). “We ask so much of these tiny humans”: Supporting beginning teachers to honor the dignity of young people as mathematical learners. Cognition and Instruction, 41(3), 291-315. Stathopoulou, C., & Appelbaum, P. (2016). Dignity, recognition, and reconciliation: Forgiveness, ethnomathematics, and mathematics education. Revista Internacional de Pesquisa em Educação Matemática, 6(1), 26-44. United Nations General Assembly (2011). United Nations Declaration on Human Rights Education and Training. General Assembly Resolution 66/137, 19 Dec. U.N.Doc. A/RES/66/137.

635 642 Wisani Hlangwani Miranda Moodley Jacob Jaftha Duncan Mhakure University of Cape Town South Africa This study investigates how mathematical sense-making manifests during the modelling process among Technical and Vocational Education and Training College mathematics students. Using a qualitative case study design, thirty third-year national certificate vocational students collaborated in groups to solve a food security modelling task. Qualitative data comprised written group solutions, modelling competency rating summaries by mathematical modelling expert raters, and transcripts of audio-recorded discussions of students solving the modelling task. Findings revealed significant disparities in sense-making across modelling competencies: stronger in assumption-making but limited in computing and interpreting, and mathematical reasoning. Biccard, P. (2018). Mathematical sense-making through learner choice. Pythagoras, 39(1), 1-9. https://hdl.handle.net/10520/EJC-11fe0312ac Cevikbas, M., Kaiser, G., & Schukajlow, S. (2022). A systematic literature review of the current discussion on mathematical modelling competencies: State-of-the-art developments in conceptualizing, measuring, and fostering. Educational Studies in Mathematics, 109(2), 205-236. https://doi.org/10.1007/s10649-021-10104-6 Edo, S. I., Hartono, Y., & Putri, R. I. I. (2013). Investigating Secondary School Students' Difficulties in Modeling Problems PISA-Model Level 5 and 6. Indonesian Mathematical Society Journal on Mathematics Education, 4(1), 41-58. Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In Stillman, G. A., Blum, W., & Biembengut, M. S. (Eds.), In Mathematical Modelling in Education Research and Practice: Cultural, Social and Cognitive Influences, (pp. 129-149). Cham Leong, K. E., & Tan, J. Y. (2020). Exploring secondary students' modelling competencies. The Mathematics Enthusiast, 17(1), 85-107. https://doi.org/10.54870/1551-3440.1481 Maaß, K. (2006). What are modelling competencies?. Zentralblatt für Didaktik der Mathematik 38(1), 113–142. https://doi.org/10.1007/BF02655885 Niss, M., Blum, W., & Galbraith, P. (2007). Introduction to modelling and applications in mathematics education. In Modelling and applications in mathematics education. 14th ICMI Study, (pp. 3-32). Cham Schoenfeld, A. H. (2013). Mathematical modeling, sense making, and the common core state standards. Journal of Mathematics Education at Teachers College, 4(2), 6-17. https://doi.org/10.7916/jmetc.v4i2.620 Schukajlow, S., Kolter, J., & Blum, W. (2015). Scaffolding mathematical modelling with a solution plan. ZDM Mathematics Education 47(1), 1241-1254. https://doi.org/10.1007/s11858-015-0707-2 Vimbelo, S., & Bayaga, A. (2024). Transforming mathematics education in TVET Colleges through humanising pedagogy: An exploration of teaching approaches, student engagement, and real-life examples. IETE Journal of Education, 65(2), 139-154. https://doi.org/10.1080/09747338.2024.2324808 Yazan, B. (2015). Three Approaches to Case Study Methods in Education: Yin, Merriam, and Stake. The Qualitative Report, 20(2), 134-152.

643 650 Jessica Hoth Ricarda Holland Carolin Wosch Tobias Rolfes University of Rostock Germany Goethe University Frankfurt Germany Arithmetical skills are essential for informed participation in society and are cumulatively developed through mathematics education up to higher education. This study examines the relationship between elementary school teacher students' skills with natural numbers, integers and rational numbers. Data from 1,711 elementary school teacher students were analyzed using item response theory and structural equation modelling. Results revealed high skills with natural numbers and rational numbers, but notably lower performance with integers. Structural equation modelling showed exceptionally strong relationships between natural number and integer skills, with both direct and indirect pathways from natural numbers to rational numbers, indicating partial mediation. Bailey, D. H., Siegler, R. S., & Geary, D. C. (2014). Early predictors of middle school fraction knowledge. Developmental Science, 17(5), 775–785. https://doi.org/10.1111/desc.12155. Bofferding, L. (2014). Negative integer understanding: Characterizing first graders' mental models. Journal for Research in Mathematics Education, 45(2), 194–245. https://doi.org/10.5964/jnc.v3i1.71 Bond, T. G., & Fox, C. M. (2015). Applying the Rasch Model. Fundamental Measurement in the Human Science. Routledge. https://doi.org/10.4324/9781315814698 Holland, R., Wosch, C., Rolfes, T., & Hoth, J. (2025). Assessment of primary school student teachers’ basic arithmetic competencies – developing a diagnostic test and a competency-level model. In J. Novotná & H. Moraová (Eds.), Elementary Mathematics: Building Strong Foundations. Proceedings of the International Symposium Elementary Mathematics Teaching (pp. 179–189). SEMT. Kultusministerkonferenz (KMK). (2022a). Bildungsstandards für das Fach Mathematik: Primarbereich (Beschluss der Kultusministerkonferenz vom 15.10.2004, in der Fassung vom 23.06.2022). Sekretariat der Ständigen Konferenz der Kultusminister der Länder. https://www.kmk.org/fileadmin/veroeffentlichungen_beschluesse/2022/2022_06_23Bista-Primarbereich-Mathe.pdf Kultusministerkonferenz. (2022b). Bildungsstandards im Fach Mathematik für den Ersten Schulabschluss (ESA) und den Mittleren Schulabschluss (MSA) (Beschluss der Kultusministerkonferenz vom 04.12.2003 in der Fassung vom 23.06.2022). https://www.kmk.org/fileadmin/veroeffentlichungen_beschluesse/2022/2022_06_23Bista-ESA-MSA-Mathe.pdf Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144–150. https://doi.org/10.1111/cdep.12077 Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101 Tatto, M. T., Schwille, J., Senk, S. L., Ingvarson, L., Rowley, G., Peck, R., Bankov, K., Rodriguez, M., & Reckase, M. (2012). Policy, Practice, and Readiness to Teach Primary and Secondary Mathematics in 17 Countries (TEDS-M). IEA. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467. https://doi.org/10.1016/j.learninstruc.2004.06.013 Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behavior, 31(3), 344–355. https://doi.org/10.1016/j.jmathb.2012.02.001 Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in "negativity". Learning and Instruction, 14(5), 469–484. https://doi.org/10.1016/j.learninstruc.2004.06.012

651 658 Xiaoshan Huang Sofia Ikonen Koen Veermans Jake McMullen Department of Teacher Education, University of Turku Finland Guided by theories of knowledge integration and adaptive expertise, this paper describes the design of number line estimation tasks that target adaptive rational number knowledge. We propose three design principles: (1) eliciting learners’ current knowledge, (2) introducing linking ideas and increasing element interactivity, and (3) embedding multiple solution routes with on-the-spot decisions. The design aims to stimulate students to coordinate five knowledge components embedded in number line estimation tasks—magnitude mapping, relational reasoning, cross-notation conversion, within-notation transformation, and arithmetic operation—and the manipulation of number line features (including non-integer-unbounded number lines) and target formats as two central mechanisms to support knowledge integration. Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1–33). Lawrence Erlbaum Associates Publishers. Braithwaite, D. W., McMullen, J., & Hurst, M. A. (2022). Cross-notation knowledge of fractions and decimals. Journal of Experimental Child Psychology, 213, 105210. https://doi.org/10.1016/j.jecp.2021.105210 Feltovich, P. J., Spiro, R. J., & Coulson, R. L. (1997). Issues of expert flexibility in contexts characterized by complexity and change. In P. J. Feltovich, K. M. Ford, & R. R. Hoffman (Eds.), Expertise in context: Human and machine (pp. 125–146). MIT Press. Hurst, M. A., & Cordes, S. (2018). Children’s understanding of fraction and decimal symbols and the notation-specific relation to pre-algebra ability. Journal of Experimental Child Psychology, 168, 32–48. https://doi.org/10.1016/j.jecp.2017.12.003 Kiili, K., Lindstedt, A., Koskinen, A., Halme, H., Ninaus, M., & McMullen, J. (2021). Flow experience and situational interest in game-based learning: Cousins or identical twins. International Journal of Serious Games, 8(3), 93-114. https://doi.org/10.17083/ijsg.v8i3.462 Linn, M. C. (2005). The knowledge integration perspective on learning and instruction. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 243–264). Cambridge University Press. https://doi.org/10.1017/CBO9780511816833.016 McMullen, J., Hannula-Sormunen, M. M., Lehtinen, E., & Siegler, R. S. (2020). Distinguishing adaptive from routine expertise with rational number arithmetic. Learning and Instruction, 68, 101347. https://doi.org/10.1016/j.learninstruc.2020.101347 McMullen, J., Koskinen, A., Kärki, T., Lindstedt, A., Määttä, S., Halme, H., Lehtinen, E., Hannula-Sormunen, M. M., & Kiili, K. (2024). A game-based approach to promoting adaptive rational number knowledge. Mathematical Thinking and Learning, 26(4), 411– 427. https://doi.org/10.1080/10986065.2023.2177818 Schiller, L. K., Abreu-Mendoza, R. A., Siegler, R. S., Rosenberg-Lee, M., & Thompson, C. A. (2024). Building integrated number sense in adults and children: Comparing fractionsonly training with cross-notation number line training. Journal of Experimental Child Psychology, 246, 106017. https://doi.org/10.1016/j.jecp.2024.106017 Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918. http://dx.doi.org/10.1037/edu0000025 Van Dooren, W. (2023). Strategy use in number line estimations of fractions—An explorative study in search for adaptive expertise. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proc. of the 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 299–306). PME

659 666 Iden Rainal Ihsan Kai-Lin Yang Samudra University Indonesia National Taiwan Normal University Taiwan Grounded in Brousseau’s theory of didactical situations, this study highlights epistemic validation, often neglected in geometry compared to arithmetic. Addressing the mechanism of "suspension of sense-making," we investigate how valid calculations can mask impossible objects. We analyzed 109 Indonesian students' reasoning regarding an overspecified task containing mutually exclusive side-angle properties. Results classify participants into "Pythagorean Pragmatists" (~80%) exhibiting metacognitive blindness to fulfill the contract, "Procedural Formalists" (~15%) experiencing dissonance, and rare "Critical Whistleblowers" (<5%) capable of validation. These findings frame errors as rational adaptations to school norms, advocating for diagnostic teaching to foster epistemic agency. Alifiani, Subanji, Permadi, H., & Irawati, S. (2023). Metacognitive Intervention : Can It Solve Suspension of Sense-Making in Integration Problem-Solving ? Mathematics Teaching Research Journal, 15(1), 44–66. As’ari, A. R., Kurniati, D., Maharani, S., & Basri, H. (2019). Ragam soal matematis untuk mengembangkan disposisi berpikir kritis. Universitas Negeri Malang. Baccaglini-frank, A., Antonini, S., Leung, A., & Alessandra, M. (2013). Reasoning by contradiction in dynamic geometry. PNA, 7(2), 63–73. Baruk, S. (2016). L’âge du capitaine. De l’erreur en mathématiques. Média Diffusion. Bonotto, C. (2002). Suspension of sense-making in mathematical word problem solving: A possible remedy. Proc. of the 2nd Int. Conf. on the Teaching of Mathematics. Brousseau, G. (2002). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990. Springer. Campbell, J. L., Quincy, C., Osserman, J., & Pedersen, O. K. (2013). Coding in-depth semistructured interviews: Problems of unitization and intercoder reliability and agreement. Sociological Methods & Research, 42(3), 294–320. Carotenuto, G., Di Martino, P., & Lemmi, M. (2021). Students’ suspension of sense making in problem solving. ZDM–Mathematics Education, 53(4), 817–830. de Corte, E., Greer, B., & Verschaffel, L. (2000). Making sense of word problems. CRC Press. Faradiba, S. S., & Alifiani, A. (2020). Metacognitive blindness in mathematics problemsolving. Journal of Education and Learning Mathematics Research, 1(2), 43–49. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162. Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1), 127–150. Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus and Giroux. Komatsu, K., Tsujiyama, Y., Sakamaki, A., & Koike, N. (2014). Proof problems with diagrams: An opportunity for experiencing proofs and refutations. For the Learning of Mathematics, 34(1), 36–42. McHugh, M. L. (2012). Interrater reliability: the kappa statistic. Biochemia Medica, 22(3), 276–282. Meijer, P. C., Verloop, N., & Beijaard, D. (2002). Multi-method triangulation in a qualitative study on teachers’ practical knowledge: An attempt to increase internal validity. Quality and Quantity, 36(2), 145–167. Muis, K. R., Psaradellis, C., Lajoie, S. P., Di Leo, I., & Chevrier, M. (2015). The role of epistemic emotions in mathematics problem solving. Contemporary Educational Psychology, 42, 172–185. O’Connor, C., & Joffe, H. (2020). Intercoder reliability in qualitative research: Debates and practical guidelines. International Journal of Qualitative Methods, 19, 1609406919899220. Potvin, P. (2023). Response of science learners to contradicting information: A review of research. Studies in Science Education, 59(1), 67–108. Schoenfeld, A. H. (2012). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In Informal reasoning and education (pp. 311–344). Routledge. Schoenfeld, A. H. (2014). Mathematical problem solving. Elsevier. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20–26.

667 674 Sofia Ikonen Jake McMullen Department of Teacher Education, University of Turku Turku Institute for Learning Analytics, University of Turku Finland Number line estimation tasks have been found to be beneficial for strengthening rational number knowledge, but there are many open questions regarding the kinds of rational number knowledge that are needed to be proficient in the task. In this study, we aimed to deepen our understanding of a previously less explored unbounded number line estimation task by examining 9th graders’ (N=95) qualitatively different answers to unbounded number line estimation tasks, as well as the connections between this and other measures of rational number knowledge. Our results show that students’ knowledge appeared fragmented and that performance in the unbounded number line estimation tasks was able to explain variance in students’ adaptive rational number knowledge. Implications are discussed. Cramer, K., Monson, D., Ahrendt, S., Wyberg, T., Pettis, C., & Fagerlund, C. (2019). Reconstructing the unit on the number line: Tasks to extend fourth graders’ fraction understandings. Investigations in Mathematics https://doi.org/10.1080/19477503.2018.1434594 Learning, 11(3), 180–194. Hatano, G. (2003). Foreword. In A. Baroody, & A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise (pp. xi–xiii). Routledge. Kim, J. (2024). Profiles of young students’ understanding of fractions on number lines. Eurasia Journal of Mathematics, Science and Technology Education, 20(5), Article em2444. https://doi.org/10.29333/ejmste/14469 Levin, M. (2018). Conceptual and Procedural Knowledge During Strategy Construction: A Complex Knowledge Systems Perspective. Cognition and Instruction, 36(3), 247–278. https://doi.org/10.1080/07370008.2018.1464003 Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221. https://doi.org/10.1016/j.dr.2015.07.008 McMullen, J., Hannula-Sormunen, M. M., Lehtinen, E., & Siegler, R. S. (2020). Distinguishing adaptive from routine expertise with rational number arithmetic. Learning and Instruction, 68, Article 101347. https://doi.org/10.1016/j.learninstruc.2020.101347 McMullen, J., Koskinen, A., Kärki, T., Lindstedt, A., Määttä, S., Halme, H., Lehtinen, E., Hannula-Sormunen, M. M., & Kiili, K. (2024). A game-based approach to promoting adaptive rational number knowledge. Mathematical Thinking and Learning, 26(4), 411– 427. https://doi.org/10.1080/10986065.2023.2177818 Saxe, G. B., Earnest, D., Sitabkhan, Y., Haldar, L. C., Lewis, K. E., & Zheng, Y. (2010). Supporting Generative Thinking About the Integer Number Line in Elementary Mathematics. Cognition Instruction, 28(4), 433–474. https://doi.org/10.1080/07370008.2010.511569 Schneider, M., Merz, S., Stricker, J., De Smedt, B., Torbeyns, J., Verschaffel, L., & Luwel, K. (2018). Associations of Number Line Estimation With Mathematical Competence: A Meta-analysis. Child Development, 89(5), 1467–1484. https://doi.org/10.1111/cdev.13068 Teppo, A., & Heuvel-Panhuizen, M. (2014). Visual representations as objects of analysis: The number line as an example. ZDM, 46(1), 45–58. https://doi.org/10.1007/s11858-0130518-2 Van Dooren, W. (2023). Strategy use in number line estimations of fractions—An explorative study in search for adaptive expertise. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proc. of the 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 299–306). PME. Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural rational numbers. Learning Instruction, 37, 1–4. https://doi.org/10.1016/j.learninstruc.2015.01.001

675 682 Pia Ilomanni Eeva Haataja Laura Niemi Jelena Radišić Anu Laine University of Helsinki Finland University of Oslo Norway Cost is a dimension of math motivation reflecting the effort and resources associated with learning mathematics, making it an important part of students’ motivation and learning experience. This study aimed to clarify the factors associated with cost by applying regression tree analysis to a data consisting of 1579 elementary students and their parents. The regression tree was trained first with a smaller dataset and then evaluated on a larger independent dataset. The analysis revealed that students perceived competence was the most prominent factor, with lower perceived competence linked to higher cost values. Mathematics performance and parental expectations further differentiated subgroups, while school-based remedial teaching and additional math clubs explained variation in smaller branches. Breiman, L., Friedman, J.H., Olshen, R.A. & Stone, C.J. (1984). Classification and Regression Trees. Belmont: Wadsworth. Brodie, K. (2022). Learning mathematics in an after-school mathematics club. African Journal of Research in Mathematics, Science and Technology Education, 26(3), 237–247. https://doi.org/10.1080/18117295.2022.2131267 Eccles, J. S., & Wigfield, A. (2020). From expectancy-value theory to situated expectancyvalue theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Eccles, J. S., & Wigfield, A. (2024). The development, testing, and refinement of Eccles, Wigfield, and colleagues’ situated expectancy-value model of achievement performance and choice. Educational Psychology Review, 36(2), 51. https://doi.org/10.1007/s10648024-09888-9 Loh, W.-Y. (2014). Fifty Years of Classification and Regression Trees. International Statistical Review, 82(3), 329–348. https://doi.org/10.1111/insr.12016 Nuutila, K., Tuominen, H., Tapola, A., Vainikainen, M.-P., & Niemivirta, M. (2018). Consistency, longitudinal stability, and predictions of elementary school students’ task interest, success expectancy, and performance in mathematics. Learning and Instruction, 56, 73–83. https://doi.org/10.1016/j.learninstruc.2018.04.003 Peixoto, F., Radišić, J., Krstić, K., Hansen, K.Y., Laine, A., Baucal, A., Sõrmus, M. & Mata, L. (2023). Contribution to the validation of the Expectancy-Value Scale for primary school students. Journal of Psychoeducational Assessment, 41(3), 343–350. https://doi.org/10.1177/07342829221144868 Peixoto, F., Mata, L., Campos, M., Caetano, T., Radišić, J., & Niemivirta, M. (2024). ‘Am I to blame because my child is not motivated to do math?’: Relationships between parents’ attitudes, beliefs and practices towards mathematics and students’ mathematics motivation and achievement. European Journal of Psychology of Education, 39(2), 1561–1586. https://doi.org/10.1007/s10212-023-00774-6 R Core Team. (2025). R: A language and environment for statistical computing (Version 4.5.2) [Computer software]. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/ Rojo, M., Gersib, J., Powell, S. R., Shen, Z., King, S. G., Akther, S. S., Arsenault, T. L., Bos, S. E., Lariviere, D. O., & Lin, X. (2024). A meta-analysis of mathematics interventions: Examining the impacts of intervention characteristics. Educational Psychology Review, 36, Article 9. https://doi.org/10.1007/s10648-023-09843-0 Simpkins, S. D., Fredricks, J., & Eccles, J. S. (2015). Families, schools, and developing achievement-related motivations and engagement. In J. E. Grusec & P. D. Hastings (Eds.), Handbook of socialization: Theory and research (2nd ed., pp. 614–636). New York: Guilford Press. Šimunović, M., & Babarović, T. (2020). The role of parents’ beliefs in students’ motivation, achievement, and choices in the STEM domain: A review and directions for future research. Social Psychology of Education, 23, 701–719. https://doi.org/10.1007/s11218020-09555-1 Therneau, T. & Atkinson, B. (2025). rpart: Recursive Partitioning and Regression Trees. R package version 4.1.24, https://doi.org/10.32614/CRAN.package.rpart. Vinni-Laakso, J., Hietajärvi, L., Juuti, K., Loukomies, A., & Salmela-Aro, K. (2025). Withinperson dynamics of science self-concept, intrinsic value, and cost in elementary school: a six-year longitudinal study. Educational Psychology, 45(5), 561– 580. https://doi.org/10.1080/01443410.2025.2489109 Virden, C., Dean, L., & Bondurant, L. (2023). The impact of after-school math club on elementary student math anxiety. Journal of Math Circles, 3(1), Article 2 Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental review, 30(1), 1–35. https://doi.org/10.1016/j.dr.2009.12.001

683 690 Pedro Ivars Ceneida Fernández Sebastian Kuntze Jens Krummenauer University of Alicante Ludwigsburg University of Education Germany Noticing is a teacher competence that implies attending, interpreting and making decisions in teaching-learning situations, and that can be developed in initial training programs analysing vignettes. Since vignettes often are contextually framed, research is needed to explore noticing of pre-service teachers from different cultural and institutional contexts. Participants are 73 Spanish and 56 German pre-service teachers (PTs) who had to plan a follow-up lesson after analysing a cartoon-based vignette depicting a teaching–learning situation about fraction operations. Results show that most PTs from the both groups were able to provide a problem according to the objective of the follow-up lesson, but they had difficulties in justifying their decisions. Barnhart, T., & van Es, E. (2015). Studying teacher noticing: Examining the relationship among pre-service science teachers' ability to attend, analyze and respond to student thinking. Teaching and Teacher Education, 45, 83-93 Buchbinder, O. & Kuntze, S. (2018). Representations of practice in teacher education and research – Spotlights on different approaches. In O. Buchbinder, & S. Kuntze, (Eds.), Mathematics teachers engaging with representations of practice. A dynamically evolving field (pp. 1–8). Springer. Dindyal, J., Schack, E., Choy, B.H., & Gamora-Sherin, M. (2021). Exploring the terrains of mathematics noticing. ZDM–Mathematics Education, 53(1), 1-16. Dreher, A., Lindmeier, A., Feltes, P., Wang, T.-Y., & Hsieh, F. J. (2021). Do cultural norms influence how teacher noticing is studied in different cultural contexts? A focus on expert norms of responding to students’ mathematical thinking. ZDM, 53, 165–179. Gibbson L.G., Kanpp M. C., & Lind, T. (2018). Coaching through focusing on student thinking. Teaching Children Mathematics, 25(1), 24-29. Herbst, P., & Kosko, K. W. (2014). Using representations of practice to elicit mathematics teachers’ tacit knowledge of practice: a comparison of responses to animations and videos. Journal of Mathematics Teacher Education, 17(6), 515-537. Ivars, P., Fernández, C., Kuntze, S., & Krummenauer, J. (2025). Portrayals of teacher noticing in two Spanish and German universities: The case of operations with fractions. ECNU Review of Education, 8(3), 903-923. Ivars, P., Fernández, C., & Llinares, S. (2019). Principles in the design of tasks to support preservice teachers’ noticing enhancement. In M. Graven, H. Venkat, A. Essien & P. Vale (Eds.) Proc. 43rd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 408-415). PME. Weyers, J., König, J., Scheiner, T., Santagata, R., & Kaiser, G. (2024). Teacher noticing in mathematics education: a review of recent developments. ZDM–Mathematics Education, 56, 249–264. Yang, X., Kaiser, G., König, J., & Blömeke, S. (2019). Professional noticing of mathematics teachers: A comparative study between Germany and China. International Journal of Science and Mathematics Education, 17, 943–963. Zhang, Q., Choy, B. H., & Lee, M. Y. (2025). Teacher noticing in diverse educational contexts: Concepts, research, and future pathways. ECNU Review of Ed., 8(3), 924-934.

691 698 Yuan Ji University of Bristol United Kingdom This study examines 9–10-year-old students’ early algebraic reasoning in representing and solving single-variable linear equations. Adopting Radford’s Theory of Objectification, a teaching–learning activity was co-designed in a Chinese primary school with concrete and iconic pan-balance representations. Video and worksheet data from 30 students revealed diverse representations of equality (numbers, letters, Chinese characters, shapes, and balance diagrams), including the culturally embedded Heavenly Stems (天干), ), a potentially novel finding in English language research. Gestures and multimodal actions supported an embodied understanding of equivalence via symmetry and compensatory operations. I argue for an extension to Radford’s framework for algebraic thinking, to include the indeterminate operations. Blanton, M., Isler-Baykal, I., Stroud, R. (2019). Growth in children’s understanding of generalizing and representing mathematical structure and relationships. Educational Studies in Mathematics, 102, 193–219. Coles, A., & Ahn, A. (2022). Developing algebraic activity through conjecturing about relationships. ZDM–Mathematics Education, 54(6), 1229–1241. Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 5–17). Lawrence Erlbaum Associates/Taylor & Francis Group. Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator, 8(1), 139–151. Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM, 54, 1131–1150. Lehmann, C. (2004). Interlinear morphemic glossing. In G. Booij, C. Lehmann, J. Mugdan, S. Skopeteas, & W. Kesselheim (Eds.), Morphologie (Vol. 2, pp. 1834–1857). McNeill, D. (1992). Hand and mind: What gestures reveal about thought. University of Chicago Press. Mellone, M., Ramploud, A., & Carotenuto, G. (2020). An experience of cultural transposition of the El’konin-Davydov curriculum. Educational Studies in Mathematics. Advance online publication. Ministry of Education of the People’s Republic of China. (2022). Mathematics curriculum standards for compulsory education. Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26, 257–277. Radford, L. (2020). Davydov’s concept of the concept and its dialectical materialist background. Educational Studies in Mathematics. Advance online publication. Radford, L. (2021). The theory of objectification: A Vygotskian perspective on knowing and becoming in mathematics teaching and learning. Brill Sense. Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov. The Mathematics Educator, 8(1), 60–87. Xie, S., & Cai, J. (2022). Fifth graders’ learning to solve equations: The impact of early arithmetic strategies. ZDM–Mathematics Education, 54, 1169–1179.

707 714 Julia Joklitschke Maike Schindler Benjamin Rott University of Duisburg-Essen Germany University of Cologne Germany As societies continue to evolve, creativity—particularly mathematical creativity—has become a crucial 21st-century skill. While prior research has primarily emphasized creative outcomes, the cognitive processes underlying mathematical creativity remain insufficiently understood. The aim of this study is to characterize the creative process of school students. In our study, 15 upper secondary school students worked on a creativity problem while wearing eye-tracking glasses and participated in a stimulated recall interview. Using a combined deductive–inductive analytic approach based in Hadamard’s stage model of creativity, we were able to characterize and develop a model of the students’ creative process. The model substantiates previous models, while also revealing additional phases as well as reinterpretations. Hadamard, J. (1945). The psychology of invention in the mathematical field. Dover Pub. Joklitschke, J., Baumanns, L., Rott, B., Schindler, M., & Liljedahl, P. (2022). Literature review on empirical findings on creativity in mathematics among secondary school students. In S. A. Chamberlin, P. Liljedahl, & M. Savić (Eds.), Mathematical creativity: A developmental perspective (1st ed. 2022, pp. 81–103). Springer. Joklitschke, J., Rott, B., & Schindler, M. (2021). Notions of creativity in mathematics education research: A systematic literature review. International Journal of Science and Mathematics Education, 20, 1161–1181. Kaufman, J. C. & Beghetto, R. A. (2009). Beyond big and little: The Four C Model of creativity. Review of General Psychology, 13(1), 1–12. https://doi.org/10.1037/a0013688 Kuckartz, U. & Rädiker, S. (2023). Qualitative content analysis: Methods, practice and software (2nd edition). Sage. Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Sense Publishers. Leikin, R. & Sriraman, B. (2022). Empirical research on creativity in mathematics (education): From the wastelands of psychology to the current state of the art. ZDM—Mathematics Education, 54(1), 1–17. https://doi.org/10.1007/s11858-022-01340-y Liljedahl, P. (2013). Illumination: An affective experience? ZDM, 45(2), 253–265. OECD. (2024). PISA 2022 results: Creative minds, creative schools (Vol. 3). OECD Pub. Schindler, M. & Lilienthal, A. J. (2020). Students’ creative process in mathematics: Insights from eye-tracking stimulated recall interview on students’ work on multiple solution tasks. International Journal of Science and Mathematics Education, 18, 1565–1586. Schindler, M. & Lilienthal, A. J. (2022). Students’ collaborative creative process and its phases in mathematics: An explorative study using dual eye tracking and stimulated recall interviews. ZDM—Mathematics Education, 54(1), 163–178. Torrance, E. P. (1974). Torrance Tests of Creative Thinking. Scholastic Testing Service.

715 722 Sara Julsgård Department of Mathematics, Linköping University Sweden The concept of logarithms can be introduced through multiple mathematically equivalent yet conceptually different definitions, which may foreground different properties of the concept. This study investigates, using interviews, how research mathematicians relate such conceptualisations of logarithms. The result is a two-level model of meaning for the concept, distinguishing formal definition variants from facets of meaning and clustering definitions according to the mathematics to which they refer. The proposed model contributes a structure for the conceptual diversity of logarithms and offers a theoretical tool for research on the teaching and learning of logarithms. Berezovski, T., & Zazkis, R. (2006). Logarithms: Snapshots from two tasks. In J. Novotná, H. Moraová, M. Krátká, & N. a. Stehlíková (Eds.), Proc. 30th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 145–152). PME. Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293– 316. Hußmann, S., Rezat, S., & Sträßer, R. (2016). Subject matter didactics in mathematics education. Journal für Mathematik-Didaktik, 37(1), 1–9. Julsgård, S. (accepted). Developing a model of meaning for the concept of logarithms based on the content of university mathematics textbooks. To appear in Proceedings of MADIF15. The fifthteenth research seminar of the Swedish Society for Research in Mathematics Education, Göteborg, January 27–28, 2026. SMDF. Lampinen, A. K., & McClelland, J. L. (2018). Different presentations of a mathematical concept can support learning in complementary ways. Journal of Educational Psychology, 110(5), 664–682. Lankeit, E., & Biehler, R. (2024). The meaning landscape of the concept of the total derivative in multivariable real analysis textbooks: An analysis based on a new model of meaning. ZDM – Mathematics Education, 56, 1361–1373. Point, S., & Baruch, Y. (2023). (Re)thinking transcription strategies: Current challenges and future research directions. Scandinavian Journal of Management, 39(2), 101272. Ricardo, H. (2022). The equivalence of definitions of the natural logarithm function. The College Mathematics Journal, 53(3), 190–196. Rozenszajn, R., Kavod, G. Z., & Machluf, Y. (2021). What do they really think? the repertory grid technique as an educational research tool for revealing tacit cognitive structures. International Journal of Science Education, 43(6), 906–927. Swedish Research Council. (2024). Good research practice. Weber, C. (2016). Making logarithms accessible – operational and structural basic models for logarithms. Journal für Mathematik-Didaktik, 37, 69–98.

723 730 Alina Kadluba Madeleine Kingan Andreas Obersteiner Jon R. Star Technical University of Munich Germany Harvard Graduate School of Education United States It is well documented that cognitive and non-cognitive variables are related to students’ academic performance. However, their specific predictive power is yet unclear. This study examined how prior performance, confidence, and calibration bias (under- and over-confidence) predict short-term subsequent mathematics performance across Grades 5–8. Students completed tasks of increasing difficulty and indicated confidence for each response. Prior performance consistently predicted later success, while confidence had an increasing effect with schooling. Under-confidence reduced subsequent performance, whereas over-confidence was slightly beneficial in older students. Findings highlight the diagnostic and motivational role of confidence and suggest that fostering self-evaluation and confidence can enhance learning outcomes. Bandura, A., & Wessels, S. (1997). Self-efficacy. Cambridge University Press Cambridge. Bouffard, T., & Narciss, S. (2011). Benefits and risks of positive biases in self-evaluation of academic competence: Introduction. Int. Journal of Educational Research 50(4), 205-208. Efklides, A. (2011). Interactions of metacognition with motivation and affect in self-regulated learning: The MASRL model. Educational Psychologist 46(1), 6-25. Fleming, S. M., & Lau, H. C. (2014). How to measure metacognition. Frontiers in Human Neuroscience 8, 443. https://doi.org/10.3389/fnhum.2014.00443 Foster, C. (2016). Confidence and competence with mathematical procedures. Educational Studies in Mathematics, 91, 271-288. https://doi.org/10.1007/s10649-015-9660-9 Foster, C. (2022). Implementing confidence assessment in low-stakes, formative mathematics assessments. Int. Journal of Science and Mathematics Education, 20(7), 1411-1429. Gardner-Medwin, A. R., & Gahan, M. (2003). Formative and summative confidence-based assessment. Proceedings of the 7th CAA Conference, Loughborough. Gigerenzer, G., Hoffrage, U., & Kleinbölting, H. (1991). Probabilistic mental models: a Brunswikian theory of confidence. Psychological Review 98(4), 506. Koriat, A. (2012). The self-consistency model of subjective confidence. Psychological Review 119(1), 80. https://doi.org/10.1037/a0025648 Kruger, J., & Dunning, D. (1999). Unskilled and unaware of it: how difficulties in recognizing one's own incompetence lead to inflated self-assessments. Journal of Personality and Social Psychology 77(6), 1121. https://doi.org/10.1037//0022-3514.77.6.1121 Lichtenstein, S., Fischhoff, B., & Phillips, L. D. (1981). Calibration of probabilities: The state of the art to 1980. Metcalfe, J. (2009). Metacognitive judgments and control of study. Current Directions in Psychological Science 18(3), 159-163. https://doi.org/10.1111/j.1467-8721.2009.01628.x Pallier, G., Wilkinson, R., Danthiir, V., Kleitman, S., Knezevic, G., Stankov, L., & Roberts, R. D. (2002). The role of individual differences in the accuracy of confidence judgments. The Journal of General Psychology 129(3), 257-299. Simonsmeier, B. A., Flaig, M., Deiglmayr, A., Schalk, L., & Schneider, M. (2022). Domainspecific prior knowledge and learning: A meta-analysis. Ed. Psychologist 57(1), 31-54. Stankov, L., Lee, J., Luo, W., & Hogan, D. J. (2012). Confidence: A better predictor of academic achievement than self-efficacy, self-concept and anxiety? Learning and Individual Differences 22(6), 747-758. https://doi.org/10.1016/j.lindif.2012.05.013 Stankov, L., Lee, J., Morony, S., Lee, Y. P., Parker, P. D., Marsh, H. W., Ciarrochi, J., Marshall, S., Abduljabbar, A. S., & Seaton, M. (2014). Noncognitive psychological processes and academic achievement Educational Psychology, 34(1). Stankov, L., Lee, J., & Paek, I. (2009). Realism of confidence judgments. European Journal of Psychological Assessment, 25(2), 123-130. https://doi.org/10.1027/1015-5759.25.2.123 Stankov, L., Morony, S., & Lee, Y. P. (2017). Confidence: the best non-cognitive predictor of academic achievement? In Noncognitive Psychological Processes and Academic Achievement (pp. 19-38). Routledge. https://doi.org/10.1080/01443410.2013.814194 Thompson, V. A., Turner, J. A. P., & Pennycook, G. (2011). Intuition, reason, and metacognition. Cognitive Psychology 63(3), 107-140. Weil, L. G., Fleming, S. M., Dumontheil, I., Kilford, E. J., Weil, R. S., Rees, G., Dolan, R. J., & Blakemore, S.-J. (2013). The development of metacognitive ability in adolescence. Consciousness and Cognition 22(1), 264-271.

731 738 Anna Kaiser Stefan Ufer LMU Munich Germany One important of primary school mathematics education is understanding of the base-ten number system, encompassing the regrouping and the place-value position principle. When extending the number space, students need to extend their understanding of these principles. Converting between manipulatives and symbolic number representations is one indicator of such emerging understanding. We investigate whether beginning third-grade students can transfer their understanding of the place-value position principle from two-digit to three-digit numbers. Students show substantial, but not perfect manipulative-based understanding for two-digit numbers, and that they can transfer it to three-digit numbers to some extent. We find no evidence, that this transfer is supported by the salience of the principle in the manipulative. Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology 105(2), 380–400. Duval, R. (2006). A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics 61, 103-131. Dehaene, S. (1992). Varieties of numerical abilities. Cognition 44, 1-42. Donovan, A. M., & Fyfe, E. R. (2022). Connecting concrete objects and abstract symbols promotes children’s place value knowledge. Educational Psychology 42(8), 1008–1026. Fromme, M. (2017). Stellenwertverständnis im Zahlenraum bis 100 - Theoretische und empirische Analysen [Place-value understanding in the number range up to 100 theoretical and empirical analyses]. Springer Spektrum. Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children’s mathematics understanding. Learning and Instruction 35, 104-120. Herzog, M., & Fritz, A. (2019). Validation of a developmental model of place value concepts. In Graven, M., Venkat, H., Essien, A. & Vale, P. (Eds). Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education Vol 2, 352-359. PME. Herzog, M., & Fritz, A. (2022). Place Value Understanding Explains Individual Differences in Writing Numbers in Second and Third Graders But Goes Beyond. Frontiers in Education, 6, 642153. Jensen, S., Gasteiger, H., & Bruns, J. (2021). Place value and regrouping as separate constructs of place value understanding. In Inprasitha, M, Changsri, N., Boonsena, N. (Eds.). Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education Vol. 3, 101-109. PME. Wittmann, G. & Lenz, K. (2025). Assessing students’ conceptual place value understanding at the end of primary school. Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14), Free University of Bozen. Moeller, K., Pixner, S., Zuber, J., Kaufmann, L., & Nuerk, H.-C. (2011). Early place-value understanding as a precursor for later arithmetic performance. A longitudinal study on numerical development. Research in Developmental Disabilities 32, 1837-1851. Resnick, M., Martin, F., Berg, R., Borovoy, R., Colella, V., Kramer, K., & Silverman, B. (1998). Digital manipulatives: new toys to think with. CHI '98: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, 281-287. Staatsinstitut für Schulqualität und Bildungsforschung (ISB) (2014). LehrplanPLUS. Uttal, D. H., Amaya, M., del Rosario Maita, M., Hand, L. L., Cohen, C. A., O’Doherty, K., & DeLoache, J. S. (2013). It Works Both Ways: Transfer Difficulties between Manipulatives and Written Subtraction Solutions. Child Development Research, 216367. Van de Walle, J., Bay-Williams, J., & Karp, K. (2023). Elementary and Middle School Mathematics: Teaching Developmentally (11 ed.). Pearson Education Limited.

739 746 Gözde Kaplan-Can Francesca Morselli Middle East Technical University Turkey University of Genova Italy The study investigates how lower secondary mathematics teachers engage with technology-supported formative assessment within a professional development program, adopting the Meta-Didactical Transposition framework as the main theoretical lens. The program was inspired by the FaSMEd Toolkit and encouraged teachers to adapt and redesign proposed activities. Data from lesson plans, reflective questionnaires, and professional development meetings were analysed in terms of teachers’ meta-didactical praxeologies. The findings show that teachers reinterpreted the Toolkit through shared and evolving praxeologies, aligning technology use with formative purposes. Aldon, G., Cusi, A., Morselli, F., Panero, M., Sabena, C. (2017). Formative Assessment and Technology: Reflections Developed Through the Collaboration Between Teachers and Researchers. In: Aldon, G., Hitt, F., Bazzini, L., Gellert, U. (eds) Mathematics and Technology. Advances in Mathematics https://doi.org/10.1007/978-3-319-51380-5_25 Education. Springer, Cham. Arzarello, F., Cusi, A., Garuti, R., Malara, N., Martignone, F., Robutti, O., & Sabena, C. (2014). Meta-didactical transposition: a theoretical model for teacher education programmes. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital era: an international perspective on technology focused professional development (pp. 347–372). Springer. Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5–31. https://doi.org/10.1007/s11092008-9068-5 Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative research in psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa Cusi, A., Morselli, F., & Sabena, C. (2017). Promoting formative assessment in a connected classroom environment: Design and implementation of digital resources. ZDM Mathematics Education, 49(5), 755–767. https:// doi.org/10.1007/s11858-017-0878-0 Cusi, A., Robutti, O., Panero, M., Taranto, E., Aldon, G. (2022). Meta-Didactical Transposition.2: The Evolution of a Framework to Analyse Teachers’ Collaborative Work with Researchers in Technological Settings. In: Clark-Wilson, A., Robutti, O., Sinclair, N. (eds) The Mathematics Teacher in the Digital Era. Mathematics Education in the Digital Era, Springer, Cham. https://doi.org/10.1007/978-3-031-05254-5_14 Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching: Theory and Practice, 15, 273–289. https://doi.org/10.1080/1354060090287534 Robutti, O. (2020). Meta-didactical Transposition. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-157890_100012 Schildkamp, K., van der Kleij, F.M., Heitink, M.C., Kippers, W.B., & Veldkamp, B.P. (2020). Formative assessment: A systematic review of critical teacher prerequisites for classroom practice. International Journal of Educational Research, 103, 101602. https://doi.org/10.1016/j.ijer.2020.101602 Shulman, L. S., & Shulman, J. H. (2004). How and what teachers learn: A shifting perspective. Journal of Curriculum Studies, 36(2), 257–271. https://doi.org/10.1177/0022057409189001-2 Wiliam, D., & Thompson, M. (2008). Integrating assessment with learning: What will it take to make it work? In C. A. Dwyer (Ed.), The Future of Assessment: Shaping Teaching and Learning, (pp.53–84). Routledge. https://doi.org/10.4324/9781315086545 Yan, Z., Li, Z., Panadero, E., Yang, M., Yang, L. & Lao, H. (2021). A systematic review on factors influencing teachers’ intentions and implementations regarding formative assessment. Assessment in Education: Principles, Policy & Practice, 28(3), 228–260. https://doi.org/10.1080/0969594X.2021.1884042

747 754 Habibe Toker Kekik Gülseren Karagöz Akar Şerife Sevinç Alanya Alaaddin Keykubat University Boğaziçi University Turkey Middle East Technical University Turkey This study provides a detailed account of advanced mathematical knowledge specifically for mathematics teachers, focusing on the concept of limits. Structured interviews were conducted with five experts who have various expertise in the fields of mathematics and mathematics education. The results demonstrate how the concept of equality is interpreted differently in the context of limits, as opposed to other mathematical contexts. The role of covariational reasoning in the context of limits is also elucidated with empirical evidence, and the significance of topology in comprehending the concept of the limit is emphasized. Results point to some important implications for future research and teacher education in the learning of limits from an advanced mathematical point of view Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14–46. Belin, M. & Karagöz Akar, G. (2020a). The effect of quantitative reasoning on pre-service mathematics teachers’ proof comprehension: The case of real numbers. Journal of Mathematical Behavior, 57. https://doi.org/10.1016/j.jmathb.2020.100757. Belin, M., & Karagöz Akar, G. (2020b). Exploring real numbers as rational number sequences with pre-service mathematics teachers. Mathematics Teacher Educator, 9(1), 63–87. https://doi.org/10.5951/MTE.2020.9999 Carlson, M. P., Larsen, S., & Jacobs, S. (2001). An investigation of covariational reasoning and its role in learning the concepts of limit and accumulation. Proceedings of the 23rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, 145–153. Choi, Y., & Do, J. (2005). Equality involved in 0.9999... and (−8)3 . For the Learning of Mathematics, 25(3), 13–15. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. Lesh (Eds), Handbook of research design in mathematics and science education (pp. 517–545). Lawrence Erlbaum Associates. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. Providence, R.I. and Washington, D.C.: American Mathematical Society and Mathematical Association of America. Retrieved from: https://www.cbmsweb.org/archive/MET2/met2.pdf Cornu, B. (2002). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153–166). Springer. https://doi.org/10.1007/0-306-47203-1_10 Harel, G. (2008). What is mathematics? A pedagogical answer to a philosophical question. In B. Gold, & R. Simons (Eds.), Proof and other dilemmas: Mathematics and philosophy (pp. 265–290). Mathematical Association of America. Hoffmann, A., & Even, R. (2018). What do mathematicians wish to teach teachers in secondary school about mathematics? Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (pp. 99–106). PME. Klein, F. (1932). Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (E. R. Hendrick & C. A. Noble, Trans., 3rd ed.). Macmillan. (Original work published 1924). Merriam S. B., & Tisdell E. J. (2015). Qualitative research: A guide to design and implementation (4th ed.). Jossey-Bass. Moru, E. K. (2009). Epistemological obstacles in coming to understand the limit of a function at undergraduate level: A case from the national university of Lesotho. International Journal of Science Mathematics Education, 7(3), 431–454. https://doi.org/10.1007/s10763-008-9143-x Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.3102/0013189X015002004 Swinyard, C., & Larsen, S. (2012). Coming to understand the formal definition of limit: Insights gained from engaging students in reinvention. Journal for Research in Mathematics Education, 43(4), 465–493. Tall, D. (2002). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 3–21). Springer. https://doi.org/10.1007/0-30647203-1_1 Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education. (pp. 421–456). National Council of Teachers of Mathematics. https://doi.org/10.1007/s11858-021-01270-1 Wasserman, N. H. (2016). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science Mathematics and Technology Education, 16(1), 28–47. https://doi.org/10.1080/14926156.2015.1093200 Yan, X., Marmur, O. & Zazkis, R (2022). Advanced mathematics for secondary school teachers: Mathematicians’ perspective. International Journal of Science and Mathematics Education, 20, 553–573. https://doi.org/10.1007/s10763-020-10146-x Zazkis, R. & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: perceptions of secondary mathematics teachers. Mathematical Thinking and Learning, 12(4), 263–281. https://doi.org/10.1080/10986061003786349 Zazkis, R. & Mamolo, A. (2025). The equality 0.999… = 1, what does it exemplify? Exploring spaces of an example. Educational Studies in Mathematics https://doi.org/10.1007/s10649-025-10463-4 Ziegler, G. M., & Loos, A. (2017). “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. In G. Kaiser (Ed.), Proceedings of the 13th International Congress on Mathematical Education (pp. 63–77). Springer International Publishing.

755 762 Myrto Karavakou Claudine Margolis National and Kapodistrian University of Athens Greece Bolton MA In this paper we report a design-based empirical study in which Grade 10 students use a digital environment, “Sketch-to-Dance”, to choreograph an avatar to “match” a song by sketching graphs. Grounded in a new-materialist perspective, we treat students’ covariational reasoning as an aesthetic–material practice distributed across an assemblage of bodies, sound, sketches, and animated motion. Using aesthetically driven covariational reasoning as our lens, we analyse how aesthetic intensities of fit/misfit organise meanings for how different quantities (e.g., vertical position, horizontal position, and orientation) change over time. Across three short episodes, we show how iterative attunement between graph shape and felt musical–dance fit actualises covariational meanings on period, direction, amplitude, and rate of change. Bagossi, S., Capone, R., & Mennuni, F. (2023). Undergraduate students' second-order covariational reasoning when conceptualizing paraboloids supported by digital tools. In Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 67-74). PME. Châtelet, G. (2000). Figuring space: Philosophy, mathematics and physics (R. Shore & M. Zagha, Trans.). Dordrecht, The Netherlands: Kluwer Academic Publishers. da Silva, R.S.R. On music production in mathematics teacher education as an aesthetic experience. ZDM Mathematics Education 52, 973–987 (2020). De Freitas, E., & Sinclair, N. (2014). Mathematics and the body: Material entanglements in the classroom. Cambridge University Press. Ferrara, F., & Ferrari, G. (2017). Agency and assemblage in pattern generalisation: A materialist approach to learning. Educational Studies in Mathematics, 94, 21–36. Gadanidis, G., Borba, M., Hughes, J., & Lacerda, H. (2016). Designing aesthetic experiences for young mathematicians: A model for mathematics education reform. International Journal of Research in Mathematics Education, 6(2), 225–244. Ferrara, F., & Sinclair, N. (2016). An early algebra approach to pattern generalisation: Actualising the virtual through words, gestures and toilet paper. Educational Studies in Mathematics, 92(1), 1-19. Karavakou, M., & Kynigos, C. (2025). Dancing with trigonometric functions: aesthetically guided covariational reasoning while creating periodic dynamic models. In M. Bosch, G. Bolondi, S. Carreira, C. Spagnolo, & M. Gaidoschik (Eds.), Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14). Free University of Bozen-Bolzano and ERME. Karavakou, M., Kynigos, C., & Sinclair, N. (2023). Bridging disciplinary aesthetics: when Mathematics meets Art through Educational Technology. Frontiers in Ed. 8, Art. 1284718. Margolis, C. (2025). The effect of digital task design features on students’ covariational reasoning and graphing activity in sketch-to-animation tasks. Doctoral dissertation. University of Michigan. Sinclair, N., De Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM: The International Journal on Mathematics Education, 45(2), 239–252. https://doi.org/10.1007/s11858-012-0465-3 Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics.

763 770 Maria Kattou Constantinos Christou Demetra Pitta-Pantazi University of Cyprus The integration of AI into mathematics education is shifting the teacher’s role from user of digital tools to designer of AI-generated apps. This study examines the criteria that prospective teachers consider when designing mathematical apps and the alignment between their design intentions and app evaluations. Forty-two prospective primary teachers (PT) participated in an AI-integrated mathematics education course, documenting their prompts, apps, and evaluations. Exploratory factor analysis identified three dimensions of criteria: pedagogical tool quality, learning experience, and assessment features. The results indicated alignment between PTs’ design intentions and their app evaluations. AI appears to support the instrumental genesis by embedding pedagogical schemes directly into app design. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274. Bokhove, C. & Drijvers, P. (2010). Digital Tools for Algebra Education: Criteria and Evaluation. International Journal of Computers for Mathematical Learning, 15, 45–62. Cayton-Hodges, G. A., Feng, G., & Pan, X. (2015). Tablet-based math assessment: What can we learn from math apps?”. Educational Technology & Society, 18(2), 3–20. Clark-Wilson, A., Robutti, O., & Thomas, M. (2020). Teaching with digital technology. ZDM Mathematics Education, 52(7), 1223–1242. Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75, 213–234. Harrison, T.R. & Lee, H.S. (2018). iPads in the mathematics classroom: Developing criteria for selecting appropriate learning apps. International Journal of Education in Mathematics, Science and Technology, 6(2), 155–172. Kieran, C. & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11, 205–263. Lee, D. & Yeo, S. (2022). Developing an AI-based chatbot for practicing responsive teaching in mathematics. Computers & Education, 191, 1–17. Taani, O. & Alabidi, S. (2024). ChatGPT in education: benefits and challenges of ChatGPT for mathematics and science teaching practices. International Journal of Mathematical Education in Science and Technology, 56, 1–30. Yeo, S., Moon, J., & Kim, D. (2024). Transforming mathematics education with AI: Innovations, implementations, and insights. The Math. Education, 63(2), 386–392.

771 778 Tandeep Kaur Hamsa Venkat Mellony Graven Angel Mukaka Corin Mathews Lawan Abdulhamid Balarabe Yushau Dublin city university Ireland Early number learning strongly predicts later mathematics achievement, yet there is limited clarity on how foundational numeracy is currently conceptualised and measured in Global South research. This scoping review maps 35 peer-reviewed academic studies on early number learning published between 2022 and 2025 in the post-COVID period. Drawing on Whitacre et al.’s (2020) framework, the analysis shows that most studies define early numeracy through symbolic performance on basic number tasks, typically assessed using short, standardised early-grade numeracy tools. Approximate number sense and mature whole number reasoning are rarely central constructs. This dominance of symbolic attainment indicators produces strong evidence for system monitoring, but offers limited insight into conceptual understanding. The findings suggest the need for a broader conception of foundational numeracy that integrates fluency with understanding. ASER Centre. (2023). Annual Status of Education Report (Rural) 2023. ASER Centre. Bezuidenhout, H. S. (2022). Associations between early numeracy and mathematics-specific vocabulary. South African Journal of Childhood Education, 12(1), a1191. Dawes, A., Biersteker, L., Snelling, M., & Tredoux, C. (2018). The Early Learning Outcomes Measure (ELOM): A South African direct assessment of early development. South African Journal of Childhood Education, 8(2). Devarajan, S. (2013). The case for education in developing countries. The World Bank Research Observer, 28(3), 435–451. Gersten, R., & Chard, D. J. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 33(1), 18–28. Kaffenberger, M., & Pritchett, L. (2020). A structured model of the sources of education system learning outcomes. International Journal of Educational Development, 75, 102– 138. Ma, L. (2013). A Critique of the Structure of U.S. Elementary School Mathematics. Notices of the American Mathematical Society, Vol 60, 10, pp. 1282-1296. Pisani, L., Seiden, J., & Wolf, S. (2022). Longitudinal evidence on the predictive validity of the International Development and Early Learning Assessment (IDELA). Educational Assessment, Evaluation and Accountability, 34(2), 173–194. Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 1102–1118). Oxford University Press. RTI International. (2014). Early Grade Mathematics Assessment (EGMA) toolkit. RTI International. Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., & De Smedt, B. (2017). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence. Developmental Science, 20(3), e12372. Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144–150. UNICEF (2022). Are children really learning? Exploring foundational skills in the midst of a learning crisis. Uwezo. (2016). Are our children learning? Uwezo learning assessment report. Twaweza East Africa. Venkat, H., & Adler, J. (2025). Mathematics education in contexts of majority marginalisation. Journal of Mathematics Teacher Education, 28(1), 1–15. Whitacre, I., Henning, R., & Atabaş, Ş. (2020). Number sense: A literature review. Mediterranean Journal for Research in Mathematics Education, 17(2), 1–30. *Note: Full list of 35 studies included in the review is available from the authors on request.

779 786 Takashi Kawakami Katsuki Akizawa Utsunomiya University Japan Utsunomiya University-Affiliated Elementary School Japan A data-driven society requires citizens, including young students, to develop the ability to validate data-based models, modelling processes, and data-generation processes. This study examined various validation approaches in a data-informed modelling task involving 30 Grade 6 students aged 11–12 years. Findings from the task-based questionnaire indicated that students validated the model itself, modelling processes, data, data-generation processes, and final products. Moreover, both deterministic and stochastic perspectives derived from data informed students’ validation practices. These findings highlight the value of integrating rich, data-informed validation activities into primary school mathematical modelling. Cobb, G. W., & Moore, D. S. (1997). Mathematics, statistics, and teaching. The American Mathematical Monthly, 104(9), 801-823. Czocher, J. A. (2018). How does validating activity contribute to the modeling process? Educational Studies in Mathematics, 99(2), 137-159. English, L., & Watson, J. (2018). Modelling with authentic data in sixth grade. ZDM– Mathematics Education, 50(1-2), 103-115. Fielding, J., Makar, K., & Ben-Zvi, D. (2025). Developing students’ reasoning with data and data-ing. ZDM–Mathematics Education, 57(1), 1-18. Gal, I., & Geiger, V. (2022). Welcome to the era of vague news: A study of the demands of statistical and mathematical products in the COVID-19 pandemic media. Educational Studies in Mathematics, 111(1), 5-28. Geisler, S. (2021). Data-based modelling with experiments–Students’ experiences with model-validation. In M. Inprasitha et al. (Eds.), Proc. 44th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 330-339). PME. Innabi, H., Marton, F., & Emanuelsson, J. (2023). Sustainable learning of statistics. In G. F. Burrill, L. de Oliveria Souza, & E. Reston (Eds.), Research on reasoning with data and statistical thinking: International perspectives (pp. 279-302). Springer. Kawakami, T., & Saeki, A. (2024). Roles of mathematical and statistical models in datadriven modelling: A prescriptive modelling perspective. In H.-S. Siller, et al. (Eds.), Researching mathematical modelling education in disruptive times (pp. 595-605). Springer. Park, J., Czocher, J. A., & White, R. T. (2025). Exploring STEM students’ model validation across two data-informed task environments. International Journal of Science and Mathematics Education. (online puplished) Wilkerson, M. H., Lanouette, K., & Shareff, R. L. (2021). Exploring variability during data preparation: a way to connect data, chance, and context when working with complex public datasets. Mathematical Thinking and Learning, 24(4), 312-330.

787 794 Mayumi Kawamura Oita University Japan This research report examines differences in the classroom practices of two high school teachers using an identity perspective. Drawing on commognitive theory, the author qualitatively analyzes teaching practices in a Japanese high school, focusing on how teachers’ identities and their interpretations of student identities shape classroom interactions. The findings suggest that differences between the two classes are influenced by how each teacher perceives their role and their students. An identity perspective is helpful for interpreting classroom practice, and teachers need to be aware of how their own identities shape how they perceive and respond to students. Graven, M., & Heyd-Metzuyanim, E. (2019). Mathematics identity research: the state of the art and future directions: Review and introduction to ZDM Special Issue on Identity in Mathematics Education. ZDM, 51(3), 361-377. Heyd-Metzuyanim, E. (2013). The co-construction of learning difficulties in mathematics– teacher–student interactions and their role in the development of a disabled mathematical identity. Educational Studies in Mathematics, 83(3), 341-368. Heyd-Metzuyanim, E., & Sfard, A. (2012). Identity struggles in the mathematics classroom: On learning mathematics as an interplay of mathematizing and identifying. International Journal of Educational Research, 51-52, 128-145. Kawamura, M. (2025). Narratives of a high school teacher in a mathematics class. Proceedings of the 9th ICMI-East Asia Regional Conference on Mathematics Education, Vol. 3, 200-203. Molfino, V., & Ochoviet, C. (2019). A mathematics teacher's identity study through their teaching practices in a postgraduate training course. The Mathematics Enthusiast, 16(1), 389-408. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge university press. Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 1422.

795 802 Johanna Hedwig Kerres Till von Monkiewitsch Miriam Lüken Bielefeld University University of Cologne Germany This literature review examines gender-specific effects on learning computational thinking (CT) in mathematics education. It analyzes thirteen intervention studies with respect to the direction of gender effects and attributional patterns. Results show inconsistent findings: most studies report no gender effects, some report gender effects favoring boys, and one reports an effect favoring girls. Furthermore, the review reveals that studies finding no gender differences mainly use external attributions, while those favoring boys rely on internal attributions; the single study favoring girls uses both. The findings underscore the potential of well-designed interventions to reduce gender disparities in CT and the need for more detailed research. Angeli, C., & Georgiou, K. (2023). Investigating the effects of gender and scaffolding in developing preschool children’s computational thinking during problem-solving with BeeBots. Frontiers in Education, 7, Article 757627. https://doi.org/10.3389/feduc.2022.757627 Angeli, C., & Valanides, N. (2020). Developing young children’s computational thinking with educational robotics: An interaction effect between gender and scaffolding strategy. Computers in Human Behavior, 105, Article 105954. https://doi.org/10.1016/j.chb.2019.03.018 Atmatzidou, S., & Demetriadis, S. (2016). Advancing students’ computational thinking skills through educational robotics: A study on age and gender relevant differences. Robotics Autonomous Systems, 75, 661–670. https://doi.org/10.1016/j.robot.2015.10.008 Del Olmo-Munoz, J., Cozar-Gutierrez, R., & Antonio Gonzalez-Calero, J. (2022). Promoting second graders’ attitudes towards technology through computational thinking instruction. International Journal of Technology and Design Education, 32(4), 2019–2037. https://doi.org/10.1007/s10798-021-09679-1 Devi, C. N., Singh, K. I., & Kumar, A. R. (2024). Gender inequality and equality: A conceptual analysis. Asian Journal of Basic Research, 27(6S), 264–266. https://doi.org/10.53555/AJBR.v27i6S.5890 Gildehaus, L. (2024). (Un)doing gender in university mathematics – theoretical challenges and practical recommendations. In A. S. González-Martín, G. Gueudet, I. Florensa, & N. Lombard (Eds.), Proceedings of the 5th Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2024) (pp. 781–790). Escola Universitaria Salesiana de Sarrià & INDRUM. Gökçe, S., & Yenmez, A. A. (2023). Ingenuity of scratch programming on reflective thinking towards problem solving and computational thinking. Education and Information Technologies, 28(5), 5493–5517. https://doi.org/10.1007/s10639-022-11385-x Guggemos, J. (2021). On the predictors of computational thinking and its growth at the high-school level. Computers & Education, 161, Article 104060. https://doi.org/10.1016/j.compedu.2020.104060 Jiang, S., & Wong, G. K. W. (2022). Exploring Age and Gender Differences of Computational Thinkers in Primary School: A Developmental Perspective. Journal of Computer Assisted Learning, 38(1), 60–75. https://doi.org/10.1111/jcal.12591 Kennedy, A. I., Srello, A., & Strieholt, R. (2025). Relationships Between Computer and Information Literacy, Computational Chinking, and Student Background. In J. Frallion (Ed.), An International Perspective on Digital Literacy: Results from ICILS 2023. Springer. https://doi.org/10.1007/978-3-031-87722-3_6 Martinot, P., Colnet, B., Breda, T., Sultan, J., Touitou, L., Huguet, P., Spelke, E., DehaeneLambertz, G., Bressoux, P., & Dehaene, S. (2025). Rapid emergence of a maths gender gap in first grade. 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Informatics in Education. https://doi.org/10.15388/infedu.2023.19 Plangg, S., & Fuchs, K. J. (2022). A Gender-Related Analysis of a Robots’ Math Class. International Journal for Technology in Mathematics Education, 29(3), 143–164. https://doi.org/10.1564/tme_v29.3.03 Polat, E., Hopcan, S., Kucuk, S., & Sisman, B. (2021). A comprehensive assessment of secondary school students’ computational thinking skills. British Journal of Educational Technology, 52(5), 1965–1980. https://doi.org/10.1111/bjet.13092 Sung, J., Lee, J. Y., & Chun, H. Y. (2023). Short-term effects of a classroom-based STEAM program using robotic kits on children in South Korea. International Journal of STEM Education, 10(1), 26. https://doi.org/10.1186/s40594-023-00417-8 Tan, W.-L., Samsudin, M. A., Ismail, M. E., Ahmad, N. J., & Abdul Talib, C. (2021). Exploring the Effectiveness of STEAM Integrated Approach via Scratch on Computational Thinking. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), Article em2049. https://doi.org/10.29333/ejmste/11403 Tamborg, A. L., Elicer, R., Misfeldt, M., & Jankvist, U. T. (2022). Computational thinking in Denmark from an anthropological theory of the didactic perspective. In C. Fernández, S. Llinares, Á. Gutiérrez, & N. Planas (Eds.), Proc. 45th Conf. of the Int. Group for the Psychology of Mathematics (Vol. 4, pp. 91–98). PME. Weiner, B. (1985). An attributional theory of achievement motivation and emotion. Psychological Review, 92(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Zhang, S., & Wong, G. K. W. (2024). Exploring the underlying cognitive process of computational thinking in primary education. Thinking Skills and Creativity, 48, Article 101314. https://doi.org/10.1016/j.tsc.2023.101314

803 810 Chia Kiesel Birke-Johanna Weber Stefanie Rach Anke Lindmeier FAU Erlangen-Nürnberg Kiel University Germany OVGU Magdeburg Germany Innovative tasks that specifically address connections between school mathematics and university mathematics have increasingly been implemented in university mathematics courses. Studies indicate that such tasks can strengthen prospective teachers’ study motivation, but it is unclear whether they have the same effect on students not enrolled in a teaching program. In a study with 146 first-year students, we investigated whether students evaluated regular and innovative tasks differently regarding task values, costs, effort, and self-efficacy. We found significant moderating effects of the study program (teaching vs. non-teaching), indicating that innovative tasks were evaluated more favourably by prospective teachers, e.g., concerning utility value. These results suggest implementing specific tasks for students in different programs. Álvarez, J.A.M., Arnold, E.G., Burroughs, E.A., Fulton, E.W., & Kercher, A. (2020). The design of tasks that address applications to teaching secondary mathematics for use in undergraduate mathematics courses. Journal of Mathematical Behavior, 60, 100814. Biehler, R., Durand-Guerrier, V. & Trigueros, M. (2024). New trends in didactic research in university mathematics education. ZDM Mathematics Education, 56, 1345–1360. https://doi.org/10.1007/s11858-024-01643-2 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students’ effort. Learning and Instruction, 47, 53–64. Eccles, J. S., & Wigfield, A. (2020). From expectancy-value theory to situated expectancyvalue theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, 101859. Eichler, A., & Isaev, V. (2023). Improving prospective teachers’ beliefs about a double discontinuity between school mathematics and university mathematics. Journal Für Mathematik-Didaktik, 44, 117–142. https://doi.org/10.1007/s13138-022-00206-w Gaspard, H., Dicke, A.-L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), 237–254. https://doi.org/10.1007/s10649-007-9100-6 Hellwig, L. & Geisler, S. (2024). Instructions in math problems: Are proof tasks considered more difficult by university students? In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 17-24). PME. Rach, S. & Heinze, A. (2017). The transition from school to university mathematics: Which influence do school-related variables have? International Journal of Science and Mathematics Education, 15(7), 1343–1363. https://doi.org/10.1007/s10763-016-9744-8 Rach, S. & Schukajlow, S. (2024). Affecting task values, costs, and effort in university mathematics courses: the role of profession-related tasks on motivational and behavioral states. International Journal of Science and Mathematics Education, 22, 1013–1035. Wasserman, N. H., Buchbinder, O., & Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. ZDM–Mathematics Education, 55(4), 719-736. Weber, B.-J., Breuer, J., & Lindmeier, A. (2023). How do school-related mathematical problems become relevant for prospective teachers in mathematics courses at university? A qualitative interview study. Research in Mathematics Education, 27(1) 114–139.

811 818 Eunji Kim Faculty of Education and Culture, Tampere University Finland This study examines how Finnish primary school teachers orchestrate Embodied Language (EL) alongside natural, pictorial, and mathematical symbolic language in mathematics classrooms. EL is conceptualized as analytically distinct domains of bodily, tactile, spatial, rhythmic, and playful participation in mathematical meaning-making. The study re-analysed classroom observation data from a qualitative multi-case study involving five lessons taught by three teachers. Qualitative episode analysis shows that EL functioned in qualitatively different ways—as a bridge to abstraction, as structured sense-making, and as procedural support—highlighting the pedagogical importance of intentional coordination of embodied action within instruction. Practical challenges and affordances in multimodal teaching are also addressed. Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The future of embodied design for mathematics teaching and learning. Frontiers in Education, 5, 147. https://doi.org/10.3389/feduc.2020.00147 Cohen, L., Manion, L., & Morrison, K. (2002). Research methods in education (5th ed.). Routledge. https://doi.org/10.4324/9780203224342 DePascale, M., & Ramani, G. B. (2025). The role of math games for children’s early math learning: A systematic review. Journal of Numerical Cognition, 11, Article e14897. Frey, K., Sproesser, U., Kowalk, S., & Pittalis, M. (2024). Supporting lower secondary students’ functional thinking with specific learning environments. In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 216–223). PME. Hannula, M. S. (2012). Exploring new dimensions of mathematics-related affect: Embodied and social theories. Research in Mathematics Education, 14(2), 137–161. Joutsenlahti, J., & Perkkilä, P. (2019). Sustainability development in mathematics educationA case study of what kind of meanings do prospective class teachers find for the mathematical symbol “2/3”? Sustainability, 11(2), Article 457. Joutsenlahti, J., & Perkkilä, P. (2022). Mathematical thinking and understanding in learning of mathematics. LUMAT: International Journal on Math, Science and Technology Education, 10(2), Article 2. https://doi.org/10.31129/LUMAT.10.2.1824 Joutsenlahti, J., & Perkkilä, P. (2024). Mastery of the concept of percentage and its representations in Finnish comprehensive school grades 7–9. Education Sciences, 14(10), Article 1043. https://doi.org/10.3390/educsci14101043 Kim, E., Perkkilä, P., & Joutsenlahti, J. (2025). Teachers’ beliefs and instructional quality of Languaging in primary school mathematics. LUMAT: International Journal on Math, Science and Technology Education, 13(1), Article 9. Krause, C., & Shvarts, A. (2022). An embodied perspective on diversity in mathematics education. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proc. 45th Conf. of the Int. Group for the Psychology of Math. Education (Vol. 1, pp. 209–210). PME. Marco, N., & Palatnik, A. (2025). Embodied argumentation: Convincing yourself, your friend, or the skeptic? In C. Cornejo, P. Felmer, M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proc. 48th Conf. of the Int. Group for the Psychology of Mathematics Education: Research Reports (Vol. 2, pp. 67–74). PME. Palatnik, A., Abrahamson, D., Baccaglini-Frank, A., Ng, O., Shvarts, A., Swidan, O., & Tabach, M. (2023). Theory and practice of designing embodied mathematics learning. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 159–189). PME. Way, J., & Ginns, P. (2024). Embodied learning in early mathematics education: Translating research into principles to inform teaching. Education Sciences, 14(7), 696. https://doi.org/10.3390/educsci1407069

819 826 Irma E. Stevens Elizabeth Kimber Kevin C. Moore University of Rhode Island University of Oxford United Kingdom University of Geogia Quantitative and covariational reasoning are crucial for understanding graphs and relationships between varying quantities in secondary and undergraduate mathematics. Research on covariational reasoning has led to a fine-grained classification of quantitative and covariational reasoning. While much of this research draws on analysis of students’ spoken and written responses to tasks, the language in these responses has rarely been analysed from a linguistic perspective. This study combines analysis of quantitative and covariational reasoning with systemic functional linguistics (SFL). We identify linguistic features of descriptions of change that foreground simultaneous change of variables and how these changes are unfolding as students account for graph shape and unpack relationships between quantities. Carlson, M. P., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378. Halliday, M. A. K., & Martin, J. R. (1993). Writing Science: Literacy And Discursive Power (1st ed.). Routledge. Halliday, M. A. K., & Matthiessen, C. M. I. M. (2004). An Introduction to Functional Grammar (3rd, Ed.). Routledge. Johnson, H. L. (2015). Together yet separate: Students’ associating amounts of change in quantities involved in rate of change. Educational Studies in Mathematics, 1-22. Karagöz Akar, G., Zembat, I. O., Arslan, S., & Thompson, P. W. (Eds.). (2022). Quantitative Reasoning in Mathematics and Science Education. Springer Cham. Kimber, E. (2024). Graphical Shape Thinking in Mathematics Classroom Discourse: Analytic Insights From Systemic Functional Linguistics (Publication Number 31871023) [Ph.D., Open University (United Kingdom)]. ProQuest One Literature. England. Kimber, E., & Smith, C. (2023). Identifying static- and emergent-shape thinking in students’ language about graphs of functions. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), Proceedings of CERME13 (pp. 1613-1620). Alfréd Rényi Institute of Mathematics and ERME. Liang, B., & Moore, K. C. (2021). Figurative and operative partitioning activity: A student’s meanings for amounts of change in covarying quantities. MTL, 23(4), 291-317. Moore, K. C., Paoletti, T., & Musgrave, S. (2013). Covariational reasoning and invariance among coordinate systems. The Journal of Mathematical Behavior, 32(3), 461-473. Morgan, C. (1996). “The language of mathematics”: Towards a critical analysis of mathematics texts. For the Learning of Mathematics, 16(3), 2-10. Morgan, C. (2006). What does Social Semiotics have to Offer Mathematics Education Research? Educational Studies in Mathematics, 61(1), 219-245. Şahin-Gür, D., & Prediger, S. (2018). “Growth goes down, but of what?” A case study on language demands in qualitative calculus. In I. E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proc. 42nd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 99–107). PME. Prediger, S., & Zindel, C. (2017). School academic language demands for understanding functional relationships: A design research project on the role of language in reading and learning. EURASIA JMSTE, 13(7b), 4157-4188. Rodriguez, J.-M. G., Bain, K., Towns, M. H., Elmgren, M., & Ho, F. M. (2019). Covariational reasoning and mathematical narratives: investigating students’ understanding of graphs in chemical kinetics [10.1039/C8RP00156A]. CERP, 20(1), 107-119. Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23(2), 139-159. Steffe, L. P., Moore, K. C., & Hatfield, L. L. (Eds.). (2014). Epistemic algebraic students: Emerging models of students’ algebraic knowing (Vol. 4). University of Wyoming. Thompson, P., & Carlson, M. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), CRME (pp. 421-456). NCTM. Yu, F. (2024). Extending the covariation framework: Connecting covariational reasoning to students’ interpretation of rate of change. JMB, 73, 101122. Zentgraf, K., & Prediger, S. (2024). Demands and scaffolds for explaining the connection of multiple representations: Revisiting the bottle-filling task. JMB, 73, 101118.

827 834 Madeleine Kingan Jon Star Harvard Graduate School of Education United States Prior work demonstrates that learners are often over-confident in their performance. However, this type of monitoring may function differently when considering its developmental trajectory in mathematics. Using item response theory, we ask: (1) What was the relationship between confidence and accuracy on a mathematics standardized test in 5th to 8th grade students? (2) Were students differentially over-confident by age, over and above proficiency, at the item level? The findings demonstrated a positive relationship between confidence and accuracy. Older students were more over-confident than younger students and were often justified in this over-confidence. These findings demonstrate both consistencies and inconsistencies with prior domain-general, mathematics-specific, and developmental work. Attali, Y., Budescu, D., & Arieli-Attali, M. (2020). An item response approach to calibration of confidence judgments. Decision, 7(1), 1–19. https://doi.org/10.1037/dec0000111 Fleming, S. M. (2024). Metacognition and Confidence: A Review and Synthesis. Annual Review of Psychology, 75(1), 241–268. https://doi.org/10.1146/annurev-psych-022423032425 Foster, C. (2022). Implementing Confidence Assessment in Low-Stakes, Formative Mathematics Assessments. International Journal of Science and Mathematics Education, 20(7), 1411–1429. https://doi.org/10.1007/s10763-021-10207-9 Kortenhoeven, A., & Serra, M. J. (2025). Experience on a prior math task affects participants’ judgments of learning (JOLs) on an unrelated memory task. Metacognition and Learning, 20(1), 1–32. https://doi.org/10.1007/s11409-025-09417-1 Lichtenstein, S., Fischhoff, B., & Phillips, L. D. (1981). Calibration of Probabilities: The State of the Art to 1980 (No. PTR1092816). Article PTR1092816. https://apps.dtic.mil/sti/html/tr/ADA101986/ Nietfeld, J. L., Cao, L., & Osborne, J. W. (2005). Metacognitive Monitoring Accuracy and Student Performance in the Postsecondary Classroom. The Journal of Experimental Education, 74(1), 7–28. Nolan, J., & Markham, R. (1998). The accuracy–confidence relationship in an eyewitness task: Anxiety as a modifier. Applied Cognitive Psychology, 12(1), 43–54. https://doi.org/10.1002/(SICI)1099-0720(199802)12:1%253C43::AIDACP487%253E3.0.CO;2-1 Rickey, N., Panadero, E., & DeLuca, C. (2025). How do students self-assess? Examining the metacognitive processes of student self-assessment. Metacognition and Learning, 20(1), 1–29. https://doi.org/10.1007/s11409-025-09430-4 Schneider, W. (2008). The Development of Metacognitive Knowledge in Children and Adolescents: Major Trends and Implications for Education. Mind, Brain, and Education, 2(3), 114–121. https://doi.org/10.1111/j.1751-228X.2008.00041.x Schneider, W., & Lockl, K. (2002). The development of metacognitive knowledge in children and adolescents. In Applied metacognition (pp. 224–257). Cambridge University Press. https://doi.org/10.1017/CBO9780511489976.011 Schraw, G. (2009). A conceptual analysis of five measures of metacognitive monitoring. Metacognition and Learning, 4(1), 33–45. https://doi.org/10.1007/s11409-008-9031-3 Weil, L. G., Fleming, S. M., Dumontheil, I., Kilford, E. J., Weil, R. S., Rees, G., Dolan, R. J., & Blakemore, S.-J. (2013). The development of metacognitive ability in adolescence. Consciousness Cognition, 22(1), 264–271. https://doi.org/10.1016/j.concog.2013.01.004

835 842 Rabia Gul Kirikcilar Megan {Mack} McCormick Vilma Mesa Laura Watkins Patrick Kimani Dexter Lim University of Michigan United States Equity has long been a foundational principle in mathematics education, yet its meaning remains contested in practice. This study examines how college algebra students in the United States conceptualize equity in their community college mathematics classrooms. The participants included 23 students from eight community colleges across different regions in the United States. The analysis yielded four distinct perspectives: equity as meeting individual needs, equity as equal opportunity in the absence of racism, equity as colorblind fairness, and equity as sameness. These findings highlight equity as a socially constructed concept and demonstrate the value of Q methodology for capturing diverse student viewpoints in the context of community college mathematics. Banasick, S. (2019) Kade: A desktop application for Q methodology. Journal of Open Source Software, 4(36): 1360. https://doi.org/10.21105/joss.01360. Banasick, S. (2023). EQ Web Sort Configurator (Version 7.0.0) [Computer software] https://doi.org/10.5281/zenodo.8337125. Dunleavy, T. K. (2015). Delegating mathematical authority as a means to strive toward equity. Journal of Urban Mathematics Education, 8(1). Esmonde, I. (2009). Mathematics learning in groups: Analyzing equity in two cooperative activity structures. The Journal of the Learning Sciences, 18(2), 247-284. Foote, M. Q., & Lambert, R. (2011). I have a solution to share: Learning through equitable participation in a mathematics classroom. Canadian Journal of Science, Mathematics and Technology Education, 11(3), 247-260. Gresalfi, M., & Cobb, P. (2006). Cultivating students’ discipline-specific dispositions as a critical goal for pedagogy and equity. Pedagogies: An International Journal, 1(1), 49–57. Gutiérrez, R. (2002). Enabling the practice of mathematics teachers in context: Toward a new equity research agenda. Mathematical Thinking and Learning, 4(2-3), 145-187. Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23(3), 187–200. Leyva, L. A., Amman, K., Wolf McMichael, E. A., Igbinosun, J., & Khan, N. (2022). Support for all? Confronting racism and patriarchy to promote equitable learning opportunities through undergraduate calculus instruction. International Journal of Research in Undergraduate Mathematics Education, 8(2), 339–364. https://doi.org/10.1007/s40753022-00177-w. Lo, M. T., & Ruef, J. L. (2020). Student or teacher? A look at how students facilitate public sensemaking during collaborative groupwork. Journal of Urban Mathematics Education, 13(1), 15-33. Mack, L. (2012). Does every student have a voice? Critical action research on equitable classroom participation practices. Language Teaching Research, 16(3), 417-434. McKeown, B., & Thomas, D. B. (2013). Q methodology, 2nd ed. Los Angeles, CA: Sage Publications. OECD. (2017). Education at a glance 2017: OECD Indicators. Paris: OECD Publishing. Reinholz, D. L. & Shah, N. (2018). Equity analytics: A methodological approach for quantifying participation patterns in mathematics classroom discourse. Journal for Research in Mathematics Education, 49(2), 140-177. Stephenson, W. (1953). The study of behavior: Q-technique and its methodology. Chicago: University of Chicago Press. UNESCO. (2017). A guide for ensuring inclusion and equity in education. Paris, France: UNESCO. United Nations. (2015). Transforming our world: The 2030 agenda for sustainable development. New York, NY: United Nations. Wilburne, J., Polly, D., Franz, D., & Wagstaff, D. A. (2018). Mathematics teachers' implementation of high-leverage teaching practices: A Q-sort study. School Science and Mathematics, 118(6), 232-243. https://doi.org/10.1111/ssm.1226.

843 850 Katharina Kirsten University of Münster Germany Proof summaries have recently been discussed from two perspectives: as a strategy to promote proof comprehension and as an instrument to measure it. To gain deeper insight into the construct of proof summaries, this study investigates their relation to prior knowledge and performance on a multiple-choice proof comprehension test. Sixty advanced undergraduates watched a proof video and completed either a summary followed by a test or the reverse. Analyses showed no meaningful correlation between summary scores and test performance, suggesting the two assessments capture different constructs. Regression analyses indicate that summarizing may be beneficial for students with higher prior knowledge, which has implications for teaching practice. Chi, M. T. H., & Wylie, R. (2014). The ICAP framework: Linking cognitive engagement to active learning outcomes. Educational Psychologist, 49(4), 219–243. https://doi.org/10.1080/00461520.2014.965823 Davies, B., Alcock, L., & Jones, I. (2020). Comparative judgement, proof summaries and proof comprehension. Educational Studies in Mathematics, 105(2), 181–197. https://doi.org/10.1007/s10649-020-09984-x Davies, B., & Jones, I. (2022). Assessing proof reading comprehension using summaries. Int. Journal of Research in Undergraduate Mathematics Education, 8(3), 469–489. Demeke, E. (2025). Investigating strategies undergraduates and mathematicians use for enriched proof-reading experience. In K. Weber & M. Savić (Eds.), Research in Mathematics Education. New Directions for Mathematics Education Research on Proving (pp. 217–237). Springer. https://doi.org/10.1007/978-3-031-85004-2_10 Donoghue, G. M., & Hattie, J. A. C. (2021). A meta-analysis of ten learning techniques. Frontiers in Education, 6, Article 581216. https://doi.org/10.3389/feduc.2021.581216 Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45(1), 62–101. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–390. https://doi.org/10.5951/jresematheduc.43.4.0358 King, A. (1992). Comparison of self-questioning, summarizing, and notetaking-review as strategies for learning from lectures. American Educational Research Journal, 29(2), 303– 323. https://doi.org/10.3102/00028312029002303 Li, C.‑H. (2016). Confirmatory factor analysis with ordinal data. Comparing robust maximum likelihood and diagonally weighted least squares. Behavior Res. Methods, 48(3), 936–949. Mejía Ramos, J. P., Weber, K., Lew, K., La Torre, J. de, & Ma, W. (2025). Student performance on proof comprehension tests in transition-to-proof courses. In K. Weber & M. Savić (Eds.), Research in Mathematics Education. New Directions for Mathematics Education Research on Proving (pp. 151–176). Springer. Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18. https://doi.org/10.1007/s10649-011-9349-7 Neuhaus, S., & Rach, S. (2019). Proof comprehension of undergraduate students and the relation to individual characteristics. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Educatio. Freudenthal Group & Freudenthal Institute, Utrecht University and ERME. Rosseel, Y. (2012). lavaan : An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2). https://doi.org/10.18637/jss.v048.i02 Samkoff, A., & Weber, K. (2015). Lessons learned from an instructional intervention on proof comprehension. The Journal of Mathematical Behavior, 39, 28–50. https://doi.org/10.1016/j.jmathb.2015.05.002 Wirth, L., Kirsten, K., Serpé, C., & Greefrath, G. (2024). Comparing different types of instructional videos in a flipped proof-based classroom. Frontiers in Education, 9, Article 1438355. Yang, K.‑L., & Lin, F.‑L. (2012). Effects of reading-oriented tasks on students’ reading comprehension of geometry proof. Mathematics Ed. Research Journal, 24(2), 215–238.

851 858 Charikleia Korompli Chronis Kynigos National and Kapodistrian University of Athens Greece This study examines the embodied mathematical exploration of students when programming digital models for 3D printing. The challenge of fitting geometrical entities within others, creates a new Conceptual Field, “Inscribability”, within which our qualitative empirical research maps students’ lived experiences, strategies and tech-mediated practices (coding, dynamic manipulation) when creating functional physical models (3D printing). Findings highlight how digital tools actively mediate expression, artefacts and perception, negotiated through the body. Abrahamson, D., & Bakker, A. (2016). Making sense of movement in embodied design for mathematics learning. Cognitive Research: Principles and Implications, 1, 33. https://doi.org/10.1186/s41235-016-0034-3 Alibali, M. W., & Nathan, M. J. (2011). Embodiment in mathematics teaching and learning: Evidence from learners’ and teachers’ gestures. Journal of the Learning Sciences, 21(2), 247–286. https://doi.org/10.1080/10508406.2011.611446 Angelides, P. (2001). The development of an efficient technique for collecting and analyzing qualitative data: The analysis of critical incidents. International Journal of Qualitative Studies in Education, 14(3), 429–442. https://doi.org/10.1080/09518390110029058 Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects: State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68. https://www.jstor.org/stable/3482238 de Freitas, E., & Sinclair, N. (2014). Mathematics and the body: Material entanglements in the classroom. Cambridge University Press. Hasse, C. (2008). Postphenomenology: Learning cultural perception in science. Human Studies, 31(1), 43–61. https://doi.org/10.1007/s10746-007-9075-4 Ihde, D. (1990). Technology and the lifeworld: From garden to earth. Indiana University Press. Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. Journal for Research in Mathematics Education, 44(2), 372– 415. https://www.jstor.org/stable/10.5951/jresematheduc.44.2.0372 Nemirovsky, R., Ferrara, F., Ferrari, G., Adamuz-Povedano, N., & Bracho López, R. (2020). Body motion, early algebra, and the colours of abstraction. Educational Studies in Mathematics, 104, 261–283. https://doi.org/10.1007/s10649-020-09955-2 Ng, O., & Chan, T. (2019). Learning as making: Using 3D computer-aided design to enhance the learning of shapes and space in STEM-integrated ways. British Journal of Educational Technology, 50(1), 294–308. https://doi.org/10.1111/bjet.12643 Ng, O., & Ferrara, F. (2020). Towards a materialist vision of ‘learning as making’: The case of 3D Printing Pens in school mathematics. International Journal of Science and Mathematics Education, 18, 925–944. https://doi.org/10.1007/s10763-019-10000-9 Ng, O., & Ye, H. (2022). Mathematics learning as embodied making: Primary students’ investigation of 3D geometry with handheld 3D printing technology. Asia Pacific Education Review, 23, 311–323. https://doi.org/10.1007/s12564-022-09755-8 Psaros, H. (2025). Toward a metatheoretical dialogue on learning with digital technologies: The case of three postcognitivist approaches. Journal of Theoretical and Philosophical Psychology. Advance online publication. https://doi.org/10.1037/teo0000331 Roth, W.-M. (2016). Growing-making mathematics: A dynamic perspective on people, materials, and movement in classrooms. Educational Studies in Mathematics, 93, 87– 103. https://doi.org/10.1007/s10649-016-9695-6 Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83– 94. https://doi.org/10.1159/000202727

859 866 Timo Kosiol Marielena Menzel Sebastian Geisler Stefanie Rach OVGU Magdeburg Germany University of Potsdam Germany Validating is considered an important process for successful modelling. It includes critically evaluating and, if necessary, revising (intermediate) models or results during a modelling process. In an experimental study with N=561 upper secondary students, we compare validation processes for students modelling the decay of beer foam using real data (either self-collected or provided) with those for students modelling smoothed data. We examine whether student characteristics (motivation and content knowledge) or task characteristics (the type of data) predict validation processes. Results indicate that a model’s critical evaluation is predicted by content knowledge, while the formulation of suggestions for improvement is predicted by the type of data. We discuss the role of student and task characteristics to stimulate validation processes. Barron, K. E. & Hulleman, C. S. (2015). Expectancy-value-cost model of motivation. In J. D. Wright (Ed.), Int. enc. of the social & beh. sciences (2nd ed., Vol. 8, pp. 503-509). Elsevier. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education (pp. 73-96). Springer. Blum, W. & Leiss, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical Modelling (pp. 222231). Woodhead Publishing. https://doi.org/10.1533/9780857099419.5.221 Carreira, S., Baioa, A. M., & Werle de Almeida, L. M. (2020). Mathematical models and meanings by school and university students in a modelling task. Avances de Investigación en Educación Matemática 17, 67-83. https://doi.org/10.35763/aiem.v0i17.308 Cevikbas, M., Kaiser, G., & Schukajlow, S. (2022). A systematic literature review of the current discussion on mathematical modelling competencies: State-of-the-art developments in conceptualizing, measuring, and fostering. Educational Studies in Mathematics, 109(2), 205-236. https://doi.org/10.1007/s10649-021-10104-6 Czocher, J. A. (2018). How does validating activity contribute to the modeling process? Educational Studies in Mathematics, 99(2), 137-159. Eccles, J. S. & Wigfield, A. (2020). From expectancy-value theory to situated expectancyvalue theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. Eichler, A. & Vogel, M. (2013). Leitidee Daten und Zufall [Key concept: Data and chance] (2 ed.). Springer. https://doi.org/10.1007/978-3-658-00118-6 Engel, J. (2009). Anwendungsorientierte Mathematik: Von Daten zur Funktion [Applicationoriented mathematics: From data to mathematical functions]. Springer. English, L. D. & Watson, J. (2018). Modelling with authentic data in sixth grade. ZDM, 50(12), 103-115. https://doi.org/10.1007/s11858-017-0896-y Goos, M. (2002). Understanding metacognitive failure. The Journal of Mathematical Behavior, 21(3), 283-302. https://doi.org/10.1016/S0732-3123(02)00130-X Hankeln, C., Adamek, C., & Greefrath, G. (2019). Assessing sub-competencies of mathematical modelling—development of a new test instrument. In G. A. Stillman & J. P. Brown (Eds.), Lines of Inquiry in Mathematical Modelling Research in Education (pp. 143-160). Springer. https://doi.org/10.1007/978-3-030-14931-4_8 Jahn, B., Friedrich, S., Behnke, J., Engel, J., Garczarek, U., Munnich, R., Pauly, M., Wilhelm, A., Wolkenhauer, O., Zwick, M., Siebert, U., & Friede, T. (2022). On the role of data, statistics and decisions in a pandemic. Advances in Statistical Analysis, 106(3), 349-382. Jonscher, M., Menzel, M., Geisler, S., & Rach, S. (2024). Modelling performance using functions: Relation to person characteristics and different solution approaches. In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 113-120). PME. Mohr, M. & Ufer, S. (2023). Measuring data-based modeling skills in a collaborative setting. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 355-362). PME. Niss, M. & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. https://doi.org/10.4324/9781315189314 Schukajlow, S., Krawitz, J., & Carreira, S. (2022). The influence and role of affective aspects in mathematical modelling. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proc. 45th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 158-161). PME. Schukajlow, S. & Krug, A. (2014). Are interest and enjoyment important for students' performance? In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 5, pp. 129.136). PME. Stillman, G. A. & Brown, J. P. (2021). Modeling the phenomenon versus modeling the data set. Mathematical Thinking and Learning, 25(3), 270-295. Zell, S. & Beckmann, A. (2009). Modelling activities while doing experiments to discover the concept of variable. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 2216-2225). INRP.

867 874 Styliani-Kyriaki Kourti Konstantinos P. Christou School of Early Childhood Education, AUTH Department of Early Childhood Education, NKUA This study presents the development and pilot testing of a vignette-based research instrument designed to capture the interplay between mathematics teachers’ emotions and decision-making. Focusing on situations involving students’ mathematical errors, the instrument elicits teachers’ emotional responses, teaching actions, and the typical and emotional resources underlying their decisions. Data from a pilot sample of secondary mathematics teachers illustrate the instrument’s capacity to capture emotion intensity, action selection, and justificatory reasoning. The findings demonstrate the analytical potential of the instrument and inform its further refinement, contributing a scalable tool for systematic quantitative investigation of emotion–decision-making processes in mathematics teaching. Engeström, Y. (2001). Making expansive decisions: An activity-theoretical study of practitioners building collaborative medical care for children. In C. M. Allwood & M. Selart (Eds.), Decision making: Social and creative dimensions (pp. 281–301). Springer. https://doi.org/10.1007/978-94-015-9827-9_14 Herbst, P., & Chazan, D. (2012). On the instructional triangle and sources of justification for actions in mathematics teaching. ZDM, 44(5), 601–612. https://doi.org/10.1007/s11858012-0438-6 Kourti, S. K. (2023). Emotional dimensions of mathematics teaching in secondary education [Doctoral dissertation, National and Kapodistrian University of Athens, Greece]. Hellenic National Archive of Doctoral Dissertations. https://www.didaktorika.gr/eadd/handle/10442/54876 Kourti, S. K., & Potari, D. (2024). Teacher emotions and in-the-moment decision making in the secondary mathematics classroom. The Journal of Mathematical Behavior, 74, 101152. https://doi.org/10.1016/j.jmathb.2024.101152 Martínez-Sierra, G., Arellano-García, Y., & Hernández-Moreno, A. (2022). Which situations trigger emotions of secondary school mathematics teachers? International Journal of Science and Mathematics Education, 20(3), 575–595. https://doi.org/10.1007/s10763-02110158-1 Plutchik, R. (2001). The nature of emotions. American Scientist, 89(4), 344–350. https://doi.org/10.1511/2001.4.344 Roth, W.-M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Sense Publishers. Shuman, V., & Scherer, K. R. (2014). Concepts and structures of emotions. In R. Pekrun & L. Linnenbrink-Garcia (Eds.), International handbook of emotions in education (pp. 13– 35). Routledge. https://doi.org/10.4324/9780203148211-3 Skilling, K., & Stylianides, G. J. (2020). Using vignettes in educational research: A framework for vignette construction. International Journal of Research & Method in Education, 43(5), 541–556.

875 882 Sindura Kularajan Cynthia Anhalt Brynja Kohler Utah State University United States University of Arizona In this conceptual paper, we share the intersections of mathematical modeling and data science through examining the role of context, data, and assumptions within these mathematical processes. We argue, in modeling context is central to decontextualizing and recontextualizing, while in data science it drives creation and connection. In modeling, assumptions guide model formulation, but must be checked before choosing analytical methods in data investigations. In modeling, data motivates and validates models, whereas they permeate every stage of the data science cycle. We discuss three major contributions: (1) a theoretical perspective for distinguishing between modeling and data science; (2) implications for modeling and data science task design within a shared context, and (3) implications for teacher-education. American Statistical Association (ASA) (2020). Pre-K-12 Guidelines for Assessment and Instruction in Statistics Education II (GAISEII): A Framework for Statistics and Data Science Education. American Statistical Association. ISBN 978-1-7342235-1-4 Bargagliotti, A., Franklin, C., Arnold, P., Gould, R., Johnson, S., Perez, L., & Spangler, D. A. (2020). The Pre-K–12 Guidelines for Assessment and Instruction in Statistics Education II (GAISE II). American Statistical Association. Baumer, B. (2015). A data science course for undergraduates: Thinking with data. The American Statistician, 69(4), 334-342. Common Core State Standards Initiative (2010). Common Core State Standards in Mathematics. Washington D.C.: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org Czocher, J., Melhuish, K., & Kandasamy, S. (2019). Building mathematics self-efficacy of STEM undergraduates through mathematical modelling. International Journal of Mathematical Education in Science Technology, 51. 1-28. https://doi.org/10.1080/0020739X.2019.1634223. Dunn, P., & Marshman, M. (2020). Teaching mathematical modelling: A framework to support teachers’ choice of resources. Teaching Mathematics and its Applications: An International Journal of the IMA, 39, 127-144. https://doi.org/10.1093/teamat/hrz008. Flatt, C., & Jacobs, R. L. (2019). Principle Assumptions of Regression Analysis: Testing, Techniques, and Statistical Reporting of Imperfect Data Sets. Advances in Developing Human Resources, 21(4), 484-502. https://doi.org/10.1177/1523422319869915 (Original work published 2019) Garfunkel, S., & Montgomery (Eds.) (2019). GAIMME: Guidelines for assessment and instruction in mathematical modeling education, second edition. Consortium for Mathematics and Its Applications (COMAP) and Society for Industrial and Applied Mathematics (SIAM). Greca, I. M., & Moreira, M. A. (2002). Mental, physical, and mathematical models in the teaching and learning of physics. Science education, 86(1), 106-121. Hicks, S. C., & Irizarry, R. A. (2018). A guide to teaching data science. The American Statistician, 72(4), 382-391. https://doi.org/10.1080/00031305.2017.1356747 Hazzan, O., and Mike, K. (2023), “What is Data Science?” in Guide to Teaching Data Science: An Interdisciplinary Approach, pp. 19–34, Springer. https://doi.org/10.1145/3575663 Lee, H. S., Mojica, G. F., and Lovett, J. N. (2020), “Examining How Online Professional Development Impacts Teachers’ Beliefs about Teaching Statistics,” Online Learning, 24, 5–27. DOI:10.24059/olj.v24i1.1992. Little, J. B. (2019). Modeling and data analysis: an introduction with environmental applications (Vol. 120). American Mathematical Society. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA, USA: NCTM. National Research Council. (2001). Adding it up: Helping children learn mathematics. In J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. National Academy Press. Neumann, D., Hood, M., & Neumann, M. (2013). Using real-life data when teaching statistics: Student perceptions of this strategy in an introductory statistics course. Statistics Education Research Journal, 12(2), 59-70. https://doi.org/10.52041/serj.v12i2.304. Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. L. Galbraith, H.W. Henn & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 1– 30). Springer. Osborne, J. W. (2013). Is Data Cleaning and The Testing of Assumptions Relevant in the 21st Century? Frontiers in Psychology, 4(370). https://doi.org/10.3389/fpsyg.2013.00370 Watson, J., Callingham, R., & Nathan, E. (2009). Probing teachers’ pedagogical content knowledge in statistics: “How will Tom get to school tomorrow?” In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides (pp. 563-570). MERGA. Zhao, K. (2025). What Are Statistical Assumptions About? An Answer From Perspectivism. Harvard Data Science Review, 7(1). https://doi.org/10.1162/99608f92.7fcd521d

883 890 Everton Lacerda Jacinto Rina Zazkis Simon Fraser University Canada We engaged undergraduate liberal arts students in investigating patterns within the Sieve of Eratosthenes, a historical method for finding prime numbers in a given interval by crossing out multiples of primes. The investigation used a digital application in which numbers appeared in a grid, and changes in grid width produced different visual displays of prime and composite numbers. The participants examined how the grid width related to the appearance of primes in specific columns and described their approaches and conclusions in the form of a dialogue between interlocutors, referred to as “scripting journeys.” The reported research examines how participants use metaphors to articulate and develop their reasoning within a mathematical investigation. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101. Chapman, O. (1997). Metaphors in the teaching of mathematical problem solving. Int. Journal of Mathematical Education in Science and Technology, 28(6), 569-575. Dietiker, L. (2015). Mathematical story: A framework for curriculum analysis. Journal for Research in Mathematics Education, 46(3), 247-278. English, L. D. (1997). Analogies, metaphors, and images: Vehicles for mathematical reasoning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 3-18). Lawrence Erlbaum. Falani, I., Supriyati, Y., Marzal, J., Iriani, D., & Simatupang, G. M. (2023). Studies investigation in mathematics education: a systematic review metaphor. Indonesian Journal of Mathematics Education, 6(1), 40–62. Grootenboer, P., & Jorgensen, R. (2009). Towards a theory of identity and agency in coming to learn mathematics. Research in Mathematics Education, 11(2), 123–139. Hansen, D., Håkansson, G., & Skott, J. (2022). Students’ agency, creative reasoning, and collaboration in mathematical problem solving. Educational Studies in Mathematics, 109(2), 203–219. Kaphesi, E. (2014). The metaphors children use in learning mathematics. Journal of Education and Practice, 5(27), 1-8. Kercher, A., Louie, N. L., & Hammer, D. (2025). Scripting journeys: Dialogues as data for studying mathematical reasoning. Journal of Mathematical Behavior, 68, 101-118. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Olsen, J., Pedersen, C., & Bremholm, J. (2020). Metaphors in advanced mathematics lectures: How instructors communicate abstract structures. International Journal of Mathematical Education in Science and Technology, 51(7), 977-995. Osman, A., Danielsen, M., & Fauskanger, J. (2022). University students’ metaphorical perceptions of mathematics. Journal of Mathematical Behavior, 65, 101-120. Pradhan, S. (2019). Using metaphors to teach prime and composite numbers. International Journal of Mathematical Education, 14(1), 74-81. Pugalee, D., & Schinck, J. (2014). Students’ metaphorical conceptions of mathematics. Mathematics Education Research Journal, 26(1), 113-131. Scheiner, T., Montes, J., & Fink, A. (2022). Prospective teachers’ metaphorical structures for mathematical and pedagogical ideas. Educational Studies in Mathematics, 109(2), 261282. Sfard, A. (2009). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press. Sfard, A. (2015). Learning, discourses, and communicational reasoning. In L. English & D. Kirshner (Eds.), Handbook of Int. Research in Math. Ed. (3rd ed., pp. 255-268). Routledge. Yee, S. P. (2017). Metaphors in mathematics classrooms: Stories we tell about learning. For the Learning of Mathematics, 37(2), 10-14.

891 898 Juulia Lahdenperä Çiğdem Haser University of Turku Finland Pre-service primary teachers’ (PSTs) views of mathematics play a central role in them becoming effective mathematics teachers. As these views are constructed through prior experiences, understanding how such experiences are formed is essential to support the development of more positive views of mathematics. The data consists of 101 PSTs’ retrospective graphs representing their relationship with mathematics and its variation over time. The analysis revealed five graph-shape categories, Positives, Uphills, Negatives, Downhills, and Zigzags, with aspects such as motivation, competence, perceived difficulty, effort, and former teachers associated with changes along the way. Overall, the study demonstrates that graphs offer a valuable alternative for solely verbal and written responses. Andrà, C., & Scalvini, C. (2023). Trait-and state-like motivations, emotions and perceived competence: a study on freshmen STEM students. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13) (pp. 1392–1399). Alfréd Rényi Institute of Mathematics and ERME. Bekdemir, M. (2010). The pre-service teachers’ mathematics anxiety related to depth of negative experiences in mathematics classroom while they were students. Educational Studies in Mathematics, 75, 311-328. Brady, P., & Bowd, A. (2005). Mathematics anxiety, prior experience and confidence to teach mathematics among pre‐service education students. Teachers and Teaching, 11(1), 37-46. Erixon, E-L., & Christiansen, I. M. (2025). Towards constructive ambiguity: Changing views during secondary mathematics teacher education? Scandinavian Journal of Educational Research, 69(5), 913-929. Guillaume, A. M., & Kirtman, L. (2010). Mathematics stories: Preservice teachers’ images and experiences as learners of mathematics. Issues in Teacher Education, 19(1), 121-143. Hannula, M. S., Kaasila, R., Laine, A., & Pehkonen, E. (2005). Structure and typical profiles of elementary teacher students' view of mathematics. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 89–96). PME. Kim, E., Mallat, E., & Joutsenlahti, J. (2023). A systematic review of primary school class teachers’ views of mathematics teaching and learning. LUMAT: International Journal on Math, Science and Technology Education, 11(2), 60-87. Lo, W. Y. (2021). Pre-service teachers’ prior learning experiences of mathematics and the influences on their beliefs about mathematics teaching. International Journal of Instruction, 14(1), 795-812. Lutovac, S. & Havia, J. (2024). Prospective mathematics teachers’ collective identity work: Navigating failure experiences and concerns about relating to students. Nordic Studies in Mathematics Education, 29(3-4), 19-40. Roesken, B., Hannula, M. S., & Pehkonen, E. (2011). Dimensions of students’ views of themselves as learners of mathematics. ZDM, 43(4), 497-506. Rule, A. C., & Harrell, M. H. (2006). Symbolic drawings reveal changes in preservice teacher mathematics attitudes after a mathematics methods course. School Science and Mathematics, 106(6), 241-258. Saldaña, Johnny (2025). The coding manual for qualitative researchers (5th ed.). Sage Publications. Satyam, V. R., Bae, Y., Smith, J. P., & Levin, M. (2022). Affect graphing: leveraging graphical representations in the study of students’ affect in mathematics. Educational Studies in Mathematics, 110, 481-501. Schreier, M. (2012). Qualitative content analysis in practice (1st ed.). Sage Publications.

899 906 Paola Lattaro Giancarlo Atriano Eliana Della Ventura Maria Mellone Tiziana Pacelli Università Federico II di Napoli, Itay Italy The paper presents a retrospective analysis of La Tombola Infernale, a project situated within Informal Mathematics Education and grounded in the Design-Based Research methodological framework. The project engages students in designing and playing a Dante-inspired tombola game, through which they explore both mathematical concepts and the Inferno canticle of the Divine Comedy. Drawing on Ferrari and Saccoletto’s (2023) theoretical perspective on argumentation as a socially situated practice, the study investigates how this specific informal learning context supports the development of students’ argumentative processes. The analysis focuses on one design cycle—implemented across different age groups and settings—highlighting the role of peer interaction, linguistic resources, and material artefacts in fostering collective meaning-making and the progressive formalisation of modular arithmetic. Bakker, A. (2018). Design Research in Education: A practical Guide for Early Career Researchers (1st ed.). Routledge. Bakker, A., Cai, J. & Zenger, L. (2021). Future themes of mathematics education research: an international survey before and during the pandemic. Educational Studies in Mathematics, 107, 1–24. Bussi, M. B., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 746–783). Routledge. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational researcher, 32(1), 9-13. Eshach, H. (2007). Bridging In-school and Out-of-school Learning: Formal, Non-Formal, and Informal Education. Journal of Science Education and Technology, 16, 171–190. Ferrari, P., & Saccoletto, M. (2023). Argumentation and interaction: discussing a new theoretical perspective. In Proceedings of the thirteen Congress of the European Society for Research in Mathematics Education (pp. 256–263). Alfréd Rényi Institute of Mathematics and ERME. Mercier, H., & Sperber, D. (2017). The enigma of reason. Harvard University Press. Nemirovsky, R., Kelton, M. L., & Civil, M. (2017). Towards a vibrant and socially significant informal mathematics education. In J. Cai (Ed.), Compendium for Research in Mathematics Education, (pp. 968-980). National Council of Teachers of Mathematics. Planas, N., & Pimm, D. (2024). Mathematics education research on language and on communication including some distinctions: Where are we now? ZDM – Mathematics Education, 56(1), 127–139. Star, S. L., & Griesemer, J. R. (1989). Institutional ecology, “translations” and boundary objects: Amateurs and professionals in Berkeley’s Museum of Vertebrate Zoology, 1907– 39. Social Studies of Science, 19(3), 387–420.

907 914 Kate le Roux Armando Solares-Rojas Alf Coles University of Cape Town South Africa CINVESTAV University of Bristol United Kingdom In this conceptual report, we develop ideas relating to a socio-ecological approach to mathematics teacher professional learning, asking, what is the role of the facilitator? We focus on the related concepts of recognition, debt, grievance, and refusal, in socio-ecological relations. We conceptualise by thinking from place, a process of reading diffractively through scholarship and narrative interviews with teaching facilitators from two geopolitical South contexts. We theorise the facilitator role as creating and holding relational, communal spaces involving multiple ways of knowing, being and teaching. These spaces function as grievances: for recognising inequities and our needing one another as qualitative debt; and for confronting inequities in refusal. We offer our theorising to others for diffractive reading in their thinking from place. Alsina, Á., & Mulà, I. (2019). Advancing towards a transformational professional competence model through reflective learning and sustainability: The case of mathematics teacher education. Sustainability, 11(15), 4039. https://doi.org/10.3390/su11154039 Barad, K. (2007). Meeting the universe halfway: Quantum physics and the entanglement of matter and meaning. Duke University Press. Brown, L., & Coles, A. (2019). Reciprocal narrative interviewing. In C. Comanducci & A. Wilkinson (Eds.), Matters of telling: The impulse of story (pp. 177-184). Brill. Coles, A., & Helliwell, T. (2023). The role of mathematics teacher educators in preparing teachers of mathematics to respond to global challenges within their classrooms. London Review of Education, 21(1), 2. https://doi.org/10.14324/LRE.21.1.02 Coles, A., Solares-Rojas, A., & le Roux, K. (2024). Socio-ecological gestures of mathematics education. Educational Studies in Mathematics, 116(2), 165-183. Escobar, A. (2006). Difference and conflict in the struggle over natural resources: A political ecology framework. Development, 49(3): 6-13. Haraway, D. (2015). A curious practice. Angelaki, 20(2), 5-14. Haraway, D. J. (2016). Staying with the trouble: Making kin in the Chthulucene. Duke University Press. Helliwell, T., & Ng, O. L. (2022). Imagining possibilities: Innovating mathematics (teacher) education for sustainable futures. Research in Mathematics Education, 24(2), 128-149. Makramalla, M., Coles, A., le Roux, K., & Wagner, D. (2025). Mathematics education for sustainable futures: a strengths-based survey of the field to invite further research action. Educational Studies in Mathematics, 119(3), 535-556. Mbembe, A. (2018). The idea of a borderless world. Africa is a Country, 11 November. https://africasacountry.com/2018/11/the-idea-of-a-borderless-world# Mbembe, A. (2021). Out of the dark night: Essays on decolonialization. Wits University Press. Mbembe, A. (2023). Pathways of tomorrow: Contribution to thinking commensurate with the planet. Education, research foresight: Working paper 32. https://unesdoc.unesco.org/ark:/48223/pf0000387026_en McGranahan, C. (2016). Theorizing refusal: An introduction. Cultural Anthropology, 31(3), 319–325. https://doi.org/10.14506/ca31.3.01 Petitmengin, C. (2006). Describing one’s subjective experience in the second person: An interview method for the science of consciousness. Phenomenology and the Cognitive Sciences, 5.3-4: 229– 269. https://doi.org/10.1007/s11097-006-9022-2 Prediger, S., Roesken-Winter, B., & Leuders, T. (2019). Which research can support PD facilitators? Journal for Mathematics Teacher Education, 22(4), 407–425. Solares-Rojas, A., Arellano-Aguilar, O., García González, M. M., López-Vargas, M. R., Coles, A. & Méndez Serrano, A. (2022). Mathematics education and social-environmental crises: an interdisciplinary proposal for didactic innovation with rural communities in Mexico. Research in Mathematics Education, 24(2), 202-223. Steffenson, L. (2025). Network of collaborations and social practices to engage students in socio-ecological issues of peatlands. In K. le Roux, A. Coles, A. Solares-Rojas, A. Bose, C. Vistro-Yu et al. (Eds.), Proceedings of the 27th ICMI Study Conference (Mathematics Education and the Socio-Ecological) (pp. 259-266). MATHTED and ICMI. Valero, P. (2007). A socio-political look at equity in the school organization of mathematics education. ZDM, 39(3), 225-233. https://doi.org/10.1007/s11858-007-0027-2 Velasco, P. (2017). Ríos de contradicción [Rivers of contradiction]. Instituto de Investigaciones Antropológicas, UNAM. http://www.iia.unam.mx/publicaciones/detalles.php?clave=498

915 922 Changsuk Lee Oh Nam Kwon Gunpo e-business High School South Korea Seoul National University South Korea This study develops design principles and an instructional model for vocational context-based mathematics instruction to enhance foundational vocational competencies in vocational high schools. Utilizing Design-Based Research, we derived 12 design principles and a six-phase model grounded in curriculum integration and situated learning theories. Validation through expert review and field enactment confirmed significant improvements in students' perception of mathematical utility and practical application skills. This study contributes by demonstrating the function of mathematics as a practical tool within vocational contexts and providing a concrete instructional design framework. Adler, J., Pournara, C., & Graven, M. (2000). Integration within and across mathematics. Pythagoras, 53(12), 2-13. Anderson, J. (2025). The design and implementation of integrated STEM student learning experiences: Challenges for mathematics teachers and school leaders. In Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1). PME. Beane, J. A. (1997). Curriculum integration: Designing the core of democratic education. Teachers College Press. Boaler, J. (2000). Exploring situated insights into research and learning. Journal for Research in Mathematics Education, 31(1), 113–119. Boddy, N., Watson, K., & Aubusson, P. (2003). A trial of the five Es: A referent model for constructivist teaching and learning. Research in science education, 33(1), 27-42. Burke, A. & Lehane, L. (2023). Conceptualising Curriculum Integration: Theoretical, Research and Practical Perspectives. NCCA. Bybee, R. W., Taylor, J. A., Gardner, A., Van Scotter, P., Powell, J. C., Westbrook, A., & Landes, N. (2006). The BSCS 5E instructional model: Origins and effectiveness. Colorado Springs, Co: BSCS, 5(88-98). Dalby, D., & Noyes, A. (2016). Locating mathematics within post-16 vocational education in England. Journal of Vocational Education & Training, 68(1), 70-86. Drake, S. M., & Burns, R. C. (2004). Meeting standards through integrated curriculum. Alexandria, VA: ASCD. Geiger, V., Goos, M., & Forgasz, H. (2015). A rich interpretation of numeracy for the 21st century: A survey of the state of the field. ZDM, 47(4), 531-548. Greeno, J. G. (1998). The situativity of knowing, learning, and research. American Psychologist, 53(1), 5–26. Kang, H. Y., & Tak, B. (2018). A Study on Mathematics Teachers' Use of Curriculum in Vocational High Schools: Developing Materials of Teaching and Learning Mathematics for Enhancing Employ Ability. Journal of the Korean School Mathematics Society, 21(1), 39–62. Kim, M., & Oh, K. (2012). Mathematics education attitude of the students in the specialized high school. Journal of the Korean Data & Information Science Society, 23(6), 1173–1181. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Ministry of Education. (2022). General Introduction to the Elementary and Secondary School Curriculum. Ministry of Education. McKenney, S., & Reeves, T. C. (2012). Conducting Educational Design Research. Routledge. OECD. (2013). OECD skills outlook 2013: First results from the Survey of Adult Skills (PIAAC). OECD Publishing. O’Sullivan, K., Goos, M., O’Meara, N., & Conway, P. F. (2019). The “N” framework: A potential solution to numeracy across the curriculum. In Proceedings of the Seventh Conference on Research in Mathematics Education in Ireland (MEI7) (pp. 235-242). Parr, B., Edwards, M. C., & Leising, J. G. (2009). Selected effects of a curriculum integration intervention on the mathematics performance of secondary students enrolled in an agricultural power and technology course: An experimental study. Journal of agricultural education, 50(1), 57-69. Roehrig, G. H., Dare, E. A., Ellis, J. A., & Ring-Whalen, E. (2021). Beyond the basics: A detailed conceptual framework of integrated STEM. Disciplinary and Interdisciplinary Science Education Research, 3(1), 11. Watters, J. J., & Christensen, C. (2014). Vocational education in science, technology, engineering and maths (STEM): Curriculum innovation through school industry partnerships. In Proceedings of the ESERA 2013 Conference (pp. 89–110). European Science Education Research Association. Wedege, T. (2010). People's Mathematics in Working Life: Why Is It Invisible?. Adults Learning Mathematics, 5(1), 89-97. Xiao, X. (2023). Research on teaching strategies of mathematics and specialized courses in secondary vocational schools. International Journal of New Developments in Education, 5(19), 195-200.

923 930 Daranee Lehtonen Esa Satomaa Jo Van Hoof Marika Parviainen Mikko-Jussi Laakso Jari Metsämuuronen Valentina Dagienė Turku Research Institute of Learning Analytics, University of Turku Finland Institute of Educational Sciences, Vilnius University The decline in global mathematics attitudes and performance raises significant concerns for educational outcomes and career choices. While evidence suggests that students' attitudes significantly influence their performance, some studies report no substantial relationship. This study examines the relationship between fourth graders' (n = 639) mathematics attitudes and performance at both the student and classroom levels. Utilizing multilevel modeling, we found that students’ positive mathematics attitudes strongly predict better performance, a relationship that remains consistent across classrooms. These findings underscore the need for effective strategies to enhance student confidence and engagement in mathematics, ultimately supporting their long-term success in STEM fields. Ajai, J., & Ogungbile, T. (2023). Relationship between laboratory method of teaching, students’ attitude and gender on students’ performance in geometry. Journal of Research in Instructional, 3(1), 1–12. Aladil, M. (2024). Female students' attitudes toward mathematics and their relationships with mathematics achievement. Journal of Umm Al-Qura University for Educational and Psychological Sciences, 16(3), 528-537. Berger, N., Mackenzie, E., & Holmes, K. (2020). Positive attitudes towards mathematics and science are mutually beneficial for student achievement: A latent profile analysis of TIMSS 2015. Australian Educational Researcher, 47(3), 409–444. Cibulskaitė, N., & Augustinovič, A. (2020). STEM education in Lithuania: Theoretical insights and practical implementation. Proc. EDULEARN20, Spain, 5769–5779. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. European Commission, Directorate-General for Research and Innovation. (2021). She figures 2021: Gender in research and innovation: Statistics and indicators. Publications Office. https://data.europa.eu/doi/10.2777/06090 Fennama, E., & Sherman, J. A. (1976). Fennema-Sherman mathematics attitude scales: Instruments designed to measure attitudes toward the learning of mathematics by females and males. Journal for Research in Mathematics Education, 7(5), 324–326. Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., ... & Fletcher, J. M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98(1), 29. Hwang, S., & Son, T. (2021). Students’ attitude toward mathematics and its relationship with mathematics achievement. Journal of Education and E-Learning Research, 8(3), 272. Johnson, H. L., Donovan, C., Knurek, R., Whitmore, K., & Bechtold, L. (2024). Attitudes toward mathematics and graphs influence graph reasoning and selection. In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 105–112). PME. Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8(1), 139–151. Laakso, M.-J., Kaila, E., & Rajala, T. (2018). ViLLE – collaborative education tool: Designing and utilizing an exercise-based learning environment. Education and Information Technologies, 23, 1655–1676. Levine, S. C., & Pantoja, N. (2021). Development of children’s math attitudes: Gender differences, key socializers, and intervention approaches. Developmental Review, 62, 100997. Metsämuuronen, J. (2012). Challenges of the Fennema-Sherman Test in the International Comparisons. International Journal of Psychological Studies, 4(3), 1–22. Niemivirta, M., Tapola, A., Tuominen, H., & Viljaranta, J. (2023). Developmental trajectories of school‐beginners' ability self‐concept, intrinsic value and performance in mathematics. British Journal of Educational Psychology, 94(2), 441–459. Organisation for Economic Co-operation and Development. (2023). Mathematics performance (PISA) (indicator). OECD Publishing. https://doi: 10.1787/04711c74-en Psyridou, M., Torppa, M., Tolvanen, A., Poikkeus, A. M., Lerkkanen, M. K., & Koponen, T. (2023). Developmental profiles of arithmetic fluency skills from Grades 1 to 9 and their early identification. Developmental Psychology, 59(12), 2379. Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV Tests of Cognitive Abilities. von Davier, M., Kennedy, A., Reynolds, K., Fishbein, B., Khorramdel, L., Aldrich, C., Bookbinder, A., Bezirhan, U., & Yin, L. (2024). TIMSS 2023 International Results in Mathematics and Science. Boston College, TIMSS & PIRLS International Study Center. Wang, M., Degol, J., & Ye, F. (2015). Math achievement is important, but task values are critical, too: Examining the intellectual and motivational factors leading to gender disparities in stem careers. Frontiers in Psychology, 6, 36. Wang, M.-T., & Degol, J. (2017). Gender gap in science, technology, engineering, and mathematics (STEM): Current knowledge, implications for practice, policy, and future directions. Educational Psychology Review, 29(1), 119–140. Wang, Y. (2023). Self-concept, learning anxiety, and performance in mathematics learning: the moderating effect of teacher cognitive activation. Eurasia Journal of Mathematics Science and Technology Education, 19(9), em2323. Wen, R., & Dubé, A. K. (2022). A systematic review of secondary students’ attitudes towards mathematics and its relations with mathematics achievement. Journal of Numerical Cognition, 8(2), 295–325.

931 938 Yew Hoong Leong Xin Li Nanyang Technological University Singapore Suzhou University of Technology In solving mathematics problems of the type that are non-routine and where the solver encounters blockages to overcome, the problem solving process is likely to require flexibility that is beyond merely alternative ways of tackling routine tasks. We thus make a distinction between procedural flexibility and heuristical flexibility in the context of problem solving. By analysing a case of a student teacher solving the “nice number problem”, we found that the insights from his procedural flexibilities can potentially be reasonably translated as heuristical flexibility in a productive way. That he did not do so led to reflections that can open up new areas of research in mathematics problem solving. Baroody, A. J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Lawrence Erlbaum. Bolat, R.C., & Arslan, C. (2024). Examination of mathematics teachers’ strategic flexibility in solving mathematical problems. Thinking Skills and Creativity, 54, 1-23. Li, X., & Leong, Y.H. (2026). Solving fractional equations: (in)flexibility and its roots. International Journal of Mathematical Education in Science and Technology, 57(1), 153-172. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561– 574. https://doi.org/10.1037/0022-0663.99.3.561 Schoenfeld, A. (1985). Mathematical problem solving. Academic Press. Schoenfeld, A. H., (2000). Models of the teaching process. Journal of Mathematical Behavior, 18(3), 243-261. Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525– 1538. https://doi.org/10.1037/a0024997 Star, J., Tuomela, D., Joglar Prieto, N. et al (2022). Exploring students’ procedural flexibility in three countries. International Journal of STEM Education, 9(4). http://dx.doi.org/10.1186/s40594-021-00322-y. Verschaffel, L. (2024). Strategy flexibility in mathematics. ZDM, 56, 115-126. Xu, L., Liu, R. -D., Star, J. R., Wang, J., Liu, Y., & Zhen, R. (2017). Measures of Potential Flexibility and Practical Flexibility in Equation Solving. Frontiers in psychology, 8, 1368. https://doi.org/10.3389/fpsyg.2017.01368 Wang, J., Liu, R. D., Star, J. R., Liu, Y., & Zhen, R. (2019). The moderating effect of regulatory focus in the relationship between potential flexibility and practical flexibility. Contemporary educational psychology, 56, 218-227. https://doi.org/10.1016/j.cedpsych.2019.01.013

939 946 Mario Lepore Roberto Capone Eleonora Faggiano University of Salerno Italy University of Bari Aldo Moro Italy Generative AI is increasingly entering mathematics classrooms, yet it may shift students’ work toward uncritical acceptance of generated outputs. To counter this risk, we analyse how learning by prompting can extend constructionist learning by making, supporting mathematical thinking within programming activities. We report a classroom‑based qualitative case study in an Italian scientific high school where small groups worked on a Python Turtle geometry task with a customised GenAI assistant. We trace how prompts evolved from vague requests to explicit mathematical specifications across prompt–debug–refine cycles. Results show that mismatches between intended and produced outputs triggered revisions that made constraints more explicit and supported decomposition of the geometric structure. Abelson, H., & diSessa, A. A. (1986). Turtle geometry: The computer as a medium for exploring mathematics. MIT Press. Bernardi, M. L., Capone, R., & Lepore, M. (2025). Empowering mathematics educators: Integrating ChatGPT as a tool for innovative teaching practices. In B. du Boulay, T. Di Mascio, E. Tovar, & C. Meinel (Eds.), Proceedings of the 17th International Conference on Computer Supported Education, (pp. 404–411). Hmelo-Silver, C. E., & Barrows, H. S. (2006). Goals and strategies of a problem-based learning facilitator. Interdisciplinary Journal of Problem-Based Learning, 1(1), 21–39. Kynigos, C. (2007). Half-baked microworlds as boundary objects in integrated design. International Journal of Computers for Mathematical Learning, 12(2), 87–111. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books. Pepin, B., Buchholtz, N., & Salinas-Hernández, U. (2025). A scoping survey of ChatGPT in mathematics education. Digital Experiences in Mathematics Education, 11(1), 9–41. Walkington, C. (2025). The implications of generative artificial intelligence for mathematics education. School Science and Mathematics. Advance online publication. Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35.

947 954 Jingyi Li Yufeng Guo Beijing Normal University China, People's Republic of Foshan University Mathematical creativity is a key goal of 21st-century mathematics education and develops through prior experience. Previous research has mainly examined group differences in creativity among students with different levels of mathematical thinking experience, but its predictive role remains unclear. This study involved 70 high school students and assessed their mathematical thinking experience and mathematical creativity. Results showed that mathematical thinking experience was significantly positively correlated with creativity. Regression analyses further indicated that mathematical thinking experience significantly predicted overall creativity and each dimension, with slightly stronger effects on fluency and flexibility, highlighting the potential value of experience-based learning in fostering mathematical creativity. Assmus, D., & Fritzlar, T. (2022). Mathematical creativity and mathematical giftedness in the primary school age range: An interview study on creating figural patterns. ZDMMathematics Education, 54(1), 113–131. https://doi.org/10.1007/s11858-022-01328-8 Eraky, A., Leikin, R., & Hadad, B. S. (2022). Relationships between general giftedness, expertise in mathematics, and mathematical creativity that associated with pattern generalization tasks in different representations. Asian Journal for Mathematics Education, 1(1), 36–51. https://doi.org/10.1177/27527263221093427 Ervynck, G. (1991). Mathematical creativity. In D. O. Tall, Advanced mathematical thinking. Kluwer Academic Publishers. Guo Y., & Shi N. (2012a). Mathematical fundamental activity experience: Putting forward, understanding and practicing. Journal of the Chinese Society of Education, 4, 42–45. Guo Y., & Shi N. (2012b). Research on mathematical fundamental activity experience: Definition and dimension division. Journal of Educational Studies, 8(5), 23–28. Hadamard, J. (1945/1954). The psychology of invention in the mathematical feld. Dover Publications. Krutetskii, V. A. (1968/1983). The psychology of mathematical abilities in schoolchildren (L. Bóshù, Trans.). Shanghai Education Press. Lee, K.-H., & Sriraman, B. (2011). Conjecturing via reconceived classical analogy. Educational Studies in Mathematics, 76(2), 123–140. https://doi.org/10.1007/s10649-0109274-1 Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu, Creativity in Mathematics and the Education of Gifted Students (pp. 129–145). BRILL. https://doi.org/10.1163/9789087909352 Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: What makes the difference? ZDM, 45(2), 183–197. Leikin, R., & Sriraman, B. (2022). Empirical research on creativity in mathematics (education): From the wastelands of psychology to the current state of the art. ZDMMathematics Education, 54(1), 1–17. https://doi.org/10.1007/s11858-022-01340-y Li, J., Guo, Y., & Zhang, J. (2025). A framework of mathematical thinking experience: Core components, hierarchical levels, and group differences in mathematical creativity. Educational Studies in Mathematics, 1-25. https://doi.org/10.1007/s10649-025-10431-y OECD. (2021). PISA 2021 creative thinking framework (Third Draft). PISA 2022. Singer, F. M., & Voica, C. (2022). Playing on patterns: Is it a case of analogical transfer? ZDM-Mathematics Education, 54(1), 211–229. Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.

955 962 Giulia Lisarelli Università di Pisa Mathematics has been widely described as a gatekeeper discipline shaping students’ educational opportunities, yet less attention has been paid to how this role is experienced and narrated by students themselves. Drawing on interviews with 40 final-year upper secondary students, this study adopts a situational and narrative approach to explore how mathematics emerges as a gatekeeping mechanism in students’ retrospective narratives of upper secondary school choice within the Italian educational system, characterized by differentiated pathways. Findings illustrate how mathematics becomes meaningful in students’ accounts through interactions with significant others, as its gatekeeping function emerges within situated processes of meaning-making rather than being assumed in advance. Boaler, J. (2000). Exploring situated insights into research and learning. Journal for Research in Mathematics Education, 31(1), 113–119. https://doi.org/10.2307/749822 Flabbi, L. (2001). Secondary schooling decisions in Italy. Rivista di Politica Economica, 91(7/8), 85–114. Lieblich, A., Tuval-Mashiach, R., & Zilber, T. (1998). Narrative research: Reading, analysis, and interpretation. Sage Publications. Lykkegaard, E., & Ulriksen, L. (2016). Choices and changes: Eccles’ Expectancy-Value model and upper-secondary school students’ longitudinal reflections about their choice of a STEM education. International Journal of Science Education, 38(5), 701–724. https://doi.org/10.1080/09500693.2016.1156782 Nolan, K., & Graham, S. (2015). Who Keeps the Gate? Pre-service Teachers’ Perceptions on Teaching and Learning Mathematics. In Proceedings of the Eight Conference Mathematics Education and Society, Vol. 3. Nolen, S., Ward, C., & Horn, I. (2012). Methods for taking a situative approach to studying the development of motivation, identity, and learning in multiple social contexts. European Journal of Psychology of Education, 27, 267–284. https://doi.org/10.1007/s10212-0110086-1 Noyes, A. (2009). Participation in mathematics: what is the problem? Improving Schools, 12(3), 277-288. https://doi.org/10.1177/1365480209342682 Panichella, N., & Triventi, M. (2014). Social inequalities in the choice of secondary school: Long-term trends during educational expansion and reforms in Italy. European Societies, 16(5), 666–693. https://doi.org/10.1080/14616696.2014.939685 Stinson, D. (2004). Mathematics as ‘gate-keeper (?)’: Three theoretical perspectives that aim toward empowering all children with a key to the gate. Mathematics Educator, 14(1), 818. Wagner, D. (2021). Gatekeeping in mathematics education. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 1–18). PME.

963 970 Emily Macmillan Jenni Ingram University of Oxford United Kingdom Children naturally use fingers or objects to count whole numbers, but there is no similarly intuitive way of representing negative numbers. Using scripting as a proxy for practice, we investigated the conceptions pre-service teachers drew on through the representations they used and the structures they highlighted when explaining how to calculate with negative numbers. We categorised the conceptions into quantity (debt, counterbalance, and relativity), movement (translation and rotation), position (order and symmetry), and symbolic (removed from context). The pre-service teachers used multiple conceptions and representations between and within explanations, rarely overtly drawing attention to mathematical structures. Alibali, M. W., & Norton, A. (2019). Synergizing research on constructing number: Themes and prospects. In A. Norton & M. W. Alibali (Eds), Constructing Number: Merging Perspectives from Psychology and Mathematics Education (pp. 341–353). Springer International Publishing. https://doi.org/10.1007/978-3-030-00491-0_16 Bofferding, L. (2019). Understanding negative numbers. In A. Norton & M. W. Alibali (Eds), Constructing number: Merging perspectives from psychology and mathematics education (pp. 251–277). Springer International Publishing. https://doi.org/10.1007/978-3-03000491-0_12 Campbell, M. P., & Baldinger, E. E. (2022). Using scripting tasks to reveal mathematics teacher candidates’ resources for responding to student errors. Journal of Mathematics Teacher Education, 25(5), 507–531. https://doi.org/10.1007/s10857-021-09505-4 Crespo, S. (2018). Generating, appraising, and revising representations of mathematics teaching with prospective teachers. In R. Zazkis & P. Herbst (Eds), Scripting approaches in mathematics education: Mathematical dialogues in research and practice (pp. 249– 264). Springer International Publishing. https://doi.org/10.1007/978-3-319-62692-5_12 Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. W. (2009). Teaching practice: A cross-professional perspective. Teachers College Record: The Voice of Scholarship in Education, 111(9), 2055–2100. https://doi.org/10.1177/016146810911100905 Harel, G. (2019). Commentary on negative numbers: Aspects of epistemology, cognition, and instruction. In A. Norton & M. W. Alibali (Eds), Constructing number: Merging perspectives from psychology and mathematics education (pp. 329–339). Springer International Publishing. https://doi.org/10.1007/978-3-030-00491-0_15 Jitendra, A. K., Nelson, G., Pulles, S. M., Kiss, A. J., & Houseworth, J. (2016). Is mathematical representation of problems an evidence-based strategy for students with mathematics difficulties? Exceptional Children, 83(1), 8–25. https://doi.org/10.1177/0014402915625062 Macmillan, E. (Forthcoming). Developing student teachers’ structural understanding of negative number arithmetic through the use of representations. Manuscript in preparation. Macmillan, E., & Ingram, J. (Forthcoming). What does a dash mean? Mathematics Teaching. Otte, M. (2006). Mathematical epistemology from a Peircean semiotic Point of View. Educational Studies in Mathematics, 61, 11–38. https://doi.org/10.1007/s10649-0060082-6 Sarma, L., Nagendra, P., & Ramasubramanian, K. (2024). The conception of negative numbers: A Study of Kṛṣṇa Daivajña’s Bījapallava. Indian Journal of History of Science, 59(3), 245–258. https://doi.org/10.1007/s43539-024-00136-1 Tsang, J. M., Blair, K. P., Bofferding, L., & Schwartz, D. L. (2015). Learning to “see” less than nothing: Putting perceptual skills to work for learning numerical structure. Cognition and Instruction, 33(2), 154–197. https://doi.org/10.1080/07370008.2015.1038539 Varma, S., Blair, K. P., & Schwartz, D. L. (2019). Cognitive science foundations of integer understanding and instruction. In A. Norton & M. W. Alibali (Eds), Constructing number: Merging perspectives from psychology and mathematics education (pp. 307–327). Springer International Publishing. https://doi.org/10.1007/978-3-030-00491-0_14 Venkat, H., Askew, M., Watson, A., & Mason, J. (2019). Architecture of mathematical structure. For the Learning of Mathematics, 39(1), 13–17. Wessman-Enzinger, N. M. (2019). Integers as directed quantities. In A. Norton & M. W. Alibali (Eds), Constructing number: Merging perspectives from psychology and mathematics education (pp. 279–305). Springer International Publishing. https://doi.org/10.1007/978-3-030-00491-0_13 Zazkis, R., & Herbst, P. (Eds). (2018). Scripting approaches in mathematics education: Mathematical dialogues in research and practice. Springer International Publishing.

971 978 Ami Maria Mamolo Rina Zazkis Ontario Tech University Canada Simon Fraser University Canada We present a case of an early-career mathematics teacher educator, Nadia, who brings real world contexts to her work with prospective teachers, focusing on issues of societal importance. Building on research that focused on teachers’ tensions, we describe tensions that the teacher educator experienced, navigating between her intention to support teachers’ mathematical knowledge and students’ attention being pulled away from mathematics by the context and pedagogy. We illustrate Nadia’s attempts to navigate tensions by amending her task design in order to support her mathematic-related learning goals. Barbosa, J.C. (2025). Bridge-building as a dimension of mathematics teacher educators’ expertise. In C. Cornejo, et al. (Eds.). Proc. of the 48th Conf. of the Int. Group for the Psychology of Mathematics Education: Research Reports, Vol. 1, pp. 83-90. PME. Bartell, T. G. (2013). Learning to teach mathematics for social justice: Negotiating social justice and mathematical goals. J. for Research in Mathematics Education, 44(1), 129-163. Barwell, R., Chapsam, L., Nkambule, T., & Phakeng, M. S. (2016). Tensions in teaching mathematics in contexts of language diversity. Mathematics education and language diversity: The 21st ICMI study, 175-192. Ben-Peretz, M., & Flores, M. A. (2018). Tensions and paradoxes in teaching: implications for teacher education. European Journal of Teacher Education, 41(2), 202–213. Berry, A. (2007). Reconceptualizing teacher educator knowledge as tensions: Exploring the tension between valuing and reconstructing experience. Studying Teacher Education, 3(2), 117–134. Beswick, K., Goos, M., & Chapman, O. (2014). Mathematics teacher educators’ knowledge. Working session presented at the 38th PME Int. Conf., Vancouver, Canada. Cuban, L. (1992). Managing dilemmas while building professional communities. Educational Researcher, 21(1), 4–11. Goodwin, A. L., Madalińska-Michalak, J., & Flores, M. A. (2023). Rethinking teacher education in/for challenging times: reconciling enduring tensions, imagining new possibilities. European Journal of Teacher Education, 46(5), 840–855. Herbst, P. G. (2003). Using Novel Tasks in Teaching Mathematics: Three Tensions Affecting the Work of the Teacher. American Educational Research Journal, 40(1), 197-238. LeSage, A. & Mamolo, A. (2023). Experiences of tension in teaching math for social justice. In (Eds.) K.M.Robinson, D.Kotsopoulos, & A.Dube, Mathematical teaching and learning: Interdisciplinary perspectives on mathematical minds in the middle school years. Springer Liljedahl, P., Andra, C., Di Martino, P., & Rouleau, A. (2015). Teacher Tension: Important Considerations for Understanding Teachers’ Actions, Intentions, and Professional Growth Needs. Proc. of the 39th Conf. of the Int. Group for the Psychology of Mathematics Education, Vol. 3, pp. 193-200. Hobart, Australia. Liljedahl, P., Andra, C., Rouleau, A., & Martino, P. (2023). Teacher tensions: Managed or resolved. International J. of Mathematical Education in Science and Technology, 54(8). Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task. Journal of Mathematics Teacher Education, 10, 239–249. Mamolo, A. (2018). Perceptions of social issues as contexts for secondary mathematics. Journal of Mathematical Behavior, 51, 28-40. Mamolo, A., Rodney, S., & Tepylo, D. (2022). Coding and climate change: Investigating prospective teachers' pathways of attention. The Journal of Mathematical Behavior, 68. Mamolo, A., & Pinto, L. (2015). Risks worth taking? Social risks and the mathematics teacher. The Mathematics Enthusiast, 12(1-3), 85–94. Mason, J. (2003). On the structure of attention in the learning of mathematics. In J. Novotná (Ed.), Proceedings, international symposium on elementary mathematics teaching, Charles University, Prague, pp. 9-16. Mason, J. (1988). Tensions. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 164– 169). London, UK: Hodder and Stoughton. Nipper, K., et al. (2011). Teacher tensions: Expectations in a professional development institute. Journal of Mathematics Teacher Education, 14, 375-392. Phillips, A., & Park Rogers, M. (2020). Examining the Tensions between Rapport with PreService Teachers and Authority in Becoming a Teacher Educator. Studying Teacher Education, 16(3), 265–285. Sparrow, L., & Frid, S. (2001). Dilemmas of beginning teachers of primary mathematics. In J. Bobis, B. Perry, & M. Mitchelmore (Eds.), Proc. of the 24th Annual Conf. of the Mathematics Education Research Group of Australasia (pp. 451–458). MERGA. Watson, A. & Ohtani, M. (2015). Task Design in Mathematics Education: An ICMI Study 22. New ICMI Study Series. Springer. Widder, M., Tabach, M., & Koichu, B. (2025). Tensions as springboards to actions in a partnership between mathematics teachers and mathematics education researchers. Journal of Mathematics Teacher Education, 28(1), 239-265. Zazkis, R., & Mamolo, A. (2016). Extending horizons: A story of a teacher educator. Proc. for the Int. Group for the Psychology of Mathematics Education- North American Chapter. Zazkis, R. & Marmur, O. (2021). Pedagogical tasks towards extending mathematical knowledge: Notes on the work of teacher educators. In M. Goos, & K. Beswick (Eds.), The learning and development of mathematics teacher educators: International perspectives and challenges (pp. 83-108). New York, NY: Springer.

979 986 Carlotta Vielmo Mirko Maracci University of Trento University of Pisa Italy This paper investigates how 10th-grade students conceptualise criteria for evaluating data-based arguments in decision-making contexts. Drawing on Habermas’ construct of rational behaviour, we analyse how evaluative criteria are articulated and negotiated in students’ talk in a whole-class discussion aimed at co-constructing validity and quality criteria. Findings show diverse students’ conceptualisations of prototypical criteria such as correctness, completeness, and clarity, and the emergence of terms as orientation, reasonableness, and significance, whose meanings cut across epistemic, teleological, and communicative dimensions. Bolstad, O. H. (2025). Mathematical literacy-What is missing?. In Bosch Casabò, M., Carreira, S., Bolondi, G., Gaidoschick, M., & Spagnolo, C. (Eds.), Proc. of the 14th Congress of the European Society for Research in Mathematics Education. ERME. Boscolo, A., Morselli, F., & Robutti, E. (2025). Implementation of formative assessment in argumentative activities for the development of algebraic thinking: indicators and where to find them. La Matematica e la sua Didattica 33(2), 75–113. Büscher, C. (2024). Adapting Habermas’ construct of communicative rationality into a framework for analyzing students’ statistical literacy. Educational Studies in Mathematics, 117(1), 121–141. Cusi, A., Morselli, F., & Sabena, C. (2017). Promoting formative assessment in a connected classroom environment: design and implementation of digital resources. ZDM Mathematics Education, 49(5), 755–767. Gómez-Blancarte, A., & Tobías-Lara, M. G. (2018). Using the Toulmin model of argumentation to validate students’ inferential reasoning. In M. A. Sorto, A. White, & L. Guyot (Eds.), Proc. of the 10th International Conference on Teaching Statistics. ISI. Grion, V., Serbati, A., Tino, C, & Nicol, D. (2017). Ripensare la teoria della valutazione e dell’apprendimento all’università: un modello per implementare pratiche di peer review. Giornale Italiano della Ricerca Educativa, 10(19), 209–226. Habermas (2003). Truth and Justification. MIT Press. Krummenauer, J., Gutensohn, F., Aichele, J., Emhart, M., & Kuntze, S. (2023). Young primary students’ difficulties related to data-based argumentation–a qualitative analysis. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), Proc. of the 13th Congress of the European Society for Research in Mathematics Education. ERME. Vielmo, C. (2025). Investigating students’ co-construction of statistical argumentation. In Bosch Casabò, M., Carreira, S., Bolondi, G., Gaidoschick, M., & Spagnolo, C. (Eds.), Proc. of the 14th Congress of the European Society for Research in Mathematics Education. ERME. Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223–265.

987 994 Nadav Marco Alik Palatnik The Hebrew University of Jerusalem Israel David Yellin Academic College of Education Israel This design-based research (DBR) tests the self–friend–skeptic framework as embodied moves for the teacher to shift spatial-geometry reasoning from measurement to proof. In five teaching experiments (TE) with 16 non-STEM education undergraduates, an instructor-guided body-sized-cube activity was iteratively refined. Naming a ‘skeptic’ often failed, whereas justifying to an absent friend without a cube reliably prompted general, non-demonstrative warrants. Progressively constraining affordances recruited an inner skeptic and semiotic resources. We propose a classroom-ready principle: escalate audiences-of-conviction framing while withholding manipulatives. Abrahamson, D., Dutton, E., & Bakker, A. (2022). Towards an enactivist mathematics pedagogy. In S. A. Stolz (Ed.), The body, embodiment, and education: An interdisciplinary approach (pp. 156–182). Routledge. Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge. https://doi.org/10.4324/9780203701010 Bell, P., Hoadley, C. M., & Linn, M. C. (2004). Design-based research in education. In M. C. Linn, E. A. Davis, & P. Bell (Eds.), Internet environments for science education (pp. 73– 85). Lawrence Erlbaum. https://doi.org/10.4324/9781410610393 Boonstra, K., Kool, M., Shvarts, A., & Drijvers, P. (2025). Primary school teachers’ practices for fostering embodied abstraction in geometry. Journal of Mathematics Teacher Education, 1–25 [online first]. https://doi.org/10.1007/s10857-025-09702-5 de Freitas, E., & Sinclair, N. (2013). New materialist ontologies in mathematics education: The body in/of mathematics. Educational Studies in Mathematics, 83(3), 453–470. Mason, J., Burton, L., & Stacey, K. (2010). Thinking Mathematically (2nd ed.). Pearson Education. Marco, N. & Palatnik, A. (2025, July). Embodied argumentation: Convincing yourself, your friend, or the skeptic?. In C. Cornejo et al. (Eds.), Proc. of the 48 Conf. of the IGPME (Vol. 2, pp. 67–74). PME. th Marco, N., & Shvarts, A. (2025). Proving in geometry as an embodied process: The case of proofs without words. Journal for Research in Mathematics Education, 56(4), 184–211. Mayring, P. (2021). Qualitative content analysis: A step-by-step guide. Sage. Palatnik, A., Abrahamson, D., Baccaglini-Frank, A., Ng, O. L., Shvarts, A., & Swidan, O. (2023). Theory and practice of designing embodied mathematics learning. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proceedings of the 46th Con. of the IGPME (Vol. 1, pp. 159–189). PME. Palatnik, A., & Marco, N. (2026). Physical interaction with manipulatives and mathematical reasoning: the case of spatial geometry. [manuscript under review] Stylianides, G., Stylianides, A., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai, (Ed.), Compendium for research in mathematics education (pp. 237–266). National Council of Teachers of Mathematics. Way, J., & Ginns, P. (2024). Embodied learning in early mathematics education: Translating research into principles to inform teaching. Education Sciences, 14(7), 696.

995 1002 Amedeo Matteucci Ilaria Veronesi Francesco Saverio Tortoriello University of Salerno Italy In this theoretical study, starting from the observation that AI systems can successfully solve mathematical problems without understanding them, we argue that this contrast forces a re-examination of the specificity of human mathematical sense making. We review influential accounts of embodied cognition, starting with Lakoff and Núñez’s proposal, and we contrast this perspective with Radford’s Theory of Objectification. Building on the latter, we focus on mathematical debate as a paradigmatic form of collective embodiment. We articulate four intertwined dimensions—objectification, subjectification, cultural-historical reenactment, and dialectical tension—showing how sense making becomes ethically and politically charged. Bakhurst, D. (2023). The heart of the matter: Ilyenkov, Vygotsky and the courage of thought. Brill. Bartolini Bussi, M. G. (1996). Mathematical discussion and perspective drawing in primary school: To Giovanni Prodi on occasion of his 70th birthday. Educational Studies in Mathematics, 31(1), 11–41. Castelvecchi, D. (2025). DeepMind AI crushes tough maths problems on par with top human solvers. Nature, 638(8051), 589. Floridi, L., Jia, Y., & Tohmé, F. (2025). A categorical analysis of large language models and why LLMs circumvent the symbol grounding problem. arXiv Preprint arXiv:2512.09117. Freeley, A. J., & Steinberg, D. L. (2013). Argumentation and debate. Cengage Learning. Gahrn-Andersen, R. (2025). Human languaging and large language models. AI & Society, 1–11. Kuhn, T. S. (2012). The structure of scientific revolutions. University of Chicago Press. Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Levant, A., & Oittinen, V. (2013). Dialectics of the ideal: Evald Ilyenkov and creative Soviet Marxism. Brill. Levant, A., Bakhurst, D., Blunden, A., Brunila, M., Camillo, J., Chaiklin, S., Cole, M., De Smet, B., Engeström, Y., Hasse, C., Rodrigues, A. M., Mattos, C., Nardi, B., Oittinen, V., Rantavuori, J., Sannino, A., Stetsenko, A., & Yamazumi, K. (2024). Activity theory: An introduction. ibidem-Verlag. Marx, K. (1987). Das Kapital. Kritik der politischen Ökonomie. Erster Band, Hamburg 1872 (6). Walter de Gruyter GmbH. (Original work published 1872) Matteucci, A., & Tortoriello, F. S. (2023). Mathematics and debate: a dialectical approach. Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Article 18. Matteucci, A., Veronesi, I., & Tortoriello, F. S. (2024a). A new agora: The role of AI-shaped debate in transforming mathematics education. In T. Evans, O. Marmur, J. Hunter, G. Leach, & J. Jhagroo (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, p. 186). PME. Matteucci, A., Tortoriello, F. S., & Veronesi, I. (2024b). Dialectical chatbots: AI-driven debate in mathematical education. In Faggiano E., Clark-Wilson A., Tabach M., Weigand H.-G. (Eds.), Proceedings of the 17th ERME Topic Conference Mathematics Education in the Digital Age – MEDA4. University of Bari Aldo Moro. Matteucci, A., Tortoriello, F. S., & Veronesi, I. (2025a). Debating the machine: Mediations and truth. Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14). Matteucci, A., Veronesi, I., & Tortoriello, F. S. (2025b). I beg to differ: The germ cell of mathematical debate. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proc. 48th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, p. 75). PME. Radford, L. (2021a). The theory of objectification: A Vygotskian perspective on knowing and becoming in mathematics teaching and learning. Brill. Radford, L. (2021b). Davydov’s concept of the concept and its dialectical materialist background. Educational Studies in Mathematics, 106(3), 327–342. Sambrotta, M. (2025). LLMs and the logical space of reasons. Minds and Machines, 35(4), 1– 23. Sriraman, B., & Wu, K. (2020). Embodied cognition. In S. Lerman (Ed.), Encyclopedia of mathematics education, (p. 266–268).

1003 1010 Martin Mayerhofer Michael Eichmair Marko Lüftenegger University of Vienna Austria Learning strategies are central for self-regulated learning and linked to achievement and motivation in learning mathematics. Since learners typically use multiple strategies simultaneously, this study used a person-centered approach, identifying four distinct combinations of learning strategies used by incoming STEM students during their first semester. The resulting profiles ranged from a maladaptive profile to highly adaptive profiles. Although exam performance did not differ significantly between profiles, the descriptive differences suggested lower achievement in the least favorable profile. Self-efficacy was higher in the adaptive profiles. These findings offer first evidence that learning-strategy profiles may help explain variations in motivation and performance and may act as a starting point for further person-centered research. Bolck, A., Croon, M., & Hagenaars, J. (2004). Estimating latent structure models with categorical variables: One-step versus three-step estimators. Political Analysis, 12(1), 3– 27. https://doi.org/10.1093/pan/mph001 Broadbent, J., & Poon, W. L. (2015). Self-regulated learning strategies & academic achievement in online higher education learning environments: A systematic review. The Internet and Higher Education, 27, 1–13. https://doi.org/10.1016/j.iheduc.2015.04.007 Heublein, U., Ebert, J., Hutzsch, C., Isleib, S., König, R., Richter, J., & Woisch, A. (2017). Zwischen Studienerwartungen und Studienwirklichkeit. Ursachen des Studienabbruchs, beruflicher Verbleib der Studienabbrecherinnen und Studienabbrecher und Entwicklung der Studienabbruchquote. Forum Hochschule. Liebendörfer, M., Göller, R., Biehler, R., Hochmuth, R., Kortemeyer, J., Ostsieker, L., Rode, J. & Schaper, N. (2021). LimSt – Ein Fragebogen zur Erhebung von Lernstrategien im mathematikhaltigen Studium. Journal für Mathematik-Didaktik, 42, 25–59. Liu, W. C., Wang, C. K. J., Kee, Y. H., Koh, C., Lim, B. S. C., & Chua, L. (2014). College students’ motivation and learning strategies profiles and academic achievement: a selfdetermination theory approach. Educational Psychology (Dorchester-on-Thames), 34(3), 338–353. https://doi.org/10.1080/01443410.2013.785067 Ning, H. K., & Downing, K. (2015). A latent profile analysis of university students’ selfregulated learning strategies. Studies in Higher Education, 40(7), 1328–1346. Organisation for Economic Co-operation and Development (2010). Mathematics Teaching Learning Strategies in PISA. https://www.oecd.org/content/dam/oecd/en/publications/reports/2010/09/mathematicsteaching-and-learning-strategies-in-pisa_g1ghbd7b/9789264039520-en.pdf Rach, S., & Heinze, A. (2011). Studying Mathematics at the University: The Influence of Learning Strategies. In Ubuz, B. (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (pp. 9–16). Middle East Technical University. Rach, S., & Heinze, A. (2017). The Transition from School to University in Mathematics: Which Influence Do School-Related Variables Have? International Journal of Science and Mathematics Education, 15(7), 1343–1363. https://doi.org/10.1007/s10763-016-9744-8 Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys. John Wiley & Sons. https://doi.org/10.1002/9780470316696 Vettori, G., Vezzani, C., Bigozzi, L., & Pinto, G. (2020). Upper secondary school students’ conceptions of learning, learning strategies, and academic achievement. The Journal of Educational Research, 113(6), 475–485. https://doi.org/10.1080/00220671.2020.1861583 Wild, K. P. & Schiefele, U. (1994). Lernstrategien im Studium: Ergebnisse zur Faktorenstruktur und Reliabilität eines neuen Fragebogens. Zeitschrift für Differentielle und Diagnostische Psychologie, 15, 185–200. Yanagida, T. (2024). Querschnittliche Latent Class Modelle in Mplus. FDZFrühjahrsakademie 2024, 29 February 2024. Zimmerman, B. J. (2000). Attaining Self-Regulation: A Social Cognitive Perspective. In Handbook of Self-Regulation (pp. 13–39). Elsevier Inc. https://doi.org/10.1016/B978012109890-2/50031-7 Zimmerman, B. J. & Schunk, D. H. (2008). Motivation. An Essential Dimension of SelfRegulated Learning. In D. H. Schunk, & B. J. Zimmerman (Eds.), Motivation and SelfRegulated Learning: Theory, Research, and Applications (1st ed., pp. 1–30). Routledge. https://doi.org/10.4324/9780203831076

1011 1018 Jake McMullen Jon Star University of Turku Finland Many students face difficulties in acquiring the skills needed for such adaptive expertise with rational numbers, a key aim of education systems worldwide. This study investigates the possibility that cross-notation knowledge, understanding the relationships between different notations of rational numbers, should be considered a feature of adaptive expertise with rational numbers. 447 high school students completed measures of adaptive expertise, including the Arithmetic Sentence Production Task. Results indicate that students who employed cross-notation solutions on this task performed well on other measures of adaptive expertise. These findings underscore the importance of cross-notation knowledge for adaptive expertise with rational numbers. Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills: Constructive Adaptive Expertise (pp. 1– 33). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410607218 Braithwaite, D. W., McMullen, J., & Hurst, M. A. (2022). Cross-notation knowledge of fractions and decimals. Journal of Experimental Child Psychology, 213. Coppersmith Garcia, J. , & Star, J. R. (2022). A Complicated Relationship: Examining the Relationship Between Flexible Strategy Use and Accuracy. Journal of Numerical Cognition, 8(3), 382–397. https://doi.org/10.5964/jnc.7601 D’Erchie, M., Forsmann, C., Schneider, M., & Obersteiner, A. (2025). Predictive relationships between conceptual and procedural knowledge across the domains of fractions and algebra. In Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 179–186). PME. Mazzocco, M. M. M., & Hanich, L. B. (2010). Math achievement, numerical processing, and executive functions in girls with Turner syndrome: Do girls with Turner syndrome have math learning disability? Learning and Individual Differences, 20(2), 70–81. McMullen, J., Hannula-Sormunen, M. M., Lehtinen, E., & Siegler, R. S. (2020). Distinguishing adaptive from routine expertise with rational number arithmetic. Learning and Instruction, 68. doi:10.1016/j.learninstruc.2020.101347 Schiller, L. K., Abreu-Mendoza, R. A., Siegler, R. S., Rosenberg-Lee, M., & Thompson, C. A. (2024). Building integrated number sense in adults and children: Comparing fractionsonly training with cross-notation number line training. Journal of Experimental Child Psychology, 246, 106017. https://doi.org/10.1016/j.jecp.2024.106017 Schiller, L. K., & Siegler, R. S. (2023). Integrated knowledge of rational number notations predicts children’s math achievement and understanding of numerical magnitudes. Cognitive Development, 68, 101380. https://doi.org/10.1016/j.cogdev.2023.101380 Schwartz, D. L., Bransford, J. D., & Sears, D. (2005). Efficiency in Innovation and Transfer. In J. Mestre (Ed.), Transfer of Learning from a Modern Multidisciplinary Perspective. Information Age Publishing. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691–697. doi:10.1177/0956797612440101 Van Dooren, W. (2023). Strategy use in number line estimations of fractions—An explorative study in search for adaptive expertise. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proc. of the 46th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 299–306). PME

1019 1026 Farouq Sessah Mensah Department of Teaching and Learning, Stockholm University Sweden This study examines how meta-didactic praxeologies are structured in mathematics teacher education to prepare pre-service teachers for teaching with instructional technology. Using the Anthropological Theory of the Didactic, a qualitative case study was conducted with two mathematics teacher educators at a Ghanaian university, drawing on course plans and interviews. The analysis focused on the types of tasks, techniques, technologies, and theories at the meta-didactic level. The findings reveal contrasting praxeologies oriented toward design-based conceptual exploration and remedial–diagnostic stabilisation. Representation and decomposition dominated as the meta-didactic technique, while approximation was limited, and justificatory discourse remained largely tacit. Arzarello, F., Robutti, O., Sabena, C., Cusi, A., Garuti, R., Malara, N., & Martignone, F. (2014). Meta-didactical transposition: A theoretical model for teacher education programmes. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital era: An international perspective on technology focused professional development (pp. 347–372). Springer Netherlands. https://doi.org/10.1007/978-94-0074638-1_15 Caniglia, J., & Meadows, M. (2018). Pre-service mathematics teachers’ use of web resources. International Journal for Technology in Mathematics Education, 25(3), 17–34. https://doi.org/10.1564/tme_v25.3.02 Chevallard. (1992). Fundamental concepts in didactics: Perspectives provided by an anthropological approach. In Research in didactique of mathematics: Selected papers (pp. 131–168). La pense´e sauvage. Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In Proceedings of the IV Congress of the European Society for Research in Mathematics Education, 21–30. Drijvers, P. (2015). Digital technology in mathematics education: Why it works (or doesn’t). In S. J. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 135–151). Springer International Publishing. https://doi.org/10.1007/978-3-319-17187-6_8 Ertmer, P. A., & Ottenbreit-Leftwich, A. T. (2010). Teacher technology change. Journal of Research on Technology in Education, 42(3), 255–284. https://doi.org/10.1080/15391523.2010.10782551 Eshetu, D., Atnafu, M., & Woldemichael, M. (2022). The effectiveness of guided inquirybased technology integration on pre-service mathematics teachers understanding of plane geometry. Journal of Pedagogical Research, 6(4), 84-100. https://doi.org/10.33902/JPR.202215241 Greene, M. D., Cheng, S.-L., & Jones, W. M. (2023). The impact of an online technology course on pre-service teachers’ technological knowledge: Strategies and design. International Journal of Professional Development, Learners and Learning, 5(2), 1–9. https://doi.org/10.30935/ijpdll/13772 Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. W. (2009). Teaching practice: A cross-professional perspective. Teachers College Record: The Voice of Scholarship in Education, 111(9), 2055–2100. https://doi.org/10.1177/016146810911100905 Hoyles, C., & Lagrange, J.-B. (Eds.). (2010). Mathematics education and technologyrethinking the terrain: The 17th ICMI Study (Vol. 13). Springer US. https://doi.org/10.1007/978-1-4419-0146-0 Kavanagh, S. S., Conrad, J., & Dagogo-Jack, S. (2020). From rote to reasoned: Examining the role of pedagogical reasoning in practice-based teacher education. Teaching and Teacher Education, 89, 1–11. https://doi.org/10.1016/j.tate.2019.102991 Lampert, M., Franke, M. L., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., Cunard, A., & Crowe, K. (2013). Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching. Journal of Teacher Education, 64(3), 226–243. https://doi.org/10.1177/0022487112473837 Mensah, F. S., Pansell, A., & Christiansen, I. M. (2024). The teaching of IT implementation in mathematics teacher education research: A critical analysis from a praxeology-informed perspective. CITE Journal, 24(4). 226–243. https://citejournal.org/volume-24/issue-424/mathematics/the-teaching-of-instructional-technology-implementation-inmathematics-teacher-education-research-a-critical-analysis-from-a-praxeology-informedperspective Nyaaba, M., Akanzire, B. N., & Nabang, M. (2024). Virtual reality in teacher education: Insights from pre-service teachers in resource-limited regions (No. arXiv:2411.10225). ArXiv, 1–10. https://doi.org/10.48550/arXiv.2411.10225 Quarder, J., Greefrath, G., Gerber, S., & Siller, H.-S. (2025). Pedagogical content knowledge for simulations and mathematical modelling with digital tools: A quasi-experimental study with pre-service mathematics teachers. ZDM–Mathematics Education, 57, 395–409. https://doi.org/10.1007/s11858-025-01673-4 Trouche, L., Gitirana, V., Miyakawa, T., Pepin, B., & Wang, C. (2019). Studying mathematics teachers interactions with curriculum materials through different lenses: Towards a deeper understanding of the processes at stake. International Journal of Educational Research, 93, 53–67. https://doi.org/10.1016/j.ijer.2018.09.002 Wilson, K., & Yonas, A. M. (2024). In search of deliberate practice: Simulating teaching in three teacher education programs. Teachers College Record, 126(9), 47–89. https://doi.org/10.1177/01614681241298842 Yildiz, E., & Arpaci, I. (2024). Understanding pre-service mathematics teachers’ intentions to use GeoGebra: The role of technological pedagogical content knowledge. Education and Information Technologies, 29(14), 18817–18838. https://doi.org/10.1007/s10639024-12614-1

1027 1034 Annamaria Miranda University of Salerno, Department of Mathematics Italy This study examines how undergraduate mathematics students developed advanced mathematical thinking and early teacher identity through embodied exploration of topological deformations. Students progressed through pre-rigorous, rigorous, and post-rigorous phases while designing teaching resources. Analysis through noticing and agency frameworks reveals how pedagogical awareness emerges when abstract mathematics is approached through embodied, multi-modal pathways. Findings show that mathematical noticing becomes pedagogical noticing when students exercise design agency, offering new perspectives on addressing Klein's second discontinuity through epistemological rather than content-based connections. Andersson, A., & Norén, E. (2011). Agency in mathematics education. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of CERME 7 (pp. 1389-1398). Rzeszów. Coles, A., & Helliwell, T. (2025). Re-conceptualising mathematics teacher noticing through critical events from an enactivist perspective. Journal of Mathematics Teacher Education. Holland, D., Lachicotte, W., Skinner, D., & Cain, C. (2001). Identity and agency in cultural worlds. Harvard University Press. Klein, F. (2016). Elementary mathematics from a higher standpoint. Springer. Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge. Mason, J. (2025). Critical noticing and its consequences. JMTE, 28, 1267-1285. Miranda A. (2023). Exploring the role of 3D printing technology in supporting undergraduate students’ topological conceptual knowledge. In D. Diamantidis, M. Karavakou, M. Grizioti, & C. Kynigos (Eds.), Proceedings of the 16th ICTMT (pp. 183-190). NKUA. Palatnik, A., Abrahamson, D., Baccaglini-Frank, A., Ng, O., Shvarts, A., & Swidan, O. (2023). Theory and practice of designing embodied mathematics learning. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel., & M. Tabach (Eds.). PME 46. (Vol. 1, pp. 159–189). Palatnik, A., & Cooper, J. (2025). Extending embodied learning to mathematics teacher professional development through multimodal learning analytics. In: Cornejo, C., Felmer, P., Gómez, D. M., Dartnell, P., Araya, P., Peri, A., & Randolph, V. (Eds.) (2025). Proceedings of the 48th Conference of the IGPME: Working Group. PME 48. Pickering, A. (1995). The mangle of practice: Time, agency, and science. Chicago Press. Sherin, M., Jacobs, V., & Philipp, R. (Eds.). (2011). Mathematics teacher noticing: Seeing through teachers’ eyes. Routledge. Tall, D. (1991). Advanced mathematical thinking. Kluwer. Tao, T. (2007). Structure and randomness: Pages from a mathematical blog. AMS. Varela, F., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. MIT Press. Wasserman, N., Buchbinder, O., & Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. ZDM Mathematics Education, 55, 719-736.

1035 1042 Jonathan Morcos Alejandro S. González-Martín Université de Montréal Canada This paper explores prospective secondary mathematics teachers’ personal relationship with combinatorics (PR_C). Existing research on the teaching and learning of combinatorics calls for an expanded role of combinatorics in secondary mathematics curricula; however, there is limited understanding of how teachers are trained in this domain and their views of combinatorics content. This study’s findings reveal a diverse range of PR_C, some of which may prevent prospective teachers from exploiting the recognised educational potential of combinatorics. Participants demonstrated difficulties applying techniques appropriate for secondary classrooms and justifying techniques learned at university. Results point to several promising avenues for strengthening mathematics teacher training related to combinatorics. Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181–199. https://doi.org/10.1023/A:1002954428327 Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266. https://revuerdm.com/1999/l-analyse-des-pratiques/ Chevallard, Y. (2003). Approche anthropologique du rapport au savoir et didactique des mathématiques. In S. Maury & M. Caillot (Eds.), Rapport au savoir et didactiques (pp. 81–104). Éditions Fabert. González-Martín, A. S., & Hernandes-Gomes, G. (2020). Mathematics in engineering programs: what teachers with different academic and professional backgrounds bring to the table. An institutional analysis. Research in Mathematics Education, 22(1), 67–86. https://doi.org/10.1080/14794802.2019.1663255 Lockwood, E., & Reed, Z. (2020). Defining and demonstrating an equivalence way of thinking in enumerative combinatorics. The Journal of Mathematical Behavior, 58, 100780. https://doi.org/10.1016/j.jmathb.2020.100780 Lockwood, E., Wasserman, N. H., & Tillema, E. S. (2020). A case for combinatorics: a research commentary. The Journal of Mathematical Behavior, 59, 100783. https://doi.org/10.1016/j.jmathb.2020.100783 Morcos, J. (2025). La formation mathématique des enseignants au secondaire. Une analyse praxéologique des contenus liés à la combinatoire au secondaire et dans la formation mathématique des futurs enseignants : liens et ruptures [masters’ thesis, Université de Montréal]. Papyrus. https://hdl.handle.net/1866/41831 Morcos, J., & González-Martín, A. S. (2025a). Examining the impact on teacher training of inconsistencies with the multiplication principle between university and secondary combinatorics. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proc. 48th Conf. of the Int. Group for the Psychology of Mathematics Education: Research Reports (Vol. 2, pp. 123–130). PME. Morcos, J., & González-Martín, A. S. (2025b). Formation mathématique universitaire des futurs enseignants au secondaire: discontinuités dans les techniques de résolution relatives à la combinatoire. In F. Venant & S. Dumont (Eds.), Actes du colloque de l’Espace Mathématique Francophone 2025 (pp. 585–595). UQÀM and EMF. Morcos, J., & González-Martín, A. S. (2025c). Permutations with indistinguishable objects: inconsistencies in the university mathematics training of prospective secondary teachers. In M. Bosch, S. Carreira, G. Bolondi, M. Gaidoschick, & C. Spagnolo (Eds.), Proc. 14th Congress European Society for Research in Math. Education (CERME14) (pp. 2301– 2308). Free Univ. of Bozen-Bolzano and ERME. https://hal.science/hal-05238725v1 Saldaña, J. (2013). The coding manual for qualitative researchers (2nd ed.). Sage. Sandefur, J., Lockwood, E., Hart, E., & Greefrath, G. (2022). Teaching and learning discrete mathematics. ZDM Mathematics Education, 54(4), 753–775. https://doi.org/qmr7

1043 1050 Kaja Mädamürk Jüri Kurvits School of Natural Sciences and Health, Tallinn University Estonia School of Digital Technologies, Tallinn University Estonia Perceived teachers’ emotional support can have a significant role in students' learning motivation and achievement. The aim of the current study is to better understand the interplay between mathematics teachers’ emotional support and students’ motivation and how both emotional support and motivation relate to achievement. The sample includes 4192 Estonian students (15-year-olds, 50% female) who filled in a questionnaire regarding perceived emotional support and mathematics motivation, and a few weeks later took the national mathematics exam. The results indicated that perceived emotional support did not predict mathematics exam results directly; however, it did predict the exam results indirectly through self-efficacy, task persistence, fixed mindset, and anxiety. Arens, A. K., Frenzel, A. C., & Goetz, T. (2020). Self-concept and self-efficacy in math: Longitudinal interrelations and reciprocal linkages with achievement. Journal of Experimental Education, 90(3), 615–633. https://doi.org/10.1080/00220973.2020.1786347 Ashcraft, M. H. (2002). Math anxiety: personal, educational, and cognitive consequences. Current Directions in Psychological Science, 11(5), 181–185. https://doi.org/10.1111/1467-8721.00196 Barroso, C., Ganley, C. M., McGraw, A. L., Geer, E. A., Hart, S. A., & Daucourt, M. C. (2020). A meta-analysis of the relation between math anxiety and math achievement. Psychological Bulletin, 147(2), 134–168. https://doi.org/10.1037/bul0000307 Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S. (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. Child Development, 78(1), 246–263. https://doi.org/10.1111/j.1467-8624.2007.00995.x Bong, M., & Skaalvik, E. M. (2003). Academic self-concept and self-efficacy: How different are they really? Educational Psychology Review, 15(1), 1–40. https://doi.org/10.1023/a:1021302408382 Dweck, C. S. (2006). Mindset: The new psychology of success. New York: Random House. Eccles, J. S., & Wigfield, A. (2020). From expectancy-value theory to situated expectancyvalue theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Kikas, E., Mädamürk, K., & Palu, A. (2019). What role do comprehension‐oriented learning strategies have in solving math calculation and word problems at the end of middle school? British Journal of Educational Psychology, 90(S1), Kikas, E., & Mägi, K. (2016). Does self-efficacy mediate the effect of primary school teachers’ emotional support on learning behavior and academic skills? Journal of Early Adolescence. https://doi.org/10.1111/bjep.12308 Muthén, L. K., & Muthén, B. O. (1998–2017). Mplus user’s guide. (8th ed.). Muthén & Muthén. Mädamürk, K., Kikas, E., & Palu, A. (2018). Calculation and word problem-solving skill profiles: relationship to previous skills and interest. Educational Psychology, 38(10), 1239–1254. https://doi.org/10.1080/01443410.2018.1495830 Roorda, D. L., Jak, S., Zee, M., Oort, F. J., & Koomen, H. M. Y. (2017). Affective Teacher– Student Relationships and Students’ Engagement and Achievement: A Meta-Analytic Update and Test of the Mediating Role of engagement. School Psychology Review, 46(3), 239–261. https://doi.org/10.17105/SPR-2017-0035.V46-3 Roorda, D. L., Koomen, H. M. Y., Spilt, J. L., & Oort, F. J. (2011). The influence of affective Teacher–Student relationships on students’ school engagement and achievement. Review of Educational Research, 81(4), 493–529. https://doi.org/10.3102/0034654311421793https://doi.org/10.17105/spr-20170035.v46-3 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a selfdetermination theory perspective: Definitions, theory, practices, and future directions. Contemporary Educational Psychology, 61, 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Sakiz, G., Pape, S. J., & Hoy, A. W. (2012). Does perceived teacher affective support matter for middle school students in mathematics classrooms? Journal of School Psychology, 50(2), 235–255. https://doi.org/10.1016/j.jsp.2011.10.005 Yang, Y., Li, G., Su, Z., & Yuan, Y. (2021). Teacher’s Emotional Support and math performance: The chain mediating effect of Academic Self-Efficacy and math Behavioral engagement. Frontiers in Psychology, 12, 651608. https://doi.org/10.3389/fpsyg.2021.651608 Yeager, D. S., & Dweck, C. S. (2012). Mindsets that promote resilience: When students believe that personal characteristics can be developed. Educational Psychologist, 47(4), 302–314. https://doi.org/10.1080/00461520.2012.722805 Zhang, X., Nurmi, J., Kiuru, N., Lerkkanen, M., & Aunola, K. (2011). A teacher-report measure of children’s task-avoidant behavior: A validation study of the Behavioral Strategy Rating Scale. Learning and Individual Differences, 21(6), 690–698. https://doi.org/10.1016/j.lindif.2011.09.007

1051 1058 Talli Nachlieli Michal Tabach Levinsky-Wingate Academic College Tel Aviv University Israel This study examines practices that promote explorative participation within a secondary mathematics M.Ed. course, focusing on the Thinking Classrooms approach. It investigates how mathematics student-teachers applied the principle of thin slicing in lesson planning. Five lesson plans were analysed through two lenses: variation theory and the type of participation the tasks afford. The findings show that all task sequences aligned with principles of variation theory and exhibited strong bondedness. Substantiation was necessary but implicit, and flexibility was limited. Across all lessons, ritual-enabling tasks preceded exploration-requiring tasks, serving as a foundation for exploration. The study offers a framework for characterizing thin slicing and highlights instructional practices that promote explorative participation. Brown, K. L. (2003). From teacher-centered to learner-centered curriculum: Improving learning in diverse classrooms. Education, 124(1), 49–55. Frazee, L. M., Cardetti, F., Cavanna, J. M., & Staples, M. (2023, October). Opportunities For Learning: Thin Slicing Content With Variation. In Conference Proceedings PME NA (Vol. 1). Proceedings of the forty-fifth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1). Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. Educational Studies in mathematics, 101(2), 153-176. Liljedahl, P. (2021). Building thinking classrooms in mathematics, grades K-12: 14 teaching practices for enhancing learning. Corwin press. Nachlieli, T., & Heyd-Metzuyanim, E. (2022). Commognitive conflicts as a learning mechanism towards explorative pedagogical discourse. Journal of Mathematics Teacher Education, 25(3), 347-369. Nachlieli, T., & Tabach, M. (2019). Ritual-enabling opportunities-to-learn in mathematics classrooms. Educational Studies in Mathematics, 101(2), 253-271. Vygotsky, L. (1962). Thought and language (E. Hanf-mann & G. Vakar, Trans.). Cambridge, MA: MIT Press.

1059 1066 Nagisa Nakawa Yusuke Uegatani Yuichiro Hattori Nur Robiah Nofikusumawati Peni Kanto Gakuin University Japan Hiroshima University High School, Fukuyama Japan Okayama University Japan Ahmad Dahlan University This paper contributes to the ethical turn in mathematics education by illustrating the mutual development of mathematics and ethics in early childhood. Using a design research approach, we implemented an authentic mathematically involved activity in which six young children distributed ten oranges for actual consumption. The findings show that children’s ethics were enacted through embodied mathematical trial and error and were inseparable from mathematical activity. The study also indicates that questionable applications of mathematics already emerge in early childhood. Developing generalisable hypotheses remains a task for future research. Bakker, A. (2018). Design Research in Education: A Practical Guide for Early Career Researchers. Routledge. Boylan, M. (2016). Ethical dimensions of mathematics education. Educational Studies in Mathematics, 92(3), 395–409. https://doi.org/10.1007/s10649-015-9678-z Dahnke, M. D. (2001). Ethics as first philosophy. Auslegung: A J. of Philosophy, 24(2), Art.2. Devlin, K. (1994). Mathematics: The science of patterns. Scientific American Library. Ernest P. (2019). Privilege, power and performativity: The ethics of mathematics in society and education. Philosophy of Mathematics Education Journal, 35. Article 6. Ernest, P. (2024). The ‘Ethical Turn’ Reaches Mathematics Education. In P. Ernest (Ed.), Ethics and Mathematics Education: The Good, the Bad and the Ugly (pp. 3–10). Springer. Foster, C. (2023). Methodological pragmatism in educational research: From qualitativequantitative to exploratory-confirmatory distinctions. International Journal of Research & Method in Education, 47(1), 1–16. https://doi.org/10.1080/1743727X.2023.2210063 Hauge, K.H., Kacerja, S., Lilland, I.E., Steffensen, L. (2024). Layers of ethical awareness: Mathematics teachers’ concerns on including socio-political issues. In: P. Ernest. (Ed.) Ethics and Mathematics Education: The Good, the Bad and the Ugly (pp. 415–439). Springer. Hedefalk, M., & Sumpter, L. (2025). Young children’s ethical reasoning about sharing: An analytical tool. For the Learning of Mathematics, 45(1), 8–13. Nakawa, N. (2024). Analysing young Japanese children’s social values, mathematical values and mathematical expressions. In Y. Dede, G. Marschall, & P. Clarkson (Eds.), Values and Valuing in Mathematics Education. Springer. Skovsmose, O. (2021). Mathematics and crises. Ed. Studies in Mathematics, 108, 369–383. Davis, J. D. (2025). An exploration of equity in international PME proceedings from 1978 through 2024. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.). Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education: Research Reports, Vol. 1, pp. 187–194. PME.

1067 1074 Melanie Neck Timo Leuders Frank Reinhold University of Education Freiburg Germany The Four-Component Instructional Design Model (4C/ID) is a well-established and evidence-based framework for the design of professional development programs. However, there is still limited research on how learners actually use the instructional components offered by 4C/ID-based learning environments. To address this gap, we implemented a digital learning environment for mathematics teachers based on the 4C/ID model. A total of 303 pre-service and in-service mathematics teachers completed the intervention focusing on diagnosing students’ mathematical problem-solving abilities and supporting them appropriately. We analysed learners’ interactions with supportive information, their engagement with feedback, and their use of part-task practice. The findings suggest that many learners did not fully exploit the opportunities for self-regulated learning provided by the learning environment. Drijvers, P. (2022). Digital Technology in Mathematics Education: Past Performance and Future Pathways. In G. Pinkernell, F. Reinhold, F. Schacht, & D. Walter (Hrsg.), Digitales Lehren und Lernen von Mathematik in der Schule (S. 399–410). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-65281-7_17 Kreutz, J., Leuders, T., & Hellmann, K. (Eds.). (2020). Professionsorientierung in der Lehrerbildung: Kompetenzorientiertes Lehren nach dem 4-Component-InstructionalDesign-Modell. Springer Fachmedien Wiesbaden. https://doi.org/10.1007/978-3-65825046-1 Lipovsky, F., & Rzejak, D. (2015). Key features of effective professional development programmes for teachers. Ricercazione, 7, 27–51. Neck, M., Leuders, T., & Reinhold, F. (2025). Implementing the four-component instructional design model in professional development programs: A systematic review with a focus on teachers. Frontiers in Education, 10. https://doi.org/10.3389/feduc.2025.1631375 Prediger, S., Quabeck, K., & Erath, K. (2022). Conceptualizing micro-adaptive teaching practices in content-specific ways: Case study on fractions. Journal on Mathematics Education, 13(1), 1–30. https://doi.org/10.22342/jme.v13i1.pp1-30 Rott, B. (2020). Teachers’ Behaviors, Epistemological Beliefs, and Their Interplay in Lessons on the Topic of Problem Solving. International Journal of Science and Mathematics Education, 18(5), 903–924. https://doi.org/10.1007/s10763-019-09993-0 Van Merriënboer, J. J. G., & Kirschner, P. A. (2017). Ten Steps to Complex Learning: A Systematic Approach to Four-Component Instructional Design (3. Aufl.). Routledge. https://doi.org/10.4324/9781315113210

1083 1090 Joy Oslund Tasnim Alshuli Hope Gerson Paul Yu Grand Valley State University United States University of Arizona This paper puts forth a model for expansive mathematics teaching for students with disabilities. We consider three lessons in which embodied mathematical curricula were used to support a blind student in collegiate mathematics courses. The goal of the lessons was to provide access and inclusion for the blind student as they worked to make sense of the mathematical concepts. What resulted were moments in which the student’s conceptions and participation in their community of practice extended beyond what was expected, leading to expansive learning opportunities. We propose a model of embodied activity that goes beyond providing access and inclusion, striving for expansive notions of mathematical sensing and learning for all. Ahmed, I., & Chao, T. (2018). Assistive learning technologies for students with visual impairments: A critical rehumanizing review. Investigations in Math Learning, 10(3). Bayram, G. İ., Corlu, M. S., Aydın, E., Ortaçtepe, D., & Alapala, B. (2015). An exploratory study of visually impaired students’ perceptions of inclusive mathematics education. The British Journal of Visual Impairment, 33(3), 212–219. Bell, E. C., & Silverman, A. M. (2019). Access to math and science content for youth who are blind or visually impaired. Jrnl of Blindness Innovation and Research, 9(1). Bishop, J. P., Hicks, M. D., Koehne, C., Bui, M., & White, A. (2025). An Activity-Based Perspective on Mathematical Authority. Journal for Research in Mathematics Education (JRME), 56(4), 159–183. CAST. (2024). Universal Design for Learning 3.0. https://udlguidelines.cast.org Davis, L. J. (2013). Disability, normality, and power. The disability studies reader (1–14). Dick, T., & Kubiak, E. (1997). Issues and aids for teaching mathematics to the blind. The Mathematics Teacher, 90(5), 344–349. D’souza, R. (2017). Ableism in mathematics education: Ideology, resistance, and solidarity. Proceedings of the 9th International MES Conference, 2, 463–470. Esmonde, I., & Booker, A. N. (2017). Power and privilege in the learning sciences: Critical and sociocultural theories of learning. Routledge. Featherstone, H., Crespo, S., Jilk, L., Oslund, J.A., Parks, A.N., & Wood, M.B. (2011). Smarter together: Collaboration and equity in the elementary math classroom. NCTM. Gerofsky, S. (2016). Approaches to embodied learning in mathematics. In Handbook of international research in mathematics education, 3rd edition. Routledge. Gerofsky, S., & Zebehazy, K. T. (2022). Enhancing mathematical noticing of graphs through movement, voice, and metaphor: An intervention with two students with visual impairment. The British Journal of Visual Impairment, 40(2), 209–221. Lambert, R. (2019). Political, relational, and complexly embodied: Experiencing disability in the mathematics classroom. ZDM, 51(2), 279–289. Lambert, R. (2021). The magic is in the margins: UDL math. Mathematics Teacher: Learning and Teaching PK-12, 114(9), 660–669. Lambert, R., Tan, P., Hunt, J., & Candela, A. G. (2018). Rehumanizing the mathematics education of students with disabilities: Critical perspectives on research and practice. Investigations in Mathematics Learning, 10(3), 129–132. Louie, N. L. (2017). The culture of exclusion in mathematics education and its persistence in equity-oriented teaching. Journal for Research in Mathematics Education, 48(5), 488– 519. Padilla, A., & Tan, P. (2019). Toward inclusive mathematics education: a metatheoretical reflection about countering ableism in mathematics standards and curriculum. International Journal of Qualitative Studies in Education, 32(3), 299–322. Sasanelli, L. D., Ranieri, G., & Campanale, R. (2025). Embodied education in the framework of Universal Design for Learning: Connections and synergies. Journal of Inclusive Methodology and Technology in Learning and Teaching, 5(2), 1–13. Sfard, A. (2025). Two metaphors for learning revisited: What did the participation metaphor do for us in the last four decades? Learning, Culture and Soc Int., 55. Shakespeare, T. (2013). The social model of disability: In The disability studies reader. Sinclair, N., & de Freitas. (2019). Body studies in mathematics education: Diverse scales of mattering. ZDM, 51(2), 227–237. Tan, P., Padilla, A., & Lambert, R. (2022). A critical review of educator and disability research in mathematics education: A decade of dehumanizing waves and humanizing wakes. Review of Educational Research, 92(6), 871–910. Tancredi, S., Chen, R. S. Y., Krause, C., Abrahamson, D., & Gomez, F. (2021). Getting up to SpEED: Special education embodied design for sensorially equitable inclusion. Education Sciences and Society, 1, 114–136. Wenger, E. (1998). Communities of practice: learning, meaning, and identity. Cambridge Yu, P. W. D., Oslund, J. A., & Alshuli, T. (2024). Understanding function transformations through tactile representations and gesturing with visually impaired and sighted students. In Kosko, et al., (Eds.), Proceedings of the 46th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (1694– 1703. Kent State. Yu, P. W. D., Oslund, J. A., & Alshuli, T. (2025a). Unexpected richness from a wooden arch puzzle: Embodied ways of learning about parabolas. Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education, 2, 427–434. Santiago, Chile. Yu, P. W. D., Oslund, J. A., & Alshuli, T. (2025b). Wikki Stix, push pins, and textured paper: Embodied ways of learning about parabolas. In R. M. Zbiek, X. Yao, A. McCloskey, & F. Arbaugh (Eds.), Proceedings of the 47th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (1466–1470). Pennsylvania State University.

1091 1098 Hiroki Otani Ishikawa Masaaki Otsuma Women’s University Japan Aichi University of Education This study investigates how the meaning of explaining becomes semantically contaminated through its curricular use in Japanese primary school mathematics. Focusing on the Japanese word setsumei, the analysis examines mathematics textbooks from Grades 1 to 6 as an operational curriculum. Drawing on the notion of semantic contamination, explanation tasks were analysed in terms of objects of explaining and types of expected responses (that, how, why). The results show that, while the word setsumei remains unchanged, its normative meaning is reorganised through repeated use: procedural and justificatory forms of explaining are foregrounded from the outset, whereas other everyday possibilities are backgrounded. These findings highlight how curricular language shapes what counts as explanation in school mathematics. Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna et al. (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 115-135). Springer. Bishop, A. (1988). Mathematics education in its cultural context. Educational Studies in Mathematics, 19(2), 179-191. Esmonde, I. (2009). Explanations in mathematics classrooms: A discourse analysis. Canadian Journal of Science, Mathematics and Technology Education, 9(2), 86-99. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5-23. Kim, D.-J., Ferrini-Mundy, J., & Sfard, A. (2012). How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity. International Journal of Educational Research, 51-52, 86-108. Ingram, J., Andrews, N. & Pitt, A. (2019). When students offer explanations without the teacher explicitly asking them to. Educational Studies in Mathematics, 101(1), 51–66. Mejía-Ramos, J. P., Alcock, L., Lew, K., Rago, P., Sangwin, C., & Inglis, M. (2019). Using corpus linguistics to investigate mathematical explanation. In E. Fischer & M. Curtis (Eds.), Methodological advances in experimental philosophy (pp. 239-264). Bloomsbury. Mejía-Ramos, J. P., & Inglis, M. (2011). Semantic contamination and mathematical proof: can a non-proof prove? The Journal of Mathematical Behavior, 30(1), 19-29. Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. Routledge & Kegan Paul. Shinno, Y., & Mizoguchi, T. (2021). Theoretical approaches to teachers’ lesson designs involving the adaptation of mathematics textbooks: Two cases from kyouzai kenkyuu in Japan. ZDM Mathematics Education, 53(6), 1387–1402. Valverde, G. A. et al. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Springer. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458-477.

1099 1106 Mickael Da Ronch Ismaïl Mili Cecile Ouvrier-Buffet University of Teacher Education Valais ²University of Fribourg Switzerland Paris-Est Creteil University, LDAR This research report addresses an underexplored yet crucial issue in teacher education: prospective primary teachers’ conceptions of mathematical proof. Based on a questionnaire administered to first-year teacher education students in French-speaking Switzerland, the study confirms the robustness and cost-effectiveness of this methodology and situates its findings within established research. The results highlight the functions future teachers attribute to proof and reveal discrepancies between their conceptions of mathematical rigor and the reasoning they mobilize when evaluating proofs, pointing to major challenges for teacher education. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). Hodder & Stoughton. Barkai, R., Tsamir, P., Tirosh, D. & Dreyfus, T. (2002). Proving or refuting arithmetic claims: The case of elementary school teachers. In A. D. Cockburn, E., Nardi (Eds), Proc. 26th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 57–64). PME. Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), 237–254. Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies in Mathematics, 44, 5–23. Healy, L., & Hoyles, C. (1998). Justifying and proving in school mathematics. Institute of Education, University of London. Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.) (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. Springer. Ouvrier-Buffet, C. (2023). Exploring students’ conceptions of proof in high-school and university: A proposal for collaborations in Europe. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), Proc. 13th Congress of the European Society for Research in Mathematics Education (pp. 240–247). ERME. Stylianides, A.J. (2016). Proving in the Elementary Mathematics Classroom. Oxford University Press. Stylianou, D.A., Blanton, M.L., & Rotou, O. (2015). Undergraduate students' understanding of proof: relationships between proof conceptions, beliefs, and classroom experiences with learning proof. IJRUME, 1(1), 91–134. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.

1107 1114 Loris Palaj Carola Manolino Alma Mater Studiorum University of Bologna Laboratorio “S. Polin”, University of Torino Italy Université Paris Cité France This study investigates how a blind learner engages with function graphs through Audiofunctions, an audio–based digital tool. Adopting the Body–Artifact Functional System (BAFS) as a theoretical framework, this contribution focuses on the transition from perception–action Loop 3, in which the artifact remains an external object of attention, to Loop 4, where it becomes functionally integrated into perception. Using a qualitative case study, a micro–episode from a task–based interview is analysed, revealing how auditory feedback shifts from being an object of exploration to a perceptually transparent means of accessing mathematical structure. These findings contribute to the development of BAFS in non–visual contexts and highlight the epistemic role of audio–based tools in inclusive mathematics education. Gibson, J. J. (1986). The Ecological Approach to Visual Perception. Routledge. Hermann, T., Hunt, A., & Neuhoff, J. G. (Eds.). (2011). The Sonification Handbook. https://sonification.de/handbook/ Manolino, C., Funghi, S., Ducci, M., Brunetto, E., & Bernareggi, C. (2025). Navigating function graphs with audio: an investigation into the narratives of a blind expert using Audiofunctions. In Bosch, M., Bolondi, G., Carreira, S., Spagnolo, C., & Gaidoschik, M. (Eds.). (2025). Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14). Free University of Bolzano and ERME. https://hal.science/hal-05299099v1 Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159–174. Pissaloux, E., Velazquez, R., & Gay, S. (2026). Haptic technologies to support spatial cognition and mobility in visually impaired people. In E. Pissaloux & R. Velazquez (Eds.), Mobility of visually impaired people. Signals and communication technology. (pp. 429– 452). Springer. https://doi.org/10.1007/978-3-031-91550-5_16 Rabardel, P. (1995). Les hommes et les technologies: une approche cognitive des instruments contemporains. https://hal.science/hal-01017462v1 Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257–277. Shvarts, A., Alberto, R., Bakker, A., Doorman, M., & Drijvers, P. (2021). Embodied instrumentation in learning mathematics as the genesis of a body artifact functional system. Educational Studies in Mathematics, 107(3), 447–469. Turvey, M. T. (2007). Action and perception at the level of synergies. Human Movement Science, 26(4), 657–697. https://doi.org/10.1016/j.humov.2007.04.002

1115 1122 Anna Ida Säfström Björn Palmberg Johan Sidenvall Umeå Mathematics Education Research Centre, Umeå university Sweden Department of Science and Mathematics Education, Umeå university Sweden Municipality of Hudiksvall Learning algebra is difficult for many students. Recent reviews highlight the need for more research on challenges students encounter when learning about equations and studies of learning in real classrooms. This paper reports on a small-scale experiment aimed at investigating how teaching design affect Grade 6 students’ equation solving, with a focus on the comparison between direct instruction and guided construction. Our results indicate that teaching design affect what approaches students learn, but to a lesser extent their proficiency in solving equations, in line with similar studies investigating other mathematical topics. Based on these findings, we provide suggestions for development of guided construction designs that support student’s equation learning. Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A. J., Barbieri, C., Augustine, A., & Koedinger, K. R. (2015). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530-551 de Jong, T., Lazonder, A. W., Chinn, C. A., Fischer, F., Gobert, J., Hmelo-Silver, C. E., Koedinger, K. R., Krajcik, J. S., Kyza, E. A., Linn, M. C., Pedaste, M., Scheiter, K., & Zacharia, Z. C. (2023). Let's talk evidence–The case for combining inquiry-based and direct instruction. Educational Research Review, 39, 100536. Elkjær, M., Kieran, C. (2025). The surprising obstacle of two unknowns on the same side of the equal sign in solving equations of the type ax ± b ± cx = d and the phenomenon of ignoring the unknown. Educational Studies in Mathematics. Ellis, A. B., & Özgür, Z. (2024). Trends, insights, and development in research on the teaching and learning of algebra. ZDM – Mathematics Education, 56, 199–210. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25. Heinze, A., Schwabe, J., Grüßing, M., & Lipowsky, F. (2015). Effects of instruction on strategy types chosen by German 3rd-graders for multi-digit addition and subtraction tasks: an experimental study. In K. Beswick, T. Muir, & J. Fielding-Wells (Eds.), Proc. 39th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 49-56). PME. Hodgen, J., Coe, R., Brown, M., & Küchemann, D. (2024). Educational performance over time: Changes in mathematical attainment between 1976 and 2009 in England. Implementation and Replication Studies in Mathematics Education, 4, 83–124. Jankvist, U.T., & Niss, M. (2015). A framework for designing a research-based “maths counsellor” teacher programme. Educational Studies in Mathematics, 90, 259–284. Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM, 54, 1131–1150. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852. Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565–579. Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49, 341–359. Xie, S., & Cai, J. (2022). Fifth graders’ learning to solve equations: The impact of early arithmetic strategies. ZDM – Mathematics Education, 54(6), 1169–1179. Zhang, L., Kirschner, P. A., Cobern, W. W., & Sweller, J. (2022). There is an evidence crisis in science educational policy. Educational Psychology Review, 34(2), 1157–1176.

1123 1130 Hanna Palmér Jessica Elofsson Maria Walla Camilla Björklund Linnaeus University Sweden Linköping University Sweden Dalarna University Sweden University of Gothenburg Sweden This study examines how preschool teachers design mathematics teaching that enables young children to mathematize, drawing on Freudenthal’s notion of mathematics as genuinely needed within an activity. Within a design-based research project in Swedish preschools (children aged 1–3), 68 video-documented activities were analysed to identify instances of mathematizing. Three approaches were identified: drawing on scripted activities, drawing on known counting chants, and drawing on an experience familiar to the children. Across these, mathematizing occurred when teachers framed and sustained mathematical necessity within play-oriented situations. The study provides empirical insight into how mathematizing can be realized in early mathematics education. Aunio, P., & Niemivirta, M. (2010). Predicting children’s mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20(5), 427–435. https://doi.org/10.1016/j.lindif.2010.06.003 Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Taylor & Francis Ltd. Björklund, C., & Palmér, H. (2022). Teaching toddlers the meaning of numbers—connecting modes of mathematical representations in book reading. Educational Studies in Mathematics, 110(3), 525–544. https://doi.org/10.1007/s10649-022-10147-3 Björklund, C., & Palmér, H. (2024). The challenges of mathematizing in Swedish early childhood education. Journal of Early Childhood Education Research, 13(2), 167–186. https://doi.org/10.58955/jecer.138122 Clements, D. H., Fuson, K. C., & Sarama, J. (2017). The research-based balance in early childhood mathematics: A response to Common Core criticisms. Early Childhood Research Quarterly, 40, 150–162. https://doi.org/10.1016/j.ecresq.2017.03.005 Freudenthal, H. (1978). Weeding and sowing. Preface to a science of mathematical education. Reidel Publishing. Freudenthal, H. (1981). Problems of mathematics education. Educational Studies in Mathematics, 12(2), 133–150. https://www.jstor.org/stable/3482361 Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. https://doi.org/10.1207/s15327833mtl0602_3 Gravemeijer, K., & Terwel, J. (2000). Hans Freudenthal: A mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777–796. https://doi.org/10.1080/00220270050167170 Leung, A., & Hung, S. (2018). Mathematizing basic addition. In V. Kinnear, M. Y. Lai, & T. Muir (Eds.), Forging connections in early mathematics teaching and learning (pp. 139– 154). Springer. https://doi.org/10.1007/978-981-10-7153-9_8 National Agency for Education. (2025). Curriculum for the preschool. Lpfö18. National Agency for Education. Palmér, H., & Björklund, C. (2023). The teaching of numbers in common preschool activities: A delicate balancing act. Early Childhood Education Journal, 51(5), 971–980. https://doi.org/10.1007/ s10643-022-01354-x Palmér, H., & Björklund, C. (2024). The real world of toddler mathematics. In H. Palmér, C. Björklund, E. Reikerås, & J. Elofsson (Eds.), Teaching mathematics as to be meaningful– Foregrounding play and children’s perspectives: Results from the POEM5 conference, 2022 (pp. 221–232). Springer. Papadakis, S., Kalogiannakis, M., & Zaranis, N. (2017). Improving mathematics teaching in kindergarten with realistic mathematical education. Early Childhood Education Journal, 45(3), 369–378. https://doi.org/10.1007/s10643-015-0768-4 Patton, M. Q. (2015). Qualitative research & evaluation methods (4th ed.). SAGE Publications. Siegler, R. S., & Braithwaite, D. W. (2017). Numerical development. Annual Review of Psychology, 68(1), 187–213. https://doi.org/10.1146/annurev-psych-010416-044101 Van den Heuvel-Panhuizen, M. (2015). Freudenthal’s work continues. In S. J. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 309–331). Springer. Van den Heuvel-Panhuizen, M. (2020). A spotlight on mathematics education in the Netherlands and the central role of Realistic Mathematics Education. In M. Van den Heuvel-Panhuizen (Ed.), National reflections on the Netherlands didactics of mathematics: Teaching and learning in the context of realistic mathematics education (pp. 1–14). Springer.

1131 1138 Mari Paloniemi Lindström Helén Sterner Faculty of Mathematics Education, Dalarna University Sweden The present study utilizes data from the Swedish TIMSS 2023 sample to investigate the relationships between Grade 4 students’ mathematics self-concept, self-efficacy, and achievement in the Number domain, as well as potential gender differences in these relationships. Structural equation modelling with latent mediation was applied. Boys outperformed girls on the Number achievement and reported higher levels of self-concept and self-efficacy. Results showed that mathematics self-concept, but not self-efficacy, was strongly and similarly associated with Number achievement for both genders. The results further indicate that gender differences in achievement were partially explained by differences in students’ self-concept. Mathematics self-concept is identified as a key factor associated with early Number achievement. Blanton, M., Stephens, A. Knuth, E., Murphy Gardiner, A., Isler, I. & Kim, J. S. (2015). The Development of Children’s Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade. Journal for Research in Mathematics Education, 46, 39–87. https://doi.org/10.5951/jresematheduc.46.1.0039 Ding, Y., (2024). The self in the school context: Mathematics self-concept and self-efficacy in PISA (Doctoral dissertation, University of Gothenburg, Gothenburg Studies in Educational Sciences, No. 489). Acta Universitatis Gothoburgensis. http://hdl.handle.net/2077/81251 Ding, Y., Klapp, A., & Hansen, K. (2024). The importance of mathematics self-concept and self-efficacy for mathematics achievement. Educational Psychology. https://doi.org/10.1080/01443410.2024.2410217 Kania, N., & Juandi, D. (2023). Does self-concept affect mathematics learning achievement? Journal of Education Learning (EduLearn), 17(3), 455–461. https://doi.org/10.11591/edulearn.v17i3.20554 Levine, S. C., & Pantoja, N. (2021). Development of children’s math attitudes: Gender differences, key socializers, and intervention approaches. Developmental review, 62, 100997. https://doi.org/10.1016/j.dr.2021.100997 Lohbeck, A. (2018). Self-concept and self-determination theory: Math self-concept, motivation, and grades in elementary school children. Early Child Development and Care, 188(8), 1031–1044. https://doi.org/10.1080/03004430.2016.1241778 Lyons, I.M., Price, G.R., Vaessen, A., Blomert, L. & Ansari, D. (2014.) Numerical predictors of arithmetic success in grades 1-6. Developmental science, 17(5), 714-726. https://doiorg.www.bibproxy.du.se/10.1111/desc.12152 Mejía-Rodríguez, A.M., Luyten, H. & Meelissen, M.R.M. (2021). Gender Differences in Mathematics Self-concept Across the World: an Exploration of Student and Parent Data of TIMSS 2015. Int J of Sci and Math Educ, 19, 1229–1250. https://doiorg.www.bibproxy.du.se/10.1007/s10763-020-10100-x Michaelides, M. P., Brown, G. T. L., Eklöf, H., & Papanastasiou, E. C. (2019). Motivational profiles in TIMSS mathematics: Exploring student clusters across countries and time. IEA Research for Education (Vol. 7). Springer. https://10.1007/978-3-030-26183-2 Nguyen, T., Watts, T.W., Duncan, G.J., Clements, D.H., Sarama, J.S., Wolfe, C., Spitler, M.E. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement? Early Childhood Research Quarterly, 36, 550-560. https://doi.org/10.1016/j.ecresq.2016.02.003. Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2018). A Longitudinal Study Revisiting the Notion of Early Number Sense: Algebraic Arithmetic AS a Catalyst for Number Sense Development. Mathematical Thinking Learning, 20(3), 222–247. https://doi.org/10.1080/10986065.2018.1474533 The National Agency of Education. (2015). TIMSS 2015. Svenska grundskolelevers kunskaper i matematik och naturvetenskap i ett internationellt perspektiv. https://www.skolverket.se/publikationer?id=3707 The National Agency of Education. (2024). TIMSS 2023. Svenska grundskoleelevers kunskaper i matematik och naturvetenskap i ett internationellt perspektiv. https://www.skolverket.se/getFile?file=13178 The National Agency of Education. (2025). Könsskillnader i skolresultat Skolverkets aktuella analyser. https://www.skolverket.se/getFile?file=13305

1139 1146 Hajin Park Wee Tiong Seah Faculty of Education, The University of Melbourne Australia This study challenges the assumption that cultural bias in assessment can be eliminated through awareness and design. Through interviews with five educators of pre-service teachers of subjects such as mathematics, this paper demonstrates that bias is a reality to be managed rather than eliminated. Findings reveal that: (1) online environments facilitate more consistent cultural integration through systematic planning than face-to-face adjustments; (2) unlimited cultural integration undermines assessment consistency; and (3) technology amplifies cultural advantages in unpredictable ways. This paper proposes a framework for systematic bias management by transparently distributing advantages across assessment sequences, making cultural assumptions explicit, and involving cultural representatives in evaluation design. Alt, D., & Raichel, N. (2021). Advancing culturally responsive assessment practices. Springer International Publishing. https://doi.org/10.1007/978-3-030-71644-8_1 Bishop, A. J. (1991). Mathematical enculturation: A cultural perspective on mathematics education. Braun, V., & Clarke, V. (2022). Thematic Analysis: a practical guide. SAGE Publications Ltd. Brown-Jeffy, S., & Cooper, J. E. (2011). Toward a conceptual framework of culturally relevant pedagogy: An overview of the conceptual and theoretical literature. Teacher Education Quarterly, 38(1), 65-84. Casinader, N., & Manathunga, C. (2020). Cultural hybridity and Australian children: speaking back to educational discourses about global citizenship. Discourse: Studies in the Cultural Politics of Education, 41(6), 940-952. de Freitas, E. (2008). Troubling teacher identity: Preparing mathematics teachers to teach for diversity. Teaching Education, 19(1), 43-55. https://doi.org/10.1080/10476210701860024 Gay, G. (2000). Culturally responsive teaching: theory, research, and practice. Teachers College Press. Hodges, C. B., Moore, S., Lockee, B. B., Bond, M. A., & Trust, T. (2024). The difference between emergency remote teaching and online learning. Brill. Lucas, T., & Villegas, A. M. (2013). Preparing linguistically responsive teachers: Laying the foundation in preservice teacher education. Theory Into Practice, 52(2), 98-109. https://doi.org/10.1080/00405841.2013.770327 Margaret, H. (2022). Formative assessment: Making it happen in the classroom (2nd ed.). Corwin. Meaney, T., Trinick, T., & Fairhall, U. (2012). Collaborating to meet language challenges in Indigenous mathematics classrooms. Springer Netherlands. Nasir, N. i. S., & Cooks, J. (2009). Becoming a hurdler: How learning settings afford identities. Anthropology & Education Quarterly, 40(1), 41-61. Paris, D. (2021). Culturally sustaining pedagogies and our futures. Educational Forum, 85(4), 364-376. https://doi.org/10.1080/00131725.2021.1957634 Shauna, W., & Mélanie, M. (2020). Moving forward or holding back? Creating a culture of academic assessment. In Canadian Journal for the Scholarship of Teaching and Learning 11(2), 1-15. https://doi.org/10.5206/cjsotl-rcacea.2020.2.8009 Shepard, L. A. (2019). Classroom assessment to support teaching and learning. The Annals of the American Academy of Political and Social Science, 683, 183-200. Solano-Flores, G., & Nelson-Barber, S. (2001). On the cultural validity of Science assessments. Journal of Research in Science Teaching, 38(5), 553-573.

1147 1154 Vishrut Patel Aniket Sule Shweta Naik Homi Bhabha Centre for Science Education, Mumbai India This study explores the role of electricity bills as "boundary objects" within an extracurricular mathematical modeling camp for Grade 9 students in CITY, India. Drawing on Cultural-Historical Activity Theory (CHAT), the research analyzes how students navigate the discontinuity between school mathematics and the real-world context of electrical power distribution. Findings indicate that while students successfully utilized school-based mathematical reasoning to reverse-engineer billing procedures, the artifact remained socioculturally opaque. Yet, such matheamtically rich artefacts have a potential for supporting boundary crossing, though a complete crossing was not observed. Engestrom, Y. (2000). Activity theory as a framework for analyzing and redesigning work. Ergonomics, 43(7), 960–974. https://doi.org/10.1080/001401300409143 Jurdak, M. (2006). Contrasting Perspectives and Performance of High School Students on Problem Solving in Real World, Situated, and School Contexts. Educational Studies in Mathematics, 63(3), 283–301. https://doi.org/10.1007/s10649-005-9008-y Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation (p. 138). Cambridge University Press. https://doi.org/10.1017/CBO9780511815355 Niss, M. (1994). Mathematics in Society: History and Epistemology of Mathematics and Mathematics Education. In R. Biehler (Ed.), Didactics of Mathematics as a Scientific Discipline. Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47204-X Venkat, H., & Winter, M. (2015). Boundary objects and boundary crossing for numeracy teaching. ZDM, 47(4), 575–586. https://doi.org/10.1007/s11858-015-0683-6 Williams, J., & Wake, G. (2007). Black Boxes in Workplace Mathematics. Educational Studies in Mathematics, 64(3), 317–343. https://doi.org/10.1007/s10649-006-9039-z Williams, J., Wake, G., & and Boreham, N. C. (2001). School or College Mathematics and Workplace Practice: An Activity Theory Perspective. Research in Math. Ed., 3(1), 69–83.

1155 1162 Melek Pesen Emine Erktin Boğaziçi University Turkey This study examines reasoning opportunities in tasks designed by prospective mathematics teachers and their conceptions of how they promote reasoning and justification. Twenty-eight prospective teachers designed curriculum-based tasks and submitted self-reports. Tasks were analysed using the Integrated Framework for Analyzing Ways of Reasoning (Sevinç et al., 2022), and self-reports were thematically analysed. Results indicate that most tasks primarily involved empirical reasoning, whereas teachers frequently described their tasks as promoting explanation and justification. Comparison of intended and enacted reasoning revealed diverse alignments, highlighting challenges in translating intentions about reasoning into task design, and have implications for teacher education. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549. Jones, K., & Pepin, B. (2016). Research on mathematics teachers as partners in task design. Journal of Mathematics Teacher Education, 19, 105–121. https://doi.org/10.1007/s10857-016-9345-z Mata-Pereira, J., & da Ponte, J. P. (2017). Enhancing students’ mathematical reasoning in the classroom: Teacher actions facilitating generalization and justification. Educational Studies in Mathematics, 96(2), 169-186. Melhuish, K., Thanheiser, E., & Guyot, L. (2020). Elementary school teachers’ noticing of essential mathematical reasoning forms: Justification and generalization. Journal of Mathematics Teacher Education, 23(1), 35-67. Mueller, M., Yankelewitz, D., & Maher, C. (2014). Teachers promoting student mathematical reasoning. Investigations in Mathematics Learning, 7(2), 1-20. Sevinç, S., Kohanovaá, I., Isiksal-Bostan, M., Kubaáček, Z., Isler-Baykal, I., Lada, M., Cakiroglu, E., Di Paola, B. (2022). Developing an integrated framework for analysing ways of reasoning in mathematics. ICERI2022 Proceedings, pp. 2082–2089. https://doi.org/10.21125/iceri.2022.0529 Steen, L. A. (1999). Twenty questions about mathematical reasoning. ERIC. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

1163 1170 Juan Luis Piñeiro Elena Castro-Rodríguez Universidad Metropolitana de Ciencias de la Educación Chile Universidad de Granada Spain This research report examines how specialized knowledge is enacted by a mathematics teacher and a special education teacher in primary mathematics classrooms organized around co-teaching. The analysis draws on six lessons taught in first, third, and sixth grade. The findings indicate that instructional practices were predominantly structured according to the “one teaches, one assists” and “one teaches, one observes” models of co-teaching. Across lessons, the knowledge enacted in classroom interaction was shaped by the teacher who assumed instructional leadership. We conclude by discussing implications for teacher education and research on teachers’ professional knowledge. Carrillo, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, Á., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model*. Research in Mathematics Education, 20(3), 236–253. https://doi.org/10.1080/14794802.2018.1479981 Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education (8th ed.). Routledge. Cook, L., & Friend, M. (1995). Co-teaching: Guidelines for creating effective practices. Focus on Exceptional Children, 28(3), 1-16. https://doi.org/10.17161/foec.v28i3.6852 Gardesten, M. (2023). How co-teaching may contribute to inclusion in mathematics education: A systematic literature review. Education Science, 13(7), 677. https://doi.org/10.3390/educsci13070677 Graziano, K., & Navarette, L. (2012). Co-teaching in a teacher education classroom: Collaboration, comprise, and creativity. Issues in Teacher Education, 21(1), 109-126. Harbour, K. E., Livers, S. D., McDaniel, S. C., Gleason, J., & Barth, J. M. (2022). Professional development to support elementary mathematics and co-teaching practices: Collaborations between general and special education. Mathematics Teacher Education and Development, 24(2), 33–56. Krippendorff, K. (2018). Content analysis: An introduction to its methodology (3rd ed.). SAGE Kuckartz, U. (2019). Qualitative text analysis: A systematic approach. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in Mathematics Education (pp. 181-198). Springer. https://doi.org/10.1007/978-3-030-15636-7_8 Lehane, P., & Senior, J. (2020). Collaborative teaching: Exploring the impact of co-teaching practices on the numeracy attainment of pupils with and without special educational needs. European Journal of Special Needs Education, 35(3), 303-317. https://doi.org/10.1080/08856257.2019.1652439 Scruggs, T. E., Mastropieri, M. A., & McDuffie, K. A. (2007). Co-teaching in inclusive classrooms: A metasynthesis of qualitative research. Exceptional Children, 73(4), 392416. https://doi.org/10.1177/001440290707300401 Shin, M., Lee, H., & McKenna, J. W. (2016). Special education and general education preservice teachers’ co-teaching experiences: A comparative synthesis of qualitative research. International Journal of Inclusive Education, 20(1), 91-107. https://doi.org/10.1080/13603116.2015.1074732

1171 1178 Georgia Pitta Xenia Vamvakoussi Department of Early Childhood Education, University of Ioannina Greece This paper reports findings from a design research study aimed at fostering early multiplicative reasoning. Kindergarten children were introduced to an estimate-and measure procedure (indirect partitioning) for identifying submultiples of discrete and continuous quantities. Pre- and post-interviews examined children’s performance, appropriation of the taught procedure, and the emergence of their own strategies. Results showed improvement across quantity and submultiple types for all children who completed the posttest. Analysis of video-recorded solution processes revealed a range of indirect and direct partitioning strategies. The findings suggest that parallel work with discrete and continuous quantities is feasible for young children and can substantially support early multiplicative reasoning. De Keersmaeker, K., Vanluydt, E., Onghena, P., & Van Dooren, W. (2024). The relation between proportional vocabulary and proportional reasoning abilities in young children. European Journal of Psychology of Education, 39(5), 2201–2221. https://doi.org/10.1007/s10212-023-00767-5 Fitzpatrick, C. L., & Hallett, D. (2019). Early understanding of fractions via early understanding of proportion and division. In K. M. Robinson, H. P. Osana, & D. Kotsopoulos (Eds.), Mathematical learning and cognition in early childhood: Integrating interdisciplinary research into practice (pp. 249–271). Springer. https://doi.org/10.1007/978-3-030-12895-1_14 Gravemeijer, K., & Prediger, S. (2019). Topic-specific design research: An introduction. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 33–57). Springer Open. https://doi.org/10.1007/978-3-030-15636-7_2 Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan. Kornilaki, K., & Nunes, T. (2005). Generalizing principles in spite of procedural differences: Children’s understanding of division. Cognitive Development, 20(3), 388–406. https://doi.org/10.1016/j.cogdev.2005.05.004 Lucas, J. R. (2000). Conceptual roots of mathematics. Routledge. Park, J. H., & Nunes, T. (2001). The development of the concept of multiplication. Cognitive development, 16(3), 763-773. https://doi.org/10.1016/S0885-2014(01)00058-2 Pitta, G., & Vamvakoussi, X. (2025). Dealing with multiplicative problems in pre-primary education: Analyzing children’s performance and strategies. In Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education. https://hal.science/hal-05236938v1 Polotskaia, E., & Savard, A. (2021). Some multiplicative structures in elementary education: A view from relational paradigm. Educational Studies in Mathematics, 106(3), 447–469. https://doi.org/10.1007/s10649-020-09979-8 Sophian, C. (2008). The origins of mathematical knowledge in childhood. Routledge. https://doi.org/10.4324/9781315085654 Steffe, L. P. (2013). On children’s construction of quantification. In R. L. Mayes & L. L. Hatfield (Eds.), Quantitative reasoning in mathematics and science education: Papers from an international STEM research symposium (pp. 13–41). University of Wyoming. Vamvakoussi, X., & Kaldrimidou, M. (2018). The multiplicative conceptual field in the early math curriculum (in Greek). In Ch. Skoumpourdi & M. Skoumios (Eds.), Proceedings of the 3rd Hellenic Conference on Educational Materials for Math and Science Education (pp. 302–311). University of the Aegean Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5, 27. https://doi.org/10.1186/s40594-018-0119-2

1179 1186 Marios Pittalis Susanne Strachota Maria Chimoni Mathías López University of Cyprus Tufts University United States This study examines Grade 5 and 6 students’ performance on and reasoning about function-graph tasks. Drawing on a synthesis of frameworks, we designed tasks addressing five conceptual categories. Data from 271 students were analyzed using mixed methods. Results show stronger performance on coordination of quantities tasks, particularly with quantitative values, and substantial challenges with tasks requiring functional inferences. Performance on covariational and graph-as-object tasks was more variable. Findings include a framework for analyzing upper elementary students’ reasoning of function graphs, with insight on how task categories foreground distinct reasoning. Carlson, M. P., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. Chimoni, M., Pitta-Pantazi, D., & Christou, C. (2018). Examining early algebraic thinking: Insights from empirical data. Educational Studies in Mathematics, 98(1), 57–76. Doorman, L. M., Drijvers, P. H. M., Gravemeijer, K. P. E., Boon, P. B. J., & Reed, H. (2013). Design research in mathematics education: The case of an ICT-rich learning arrangement for the concept of function. In Educational design research-Part B. Illustrative cases (pp. 425–446). SLO: Netherlands institute for curriculum development. Duijzer, C., Van den Heuvel-Panhuizen, M., Veldhuis, M., & Doorman, M. (2019). Supporting primary school students’ reasoning about motion graphs through physical experiences. ZDM – Mathematics Education, 51(6), 899–913. Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM – Mathematics Education, 54(6), 1131–1150. Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of educational research, 60(1), 1–64. Pinto, E., & Cañadas, M. C. (2021). Generalizations of third and fifth graders within a functional approach to early algebra. Mathematics Education Research Journal, 33, 113– 134. Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2020). Young students’ functional thinking modes: The relation between recursive patterning, covariational thinking, and correspondence relations. Journal for Research in Mathematics Education, 51(5), 631– 674. Radford, L. (2025). Cartesian graphs and co-variational thinking in early algebra. Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14), Free University of Bozen-Bolzano; ERME, BozenBolzano, Italy. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 386–420). National Council of Teachers of Mathematics. Strachota, S., López, M. A., Brizuela, B. M., Pérez-Martos, M. C., Murphy Gardiner, A., & Blanton, M. (2024). Young Students’ Understandings of Function Graphs. In Lindgren, R., Asino, T. I., Kyza, E. A., Looi, C. K., Keifert, D. T., & Suárez, E. (Eds.), Proceedings of the 18th International Conference of the Learning Sciences - ICLS 2024 (pp. 330–337). Buffalo, USA: International Society of the Learning Sciences. Tasova, H. I., & Moore, K. C. (2025). Middle School Students’ Meanings of Points From Quantitative and Covariational Reasoning Perspectives. Journal for Research in Mathematics Education, 56(5), 233-258.

1187 1194 Jerome Proulx King’s College London United Kingdom This Research Report presents preliminary findings on a study about students’ ways of making-sense of %. Grade-8 students’ strategies for solving one arithmetic task is looked into: the 3+50% task. The initial analyses of students’ strategies for ‘3+50%’ represent a first step in better understanding varieties of conceptions of % that students engage with, as well as the (implicit or explicit) roles they give to the %-referent. Baratta, W. et al. (2010). Percentages. Proceedings of MERGA-33 (pp. 61-68). MERGA. Berry, J. et al. (1999). Dictionary of mathematics. Fitzroy Dearborn. Champlain, P. et al. (1996). Lexique mathématique. Triangle d’Or. Chen, H., & Rao, A.R. (2007). When two plus two is not equal to four. Journal of Consumer Research, 34, 327-340. Douady, R. (1994). Ingénierie didactique et évolution du rapport au savoir. Repères, 15, 37-61. Erdem, E., Gökkurt, B., & Soylu, Y. (2023). Seventh graders’ strategies for shading the percentages of geometric shapes. Int. J. of Math. Ed. in Sc. and Tech., 54, 801-826. Gauvrit, N. (2014). Statistiques, méfiez-vous! Ellipses. Glatzer, D. (1984). Percentage: ideas and suggestions. Arithmetic Teacher, 31, 24-26. Hahn, C. (1999). Proportionalité et pourcentage chez des apprentis vendeurs. Educational Studies in Mathematics, 39, 229-249. Heuvel-Panhuizen, M. (1994). Improvement of (didactical) assessment by improvement of problems. Educational Studies in Mathematics, 27, 341-372. Janvier, C. (1983). Représentation et compréhension. Bulletin AMQ, 23, 22-28. Lembke, L., & Reys, B. (1994). The development of and interaction between intuitive and schooltaught ideas about percent. Journal for Res. in Math. Ed., 25, 237-259. Packard, G., & Boardman T. (1988). The misuse of ratios, indices and percentages in ecophysiological research. Physiological zoology, 61, 1-9. Parker, M., & Leinhardt, G. (1995). Percent: a privileged proportion. Review of Educational Research, 65, 421-481. Proulx, J. (2019). Mental mathematics as oral mathematics. Proceedings of PME-43 (vol. 3, pp. 249-256). PME. Proulx, J. (2025). Toward a conceptualization of %. Proceedings of CERME-14 (pp. 789-796). CERME. Streefland, L., & Heuvel-Panhuizen, M. (1992). Evoking pupils’ informal knowledge on percent. Proceedings of PME-16 (vol. 3, pp. 51-57). PME. 1 - 1194

1195 1202 Ratu Ilma Indra Putri Dhea Anisah Putri Zulkardi Zulkardi MIchiel Doorman universitas Sriwijaya Indonesia This study explores how sixth-grade students make sense of ratio situations, as reflected in the problems they pose in response to a given context. Adopting a sense-making perspective that views understanding as the process of structuring a situation, we analyze students’ posed problems and interviews in a familiar mixture context. The analysis shows that different framings and interpretations of the same situation can lead to different forms of ratio reasoning and shape the ratio structure students construct. In particular, some framings afford multiplicative and unit-rate reasoning, while others remain oriented toward direct quantitative comparison grounded in the context. These findings contributed to ratio learning by showing how problem posing reveals both the possibilities and limits of students’ emerging sense-making of ratio. Bevan, D., & Capraro, M. M. (2021). More Posing creative problems: A study of elementary students’ mathematics understanding. Int. Electronic J. of Math. Ed., 16(3), em0654. Cai, J., & Rott, B. (2024). On understanding mathematical problem-posing processes. ZDM– Mathematics Education, 56(1), 61-71. https://doi.org/10.1007/s11858-023-01536-w Cai, J., & Hwang, S. (2023). Making Mathematics Challenging Through Problem Posing in the Classroom. In: Leikin, R. (eds) Mathematical Challenges For All. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-18868-8_7 Cai, J. (2022). What research says about teaching mathematics through problem posing. Éducation & didactique, 16(3), 31-50. https://doi.org/10.4000/educationdidactique.10642 Cai, J., & Hwang, S. (2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 101420. https://doi.org/10.1016/j.ijer.2019.01.001 Carmona-Medeiro, E., Martín-Díaz, J. P., & Climent, N. (2024). Variables in planning and carrying out a problem-posing task in early childhood education. The Journal of Mathematical Behavior, 73, 101131. https://doi.org/10.1016/j.jmathb.2024.101131 Cengiz, N., & Rathouz, M. (2018). Making sense of equivalent ratios. Mathematics Teaching in the Middle School, 24(3), 148-155. Chen, T., & Cai, J. (2020). An elementary mathematics teacher learning to teach using problem posing: A case of the distributive property of multiplication over addition. International Journal of Educational Research, 102, 101420. Christou, C., Pitta-Pantazi, D., & Chimoni, M. (2024). Teachers' mathematical problem posing: The role of processes and complexity levels in posing problems on the fraction part-whole concept. The Journal of Mathematical Behavior, 73, 101134. Liljedahl, P., & Cai, J. (2021). Empirical research on problem solving and problem posing: A look at the state of the art. ZDM–Mathematics Education, 53(4), 723-735. Lobato, J., Ellis, A., & Zbiek, R. M. (2010). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-8. National Council of Teachers of Mathematics. 1906 Ass. Drive, Reston, VA 20191-1502. McCallum, W. (2023). Making sense of mathematics and making mathematics make sense. In Mathematics Curriculum Reforms Around the World: The 24th ICMI Study (pp.511521).Cham:Springer International Publishing. https://doi.org/10.1007/978-3-031-13548-4 Pinto, M. M. F. (2018, November). Making sense of students’ sense making of mathematics. In Proceedings of the first PME regional conference: South America. PME, Rancagua (pp. 31-45). Possamai, J. P., & Allevato, N. S. G. (2024). Teaching mathematics through problem posing: Elements of the task. The Journal of Mathematical Behavior, 73, 101133. Ran, H., Cai, J., Muirhead, F., & Hwang, S. (2025). An analysis of unexpected responses in middle school students’ mathematical problem posing from the perspective of problemposing processes. Educational Studies in Mathematics, 1-20. Sawatzki, C., Downton, A., & Cheeseman, J. (2019). Stimulating proportional reasoning through questions of finance and fairness. Mathematics Ed. Res. Journal, 31(4), 465-484. Silver, E. A., & Cai, J. (2005). Assessing students' mathematical problem posing. Teaching children mathematics, 12(3), 129-135. https://doi.org/10.5951/TCM.12.3.0129 Van den Heuvel-Panhuizen, M., & Drijvers, P. (2020). Realistic mathematics education. In Encyclopedia of mathematics education (pp. 713-717). Cham: Springer International Publishing. Walkington, C., & Bernacki, M. (2015). Students authoring personalized “algebra stories”: Problem-posing in the context of out-of-school interests. J. of Math. Beh., 40, 171-191. Xu, B., Cai, J., Liu, Q., & Hwang, S. (2020). Teachers’ predictions of students’ mathematical thinking related to problem posing. Int. Journal of Educational Research, 102, 101427. Zhang, L., Cai, J., Song, N., Zhang, H., Chen, T., Zhang, Z., & Guo, F. (2022). Mathematical problem posing of elementary school students: The impact of task format and its relationship to problem solving. ZDM–Mathematics Education, 54(3), 497-512. Zulkardi, Z., Putri, R. I. I., & Wijaya, A. (2020). Two decades of realistic mathematics education in Indonesia. Int. reflections on the Netherlands didactics of math, 325-340.

1203 1210 Stefanie Rach Marielena Menzel Sebastian Geisler Timo Kosiol OVGU Magdeburg Germany University of Potsdam Germany Mathematizing real-world problems is a crucial skill in modern society, yet challenging for many students. We examine variations in students’ mathematization processes and the conditions under which they successfully formulate a mathematical function for a real-world problem represented by data. In this experimental study, 561 upper secondary students model the decay of beer foam using one of the following data types: self-collected experimental data, provided experimental data, or smoothed data. Additionally, we collected students’ mathematics grades, and assessed their content knowledge about functions. The findings suggest significant differences in students’ mathematization approaches, partially related to their content knowledge and the data type provided. We discuss strategies to enhance this crucial skill. Barbato, G., Barini, E. M., Genta, G., & Levi, R. (2011). Features and performance of some outlier detection methods. Journal of Applied Statistics, 38(10), 2133-2149. Cevikbas, M., Kaiser, G., & Schukajlow, S. (2022). A systematic literature review of the current discussion on mathematical modelling competencies: State-of-the-art developments in conceptualizing, measuring, and fostering. Educational Studies in Mathematics, 109(2), 205-236. https://doi.org/10.1007/s10649-021-10104-6 Djepaxhija, B., Vos, P. & Fuglestad, A. B. (2015). Exploring grade 9 students’ assumption making when mathematizing. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 848854). CERME 9. Engel, J. (2018). Anwendungsorientierte Mathematik: Von Daten zur Funktion: Eine Einführung in die mathematische Modellbildung für Lehramtsstudierende. [Applicationoriented mathematics: From data to function: An introduction to mathematical modelling for student teachers.] Springer Berlin Heidelberg. Frejd, P., & Ärlebäck, J. B. (2011). First results from a study investigating Swedish upper secondary students’ mathematical modelling competencies. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in Teaching and Learning of Mathematical Modelling: ICTMA14 (pp. 407-416). Springer. Hankeln, C. & Greefrath, G. (2021). Mathematische Modellierungskompetenz fördern durch Lösungsplan oder Dynamische Geometrie-Software? Empirische Ergebnisse aus dem LIMo-Projekt [Fostering Mathematical Modelling Competency via Solution Plan or Dynamic Geometry Software? Empirical Results of the LIMo-Project]. Journal für Mathematik-Didaktik, 42, 367-394. https://doi.org/10.1007/s13138-020-00178-9 Jankvist, U. T. & Niss, M. (2020) Upper secondary school students’ difficulties with mathematical modelling. International Journal of Mathematical Education in Science and Technology, 51(4), 467-496. https://doi.org/10.1080/0020739X.2019.1587530 Jonscher, M., Menzel, M., Geisler, S., & Rach, S. (2024). Modelling performance using functions: Relation to person characteristics and different solution approaches. In T. Evans, O. Marmur, J. Hunter, & G. Leach (Eds.), Proc. 47th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 113-120). PME. Kaiser, G., & Schukajlow, S. (2022). Innovative Perspectives in Research in Mathematical Modelling Education. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proc. 45th Conf. of the Int. Group for Psychology of Math. Education, Vol. 1 (pp. 147-176). PME. Kawakami T., & Mineno K. (2021). Data-Based Modelling to Combine Mathematical, Statistical, and Contextual Approaches: Focusing on Ninth-Grade Students. In F.K.S. Leung, G.A. Stillman, G. Kaiser, & K.L. Wong (Eds.), Mathematical Modelling Education in East and West (pp. 389-400). Springer. Leiss, D., Ehmke, T., & Heine, L. (2024). Reality-based tasks for competency-based education: The need for an integrated analysis of subject-specific, linguistic, and contextual task features. Learning and Individual Differences, 114, 102518. Michelsen, C. (2006). Functions: a modelling tool in mathematics and science. Zentralblatt für Didaktik der Mathematik, 38, 269-280. https://doi.org/10.1007/BF02652810 Mohr, M., & Ufer, S. (2023). Measuring Data-based Modeling Skills in a Collaborative Setting. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proc. 46th Conf. of the Int. Group for the Psychology of Math. Education (Vol. 3, pp. 355-362). PME. Stillman, G. A., & Brown, J. P. (2021). Modeling the phenomenon versus modeling the data set. Mathematical Thinking and Learning, 25(3), 270-295.

1211 1218 Daniela Assmus Nina Bohlmann Kerstin Bräuning Caren Feskorn Torsten Fritzlar Heike Hahn Simone Reinhold Denise Resche University of Halle-Wittenberg Germany Leipzig University Germany University of Erfurt Freie Universität Berlin Germany Teachers’ specialized content knowledge on arithmetic operations is considered to play a meaningful role for teaching as learners’ development depends on their teachers’ understanding of relevant concepts, especially concerning the variety of strategies and representations for multiplication. This paper reports on a study with 863 preservice primary teachers from three different universities at the beginning of their university studies for teaching mathematics, using a written survey. Results of the study which included drawing tasks and tasks requiring preservice teachers’ comments, illustrate a surprisingly wide variety of iconical representations, and moreover reveal the novices’ challenges in finding corresponding comments to their drawings. Implications and obstructive details from the results are discussed. Aßmus, D., Bohlmann, N., Bräuning, K., Feskorn, C., Fritzlar, T., Hahn, H., Lenz, D., & Reinhold, S. (2025). Vorstellungen von Studienanfänger*innen zur Multiplikation. In L. Schick, M. Platz & A. Lambert (Hrsg.), Beiträge zum Mathematikunterricht 2025 (p. 1444). WTM. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. Barmby, P., & Milinković, J. (2011). Pre-service teachers’ use of visual representations of multiplication. In B. Ubuz (Ed.), Proceedings of 35th PME (Vol. 2, pp. 105–113). PME. Bruner, J. S. (1966). Toward a theory of instruction. Belkapp Press. Bruner, J. S., & Kenney, H. J. (1965). Representation and Mathematics Learning. Monographs of the Society for Research in Child Development, 30(1), 50–59. Cheeseman, J., Downton, A., Ferguson, S., & Roche, A. (2023). Meeting multiplicative thinking through thought-provoking tasks. Math. Ed. Research Journal, 35, 789–820. Dabić, M., & Milinković, J. (2015). Teachers’ representations of multiplication – Do children understand them? In J. Novotna & H. Moraova (Eds.), Developing mathematical language and reasoning (pp. 99–107), Charles University. Fetzer, M., & Tiedemann, K. (2022). Multiplication as a matter of Grundvorstellungen, strategies and representations. Twelfth Congress of the European Society for Research in Mathematics Education (CERME12), Feb 2022, Bozen-Bolzano, Italy. Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3–17. Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan. Harries, T., & Barmby, P. (2007). Representing and understanding multiplication. Research in Mathematics Education, 9(1), 33–46. Harries, T., & Sutherland, R. (2000). The representation of mathematical concepts in primary mathematics textbooks: a focus on multiplication. In: A. Rogerson (Ed.), Proceedings of the International Conference on Mathematics Education into the 21st Century: Mathematics for Living. Amman. https://sites.unipa.it/grim/jourdain.htm Lamprecht, X. (2020). Multiplikatives Verständnis fördern: Entwicklung und Evaluation eines Förderkonzepts in differenten Rahmenbedingungen. University of Bamberg Press. Maier, H. (1991). Interpretative Forschung im Bereich der Mathematikdidaktik. In: Beiträge zum Mathematikunterricht (pp. 97–107). Bad Salzdetfurth: Franzbecker. Mayring, P. (2015). Qualitative Content Analysis: Theoretical Background and Procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to Qualitative Research in Mathematics Education. Advances in Mathematics Education (pp. 365–380). Berlin: Springer. https://doi.org/10.1007/978-94-017-9181-6_13 Milinković, J. (2012). Pre-service teachers’ representational preferences. In Tai-Yih Tso (Ed.), Opportunities to Learn in Mathematics Education Proceedings of the 36th Conference of the IGPME (Vol. 3, pp. 209–215). PME. Mulligan, J. T. (1992). Children’s solutions to multiplication and division word problems: A longitudinal study. Mathematics Education Research Journal, 4, 23–42. Mulligan, J. T., & Mitchelmore, M. C. (1997). Young children's intuitive models of multiplication and division. J. for Research in Mathematics Education, 28(3), 309–330. Novikasari, I., & Dede, Y. (2021). Turkish Pre-service mathematics teachers' beliefs in multiplication. Journal on Mathematics Education, 12(3), 469–486. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Tirosh, D., & Graber, A. O. (1989). Preservice elementary teachers’ explicit beliefs about multiplication. Educational Studies in Mathematics, 20, 79–96. vom Hofe, R. (1995). Grundvorstellungen mathematischer Inhalte. Spektrum. vom Hofe, R., & Roth, J. (2023). Grundvorstellungen aufbauen. Mathematik lehren (236), 2– 7.

1219 1226 Torsten Fritzlar Denise Resche Martin Luther University Halle-Wittenberg Germany Leipzig University Germany Bayesian reasoning is a core element of statistical literacy but remains difficult even for adults. This qualitative exploratory study examines proto-Bayesian reasoning of 9–12-year-old students in a simple Bayesian model selection task with two bags and sequential draws of cubes with replacement. Using clinical interviews and probability scales, third- to fifth-grade students estimated and updated likelihoods after 5, 10, and 15 draws. Results show that many students intuitively adjust likelihoods in response to new information. The findings indicate that successful proto-Bayesian reasoning is possible in the age group studied and probability calculation skills are not necessary for proto-Bayesian reasoning. Fornacon-Wood, I., Mistry, H., Johnson-Hart, C., Faivre-Finn, C., O'Connor, J. P. B., & Price, G. J. (2022). Understanding the differences between Bayesian and frequentist statistics. International Journal of Radiation Oncology, Biology, Physics, 112(5), 1076– 1082. Gal, I., & Geiger, V. (2022). Welcome to the era of vague news: A study of the demands of statistical and mathematical products in the COVID-19 pandemic media. Educational Studies in Mathematics, 111(1), 5–28. Gallistel, C. R., Krishan, M., Liu, Y., Miller, R., & Latham, P. E. (2014). The perception of probability. Psychological Review, 121(1), 96–123. Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704. Gigerenzer, G., Multmeier, J., Föhring, A., & Wegwarth, O. (2021). Do children have Bayesian intuitions? Journal of Experimental Psychology. General, 150(6), 1041–1070. Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice. The Journal of Mathematical Behavior, 16(2), 145–165. KMK. (2022). Bildungsstandards für das Fach Mathematik Primarbereich. Laplace, P. S. (1902). A philosophical essay on probabilities. John Wiley and Sons (Original work published 1814). Peterson, C. R., & Beach, L. R. (1967). Man as an intuitive statistician. Psychological Bulletin, 68(1), 29–46. Tversky, A., & Kahneman, D. (1973). Availability: A heuristic for judging frequency and probability. Cognitive Psychology, 5(2), 207–232.

1227 1234 Hans Richter Sebastian Geisler University of Potsdam Germany The transition from school to a university mathematics program poses substantial challenges for many students and is associated with high dropout rates. Previous research has shown that students’ interest and self-concept are relevant for study satisfaction, although evidence has often been based on single lectures. The present study aims to replicate these findings using a larger sample comprising multiple university mathematics lecture courses and examines whether satisfaction differs between courses. To account for the hierarchical data structure, multilevel regression analyses were conducted. The results indicate that interest and self-concept related to university mathematics are positively associated with study satisfaction across courses, whereas school-related interest and self-concept show no clear associations. Bong, M., & Skaalvik, E. M. (2003). Academic self-concept and self-efficacy: How different are they really? Educational Psychology Review, 15(1), 1–40. Dieter, M., & Törner, G. (2012). Vier von fünf geben auf. Fors und Lehre, 19(10), 826–827. Engelbrecht, J. (2010). Adding structure to the transition process to advanced mathematical activity. International Journal of Mathematical Education in Science and Technology, 41(2), 143-154. Geisler, S., & Rach, S. (2019). Interest development and satisfaction during the transition from school to university. In M. Graven, H. Venkat, A. A. Essien & P. Vale (Eds.), Proc. of the 43rd Conf. of the Int. Group for the Psyc. of Math. Ed. (Vol. 2, pp. 264–271). PME. Geisler, S., Rach, S., & Rolka, K. (2023). The relation between attitudes towards mathematics and dropout from university mathematics—the mediating role of satisfaction and achievement. ESM, 112(3), 359–381. González-Romá, V., & Hernández, A. (2017). Multilevel modeling: Research-based lessons for substantive researchers. Annual Review of Organizational Psychology and Organizational Behavior, 4, 183–210. Gueudet, G. (2008). Investigating the secondary-tertiary transition. ESM, 67, 237-254. Heublein, U., Hutzsch, C., & Schmelzer, R. (2022). Die Entwicklung der Studienabbruchquoten in Deutschland. DZHW. Kirsten, K., Greefrath, G., & Geisler, S. (2025). Development of interest in mathematics and mathematical self-concept at the transition from school to university and its relation to study success. ESM. https://doi.org/10.1007/s10649-025-10469-y Kosiol, T., Rach, S., & Ufer, S. (2019). (Which) mathematics interest is important for a successful transition to a university study program? International Journal of Science and Mathematics Education, 17, 1583–1608. Krapp, A. (2007). An educational-psychological conceptualisation of interest. International Journal for Educational and Vocational Guidance, 7(1), 5–21. Lubinski, D., & Benbow, C. (2000). States of excellence. Am. Psychologist, 55(1), 137–150 Rach, S., Ufer, S., & Kosiol, T. (2021). Die Rolle des Selbstkonzepts im Mathematikstudium – Wie fit fühlen sich Studierende in Mathematik? Zeitschrift für Erziehungswissenschaft, 24, 1549–1571. Schiefele, U., & Jacob-Ebbinghaus, L. (2006). Lernermerkmale und Lehrqualität als Bedingungen der Studienzufriedenheit. Zeitschrift für Päd Psychologie, 20(3), 199–212. Shavelson, R. J., Hubner, J. J., & Stanton, G. C. (1976). Self-concept: Validation of construct interpretations. Review of Educational Research, 46(3), 407–441. Ufer, S., Rach, S., & Kosiol, T. (2017). Interest in mathematics = interest in mathematics? What general measures of interest reflect when the object of interest changes. ZDM Mathematics Education, 49(3), 397–409.

1235 1242 Sophia Rothering Bettina Roesken-Winter Malte Lehmann Lars Meyer-Jenßen University of Münster Germany Health and Medical University Erfurt Although student ratings are increasingly used to evaluate teaching quality, prior research has focused primarily on teaching in general or on secondary school students. This study therefore developed a questionnaire to capture primary school students’ perspectives on mathematics teaching. The process involved two phases of cognitive interviews with 25 primary school students and one phase of expert ratings provided by ten experts in mathematics education. The resulting 20-item questionnaire was broadly comprehensible to the students and covered the intended constructs, highlighting the importance of integrating both student and expert perspectives in a unified procedure for developing comprehensible, theoretically grounded items. American Educational Research Association [AERA], American Psychological Association [APA], & National Council on Measurement in Education [NCME]. (2014). Standards for educational and psychological testing. American Educational Research Association. Bell, A. (2007). Designing and testing questionnaires for children. Journal of Research in Nursing 12(5), 461–469. Bijlsma, H. (2021). The quality of student perceptions questionnaires: A systematic review. In W. Rollett, H. Bijlsma, & S. Röhl (Eds.), Student feedback on teaching in schools: Using student perceptions for the development of teaching and teachers (pp. 47–72). Springer. Bruner, J. (1966). Toward a theory of instruction. Harvard University Press. Fauth, B., Decristan, J., Rieser, S., Klieme, E., & Büttner, G. (2014). Student ratings of teaching quality in primary school. Dimensions and prediction of student outcomes. Learning and Instruction, 29, 1–9. Hattie, J. (2008). Visible learning. A synthesis of over 800 meta-analyses relating to achievement. Routledge. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 65–97). Macmillan. Hill, H. C., Charalambous, C. Y., & Kraft, M. A. (2012). When rater reliability is not enough: Teacher observation systems and a case for the generalizability study. Educational Researcher, 41(2), 56–64. Klieme, E., Pauli, C., & Reusser, K. (2009). The Pythagoras study: Investigating effects of teaching and learning in Swiss and German mathematics classrooms. In T. Janik & T. Seides (Eds.), The power of video studies in investigating teaching and learning in the classroom (pp. 137–160). Waxmann. Liu, X., Yang Hansen, K., De Neve, J., & Valcke, M. (2024). Teacher versus student perspectives on instructional quality in mathematics education across countries. Instructional Science, 52, 477–513. Messick, S. (1995). Validity of psychological assessment. Validation of inferences from persons’ responses and performances as scientific inquiry into score meaning. American Psychologist, 50(9), 741–749. Prediger, S., Götze, D., Holzäpfel, L., Rösken-Winter, B., & Selter, C. (2022). Five principles for high quality mathematics teaching: Combining normative, epistemological, empirical, and pragmatic perspectives for specifying the content of professional development. Frontiers in Education, 7(969212), 1–15. Renkl, A. (2015). Different roads led to Rome: the case of principle-based cognitive skills. Learning: Research and Practice, 1(1), 79–90. Röhl, S., & Rollett, W. (2021). Student perceptions of teaching quality: Dimensionality and halo effects. In W. Rollett, H. Bijlsma, & S. Röhl (Eds.), Student feedback on teaching in schools: Using student perceptions for the development of teaching and teachers (pp. 31– 46). Springer. Wagner, W., Göllner, R., Helmke, A., Trautwein, U., & Lüdtke, O. (2013). Construct validity of student perceptions of instructional quality is high, but not perfect: Dimensionality and generizability of domain-independent assessments. Learning and Instruction, 28, 1–11. Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms. Review of Educational Research, 78(3), 516– 551. Van der Lans, R. M., Maulana, R., Helms-Lorenz, M., Fernándenz-García, C.-M., Chun, S., de Jager, T., Irnidayanti, Y., Inda-Caro, M., Lee, O., Coetzee, T., Fadhilah, N., Jeon, M., & Moorer, P. (2021). Student Perceptions of Teaching Quality in Five Countries: A Partial Credit Model Approach to Assess Measurement Invariance. Sage Open, 11(3), 1–20. Van der Scheer, E. A., Bijlsma, H. J. E., & Glas, C. A. W. (2019). Validity and reliability of student perceptions of teaching quality in primary education. School Effectiveness and School Improvement, 30(1), 30–50.

1243 1250 Benjamin Rott André Bresges University of Cologne Germany This longitudinal qualitative study investigates how pre-service teachers’ experiences with and beliefs about artificial intelligence (AI) tools develop during a semester-long university seminar. Fifteen pre-service teachers participated in a seminar on mathematics and physics education with weekly sessions. They explored and reflected upon educational applications of AI chatbots and reported on their experiences. Data were collected at three time points through open questions and analysed using qualitative content analysis. Results show clear development from limited prior experience toward confident and reflective use, accompanied by a shift from vague perceptions to more nuanced views emphasizing opportunities while acknowledging risks. Implications are discussed. Almarashdi, H. S., Jarrah, A. M., Abu Khurma, O., & Gningue, S. M. (2024). Unveiling the potential: A systematic review of ChatGPT in transforming mathematics teaching and learning. Eurasia Journal of Mathematics, Science and Technology Education, 20(12), 1– 14. https://doi.org/10.29333/ejmste/15739 Buchholtz, N., Schorcht, S., Baumanns, L., Huget, J., Noster, N., Rott, B., Siller, H.-S., & Sommerhoff, D. (2024). Damit rechnet niemand! Sechs Leitgedanken zu Implikationen und Forschungsbedarfen zu KI-Technologien im Mathematikunterricht [No one expects that! Six key ideas on the implications and research needs of AI technologies in mathematics education]. Mitteilungen der GDM, 117, 15–24. Danyaro, K. U., Abdullahi, S., Abdallah, A. S., & Chiroma, H. (2025). Hallucinations in Large Language Models for Education: Challenges and Mitigation. International Journal of Teaching, Learning and Education, 4(6), 13–19. https://dx.doi.org/10.22161/ijtle.4.6.2 Jancarik, A., & Dusek, O. (2024). The problem of AI hallucination and how to solve it. In F. Moriera (Ed.), Proceedings of the 23rd European Conference on e-Learning – ECEL 2024 (pp. 122–128). ACI, UK. Jee, H. (2023). Emergence of artificial intelligence chatbots in scientific research. Journal of exercise rehabilitation, 19(3), 139–140. https://doi.org/10.12965/jer.2346234.117 Lawson, B. (2006). How Designers Think. Routledge. Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. Klagenfurt. https://nbn-resolving.org/urn:nbn:de:0168-ssoar395173 Rott, B. (2021). Epistemological beliefs and critical thinking in mathematics. Springer. https://doi.org/10.1007/978-3-658-33539-7 Sawyer, A. G. (2024). Artificial Intelligence Chatbot as a Mathematics Curriculum Developer: Discovering Preservice Teachers’ Overconfidence in ChatGPT. International Journal on Responsibility 2024, 7(1), 1–26. https://doi.org/10.62365/2576-0955.1106 Schorcht, S., Buchholtz, N., & Baumanns, L. (2024). Prompt the problem – investigating the mathematics educational quality of AI-supported problem solving by comparing prompt techniques. Frontiers in Education, 9, 1–15. https://doi.org/10.3389/feduc.2024.1386075 Zhuang, Y., & Zhang, S. (2025). Pre-service mathematics teachers’ perceptions of using GenAI for practicing teacher questioning: A semester-long study. EURASIA Journal of Mathematics, Science Technology Education, 21(9). 1–16. https://doi.org/10.29333/ejmste/16764

1251 1258 Laurie H Rubel Shimrit Goidel University of Haifa Israel We investigate ChatGPT-4's pedagogical approach by studying mathematics lessons it generates, with respect to school context descriptors. We generated 45 eighth-grade probability lesson plans across three conditions: a baseline prompt, a "Successful School" context, and a "Struggling School" context. Content analysis of the lesson plans and 368 teacher-action verbs suggest that the LLM tends towards direct instruction and shifts its responses to school descriptors. "Successful School" prompts generated lessons with additional mathematics concepts and advanced technology integration, whereas "Struggling School" prompts produced lessons with more monitoring of students’ progress, reinforcement activities, and attention to socioemotional support. Akgun, S., Greenhow, C. (2022). Artificial intelligence in education: Addressing ethical challenges in K-12 settings. AI Ethics, 2, 431–440. https://doi.org/10.1007/s43681-02100096-7 Biton, Y., & Segal, R. (2025). Learning to craft and critically evaluate prompts: The role of generative AI (ChatGPT) in enhancing pre-service mathematics teachers' TPACK and problem-posing skills. International Journal of Education in Mathematics, Science, and Technology, 13(1), 202–223. https://doi.org/10.46328/ijemst.4654 Broutin, M. S. T. (2024). Exploring mathematics teacher candidates' instrumentation process of generative artificial intelligence for developing lesson plans. Yükseköğretim Dergisi, 14(1), 165–176. https://doi.org/10.53478/yuksekogretim.1347061 Buchholtz, N., & Huget, J. (2024). ChatGPT as a reflection tool to promote the lesson planning competencies of pre-service teachers. In E. Faggiano, A. Clark-Wilson, M. Tabach, & H.-G. Weigand (Eds.), Proceedings of the 17th ERME Topic Conference MEDA 4 (pp. 129–136). University of Bari Aldo Moro. Cameron, S., & Mesiti, C. (2024). What kind of mathematics teacher is ChatGPT? Identifying the pedagogical practices referenced by generative AI tools when preparing lesson plans. In J. Višňovská, E. Ross, & S. Getenet (Eds.), Surfing the waves of mathematics education: Proceedings of the 46th annual conference of the Mathematics Education Research Group of Australasia (pp. 135–142). MERGA. Diversity in Mathematics Education Center for Learning and Teaching. (2007). Culture, race, power and mathematics education. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 405–433). Information Age Publishing. Etgar, R., Guttmann, R., Taub-Tabib, H., & Shmueli, E. (2024). Implicit bias in LLMs: Bias in financial advice based on implied gender (SSRN Scholarly Paper No. 4930361). https://doi.org/10.2139/ssrn.4930361 Gurl, T., Markinson, M., & Artzt, A. F. (2025). Using ChatGPT as a lesson planning assistant in secondary mathematics teacher education. Digital Experiences in Mathematics Education. https://doi.org/10.1007/s40751-024-00162-9 Munter, C., Stein, M. K., & Smith, M. S. (2015). Dialogic and direct instruction: Two distinct models of mathematics instruction and the debate(s) surrounding them. Teachers College Record, 117(11), 1–32. https://doi.org/10.1177/016146811511701102 Naresh, N., Yilmaz, Y., & Galanti, T. (2024). Leveraging the potential of AI as a partner in teaching. AMTE Connections https://amte.net/connections/2024/05/connectionsthematic-articles-artificial-intelligence-mathematics-teacher O'Connor, C., Michaels, S., Chapin, S., Harbaugh, A.G. (2017). The silent and the vocal: Participation and learning in whole-class discussion. Learning and Instruction, 48, 5-13. https://doi.org/10.1016/j.learninstruc.2016.11.003 Pepin, B., Buchholtz, N. & Salinas-Hernández, U. (2025). A scoping survey of ChatGPT in mathematics education. Digital Experiences in Mathematics Education, 11, 9–41. https://doi.org/10.1007/s40751-025-00172-1 Reinholz, D., Johnson, E., Andrews-Larson, C., Stone-Johnstone, A., Smith, J., Mullins, B., Fortune, N., Keene, K., & Shah, N. (2022). When active learning is inequitable: Women’s participation predicts gender inequities in mathematical performance. Journal for Research in Mathematics Education, 53(3), 204–226. https://doi.org/10.5951/jresematheduc-2020-0143 Remillard, J. T. & Kim, O. K. (2020). Elementary mathematics curriculum materials: Designs for student learning and teacher enactment. Springer. https://doi.org/10.1007/978-3-030-38588-0_4 Sapkota, B. & Bondurant, L. (2024). Assessing concepts, procedures, and cognitive demand of ChatGPT-generated mathematical tasks. International Journal of Technology in Education (IJTE), 7(2), 218-238. https://doi.org/10.46328/ijte.677 Şimşek, N. (2025). Integration of ChatGPT in mathematical story-focused 5E lesson planning: Teachers and pre-service teachers' interactions with ChatGPT. Education and Information Technologies, 30, 11391–11462. https://doi.org/10.1007/s10639-024-13258x Walkington, C. (2025). The implications of generative artificial intelligence for mathematics education. School Science and Mathematics, 1–10. https://doi.org/10.1111/ssm.18356 Yanar, B., & Ergene, Ö. (2025). Preservice teachers’ views on using ChatGPT for mathematics lesson planning: Opportunities and threats. Hacettepe Journal of Education, 40(1), 199–219. https://doi.org/10.16896/HUJE.2024127226

1259 1266 Rebecca Rudack Julia Bruns Paderborn University Germany The recognition of patterns and structures is a central component of early mathematical learning and development, yet empirically grounded insights into children’s learning trajectories in patterning remain limited. This study explores how first-grade students approach repeating patterning tasks and how their learning pathways relate to a theoretically grounded intended learning trajectory (LT). Using a qualitative study design, eye-tracking interviews were conducted before and after video-recorded support sessions. The analysis combines gaze data and verbal explanations to capture students’ patterning strategies. Results indicate a shift from unsystematic and pattern-based approaches towards a structural focus on the repeating pattern, evident in both eye movements and justifications, and consistent with the posited intended LT. Baumanns, L., Pitta-Pantazi, D., Demosthenous, E., Christou, C., Lilienthal, A., & Schindler, M. (2024). Pattern-Recognition Processes of First-Grade Students: An Explorative EyeTracking Study. International Journal of Science and Mathematics Education, 22, 1663– 1682. Clements, D. H., & Sarama, J. (2021). Learning and Teaching Early Math: The learning trajectory approach. Studies in Mathematical Thinking and Learning. Lüken, M. M., & Sauzet, O. (2021). Patterning strategies in early childhood: a mixed methods study examining 3- to 5-year-old children’s patterning competencies. Mathematical Thinking and Learning, 23(1), 28–48. Mayring, P. (2000). Qualitative content analysis. Forum: Qualitative Social Research, 1(2), Art. 20. Mulligan, J., & Mitchelmore, M. (2009). Awareness of Pattern and Structure in Early Mathematical Development. Mathematics Education Research Journal, 21(2), 33-49. Mulligan, J., & Mitchelmore, M. (2018). Promoting early mathematical structural development through an integrated assessment and pedagogical program. In I. Elia, J. Mulligan, A. Anderson, A. Baccaglini-Frank, & C. Benz (Eds.), Contemporary research and perspectives on early childhood mathematics education (pp. 17-33). ICME. Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers’ mathematical patterning. Journal for Research in Mathematics Education, 42(3), 237–269. Reuter, D., & Reinhold, F. (2025). Investigating the complexity of pattern recognition in primary students. Educational Studies in Mathematics. Threlfall, J. (1999). Repeating Patterns in the Early Primary Years. In A. Orton (Ed.), Patterns in the Teaching and Learning of Mathematics (pp. 18-30). Wijns, W., De Smedt, B., Verschaffel, L., & Torbeyns, J. (2020). Are preschoolers who spontaneously create patterns better in mathematics? British Journal of Educational Psychology, 90, 753-769.

1267 1274 Beatriz Ruiz Ordóñez Universidad Iberoamericana Mexico Math anxiety, linked to avoidance of numerical tasks and problem solving, remains under-researched in Mexican primary education. This study examines math anxiety and its emotional dimension in third- and fourth-grade students using a mixed-methods design with 1,300 students from 11 schools. Associations between math anxiety, emotional skills, and math performance were analysed. Results show strong negative relationships between emotional skills and math anxiety, and between math anxiety and performance. Students with lower perceived self-efficacy and more negative beliefs reported higher anxiety and lower achievement. High math anxiety was observed among high-performing students, suggesting future avoidance. The findings highlight the importance of the emotional dimension of math anxiety in early education. Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General, 130(2), 224–237. https://doi.org/10.1037/0096-3445.130.2.224 Ashcraft, M., & Krause, J. (2007). Working memory, maths performance, and maths anxiety. Psychonomic Bulleting & Review, 14(2), 243–248. Ashcraft, M. H., & Moore, A. M. (2009). Mathematics anxiety and the affective drop in performance. Journal of Psychoeducational Assessment, 27(3), 197–205. https://doi.org/10.1177/0734282908330580 Bandura, A. (1993). Perceived Self-Efficacy in Cognitive Development and Functioning. Educational Psychologist, 28(2), 117–148. https://doi.org/10.1207/s15326985ep2802_3 Conesa, P., & Duñabeitia, J. (2022). Adaptación y validez del Academic Self-Regulation Questionnaire ( SRQ-A ) en estudiantes españoles de Primaria. Electronic Journal of Research in Education Psychology, 20(57), 403–426. Dowker, A., Cheriton, O., Horton, R., & Mark, W. (2019). Relationships between attitudes and performance in young children’s mathematics. Educational Studies in Mathematics, 100(3), 211–230. https://doi.org/10.1007/s10649-019-9880-5 Davies, K. A., Lane, A. M., Devonport, T. J., & Scott, J. A. (2010). Validity and Reliability of a Brief Emotional Intelligence Scale (BEIS-10). Journal of Individual Differences, 31(4), 198–208. https://doi.org/10.1027/1614-0001/a000028 Ruiz Ordóñez Gillis, J. S. (2003). Cuestionario de Ansiedad Infantil (CAS) (T. Ediciones (ed.); 4a ed.). Hunt, T., Clark-Carter, D. & Sheffield, D. (2014). Math anxiety, intrusive thoughts, and performance. Journal of Education, Psychology and Social Sciences, 2(2), 69–75. https://www.researchgate.net/publication/264905347 Lau, N. T. T., Hawes, Z., Tremblay, P., & Ansari, D. (2022). Disentangling the individual and contextual effects of math anxiety: A global perspective. Proceedings of the National Academy of Sciences of the United States of America, 119(7). https://doi.org/10.1073/pnas.2115855119 Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents' math anxiety on children's math achievement and anxiety. Psychological Science, 26(9), 1480–1488. https://doi.org/10.1177/0956797615592630 OECD. (2010). The High Cost of Low Educational Performance: The Long-run Economic Impact of Improving PISA Outcomes, PISA, OECD Publishing. https://doi.org/10.1787/9789264077485-en. Pajares, F., & Graham, L. (1999). Self-efficacy, motivation constructs, and mathematics performance of entering Middle School students. Contemporary Educational Psychology, 24(2), 124–139. https://doi.org/10.1006/ceps.1998.0991 Pekrun, R., Frenzel, A. C., Goetz, T., & Perry, R. P. (2007). The control-value theory of achievement emotions. An integrative approach to emotions in education. Emotion in Education (pp. 13–36). Academic Press. https://doi.org/10.1016/B978-0123725455/50003-4 Richardson, F. C., & Suinn, R. M. (1972). The mathematics anxiety rating scale: Psychometric data. Journal of Counseling Psychology, 19(6), 551–554. Rossnan, S. (2006). Overcoming math anxiety. Mathitudes, 1, 1–4. Sánchez-Pérez, N., Fuentes, L. J., & González-Salinas, C. (2021). Assessing math anxiety in elementary schoolchildren through a Spanish version of the Scale for Early Mathematics Anxiety (SEMA). PLoS ONE, 16(8 August). https://doi.org/10.1371/journal.pone.0255777 Secretaría de Educación Pública. (2013). Enlace 2013. Wigfield, A., & Eccles, J. S. (2000). Expectancy-value theory of achievement motivation. Contemporary Educational Psychology, 25(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 Wu, S. S., Barth, M., Amin, H., Malcarne, V., & Menon, V. (2012). Math anxiety in second and third graders and its relation to mathematics achievement. Frontiers in Psychology, 3(JUN), 1–11. https://doi.org/10.3389/fpsyg.2012.00162 Zimmerman, B. J., Bandura, A., & Martinez-Pons, M. (1992). Self-Motivation for Academic Attainment: The Role of Self-Efficacy Beliefs and Personal Goal Setting. American Educational Research Journal, 29(3), 663–676. https://doi.org/10.3102/00028312029003663

1275 1282 Soryna Sabbah Nadav Ehrenfeld Weizmann Institute of Science Israel Mathematics education research increasingly examines facilitators’ decision-making through “lifting strategies” by which frameworks for researching mathematics teachers’ sensemaking are being lifted to the facilitator level. One such central example is the ROGI framework (Resources, Orientations, Goals, Identity; Karsenty et al., 2021). In this paper, we explore the possibility of lifting an ecological perspective on mathematics teachers’ sensemaking and agency, combining it with the ROGI framework, to explore sensemaking and decision-making processes of one facilitator leading a year-long professional learning community (PLC) for mathematics teachers. The study conceptualizes and illustrates facilitator agency as an ongoing accomplishment achieved through decision-making across the PLC ecologies. Biesta, G., & Tedder, M. (2007). Agency and learning in the lifecourse: Towards an ecological perspective. Studies in the Education of Adults, 39(2), 132–149. https://doi.org/10.1080/02660830.2007.11661545 Borko, H., Koellner, K., & Jacobs, J. (2014). Examining novice teacher leaders’ facilitation of mathematics professional development. The Journal of Mathematical Behavior, 33, 149–167. https://doi.org/10.1016/j.jmathb.2013.11.003 Cobb, P., Jackson, K., Henrick, E., & Smith, T. M. (2018). Systems for instructional improvement: Creating coherence from the classroom to the district office. Cambridge, MA: Harvard Education Press. Ehrenfeld, N. (2022). Framing an ecological perspective on teacher professional development. Educational Researcher, 51(7), 489–495. https://doi.org/10.3102/0013189X221112113 Karsenty, R., Pöhler, B., Schwarts, G., Prediger, S., & Arcavi, A. (2021). Processes of decision-making by mathematics PD facilitators: The role of resources, orientations, goals, and identities. Journal of Mathematics Teacher Education, 26(1), 27–51. https://doi.org/10.1007/s10857-021-09518-z Priestley, M. R., Biesta, G., & Robinson, S. (2015). Teacher agency: An ecological approach. Bloomsbury Publishing.

1283 1290 Raffaele Casi Cristina Sabena Carlotta Soldano University of Torino Italy Grounding on Variation Theory, this study examines how four-year-old children identify and manage critical aspects of visual stimuli during gameplay in a kindergarten setting, and how teacher scaffolding supports this process. Data were drawn from the implementation of the Betta-the-Bee card game, a mathematically rich game designed to engage children with the discernment of numeric and spatial variables depicted on the game cards. Findings from a case-study analysis indicate that structured, variation-rich games can effectively foster dynamic discernment processes, with teacher guidance playing a crucial role in supporting these emerging skills. Casi, R., Sabena, C., & Soldano, C. (in press). Fostering logical reasoning with the “Bettathe-Bee” game. In T. Meaney et al. (Eds.), Engaging with Mathematics in the Early Years. Springer. 1 - 1289 Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131. Huang, R., & Li, Y. (Eds.). (2017). Teaching and learning mathematics through variation: Confucian heritage meets Western theories. Sense Publishers. Kullberg, A., Runesson Kempe, U., & Marton, F. (2017). What is made possible to learn when using the variation theory of learning? ZDM—Mathematics Education, 49, 559–569. https://doi.org/10.1007/s11858-017-0858-4 Marton, F. (2015). Necessary conditions of learning. Routledge. Marton, F., & Booth, S. (1997). Learning and awareness. Lawrence Erlbaum Associates. Marton, F., & Pang, M. F. (2013). Meanings are acquired from experiencing differences against a background of sameness, rather than from experiencing sameness against a background of difference: Putting a conjecture to the test by embedding it in a pedagogical tool. Frontline Learning Research, 1(1), 24–41. https://doi.org/10.14786/flr.v1i1.16 McKenney, S., & Reeves, T. C. (2019). Conducting educational design research. Routledge. Pang, M. F., Bao, J., & Ki, W. W. (2017). ‘Bianshi’ and the Variation Theory of Learning: Illustrating two frameworks of variation and invariance in the teaching of mathematics. In R. Huang & Y. Li (Eds.), Teaching and learning mathematics through variation: Confucian heritage meets Western theories (pp. 43–67). Sense Publishers. Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237–268. Runesson, U. (2006). What is it possible to learn? On variation as a necessary condition for learning. Scandinavian Journal of Educational Research, 50(4), 397–410. https://doi.org/10.1080/00313830600823753 Russo, J., Russo, T., & Bragg, L. A. (2018). Five principles of educationally rich mathematical games. Australian Primary Mathematics Classroom, 23(3), 30–34. Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111. 1 - 1290

1291 1298 Barana Alice Boetti Giulia Marchisio Conte Marina Perrotta Adamaria Matteo Sacchet Università degli Studi di Torino University College Dublin This paper investigates collaborative problem solving in the context of an online international experience in higher education. The study aims to examine how students interact while working in mixed international groups on application-oriented linear algebra tasks. Data were collected through students’ responses to individual reflective questionnaires focusing on their perceptions of collaborative learning. The responses were analysed using the adapted Interaction Analysis Model to identify phases of knowledge co-construction. The results show that all phases of the model emerged across all group tasks. Although some differences were observed among groups, the findings suggest that international online collaborative activities can effectively support collaboration in solving applied linear algebra problems. Barana, A., Boetti, G., Marchisio, M., Perrotta, A., & Sacchet, M. (2023a). Investigating the Knowledge Co-Construction Process in Homogeneous Ability Groups during Computational Lab Activities in Financial Mathematics. Sustainability, 15(18), 13466. Barana, Boetti, Marchisio Conte, Perrotta, Sacchet Barana, A., Boetti, G., Marchisio, M., Perrotta, A., & Sacchet, M. (2025a). Mathematics for interculturality: Collaborative online international learning in linear algebra for finance, security and defence strategies. In. S. Nic Mhuirí, L. Harbison & M. Ryan (Eds.). Proc. of the 10th Conference on Research in Mathematics Education in Ireland (MEI 10), 29-34. Barana, A., Chatzea, V., Henao, K., Hildebrandt, A., Marchisio Conte, M., Samoilovich, D., Triantafyllidis, G., & Vidakis, N. (2025b). Designing an online training module to develop virtual and blended international modalities in higher education. Proceedings of the Int. Conferences Mobile Learning 2025 and Ed.ucational Technologies 2025, IADIS Press, 117124. Barana, A., Marchisio, M., Perrotta, A., & Sacchet, M. (2023b). Collaborative Knowledge Construction during Computational Lab Activities in Financial Mathematics. In Proceedings of the 9th International Conference on Higher Education Advances (HEAd’23), Editorial Universitat Politècnica de València, 1021-1028. Cobb, P., Yackel, E. & Wood, T. (1992). Interaction and learning in mathematics classroom situations. Educ Stud Math 23, 99-122. Felmer, P. (2023a). Collaborative problem-solving in mathematics, Current Opinion in Behavioral Sciences, 52, 101296. Felmer, P. (2023b). Implementation of Collaborative Problem Solving: Experiences in Chile. In: M. Ayalon, B. Koichu, R. Leikin, L. Rubel & M. Tabach (Eds.). Proc. 46th Conf. of the Int. Group for the Psychology of Mathematics Education, (Vol. 1, pp. 35-51). PME. Gillies, R. M. (2016). Cooperative learning: Review of research and practice. Australian Journal of Teacher Education, 41(3), 39-54. Gunawardena, C. N., Lowe, C. A., & Anderson, T. (1997). Analysis of a Global Online Debate and the Development of an Interaction Analysis Model for Examining Social Construction of Knowledge in Computer Conferencing. Journal of Educational Computing Research, 17(4), 397-431. Hannula, M. S. (2022). Explorations on visual attention during collaborative problem solving. In: Fernández, Ceneida, et al. (Eds.). Proc. 45th Conf. of the Int. Group for the Psychology of Mathematics Education, (Vol. 1, pp. 19-34). PME. Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom. Journal of Mathematical Behavior, 26(1), 60-82. Lin, L., Mills, L.A., & Ifenthaler, D. (2015). Collaborative problem solving in shared space. Proceedings of the 12th International Conference on Cognition and Exploratory Learning in Digital Age (CELDA 2015), 233-239. Retnowati, E., Ayres, P., & Sweller, J. (2017). Can collaborative learning improve the effectiveness of worked examples in learning mathematics? Journal of Educational Psychology, Vol 109(5), Jul 2017, 666-679. Sfard, A. (2001). Learning mathematics as developing a discourse. In R. Speiser, C. Maher, C. Walter (Eds), Proc. 23rd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 23-44). PME. Stahl, G. (2006). Group cognition: Computer support for building collaborative knowledge. MIT Press.

1299 1306 Laura Schmidt Krawitz Janina Krawitz University of Cologne Germany In real life, problems arise and have to be posed before they can be solved. While interest in research on mathematical problem posing is growing, affective aspects, particularly emotions, have received little attention. To contribute to closing this gap, we examined (RQ1) whether students’ enjoyment and anger differed across problem-posing activities and (RQ2) which events triggered these emotions during problem posing. We analyzed the problem-posing processes of two tenth-grade students by applying facial expression analysis to assess their emotions. Students expressed more enjoyment in problem evaluation compared with problem solving and more anger in problem solving compared with all other activities. Among various emotional triggers, enjoyment tended to occur when students set up and adjusted real-world conditions. Baumanns, L., & Rott, B. (2022). The process of problem posing: development of a descriptive phase model of problem posing. Educational Studies in Mathematics, 110(2), 251–269. Bishay, M., Preston, K., Strafuss, M., Page, G., Turcot, J., & Mavadati, M. (2022). AFFDEX 2.0: A Real-Time Facial Expression Analysis Toolkit. Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-Posing Research in Mathematics Education: Some Answered and Unanswered Questions. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing (pp. 3–34). Springer New York. Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: conceptualization, advances, and future directions for research. Educational Studies in Mathematics, 105(3), 287–301. Hannula M. S. (2015). Emotions in Problem Solving. In S.J. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education, pp. 269-288. Springer. Krawitz, J., Schukajlow, S., & Hartmann, L. (2025). Does problem posing affect self-efficacy, task value, and performance in mathematical modelling? Educational Studies in Mathematics, 119(3), 445–466. Linnenbrink-Garcia, L., Patall, E. A., & Pekrun, R. (2016). Adaptive Motivation and Emotion in Education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 228–236. Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. Pekrun, R. (2024). Control-Value Theory: From Achievement Emotion to a General Theory of Human Emotions. Educational Psychology Review, 36(83). Schindler, M. & Bakker, A. (2020) Affective field during collaborative problem posing and problem solving: a case study. Educational Studies in Mathematics, 105, 303–324. Silver, E. A. (1994). On Mathematical Problem Posing. For the Learning of Mathematics, 14(1), 19–28. Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In P. C. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Mathematics Education Research Group of Australasia. van Bommel, J., & Palmér, H. (2021). Young students’ views on problem solving versus problem posing. Journal of Childhood, Education & Society, 2(1), 1–13.

1307 1314 Stanislaw Schukajlow Janina Krawitz University of Münster Germany University of Cologne Germany Mathematical modelling comprises various activities and focuses on diverse research foci. However, overviews of these aspects are largely missing. In a systematic literature review of 105 papers published between 2020 and 2024, we asked: (1) What modelling activities were addressed in research? (2) What research foci were explored, and what activities were analyzed within these foci? Analysis revealed that most studies examined the modelling process holistically. Among specific activities, research studies most frequently analyzed mathematizing, working mathematically, and validating. Modelling performance was the most frequently examined research focus, alongside affect, technology, and STEM. Creativity was analyzed holistically, whereas other foci (e.g., problem posing) were assessed through activities such as simplifying. Babb, A. P. P., Acuña, F. P., Rocha, Y. A. O., & Rojas, A. S. (2023). Diversity of perspectives on mathematical modelling: A review of the international landscape. Advancing and Consolidating Mathematical Modelling, 43–57. https://doi.org/10.1007/978-3-031-271151_3 Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical modelling problems? The example sugarloaf and the DISUM project. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical Modelling (ICTMA 12): Education, Engineering Economics (pp. 222–231). Horwood. https://doi.org/10.1533/9780857099419.5.221 Borromeo Ferri, R. (2018). Learning how to teach mathematical modeling in school and teacher education. Springer. https://doi.org/10.1007/978-3-319-68072-9 Cevikbas, M., Kaiser, G., & Schukajlow, S. (2022). A systematic literature review of the current discussion on mathematical modelling competencies: state-of-the-art developments in conceptualizing, measuring, and fostering. Educational Studies in Mathematics, 109(2), 205–236. https://doi.org/10.1007/s10649-021-10104-6 Czocher, J. A. (2018). How does validating activity contribute to the modeling process? Educational Studies in Mathematics, 99, 137–159. Ferrando, I., & Albarracín, L. (2021). Students from grade 2 to grade 10 solving a Fermi problem: Analysis of emerging models. Mathematics Education Research Journal, 33(1), 61–78. https://doi.org/10.1007/s13394-019-00292-z Geiger, V., & Frejd, P. (2015). A reflection on mathematical modelling and applications as a field of research: Theoretical orientation and diversity. In Mathematical Modelling in Education Research and Practice (pp. 161–171). Springer. Krawitz, J., Schukajlow, S., Yang, X., & Geiger, V. (2025). A systematic review of international perspectives on mathematical modelling: Modelling goals and task characteristics. ZDM – Mathematics Education, 57, 193–212. Mayring, P. (2015). Qualitative content analysis: theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 365–380). Springer. Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. https://doi.org/10.4324/9781315189314 Page, M. J., McKenzie, J. E., Bossuyt, P. M., Boutron, I., Hoffmann, T. C., Mulrow, C. D., Shamseer, L., Tetzlaff, J. M., Akl, E. A., & Brennan, S. E. (2021). The PRISMA 2020 statement: an updated guideline for reporting systematic reviews. bmj, 372(71), 1–9. Schukajlow, S., Kaiser, G., & Stillman, G. (2023). Modeling from a cognitive perspective: theoretical considerations and empirical contributions. Mathematical Thinking and Learning, 25(3), 259–269. https://doi.org/10.1080/10986065.2021.2012631 Stillman, G., & Brown, J. P. (2014). Evidence of implemented anticipation in mathematising by beginning modellers. Mathematics Education Research Journal, 26(4), 763–789.

1315 1322 Uwe Schürmann Georg Bruckmaier Alexander Repenning University of Applied Sciences and Arts Northwestern Switzerland University of Basel University of Colorado Boulder United States This exploratory study investigates learning processes related to problem solving and probabilistic thinking through the programming of computer games with random elements – based on Seymour Papert’s constructionist approach. The data consist of video recordings of programming phases and video-stimulated recall interviews with four student pairs (Year 6) from two Swiss primary schools. The results show cyclical design processes reflecting key components of computational thinking. Probabilistic preconceptions emerge and partly develop during the game design process. Bruckman, A. (1999). Can educational be fun. Game Developers Conference, 99, 75–79. Buckley, P., & Doyle, E. (2016). Gamification and student motivation. Interactive Learning Environments, 24(6), 1162–1175. https://doi.org/10.1080/10494820.2014.964263 Chen, P., Yang, D., Metwally, A. H. S., Lavonen, J., & Wang, X. (2023). Fostering computational thinking through unplugged activities: A systematic literature review and meta-analysis. International Journal of STEM Education, 10(1), 47. https://doi.org/10.1186/s40594-023-00434-7 EDK. (2011). Grundkompetenzen für die Mathematik. Nationale Bildungsstandards. https://edudoc.ch/record/96784/files/grundkomp_math_d.pdf Kafai, Y. B. (2018). Constructionist visions: Hard fun with serious games. International Journal of Child-Computer Interaction, 18, 19–21. https://doi.org/10.1016/j.ijcci.2018.04.002 Kafai, Y. B., Franke, M. L., Ching, C. C., & Shih, J. C. (1998). Game Design as an Interactive Learning Environment for Fostering Students’ and Teachers’ Mathematical Inquiry. International Journal of Computers for Mathematical Learning, 3(2), 149–184. https://doi.org/10.1023/A:1009777905226 Kallia, M., van Borkulo, S. P., Drijvers, P., Barendsen, E., & Tolboom, J. (2021). Characterising computational thinking in mathematics education: A literature-informed Delphi study. Research in Mathematics Education, 23(2), 159–187. https://doi.org/10.1080/14794802.2020.1852104 Kuckartz, U., & Radiker, S. (2023). Qualitative Content Analysis: Methods, Practice and Software (2nd ed.). SAGE. Mayring, P., & Fenzl, T. (2022). Qualitative Inhaltsanalyse. In N. Baur & J. Blasius (Eds.), Handbuch Methoden der empirischen Sozialforschung (pp. 691–706). Springer. https://doi.org/10.1007/978-3-658-37985-8_43 Nyimbili, F., & Nyimbili, L. (2024). Types of Purposive Sampling Techniques with Their Examples and Application in Qualitative Research Studies. British Journal of Multidisciplinary Advanced Studies, 5(1), 90–99. https://doi.org/10.37745/bjmas.2022.0419 Papert, S. (1986). Constructionism: A New Opportunity for Elementary Science Education. Massachusetts Institute of Technology, Media Laboratory, Epistemology and Learning Group. Papert, S. (1996). An exploration in the space of mathematics educations. International Journal of Computers for Mathematical Learning, 1(1), 95–123. https://doi.org/10.1007/BF00191473 Plass, J. L., Homer, B. D., & Kinzer, C. K. (2015). Foundations of Game-Based Learning. Educational Psychologist, 50(4), 258–283. https://doi.org/10.1080/00461520.2015.1122533 Resnick, M., & Rosenbaum, E. (2013). Designing for Tinkerability. In M. Honey & D. Kanter (Eds.), Design, Make, Play: Growing the Next Generation of STEM Innovators (pp. 163– 181). Routledge. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 334–370). Macmillan. Schreier, M. (2012). Qualitative Content https://doi.org/10.4135/9781529682571 Analysis in Practice. SAGE. Shute, V. J., Sun, C., & Asbell-Clarke, J. (2017). Demystifying computational thinking. Educational Research Review, 22, 142–158. https://doi.org/10.1016/j.edurev.2017.09.003 Stacey, K., Price, B., Gvozdenko, E., & Steinle, V. (2013). Specific Mathematics Assessments that Reveal Thinking. http://www.smartvic.com/teacher/ Weintrop, D., Beheshti, E., Horn, M., Orton, K., Jona, K., Trouille, L., & Wilensky, U. (2016). Defining Computational Thinking for Mathematics and Science Classrooms. Journal of Science Education Technology, 25(1), 127–147. https://doi.org/10.1007/s10956-015-9581-5 Wing, J. M. (2006). Computational thinking. Commun. https://doi.org/10.1145/1118178.1118215 ACM, 49(3), 33–35. Woo, K., & Falloon, G. (2022). Problem solved, but how? An exploratory study into students’ problem solving processes in creative coding tasks. Thinking Skills and Creativity, 46, 101193. https://doi.org/10.1016/j.tsc.2022.101193

1323 1330 Ruti Segal Yaniv Biton Olga Fellus Oranim Academic College of Education Centre for Educational Technology (CET) Israel Brock University This paper is part of a larger mixed-methods study that focuses on 152 students’ and 24 teachers’ perspectives of the experience of learning mathematics via anonymous WhatsApp posts. The study was conducted within the Bagroup program, a statewide initiative supporting Grade 11-12 students in preparing for their matriculation exams in mathematics. Following registration, students were assigned to 40 WhatsApp groups with approximately 100 students each. The groups engage in solving mathematical problems for three months before the exam. Social-emotional aspects were the objects of investigation. Data comprised the WhatsApp posts and questionnaires. Findings suggest that learning through WhatsApp occasions makes readily available emotional support and an increased sense of confidence. Biton, Y., & Segal, R. (2024). Learning mathematics via the WhatsApp social network in the Bagroup national project: Students’ perceptions. Eurasia Journal of Mathematics, Science and Technology Education, 20 (8), em2484. https://doi.org/10.29333/ejmste/14876.Biton, Y., Fellus, O., Hershkovitz, S., & Hoch, M. (2015/April). Using the lens of complexity theory to examine a second chance at school mathematizing through Facebook. Paper presented at the Annual Meeting of the American Educational Research Association (AERA), Chicago, Illinois. Biton, Y., Segal, R. (2021). Learning and teaching mathematics with online social networks: The case of Facebook [Online First], IntechOpen, DOI: 10.5772/intechopen.95998. Boaler, J. (2016). Designing mathematics classes to promote equity and engagement. The Journal of Mathematical Behavior, 41, 172-178. Collaborative for Academic, Social, and Emotional Learning (CASEL) (2020). CASEL's SEL Framework. https://casel.org/casel-sel-framework-11-2020/ Corbin, J., & Strauss, A. (2014). Basics of qualitative research: Techniques and procedures for developing grounded theory. Sage Publications. Creswell, J. W. (2014). Qualitative, quantitative, and mixed methods approaches. Sage. Durgungoz, A., & Durgungoz, F. C. (2022). “We are much closer here”: Exploring the use of WhatsApp as a learning environment in a secondary school mathematics class. Learning Environments Research, 25(2), 423–444. Dyson, B., Vickers, K., Turtle, J., Cowan, S., & Tassone, A. (2015). Evaluating the use of Facebook to increase student engagement and understanding in lecture-based classes. Higher Education, 69, 303-313. Fellus, O. (2019). Connecting the dots: Toward a four-dimensional conceptualization and operationalization of identity in mathematics education. ZDM Mathematics Education, 51(3), 445-455. Greenhow, C., & Askari, E. (2017). Learning and teaching with social network sites: A decade of research in K-12 related education. Education and Information Technologies, 22(2), 623-645. Greenhow, C., & Lewin, C. (2016). Social media and education: Reconceptualizing the boundaries of formal and informal learning. Learning, Media and Technology, 41(1), 630. Madge, M., Meek, J., Wellens, J., & Hooley, T. (2009). Facebook, social integration and informal learning at university: ‘It is more for socialising and talking to friends about work than for actually doing work.’ Learning, Media and Technology, 34(2), 141-155. Malka, V., Ariel, Y., & Avidar, R. (2015). Fighting, worrying and sharing: Operation ‘Protective Edge’ as the first WhatsApp war. Media, War & Conflict, 8(3), 329-344. National Council of Teachers of Mathematics (NCTM). (2021). Continuing the journey: Mathematics learning 2021 and beyond. NCTM. Nida, N. K., Usodo, B., & Saputro, D. R. S. (2020). The blended learning with WhatsApp media on mathematics creative thinking skills and math anxiety. Journal of Education and Learning (EduLearn), 14(2), 307–314. http://dx.doi.org/10.11591/edulearn.v14i2.16233 Safir, S., & Dugan, J. (2021). Street data: A next-generation model for equity, pedagogy, and school transformation. Corwin. Sears, R., Bay-Williams, J., Willingham, J. C., & Cullen, A. (2022). Symbiosis: Social and emotional learning & mathematics learning. Mathematics Teacher: Learning and Teaching PK-12, 115(11), 770–780. http://dx.doi.org/10.5951/MTLT.2022.0081 Segal, R., & Biton, Y. (2024). Teaching and learning high-school mathematics via WhatsApp: Teachers’ perspectives. Learning Environment Research, 1-29. Tella, A. (2014). Globalisation, blended learning, and mathematics education: Implications for pedagogy in tertiary institutions. In N. P. Ololube (Ed.), Advancing technology and educational development through blended learning in emerging economies, (pp. 190-211). IGI Global. Tong, D. H., Uyen, B. P., & Quoc, N. V. A. (2021). The improvement of 10th students’ mathematical communication skills through learning ellipse topics. Heliyon, 7(1), e08282. Yeo, M. M. L. (2014). Social media and social networking applications for teaching and learning. European Journal of Science and Mathematics Education, 2(1), 53-62. Wilkie, K. J., & Sullivan, P. (2018). Exploring intrinsic and extrinsic motivational aspects of middle school students’ aspirations for their mathematics learning. Educational Studies in Mathematics, 97(3), 235–254. https://doi.org/10.1007/s10649-017-9795-y

1331 1338 Alexander Jonas Viktor Selling Guri A. Nortvedt University of Oslo Norway This paper reports from a study investigating collaborative problem solving in upper secondary mathematics classrooms following the principles of the Thinking class-rooms approach, focusing on teacher scaffolding strategies. In the observed lessons, students worked in groups of three attempting to solve problems involving solids of revolution and trigonometry. Overall, conceptual scaffolding was the most often observed scaffold. In addition, procedural, metacognitive, participatory, and epistemic scaffolding was observed. The most common scaffolding practice was questioning. Throughout the lesson the teacher managed to maintain the complexity of the task by withdrawing scaffolding at an early stage, thereby transferring responsibility to the students and allowing them to struggle with the problem. Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9(1), 33-52. Calder, N. (2015). Student wonderings: Scaffolding student understanding within studentcentred inquiry learning. ZDM, 47(7), 1121-1131. Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational studies in Mathematics, 49(2), 193-223. Gou, P., Li, W., & Teng, T. (2025). How can Scaffolding Effectively Promote Students’ Problem-Solving Ability: A Meta-Analysis. The Asia-Pacific Education Researcher, 1-14. Klette, K. (2023). Towards programmatic research when studying classroom teaching and learning. In Didactics in a changing world: European perspectives on teaching, learning and the curriculum (pp. 137-158). Cham: Springer International Publishing. Liljedahl, P. (2021). Building thinking classrooms. Corwin Press. Muir, T. (2025). Building a thinking classroom: a case study of implementation and professional growth. Journal of Mathematics Teacher Education, 1-29. Schoenfeld, A. (1987). What’s all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Lawrence Erlbaum A. Thompson, J. (2022). A guide to abductive thematic analysis. The qualitative report, 27(5), 1410-1421. Van de Pol, J., Volman, M., & Beishuizen, J. (2010). Scaffolding in teacher–student interaction: A decade of research. Educational psychology review, 22(3), 271-296. Webb, N. M., Franke, M. L., Ing, M., Turrou, A. C., Johnson, N. C., & Zimmerman, J. (2019). Teacher practices that promote productive dialogue and learning in mathematics classrooms. International Journal of Educational Research, 97, 176-186. Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of child psychology and psychiatry, 17(2), 89-100.

1339 1346 Jisoo Seo Gabriel J. Stylianides Maria Evangelou University of Oxford United Kingdom International Hellenic University Despite a focus on mechanisms underlying the spatial–mathematical link, few studies have examined processes mediating spatial ability and young pupils’ one-step additive word-problem solving. Adopting the Potential Outcomes Framework, this study investigates if ‘Correct Number-Fact’ strategy use—a mental strategy relying on the accurate representation of a problem’s quantitative structure—mediates the relationship between spatial visualization and problem-solving accuracy. Using data from 137 pupils (6- to 7-year-olds) in England while controlling for key covariates (visuospatial working memory, fluid reasoning, language ability, numerical ability, age and gender), the analyses showed that spatial visualization exerted a significant indirect effect on problem-solving accuracy via strategy use. Atit, K., Power, J.R., Pigott, T., Lee, J., Geer, E. A., Uttal, D. H., Ganely, C. M., & Sorby, S. A. (2022). Examining the relations between spatial skills and mathematical performance: A meta-analysis. Psychonomic Bulletin & Review 29, 699–720. https://doi.org/10.3758/s13423-021-02012-w Bailey, D. H. (2017). Causal inference and the spatial–math link in early childhood. Monographs of the Society for Research in Child Development, 82(1), 127–136. https://doi.org/10.1111/mono.12288 Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51(6), 1173–1182. https://doi.org/10.1037/00223514.51.6.1173 Boonen, A. J. H., van der Schoot, M., van Wesel, F., de Vries, M. H., & Jolles, J. (2013). What underlies successful word problem solving? A path analysis in sixth grade students. Contemporary Educational Psychology, 38(3), 271–279. https://doi.org/10.1016/j.cedpsych.2013.05.001 Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. In T. P. Carpenter, J. M. Moser & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 9–24). Erlbaum. Casey, B. M., Lombardi, C. M., Pollock, A., Fineman, B., & Pezaris, E. (2017). Girls’ spatial skills and arithmetic strategies in first grade as predictors of fifth-grade analytical math reasoning. Journal of Cognition Development, 18(5), 530–555. https://doi.org/10.1080/15248372.2017.1363044 Chi, W. E., Huang, S., Jeon, M., Park, E. S., Melguizo, T., & Kezar, A. (2022). A practical guide to causal mediation analysis: Illustration with a comprehensive college transition program and nonprogram peer and faculty interactions. Frontiers in Education, 7, Article 886722. https://doi.org/10.3389/feduc.2022.886722 Cuartas, J., & McCoy, D. C. (2021). Causal mediation in developmental science: A primer. International Journal of Behavioral Development, 45(3), 269–274. https://doi.org/10.1177/0165025420981640 Harrell, F. E., Jr. (2015). Regression modeling strategies: With applications to linear models, logistic and ordinal regression, and survival analysis (2nd ed.). Springer. Hawes, Z., & Ansari, D. (2020). What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior. Psychonomic Bulletin & Review, 27(3), 465–482. https://doi.org/10.3758/s13423-019-01694-7 Hawes, Z., Moss, J., Caswell, B., Seo, J., & Ansari, D. (2019). Relations between numerical, spatial, and executive function skills and mathematics achievement: A latent-variable approach, Cognitive Psychology, 109, 68–90. https://doi.org/10.1016/j.cogpsych.2018.12.002 Imai, K., Keele, L., & Tingley, D. (2010). A general approach to causal mediation analysis. Psychological Methods, 15(4), 309–334. https://doi.org/10.1037/a0020761 Möhring, W., Moll, L., & Szubielska, M. (2024). Unpacking associations among children’s spatial skills, mathematics, and arithmetic strategies: Decomposition matters. Psychological Research, 88(5), 1550–1564. https://doi.org/10.1007/s00426-024-01952-x Nunes, T., & Bryant, P. (1996). Children doing mathematics. Blackwell Publishing. Nunes, T., Bryant, P., Evans, D., & Barros, R. (2015). Assessing quantitative reasoning in young children. Mathematical Thinking and Learning, 17(2-3), 178–196. https://doi.org/10.1080/10986065.2015.1016815 Nunes, T., Bryant, P., Sylva, K., & Barros, R. (2009). Development of maths capabilities and confidence in primary school. Dept. for Children, Schools and Families. Seo, J., Stylianides, G. J., & Evangelou, M. T. (2025). Spatial ability and one-step additive word problems in early childhood. In M. Bosch, G. Bolondi, S. Carreira, C. Spagnolo, & M. Gaidoschik (Eds.), Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14). (pp. 2120–2127). Free University of Bozen-Bolzano and ERME. Spencer, M., Fuchs, L. S., Geary, D. C., & Fuchs, D. (2022). Connections between mathematics and reading development: Numerical cognition mediates relations between foundational competencies and later academic outcomes. Journal of Educational Psychology, 114(2), 273–288. https://doi.org/10.1037/edu0000670 Strohmaier, A. R., Reinhold, F., Hofer, S., Berkowitz, M., Vogel-Heuser, B., & Reiss, K. (2022). Different complex word problems require different combinations of cognitive skills. Educational Studies in Mathematics, 109(1), 89–114. https://doi.org/10.1007/s10649-021-10079-4 Tingley, D., Yamamoto, T., Hirose, K., Keele, L., & Imai, K. (2014). mediation: R Package for Causal Mediation Analysis. Journal of Statistical Software, 59(5), 1–38. https://doi.org/10.18637/jss.v059.i05 Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In T. P. Carpenter, J. M. Moser & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 39–59). Erlbaum. Verschaffel, L., Schukajlow, S., Star, J., & Van Dooren, W. (2020). Word problems in mathematics education: a survey. ZDM - Mathematics Education, 52(1), 1–16. https://doi.org/10.1007/s11858-020-01130-4

1347 1354 Sashi Sharma University of Waikato New Zealand University of Canterbury Norwegian University of Science and Technology Norway In multilingual mathematics classrooms, language is often conceptualised as a resource, a supportive tool that teachers may deploy to assist learners in accessing mathematical content. While this perspective has contributed significantly to equity-oriented practice, it risks treating students’ linguistic repertoires instrumentally and fails to recognise the epistemic functions of language in constructing mathematical meanings. Drawing on classroom empirical illustrations from probability lessons, this paper advances a theoretical and practice-oriented shift from understanding language as a resource to understanding languages as sources of meaning (Barwell, 2018). Implications for teaching practice are discussed, emphasizing the centrality of multilingual meaning-making in shaping mathematical thinking in statistics classrooms Barwell, R. (2018). From language as a resource to sources of meaning in multilingual mathematics classrooms. The Journal of Mathematical Behavior, 50, 155-168. Barton, B., Fairhall, U., & Trinick, T. (2007). Mathematics education and language: Lessons from the Māori-medium classroom. Springer. Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on teaching and learning probability. Springer. Bishop, A. (1988). Mathematical enculturation: A cultural perspective on mathematics education. Kluwer. Bishop, R., & Glynn, T. (1999/ 2003). Culture counts: Changing power relations in education. Zed Books. Borovcnik, M., & Kapadia, R. (2014). A historical and philosophical perspective on probability. In E. Chernoff & B. Sriraman (Eds.), Probabilistic thinking (pp. 7–34). Springer. https://doi.org/10.1007/978-94-007-7155-0_2 Farsani, D. (2022). The name game: The role of cultural affordances in learning probability. For the Learning of Mathematics, 42(2), 17–18. Hunter, R., Hunter, J., & Bills, T. (2022). Learning mathematics through home languages. NZCER Press. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from. Basic Books. Moschkovich, J. (2018). Talking to learn mathematics with understanding: Supporting academic literacy in mathematics for English learners. In A. Bailey, C. Maher, & L. C. Wilkinson (Eds.), Language, literacy, and learning in the STEM disciplines: Language counts for English learners (pp. 13–34). Routledge Taylor Francis. Planas, N. (2018). Language as resource: A key notion for understanding the complexity of mathematics learning. Educational Studies in Mathematics, 98(3), 215–229. http://doi.org/10.1007/s10649-018-9810-y Prediger, S., & Uribe, A. (2021). Exploiting the epistemic role of multilingual resources in superdiverse mathematics classrooms: Design principles and insights into students’ learning processes. In A. Fritz, E. Gursoy, & M. Herzog (Eds.), Diversity Dimensions in Mathematics and Language Learning (pp. 80–97). De Gruyter/Mouton. Radford, L. (2014). Gesture–speech unity in mathematical thinking. Springer. Ryan, U., & Parra, A. (2019). Epistemological aspects of multilingualism in mathematics education. Research in Mathematics Education, 21, 1–17. Sfard, A. (2008). Thinking as communicating. Cambridge University Press. Sharma, S. (2014). Influence of culture on high school students’ understanding of statistics: A Fijian perspective. Statistics Education Research Journal, 13(2), 104–117. Sharma, S. (2024). Using home languages as a resource to enhance statistical thinking in a multicultural classroom. (TLRI Final Report). New Zealand Council for Ed. Research. In-Confidence

1355 1362 Victoria Shure Bettina Roesken-Winter University of Münster Germany Mathematics teacher professional development (TPD) has received increasing attention in recent years, with a focus on supporting teacher learning through discussion. Through a situated examination with a fictional TPD situation, this study explores how facilitators (N = 116) envision supporting teacher learning through their discussion moderation strategies. Findings highlight that the facilitators overwhelmingly favoured probing teachers’ thinking and eliciting teachers’ thinking as most supportive and viewed making their own contributions and, to a lesser extent, orienting the teachers to the thinking of others as less favourable. The study provides insights into facilitators’ discussion-leading with implications for the preparation of facilitators for future leading of discussions in TPD. Barzel, B., & Selter, C. (2015). Die DZLM-Gestaltungsprinzipien für Fortbildungen [The DZLM design principles for professional development]. Journal für Mathematik-Didaktik, 36(2), 259–284. https://doi.org/10.1007/s13138-015-0076-y Borko, H., Koellner, K., & Jacobs, J. (2014). Examining novice teacher leaders’ facilitation of mathematics professional development. The Journal of Mathematical Behavior, 33, 149-167. https://doi.org/10.1016/j.jmathb.2013.11.003 Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM - Mathematics Education, 43(1), 175–187. https://doi.org/10.1007/s11858-010-0302-5 Cohen-Nissan, O., & Kohen, Z. (2023). Mathematics coordinators as school team PD leaders before during COVID-19. Teaching Teacher Education. https://doi.org/10.1016/j.tate.2022.103921. Creswell, J. W., & Creswell, J. D. (2022). Research design: Qualitative, quantitative and mixed methods approaches. (6th. Ed.). Sage. Desimone, L. M. (2009). Improving Impact Studies of Teachers’ Professional Development: Toward Better Conceptualizations and Measures. Educational Researcher, 38(3), 181– 199. https://doi.org/10.3102/0013189X08331140 Elliott, R., Kazemi, E., Lesseig, K., Mumme, J., Carroll, C., & Kelley-Petersen, M. (2009). Conceptualizing the Work of Leading Mathematical Tasks in Professional Development. Journal of Teacher Education, 60(4), 364–379. https://doi.org/10.1177/0022487109341150 Joyce, B., & Showers, B. (2002). Student achievement through staff development. Longman. Lipowsky, F., & Rzejak, D. (2015). Key features of effective professional development programs for teachers. Ricercazione — Six-monthly Journal on Learning. Research and Innovation in Education, 7(2), 27–51. Onrubia, J., Roca, B., & Minguela, M. (2022). Assisting teacher collaborative discourse in professional development: An analysis of a facilitator’s discourse strategies. Teaching and Teacher Education, 113, 1–18. https://doi.org/10.1016/j.tate.2022.103667 Pöhler-Friedrich, B., Prediger, S., Schwarts, G., & Karsenty, R. (2024). Learning to leverage teachers’ ideas: Facilitators’ growth in moderation practices with simultaneously strong goal focus and participant focus. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-024-09660-4 Prediger, S., & Selter, C. (2024). Establish shared visions and support productive adaptations on all levels: Aims, strategies, and architecture of a nationwide implementation program. Implementation and Replication Studies in Mathematics Education, 4(1), 15– 49. https://doi.org/10.1163/26670127-bja10020 Roesken-Winter, B., Stahnke, R., Prediger, S., & Gasteiger, H. (2021). Towards a research base for implementation strategies addressing mathematics teachers and facilitators. ZDM - Mathematics Education, 53(5), 1007-1019. https://doi.org/10.1007/s11858-021-01220-x Shaughnessy, M., Garcia, N. M., O’Neill, M. K., Selling, S. K., Willis, A. T., Wilkes, C. E., Salazar, S. B., & Ball, D. L. (2021). Formatively assessing prospective teachers’ skills in leading mathematics discussions. Educational Studies in Mathematics, 108(3), 451–472. Shenton, A. (2004). Strategies for ensuring trustworthiness in qualitative research projects. Education for Information, 22, 63–75. https://doi.org/10.3233/EFI-2004-22201 Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell. Mathematical Thinking and Learning, 10(4), 313–340. van Es, E.A., Tunney, J., Goldsmith, L., & Seago, N. (2014). A Framework for the Facilitation of Teachers’ Analysis of Video. Journal of Teacher Education, 65, 340 - 356.

1363 1370 Fabio Luiz Borges Simas Michael Fischer Rúbia Barcelos Amaral Federal University of State of Rio de Janeiro - Brazil University of Graz - Austria São Paulo State University - Brazil Proofs are often conceptualised as sequences of deductively connected assertions, yet in practice they frequently include instructions that guide the reader’s actions. Little is known about how culturally or linguistically dependent such proof language may be. This study examines how instructional language is used to structure written proofs in two school textbook traditions, by analysing multiple versions of the same theorem. In doing so, we refine an analytical framework for proof discourse. In one tradition, proofs are densely expressed through symbolic statements with few instructions, whereas in another they rely on frequent verbal instructions that guide the reader step by step. These contrasts identify culturally contingent features of proof language and their potential implications for students’ engagement with proving. Balacheff, N. (1987). Processus de preuve et situations de validation [Processes of proving and validation situations]. Educational Studies in Mathematics, 18(2), 147–176. Beer, R., Chelly, A., Lilias, P., Jilka, S., Stefan, C., & Varelija, G. (2024). Genial! Mathematik 2: Lehr- und Arbeitsbuch für die 2. Klasse [Genial! Mathematics 2: Textbook and workbook for grade 2]. Bildungsverlag Lemberger. Dawkins, P. C., & Vroom, K. (2025). Researching Proof Viewed as a Genre of Text. In K. Weber & M. Savić (Eds), New Directions for Mathematics Education Research on Proving: Honoring the Legacy of John and Annie Selden (pp. 293–318). Springer Nature Switzerland. https://doi.org/10.1007/978-3-031-85004-2_13 Ernest, P. (1998). Social constructivism as a philosophy of mathematics. State University of New York Press. Fischer, M. & Oldenburg, R. (submitted for publication). Computational Thinking als Bindeglied zwischen Algorithmen und Rezeptbeweisen [Computational thinking as a link between algorithms and recipe proofs] Fujita, T., & Jones, K. (2014). Reasoning-and-proving in geometry in school mathematics textbooks in Japan. International Journal of Educational Research, 64, 81–91. Herbst, P. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283–312. https://doi.org/10.1023/A:1020264906740 Jahnke, H. N., Sommerhoff, D., & Ufer, S. (2023). Argumentieren, Begründen und Beweisen [Argumentation, Reasoning and Proof]. In R. Bruder, A. Büchter, H. Gasteiger, B. Schmidt-Thieme, & H.-G. Weigand (Eds), Handbuch der Mathematikdidaktik (pp. 369–398). Springer Berlin Heidelberg. Knipping, C., Bredow, F., Miyakawa, T., Otani, H., & Shinno, Y. (2025). Analytic and narrative perspectives on proof and proving in mathematics classrooms: A cultural comparative approach. In Proceedings of the 14th Congress of the European Society for Research in Mathematics Ed. (CERME14). Free University of Bozen-Bolzano / ERME. Knipping, C., & Reid, D. A. (2019). Argumentation analysis for early career researchers. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 3–31). Springer. https://doi.org/10.1007/978-3-030-15636-7 Kohanová, I., Slavíčková, M., Rosa, S., Di Paola, B., Michal, J., & Çakıroğlu, E. (2025). Exploring teachers’ resource utilization practices and beliefs in mathematics education: A cross-national study on reasoning and proving. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-025-10577-4 Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. Routledge & Kegan Paul. Rotman, B. (2006). Towards a Semiotics of Mathematics. In Semiotica, 72, 97–127. Simas, F., & Amaral, R. (2025). Reasoning in textbooks post-curriculum reform. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proc. 48th Conf. of the Int. Group for the Psychology of Mathematics Education (p. 400). Skovsmose, O. (2024). Critical philosophy of mathematics. Springer. Tanswell, F.S., & Inglis, M. (2024). The language of proofs: A philosophical corpus linguistics study of instructions and imperatives in mathematical texts. In B. Sriraman (Ed.), Handbook of the history and philosophy of mathematical practice (pp. 1–28). Springer. Weber, K. (2023). Instructions and constructions in set theory proofs. Synthese, 202(34). Weber, K., & Tanswell, F. S. (2022). Instructions and recipes in mathematical proofs. Educational Studies in Mathematics, 111(1), 73–87. Wittmann, E. C. (2014). Operative Beweise in der Schul- und Elementarmathematik [Operative proofs in school and elementary mathematics]. mathematica didactica, 37(2), Article 2.

1371 1378 Anna Lisa Simon Florian Schindler Maike Schindler University of Cologne Germany Heidelberg University of Education The number line (NL) is an important tool in mathematics education for developing students’ understanding of numbers, yet its use is often not immediately accessible to students. To date, little is known about how students can be supported in learning to use the NL and how their related learning processes unfold. In this paper, we trace the learning process of Emir, a fourth-grade student with difficulties using the NL. Based on Emir’s verbalizations, eye movements, gestures, and actions, we qualitatively follow his learning process over the course of a school support study, focusing on reference-point use and distance interpretations. We use Emir’s case to showcase the volatile learning process of coming to use the NL and its reference points, and identify prominent challenges and turning points in supporting students’ use of the NL. Fitzsimmons, C. J., Morehead, K., Thompson, C. A., Buerke, M., & Dunlosky, J. (2023). Can feedback, correct, and incorrect worked examples improve numerical magnitude estimation precision?. The Journal of Experimental Education, 91(1), 20–45. Fuson, K. (1984). More complexities in subtraction. Journal for Research in Mathematics Education, 15(3), 214–225. Gaidoschik, M. (2024). Number line estimation tasks: Ein fachdidaktischer Blick auf einen in den kognitiven Neurowissenschaften prominenten Aufgabentyp. JMD, 45, 1. Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schönmann, C., Plangger, F., Gälli, M., Martin, E., & von Aster, M. (2011). Mental number line training in children with developmental dyscalculia. NeuroImage, 57(3), 782–795. Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. Beltz. Peeters, D., Verschaffel, L., & Luwel, K. (2017). Benchmark-based strategies in whole number line estimation. British Journal of Psychology, 108, 668–686. Scherer, P. & Moser Opitz, E. (2010). Förderung mathematischer Kompetenzen bei Kindern mit Lernschwierigkeiten. Kohlhammer. Simon, A. L., Rott, B., & Schindler, M. (2025). Number line strategies of students with mathematical learning difficulties and students with general learning difficulties: Findings through eye tracking. Education Sciences, 15(11), 1461. Teppo, A., & van den Heuvel-Panhuizen, M. (2014). Visual representations as objects of analysis: The number line as an example. ZDM Mathematics Education, 46, 45–58. van’t Noordende, J. E., van Hoogmoed, A. H., Schot, W. D., & Kroesbergen, E. H. (2016). Number line estimation strategies in children with mathematical learning difficulties measured by eye tracking. Psychological Research, 80, 368–378.

1379 1386 Daniel Sommerhoff IPN - Leibniz Institute for Science and Mathematics Education, Kiel Germany Mental addition and subtraction are key learning goals in primary education, yet they remain challenging for learners. Unfortunately, there is limited quantitative evidence on characteristics that influence the difficulty of these tasks (DGICs). Using a dataset with 1,085,842 task performances from 7,307 students across 3,434 tasks, this study quantitatively explores more than 10 DGICs. Results (i) show that all proposed DGICs significantly predict item difficulty, (ii) quantify main and interaction effects, and (iii) underline that the DGICs explain over 50% of the variance in student performance, even without considering students’ backgrounds or solution strategies. Overall, this study provides important insights into the difficulty of mental addition and subtraction tasks, informing task design and guiding future research in mathematics education. Anderson, S., Sommerhoff, D., Schurig, M., Ufer, S., & Gebhardt, M. (2022). Developing learning progress monitoring tests using difficulty-generating item characteristics: An example for basic arithmetic operations in primary schools. Journal for educational research online, 2022(1), 122-146. Beishuizen, M., Van Putten, C. M., & Van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction, 7(1), 87106. Benz, C. (2005). Erfolgsquoten, Rechenmethoden, Lösungswege und Fehler von Schülerinnen und Schülern bei Aufgaben zur Addition und Subtraktion im Zahlenraum bis 100 [Students’ success rates, calculation methods, solutions and mistakes in addition and subtraction tasks in the range up to 100]. Franzbecker. Deno, S. L., Reschly, A. L., Lembke, E. S., Magnusson, D., Callender, S. A., Windram, H., & Stachel, N. (2009). Developing a school-wide progress-monitoring system. Psychology in the Schools, 46(1), 44-55. Fuchs, A., Radkowitsch, A., & Sommerhoff, D. (2025). Using learning progress monitoring to promote academic performance? A meta-analysis of the effectiveness. Educational Research Review, 46, 100648. Heirdsfield, A. M., & Cooper, T. J. (2004). Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers. The Journal of Mathematical Behavior, 23(4), 443-463. Hickendorff, M., Torbeyns, J., & Verschaffel, L. (2019). Multi-digit Addition, Subtraction, Multiplication, and Division Strategies. In International Handbook of Mathematical Learning Difficulties (pp. 543-560). Kultusministerkonferenz (KMK). (2004). Bildungsstandards im Fach Mathematik für den Primarbereich. Beschluss vom 15.10.2004. (Beschlüsse der Kultusministerkonferenz). National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics. Nuerk, H.-C., Moeller, K., & Willmes, K. (2015). Multi-digit Number Processing: overview, conceptual clarifications, and language influences. In R. Cohen Kadosh & A. Dowker (Eds.), The Oxford Handbook of Numerical Cognition (pp. 106-139). OUP. Vanbinst, K., Ceulemans, E., Ghesquiere, P., & De Smedt, B. (2015). Profiles of children's arithmetic fact development: a model-based clustering approach. J Exp Child Psychol, 133, 29-46. Varol, F., & Farran, D. (2007). Elementary School Students’ Mental Computation Proficiencies. Early Childhood Education Journal, 35(1), 89-94. Zbrodoff, N. J., & Logan, G. D. (2005). What everyone finds: The problem-size effect. In The handbook of mathematical cognition (pp. 331-345). Psychology Press.

1387 1394 Christos Souralis Chrissavgi Triantafillou Konstantinos Stouraitis National and Kapodistrian University of Athens Greece This study explores how an experienced practicing teacher, in a reform curriculum context, interpreted and responded to the expected learning outcomes regarding the use of a realistic context for pattern generalization. The data came from two professional development meetings and was analyzed according to the curriculum noticing framework. The results highlight the importance of generalization in teaching patterns, supported by visualization and manipulatives. The realistic context was not emphasized. Demonty, I., Vlassis, J., & Fagnant, A. (2018). Algebraic thinking, pattern activities and knowledge for teaching at the transition between primary and secondary school. Educational Studies in Mathematics, 99, 1-19. https://doi.org/10.1007/s10649-018-98209 Dietiker, L., Males, L. M., Amador, J. M., & Earnest, D. (2018). Research commentary: Curricular noticing: A framework to describe teachers' interactions with curriculum materials. Journal for Research in Mathematics Education, 49(5), 521-532. https://doi.org/10.5951/jresematheduc.49.5.0521 Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for research in mathematics education, 41(2), 169-202. https://doi.org/10.5951/jresematheduc.41.2.0169 Lee, M. Y., & Lee, J. E. (2025). Investigating Pre-Service Teachers’ Problem-Solving Ability and Their Curricular Noticing Ability Through Pedagogical Sequencing Activities. International Journal of Science and Mathematics Education, 23(5), 12111236. https://doi.org/10.1007/s10857-021-09520-5 Males, L. M., Earnest, D., Dietiker, L., & Amador, J. M. (2015). Examining K-12 Prospective Teachers' Curricular Noticing. North American Chapter of the International Group for the Psychology of Mathematics Education. Males, L. M., & Setniker, A. (2019, April). An examination of prospective secondary teachers’ curricular noticing when planning. In Paper presentation at the annual meeting of the American Educational Research Association. Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge Özyildirim Gümüş, F. (2022). Preservice elementary mathematics teachers’ use of patterns and pattern problems when planning and implementing lessons. International Journal of Mathematical Education in Science and Technology, 53(8), 2152-2175. https://doi.org/10.1080/0020739X.2021.1952325 Rotem, S. H., & Ayalon, M. (2022). Building a model for characterizing critical events: Noticing classroom situations using multiple dimensions. The Journal of Mathematical Behavior, 66, 100947. https://doi.org/10.1016/j.jmathb.2022.100947 Yannis Liakos, Saba Gerami, Vilma Mesa, Thomas Judson & Yue Ma (2022) How an inquiry-oriented textbook shaped a calculus instructor’s planning. International Journal of Mathematical Education in Science and Technology, 53:1, 131-150, https://doi.org/10.1080/0020739X.2021.1961171 van Es, E. A., & Sherin, M. G. (2021). Expanding on prior conceptualizations of teacher noticing. ZDM–Mathematics Education, 53(1), 17-27. https://doi.org/10.1007/s11858020-01211-4 Wöller, S. (2025). Noticing of preservice primary school teachers in mathematics while watching own video vignettes. In Cornejo, C., Felmer, P., Gómez, D. M., Dartnell, P., Araya, P., Peri, A., & Randolph, V. (Eds), Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 411–418).

1395 1402 Marieke Spijkstra Filip Moons Utrecht University, Freudenthal Institute Netherlands Probability literacy is a vital skill for vocational students, yet it remains a neglected area in research, despite its relevance to real-world decision-making. While probability education is standard in pre-university tracks, vocational students, who make up 50% of Dutch secondary education, receive little formal instruction, leaving them vulnerable to risky behaviours like gambling. This study aims to explore how vocational students reason about probability and risk, and how emotions and heuristics influence their decisions. Using a questionnaire with six risk-literacy items, we surveyed 355 students (ages 13–15) and found that many declined games with positive expected values, often due to fear of losing or aversion to gambling. The qualitative explanations revealed that emotional and intuitive reasoning frequently overrode analytical thinking, even in low-stakes scenarios. These findings highlight the need to address cognitive and affective biases in probability education for vocational students. Borovcnik, M., & Kapadia, R. (2018). Reasoning with Risk: Teaching Probability and Risk as Twin Concepts. In C. Batanero & E. J. Chernoff (Eds.), Teaching and Learning Stochastics: Advances in Probability Education Research (pp. 3–22). Springer International Publishing. https://doi.org/10.1007/978-3-319-72871-1_1 Casey, B. J., Getz, S., & Galvan, A. (2008). The adolescent brain. Developmental Review, 28(1), 62–77. https://doi.org/10.1016/j.dr.2007.08.003 Donati, M. A., Chiesi, F., Iozzi, A., Manfredi, A., Fagni, F., & Primi, C. (2018). GamblingRelated Distortions and Problem Gambling in Adolescents: A Model to Explain Mechanisms and Develop Interventions. Frontiers in Psychology, 8. https://doi.org/10.3389/fpsyg.2017.02243 Hansen, J., & Hammann, M. (2017). Risk in Science Instruction. Science & Education, 26(7), 749–775. https://doi.org/10.1007/s11191-017-9923-1 Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus and Giroux. Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), 263–291. https://doi.org/10.2307/1914185 Meder, B., & Gigerenzer, G. (2014). Statistical Thinking: No One Left Behind. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking: Presenting Plural Perspectives (pp. 127–148). Springer Netherlands. https://doi.org/10.1007/978-94-007-7155-0_8 Radakovic, N. (2015). Pedagogy of Risk: Why and How Should We Teach Risk in High School Math Classes? The Mathematics Enthusiast, 12(1–3), 307–329. https://doi.org/10.54870/1551-3440.1350 Till, C. (2014). Fostering Risk Literacy in Elementary School. International Electronic Journal of Mathematics Education, 9(2), 83–96. https://doi.org/10.29333/iejme/283 Tversky, A., & Kahneman, D. (1974). Judgment under Uncertainty: Heuristics and Biases: Biases in judgments reveal some heuristics of thinking under uncertainty. Science, 185(4157), 1124–1131. https://doi.org/10.1126/science.185.4157.1124

1403 1410 Femke Sporn Kristin Litteck Irene Neumann Aiso Heinze IPN - Leibniz Institute for Science and Mathematics Education, Kiel Germany Mathematics is encountered both as a deductive discipline, where proofs are central, and as an applied discipline, where reasoning is used to model and solve real-world problems. While arguments based on examples, measurements, or drawings are generally invalid in proofs, they are often legitimate when mathematics is applied in context. This study investigates whether high school students can distinguish between these normative frameworks and apply appropriate sociomathematical norms flexibly. Data from 436 Grade 11 students show that students’ knowledge of proof-related norms does not systematically predict their evaluation of applied mathematical context. However, the results suggest that some students are familiar with proof-related norms but seem to transfer them inappropriately to applied contexts. Ärlebäck, J. B. (2009). On the use of Realistic Fermi problems for introducing mathematical modelling in school. The Mathematics Enthusiast, 6(3), 331–364. https://doi.org/10.54870/1551-3440.1157 Blum, W., & Borromeo Ferri, R. (2009). Mathematical Modelling: Can It Be Taught And Learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects ? State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68. https://doi.org/10.1007/BF00302716 Buchbinder, O., & Zaslavsky, O. (2019). Strengths and inconsistencies in students’ understanding of the roles of examples in proving. The Journal of Mathematical Behavior, 53, 129–147. https://doi.org/10.1016/j.jmathb.2018.06.010 Dawkins, P. C., & Weber, K. (2017). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics, 95(2), 123–142. https://doi.org/10.1007/s10649-016-9740-5 Fesser, P., & Rach, S. (2022). Meta-scientific reflection of undergraduate students: Is mathematics a natural science? In C. Fernández, S. Llinaress, A. Gutiérrez, & N. Plana (Eds.), Proceedings of the 45th Congress of the International Group for the Psychology in Mathematics Education (Vol. 2, pp. 259–266). PME. Hansen, V. L. (2008). The dual nature of mathematics. In M. Niss (Ed.), Proceedings of the 10th International Congress on Mathematical Education. Roskilde University Press. Healy, L., & Hoyles, C. (2000). A Study of Proof Conceptions in Algebra. Journal for Research in Mathematics Education, 21(4), 396–428. https://doi.org/10.2307/749651 Heinze, A., & Reiss, K. (2003). Reasoning and Proof: Methodological Knowledge as a Component of Proof Competence. In M. A. Mariotti (Ed.), Proceedings of the Third Conference of the European Society for Research in Mathematics Education (Vol. 4, pp. 1–10). Bellaria, Italy. Komatsu, K., Jones, K., Ikeda, T., & Narazaki, A. (2017). Proof validation and modification in secondary school geometry. The Journal of Mathematical Behavior, 47, 1–15. https://doi.org/10.1016/j.jmathb.2017.05.002 Manin, Y. I. (2010). A course in mathematical logic for mathematicians (2nd ed.). Springer. Rellensmann, J., Schukajlow, S., & Leopold, C. (2017). Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53–78. https://doi.org/10.1007/s10649-016-9736-1 Sporn, F., Sommerhoff, D., & Heinze, A. (2022). Students’ Knowledge About Proof and Handling Proof. In C. Fernández, S. Llinares, Á. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 27–34). Alicante, Spain: PME. Sporn, F., Sommerhoff, D., & Heinze, A. (2023). Fostering Students’ Knowledge About Proof. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 235–242). Israel: PME. Stylianides, A. J. (2007). Proof and Proving in School Mathematics. Journal for Research in Mathematics Education, 38(3), 289–321. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.

1411 1418 Susanne Digel Ute Sproesser RPTU University of Kaiserslautern-Landau Ludwigsburg University of Education Germany This study examines the structure and relationships of functional thinking facets, i. e., conceptions of functions and representational competencies. A sample of 678 students completed assessments on both facets. Rasch model comparisons identified all seven theory-based dimensions and showed varied correlations between them. The strongest correlations were found between changes involving situational representation and covariation/object and the weakest between changes with algebraic expressions and correspondence. This suggests that changes with situational representations rely on function conceptions, whereas changes with algebraic expressions often are performed algorithmically. Hence, instruction should cultivate function conceptions and comprehensible representational changes, while deemphasising fragile algorithms. Adu-Gyamfi, K., Stiff, L. V., & Bossé, M. J. (2012). Lost in translation: examining translation errors associated with mathematical representations. School Science and Mathematics, 112(3), 159–170. https://doi.org/10.1111/j.1949-8594.2011.00129.x Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal of Mathematics Education, 6(3), 113–133. https://doi.org/10.29333/iejme/264 Digel, S., & Roth, J. (2020). A qualitative-experimental approach to functional thinking with a focus on covariation. In A. Donevska-Todorova, E. Faggiano, J. Trgalova, Z. Lavicza, R. Weinhandl, A. Clark-Wilson, & H.-G. Weigand (Eds.), Proceedings of the 10th ERME Topic Conference on Mathematics Education in the Digital Age (MEDA) 2020 (pp. 167– 174). Johannes Kepler University. Digel, S., & Roth, J. (2022). Fostering a concept of function with combined experiments in distance and in-class learning. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 227–234). PME. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131. https://doi.org/10.1007/s10649-006-0400-z Krüger, K. (2019). Functional thinking: The history of a didactical principle. In H.-G. Weigand, W. McCallum, M. Menghini, M. Neubrand, & G. Schubring (Eds.), The legacy of Felix Klein (pp. 35–54). Springer International Publishing. Lichti, M., & Roth, J. (2019). Functional thinking − A three-dimensional construct? Journal für Mathematik Didaktik, 40(2), 169–195, DOI: 10.1007/s13138-019-00141-3 Niss, M. (2020). Functions Learning and Teaching. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 303–306). Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_96 Nitsch, R (2015). Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge. [Diagnosing learning difficulties in the area of functional relationships] Springer. Sproesser, U., Vogel, M., Dörfler, T., & Eichler, A. (2020). Typische Lernschwierigkeiten mit Darstellungswechseln bei elementaren Funktionen – Welche Schwierigkeiten kennen Lehrkräfte und wie schätzen sie Aufgabenbearbeitungen ihrer Klassen ein? [Typical learning difficulties with representational changes in elementary functions – What difficulties are teachers aware of and how do they judge their classes' performance?] Mathematica Didactica, 43(2020)2, 175–198. https://doi.org/10.18716/ojs/md/2020.1154 Sproesser, U., Vogel, M., Dörfler, T., & Eichler, A. (2022). Changing between representations of elementary functions: students’ competencies and differences with a specific perspective on school track and gender. International Journal for STEM Education, 9(33). https://doi.org/10.1186/s40594-022-00350-2 Thompson, P.W., & Carlson, M.P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics. Vollrath, H.-J. (1989). Funktionales Denken. [Functional Thinking]. Journal für Mathematik Didaktik, 10, 3–37. vom Hofe, R., Lotz, J., & Salle, A. (2015). Analysis: Leitidee Zuordnung und Veränderung. [Calculus: Guiding principle Correspondence and Change] In R. Bruder, L. HefendehlHebeker, B. Schmidt-Thieme, & H.-G. Weigand (Hrsg.), Handbuch der Mathematikdidaktik [Handbook of Mathematics Education] (pp. 149–184). Springer vom Hofe, R., & Blum, W. (2016). “Grundvorstellungen” as a Category of Subject-Matter Didactics. Journal für Mathematik Didaktik 37, 225–254. https://doi.org/10.1007/s13138016-0107-3 Zindel, C. (2018). Den Kern des Funktionsbegriffs verstehen. [Understanding the core of the functional concept]. Springer Spektrum.

1419 1426 Hendrik Streit Benjamin Rott University of Cologne Germany An important goal of mathematics education is for students to become competent in solving problems. Therefore, it is essential that research clarifies what is required of teachers to appropriately integrate Problem Solving into their mathematics teaching. For this reason, we conducted a Scoping Review to provide a broad and well-founded overview of research on teaching Problem Solving, resulting in the identification of 11 research foci. In this paper, we focus primarily on one of these foci, namely Teachers’ Knowledge. By using connections to further research foci, we show how the overview allows us to complement and expand existing concepts developed by research. A key insight is that knowledge of classroom environments for Problem Solving is a category that could complement existing knowledge frameworks for teaching Problem Solving. Chapman, O. (2015). Mathematics teachers’ knowledge for teaching problem solving. LUMAT, 3(1), 19–36. https://doi.org/10.31129/lumat.v3i1.1049 Chapman, O. (2017). Mathematics Teachers’ Ways of Supporting Students’ Learning of Problem Solving. In M. Stein (Ed.), A Life’s Time for Mathematics Education and Problem Solving (pp. 45–69). WTM. https://doi.org/10.37626/GA9783959870641.0.04 Foster, C., Wake, G. & Swan, M. (2014). Mathematical Knowledge for Teaching Problem Solving: Lessons from Lesson Study. In P. Liljedahl, C. Nicol, S. Oesterle & D. Allan (Eds.), Proc. of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 3, pp. 97–104). PME. Heinrich, F. (2004). Strategische Flexibilität beim Lösen mathematischer Probleme. Theoretische Analysen und empirische Erkundungen über das Wechseln von Lösungsanläufen [Strategic flexibility when solving mathematics problems. Theoretical analyses and empirical explorations of switching solution approaches]. Dr. Kovač. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400. https://doi.org/10.5951/jresematheduc.39.4.0372 Koichu, B. (2019). A Discursively Oriented Conceptualization of Mathematical Problem Solving. In P. Felmer, P. Liljedahl & B. Koichu (Eds.), Problem Solving in Mathematics Instruction and Teacher Professional Development (pp. 43–66). Springer. Laine, A. (2019). How to promote learning in problem-solving? In A. Kuzle, I. Gebel & B. Rott (Eds.), Implementation Research on Problem Solving in School Settings (pp. 3–18). WTM. https://doi.org/10.37626/GA9783959871167.0.01 Lester, F. K., & Cai, J. (2016). Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research. In P. Felmer, E. Pehkonen & J. Kilpatrick (Eds.), Posing and Solving Mathematical Problems (pp. 117–135). Springer. Lester, F. K. (2013). Thoughts About Research On Mathematical Problem-Solving Instruction. The Mathematics Enthusiast, 10(1), 245–278. Liljedahl, P., & Cai, J. (2021). Empirical research on problem solving and problem posing: a look at the state of the art. ZDM, 53(4), 723–735. McIntosh, R., & Jarrett, J. (2000). Teaching mathematical problem solving: Implementing the vision. A Literature Review. NWREL. Stephan, M., & Whitenack, J. (2005). Establishing Classroom Social and Sociomathematical Norms for Problem Solving. In F. K. Lester (Ed.), Teaching mathematics through problem solving: Prekindergarten – Grade 6 (pp. 149–162). NCTM. Tricco, A. C. et al. (2018). PRISMA Extension for Scoping Reviews (PRISMA-ScR): Checklist and Explanation. Annals of Internal Medicine, 169(7), 467–473. Tzohar-Rozen, M., & Kramarski, B. (2017). Metacognition and Meta-Affect in Young Students: Does it Make a Difference in Mathematical Problem Solving? Teacher College Record, 119(13), Article 130314. https://doi.org/10.1177/016146811711901308

1427 1434 Anselm Strohmaier Samira Bödefeld Frank Reinhold University of Education Freiburg Germany The capabilities of generative AI in mathematics education are rapidly evolving, posing significant challenges for research to keep pace. Research syntheses remain scarce and risk being outdated by the time of publication. To address this issue, we present a Living Meta-Analysis (LIMA) on the effects of generative AI–based interventions for learning mathematics. Following PRISMA-LSR guidelines, we continuously update the literature base, apply a Bayesian multilevel meta-regression model to account for cumulative data, and publish updated versions on a preprint server at regular intervals. This paper reports results from the first version, including 14 studies. The analyses indicate a small positive effect (g = 0.28) with a wide credible interval [0.01, 0.59], reflecting the still limited evidence base. Akl, E. A., Khabsa, J., Iannizzi, C., Piechotta, V., Kahale, L. A., Barker, J. M., McKenzie, J. E., Page, M. J., & Skoetz, N. (2024). Extension of the PRISMA 2020 statement for living systematic reviews (PRISMA-LSR): checklist and explanation. BMJ, 387, e079183. Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to Meta-Analysis. Chichester: Wiley. Bürkner, P.-C. (2017). brms: An R package for Bayesian multilevel models using Stan. Journal of Statistical Software, 80, 1-28. Elliott, J. H., Turner, T., Clavisi, O., Thomas, J., Higgins, J. P., Mavergames, C., & Gruen, R. L. (2014). Living systematic reviews: an emerging opportunity to narrow the evidencepractice gap. PLoS Med, 11(2). Fock, A., & Siller, H.-S. (2025). The Impact of Generative AI on student learning achievement in secondary STEM education: A meta-analysis. Proceedings of the 17th International Conference on Technology in Mathematics Teaching, London, UK. Harrer, M., Cuijpers, P., Furukawa, T. A., & Ebert, D. D. (2021). Doing Meta-Analysis With R: A Hands-On Guide (1st ed.). Boca Raton, FL and London: Chapman & Hall/CRC Press. Hattie, J. (2008). Visible learning (1st ed.). Routledge. Hillmayr, D., Ziernwald, L., Reinhold, F., Hofer, S. I., & Reiss, K. M. (2020). The potential of digital tools to enhance mathematics and science learning in secondary schools: A context-specific meta-analysis. Computers & Education, 153, 103897. Howard, J. L., & Slemp, G. R. (2025). Why we need a living meta-analysis of selfdetermination theory: an illustration examining temporal changes in need supportive education. Motivation and Emotion, 1-18. Huisman, J., & Smits, J. (2017). Duration and quality of the peer review process: the author's perspective. Scientometrics, 113(1), 633–650. Lipsey, M. W., & Wilson, D. B. (2000). Practical Meta-Analysis. Thousand Oaks, CA: Sage. Pepin, B., Buchholtz, N., & Salinas-Hernández, U. (2025). A Scoping Survey of ChatGPT in Mathematics Education. Digital Experiences in Mathematics Education, 11(1), 9-41. Schoenherr, J., Strohmaier, A. R., & Schukajlow, S. (2024). Learning with visualizations helps: A meta-analysis of visualization interventions in mathematics education. Educational Research Review, 45, 100639. Strohmaier, A. R., Van Dooren, W., Seßler, K., Greer, B., & Verschaffel, L. (2025). Large Language Models Don't Make Sense of Word Problems. A Scoping Review from a Mathematics Education Perspective. arXiv preprint arXiv:2506.24006. Tlili, A., Saqer, K., Salha, S. & Huang, R. (2025). Investigating the effect of artificial intelligence in education (AIEd) on learning achievement: A meta-analysis and research synthesis. Information Development, 41(3), 825–842.

1435 1442 Michal Tabach Einat Heyd-Metzuyanim Tamar Aviram Tel Aviv University, School of Education Israel Technion - Israel Institute of Technology Israel This study employs an integration of commognition and complex dynamic systems (CDS) theory to investigate how beginning algebra students learn in an Excel environment. Through a case study of two low achieving seventh grade students, we trace how noncanonical and canonical mathematical narratives emerge, stabilize, and shift during spreadsheet mediated activity. Findings show that Excel’s feedback loops reinforced students’ prevailing meta rules—whether correct or not. A phase shift occurred only when the interviewer explicitly targeted the underlying non-canonical meta rule, enabling reorganization toward a canonical discourse. We claim that the CDS lens offers terms for describing difficult-to-capture processes of stabilization and perturbation in students’ learning with technological tools. Ben-Dor, N., & Heyd-Metzuyanim, E. (2025). Unifying dialogic learning and mathematics learning: A discursive lens for the study of dialogic mathematics peer learning. Journal of the Learning Sciences, 34(1), 1-34. Byrne, D. (1998). Complexity theory and the social sciences: An introduction. Routledge. Chan, M. C. E., & Sfard, A. (2020). On learning that could have happened: The same tale in two cities. The Journal of Mathematical Behavior, 60, Article 100815. Heyd-Metzuyanim, E. (2025). Failure to teach/learn mathematics: A complexity-discursive perspective. Educational Studies in Mathematics, 119(3), 515–533. Higgins, K., Huscroft-D’Angelo, J., & Crawford, L. (2019). Effects of technology in mathematics on achievement, motivation, and attitude: A meta-analysis. Journal of Educational Computing Research, 57(2), 283-319. Hilpert, J. C., & Marchand, G. C. (2018). Complex systems research in educational psychology: Aligning theory and method. Educational Psychologist, 53(3), 185–202. Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. Educational Studies in mathematics, 101(2), 153-176. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge university press. Shvarts, A., Bos, R., Doorman, M., & Drijvers, P. (2024). Reifying actions into artifacts: process–object duality from an embodied perspective on mathematics learning. Educational Studies in Mathematics, 117(2), 193-214. Tabach, M., Friedlander, A. (2008). Understanding equivalence of symbolic expressions in a spreadsheet-based environment. International Journal of Computers in Mathematics Learning, 13, 27–46.

1443 1450 Fatih Taş Yüksel Dede Pınar Akyıldız Taş Bartın University Turkey Gazi University Turkey Bartın University Turkey This study examines how pre-service mathematics teachers' (PMTs') beliefs about the nature of mathematics impact their interactions with Artificial Intelligence (AI) during task design. Using a multiple case study approach, 16 PMTs were initially divided into three belief profiles (Integrated Connectionists, Balanced Relationalists, and Procedural-Hybridists). Qualitative analysis of orientation strategies and task outcomes from nine focus cases revealed that PMTs with high relational beliefs utilized AI as a critical co-designer to enhance cognitive demand; conversely, those with stronger transactional beliefs employed AI primarily as a tool for task automation and shaping. The results suggest that mathematical beliefs act as a significant filter for pedagogical action in AI-integrated lesson planning. Akyıldız, P., & Dede, Y. (2019). A Belief Questionnaire about the Nature of Mathematics for Turkish Preservice Mathematics Teachers: An Exploratory Mixed Methods Study. Adıyaman University Journal of Educational Sciences, 9(1), 69-98. https://doi.org/10.17984/adyuebd.539351 Bernardi, M. L., Capone, R., Faggiano, E., & Rocha, H. (2025). Generative AI in mathematics education: pre-service teachers’ knowledge and implications for their professional development. International Journal of Mathematical Education in Science and Technology, 1-18. https://doi.org/10.1080/0020739X.2025.2490104 Cheung, K. K. C., Long, Y., Liu, Q., & Chan, H. Y. (2025). Unpacking epistemic insights of artificial intelligence (AI) in science education: A systematic review. Science & Education, 34(2), 747–777. https://doi.org/10.1007/s11191-024-00511-5 Dede, Y., & Karakus, F. (2014). The Effect of Teacher Training Programs on Pre-Service Mathematics Teachers' Beliefs towards Mathematics. Educational Sciences: Theory and Practice, 14(2), 804-809. DOI:10.12738/estp.2014.2.1787 Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. Mathematics teaching: The state of the art, 249, 254. Hair, J. F., Black, W. C., Babin, B. J., Anderson, R. E., & Tatham, R. L. (2006). Multivariate data analysis. (6th ed.). Upper Saddle River, NJ: Pearson/Prentice Hall. Li, D., Osman, S., Alhassora, N. S. A., Kumar, J. A., & Husain, S. K. S. (2025). Empowering K-12 Mathematics Teachers with Artificial Intelligence: A Systematic Review of Insights, Applications, Challenges. IEEE Access. https://doi.org/10.1109/ACCESS.2025.3585784 Lin, T., Zhang, J., & Xiong, B. (2025). Effects of Technology Perceptions, Teacher Beliefs, and AI Literacy on AI Technology Adoption in Sustainable Mathematics Education. Sustainability, 17(8), 3698. https://doi.org/10.3390/su17083698 Mayring, P. (2014). Qualitative content analysis: theoretical foundation, basic procedures and software solution. Molenaar, I. (2022). Towards hybrid human‐AI learning technologies. European Journal of Education, 57(4), 632-645. https://doi.org/10.1111/ejed.12527 Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of educational research, 62(3), 307-332. https://doi.org/10.3102/00346543062003307 Ruthven, K. (2009). Towards a naturalistic conceptualisation of technology integration in classroom practice: The example of school mathematics. Éducation et didactique, (3-1), 131-159. https://doi.org/10.4000/educationdidactique.434 Silvola, A., Kajamaa, A., Merikko, J., & Muukkonen, H. (2025). AI‐mediated sensemaking in higher education students’ learning processes: Tensions, sensemaking practices, and AI‐ assigned purposes. British Journal of Educational Technology. https://doi.org/10.1111/bjet.13606 Sui, C. J., & Chang, C. Y. (2026). Generative AI and education: Exploring the interplay between teachers' epistemic beliefs and self-regulated learning. Educational Technology & Society, 29(3), 34-50. https://doi.org/10.30191/ETS.202607_29(3).RP03 Thompson, A. (1992). Teachers’ beliefs and conceptions. Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics, 127-146. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for mathematical learning, 9(3), 281-307. https://doi.org/10.1007/s10758-004-3468-5 Trouche, L., Drijvers, P., Gueudet, G., & Sacristan, A. I. (2012). Technology-driven developments and policy implications for mathematics education. In Third international handbook of mathematics education (pp. 753-789). New York, NY: Springer New York. https://doi.org/10.1007/978-1-4614-4684-2_24 Warshauer, H. K. (2015). Strategies to support productive struggle. Mathematics teaching in the middle school, 20(7), 390-393. https://doi.org/10.5951/mathteacmiddscho.20.7.0390 Yin, R. K. (2003). Case study research: Design and methods. Sage

1451 1458 Charlott Thomas Birte Pöhler University of Potsdam Germany This study investigates the strategies that students employ when solving combinatorial problems. Therefore, a written test with 26 items was administered to 282 secondary students at the end of their school career. This paper focuses on higher-performing students who solved at least 75% of the tasks correctly, as their strategy use provides insights into successful approaches. The application of strategies was (overall and correctly applied) as well as the frequency and variance of strategies per person were analyzed. The findings indicate that higher-performing students predominantly rely on formulas, arithmetic rules, and systematic enumeration compared to other strategies and highlight the importance of fostering different combinatorial strategies in instruction and provide implications for the design of teaching learning environments. Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181–199. Berliner Senatsverwaltung für Bildung, Jugend und Familie sowie dem Ministerium für Bildung, Jugend und Sport des Landes Brandenburg (MBJS) (Eds.) (2022). Hinweise zur Vorbereitung auf die schriftliche Abiturprüfung 2024 im Land Brandenburg [Information on preparing for the 2024 written final examination in the state of Brandenburg]. Retrieved October 21, 2025, from https://bildungsserver.berlinbrandenburg.de/fileadmin/bbb/unterricht/pruefungen/abitur_bb/RS_ZA_2024/PS_2024/P S_Mathematik_2024_GK-BB.pdf Eizenberg, M. M., & Zaslavsky, O. (2003). Cooperative problem solving in combinatorics: The interrelations between control processes and successful solutions. The Journal of Mathematical Behavior, 22(4), 389–403. English, L. D. (2005). Combinatorics and the development of children’s combinatorial reasoning. In G. A. Jones (Ed.), Exploring probability in school (pp. 121–141). Springer. Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Reidel. Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt für Didaktik der Mathematik, 20(5), 193–198. Godino, J. D., Batanero, C., & Roa, R. (2005). An onto-semiotic analysis of combinatorial problems and the solving processes by university students. Educational Studies in Mathematics, 60(1), 3–36. Hadar, N., & Hadass, R. (1981). The road to solving a combinatorial problem is strewn with pitfalls. Educational Studies in Mathematics, 12(4), 435–443. Kapur, J. N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111–127. Lamanna, L., Gea, M. M., & Batanero, C. (2022). Do Secondary School Students’ Strategies in Solving Permutation and Combination Problems Change with Instruction? Canadian Journal of Science, Mathematics and Technology Education, 22(3), 602–616. Landis, J. R., & Koch, G. G. (1977). The Measurement of Observer Agreement for Categorical Data. Biometrics, 33(1), 159. Lockwood, E. (2013). A model of students’ combinatorial thinking. The Journal of Mathematical Behavior, 32(2), 251–265. Mayring, P. (2015). Qualitative Content Analysis: Theoretical Background and Procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods (pp. 365– 380). Springer. Pólya, G. (1995). Schule des Denkens: Vom Lösen mathematischer Probleme [School of Thought: Solving Math Problems]. Francke. Roa, R. (2000). Razonamiento combinatorio en estudiantes con preparación matemática avanzada [Combinatorial reasoning in students with advanced mathematical preparation] (Unpublished Doctoral dissertation). Universidad de Granada. Rott, B., Bruder, R., Heinrich, F., & Bauer, C. (2023). Problemlösen lernen [Learning problem-solving]. In R. Bruder, A. Büchter, H. Gasteiger, B. Schmidt-Thieme, & H.G.Weigand (Eds.), Handbuch der Mathematikdidaktik (pp. 313–339). Springer. Thomas, C., & Pöhler, B. (submitted). Combinatorial learning opportunities in school textbooks: A systematic analysis of operations, models, and the nature of elements.

1459 1466 Franziska Tilke Karina Höveler University of Münster Germany Mathematical structures are crucial for learning mathematics. Patterns can provide access to these. But studies show that recognizing patterns is not enough to understand the underlying mathematical structures. This paper presents a design research study with 42 elementary school learners who explore the mathematical structures of constancy properties of operations (e.g., 16 + 29 = 15 + 30). Based on reconstructed interpretations, the results provide a more detailed differentiation of the connection between recognizing the pattern at the process level and knowledge of structures at the content level. Akinwunmi, K., & Steinweg, A. S. (2022). Analysis of children’s generalisations with a focus on patterns and with a focus on structures. In J. Hodgen, E. Geraniou, G. Bolondi, & F. Ferretti (Eds.), Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) (pp. 465–472). Free University of Bozen-Bolzano and ERME. https://hal.science/hal-03744472 Kieran, C. (2018). Seeking, Using, and Expressing Structure in Numbers and Numerical Operations: A Fundamental Path to Developing Early Algebraic Thinking. In C. Kieran (Ed.) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds (pp. 79-105). Springer. https://doi.org/10.1007/978-3-319-68351-5 Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM – Mathematics Education, 54(6), 1131– 1150. https://doi.org/10.1007/s11858-022-01435-6 Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10–32. https://doi.org/10.1007/BF0321754 Mulligan, J. T., & Mitchelmore, M. C. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49. https://doi.org/10.1007/BF03217544 Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes: an overview on achievements and challenges. ZDM Mathematics Education 47, 877–891. https://doi.org/10.1007/s11858-015-0722-3 Schifter, D. (2018). Early Algebra as Analysis of Structure: A Focus on Operations. In C. Kieran (Ed.), Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds (pp. 309–327). Springer. https://doi.org/10.1007/978-3-319-68351-5 Söbbeke, E. (2005). Building visual structures in arithmetical knowledge – A theoretical characterization of young students' "Visual Structuring Ability (ViSA)". In M. Bosch (Ed.), Proceedings of Fourth Congress of the European Society for Research in Mathematics Education: CERME 4 (pp. 132–142). CERME. Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction. An epistemological perspective. Springer. https://doi.org/10.1007/b104944 Tilke, F. (2026). Von Mustern zu Strukturen im inklusiven Mathematikunterricht. Eine fachdidaktische Entwicklungsstudie zur Erkundung der Konstanzeigenschaften [From patterns to structures in inclusive mathematics teaching. A topic-specific design research study exploring constancy properties]. Springer. https://doi.org/10.1007/978-3-65851016-9 Venkat, H., Askew, M., Watson, A., & Mason, J. (2019). Architecture of mathematical structure. For the Learning of Mathematics, 31(1), 13-17. Wilkie, K. J., & Hopkins, S. (2024). Generalizing actions with the subtraction-compensation property: primary students’ algebraic thinking with tasks involving vertical towers of blocks. Educ Stud Math 115, 433–458. https://doi.org/10.1007/s10649-024-10303-x

1467 1474 Margarita Trifonov Jason Cooper Boris Koichu Weizmann Institute of Science Israel Curiosity is central to meaningful mathematics learning. This study examines the emergence of situational curiosity in middle-school students during mathematics lessons, using an emoji-based questionnaire (five time points, seven emotions) alongside analysis of classroom events and interactions. Across a lesson co-designed with the researcher, taught by a teacher in a Grade 7 class, we identified “curiosity windows”: episodes where curiosity intensifies in relation to specific classroom events. In a lesson analysed, curiosity rose during open inquiry, peaked when students’ propositions were tested and formalized into a canonical formulation of the Pythagorean theorem, and remained high when the theorem was framed historically. The study offers a new method for capturing curiosity in situ as a dynamic emotion. Berlyne, D. E. (1954). A theory of human curiosity. British Journal of Psychology, 45(3), 180–191. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa Engel, S. (2011). Children’s need to know: Curiosity in schools. Harvard Educational Review, 81(4), 625–645. https://doi.org/10.17763/haer.81.4.h054131316473115 Evans, N. S., Burke, R., Vitiello, V., Zumbrunn, S., & Jirout, J. J. (2023). Curiosity in classrooms: An examination of curiosity promotion and suppression in preschool math and science classrooms. Thinking Skills Creativity, 49, 101333. https://doi.org/10.1016/j.tsc.2023.101333 Leikin, R., Cai, J., & Karp, A. (Eds.) (2026). Mathematical curiosity: Conceptualizations and multiple perspectives. Springer. Leikin, R., & Pitta-Pantazi, D. (2025). Curiosity in mathematics: Theoretical, methodological, and practical lenses [Research Forum summary]. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education: General Contributions (PME48, Santiago, Chile). International Group for the Psychology of Mathematics Education. Loewenstein, G. (1994). The psychology of curiosity: A review and reinterpretation. Psychological Bulletin, 116(1), 75–98. https://doi.org/10.1037/0033-2909.116.1.75 Muis, K. R., Chevrier, M., & Singh, C. A. (2018). The role of epistemic emotions in personal epistemology and self-regulated learning. Educational Psychologist, 53(3), 165–184. Pekrun, R. (2019). The control-value theory of achievement emotions. In R. Pekrun & L. Linnenbrink-Garcia (Eds.), Handbook of emotions in education (2nd ed., pp. 116–142). Routledge. Rotem, S. H., & Ayalon, M. (2022). Characterizing critical teaching events: A multidimensional approach to lesson analysis. Teaching and Teacher Education, 117, 103798. https://doi.org/10.1016/j.tate.2022.103798 Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press.

1475 1482 Federica Troilo Eleonora Faggiano Melih Turgut University of Bari Aldo Moro Italy Norwegian University of Science and Technology Norway Student agency has become a central concept in contemporary mathematics education. This study examines how interaction with Generative Artificial Intelligence (GenAI) reshapes undergraduate students’ agency, particularly concerning perception of themselves as ‘actors’ or ‘executors’ in their learning process. Using a mixed-methods approach, the analysis revealed different individual trajectories, ranging from an increased sense of active involvement to reliance on GenAI for decision-making, when tasks become complex. The findings suggests that GenAI does not influence agency unilaterally but rather acts as a resource whose outcomes depend on how students choose to engage with the tool. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171– 200). Ablex Publishing. Brown, R. (2009). Teaching for social justice: Exploring the development of student agency through participation in the literacy practices of a mathematics classroom. Journal of Mathematics Teacher Education, 12(3), 171–185. Carlsen, M., Erfjord, I., Hundeland, P. S., & Monaghan, J. (2016). Kindergarten teachers’ orchestration of mathematical activities afforded by technology: agency and mediation. Educational Studies in Mathematics, 93(1), 1–17. Jääskelä, P., Poikkeus, A. M., Vasalampi, K., Valleala, U. M., & Rasku-Puttonen, H. (2017). Assessing agency of university students: Validation of the AUS Scale. Studies in Higher Education, 42(11), 2061–2079. https://doi.org/10.1080/03075079.2015.1130693 Nieminen, J. H., Chan, M. C. E., & Clarke, D. (2022). What affordances do open-ended reallife tasks offer for sharing student agency in collaborative problem-solving? Educational Studies in Mathematics, 109(1), 115–136. https://doi.org/10.1007/s10649-021-10074-9 OECD. (2019). An OECD learning framework 2030. In G. Bast, E.G. Carayannis & D.F.J. Campbell (Eds), The Future of Education and Labor. Arts, Research, Innovation and Society (pp. 23–35). Springer. https://doi.org/10.1007/978-3-030-26068-2_3 Pepin, B., Kock, Z. J., & Rezat, S. (2024). Toward student agency in the selection and use of digital resources for learning and studying mathematics. In B. Pepin, G. Gueudet & J. Choppin (Eds.), Handbook of digital resources in mathematics education (pp. 647–674). Springer International Publishing. https://doi.org/10.1007/978-3-030-95060-6_26-2 Pepin, B., Buchholtz, N., & Salinas-Hernández, U. (2025). A scoping survey of ChatGPT in mathematics education. Digital Experiences in Mathematics Education, 1–33. Pickering, A. (1995). The mangle of practice: Time, agency, and science. University of Chicago Press. Trouche, L. (2004). Managing the complexity of human–machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307. https://doi.org/10.1007/s10758-004-3468-5 Wagner, D. (2007). Students’ critical awareness of voice and agency in mathematics classroom discourse. Mathematical Thinking and Learning, 9(1), 31–50.

1483 1490 Larissa Tröbelsberger Alissa Fock Reinhard Oldenburg Hans-Stefan Siller University of Augsburg Germany Julius-Maximilians-University Würzburg Germany Artificial Intelligence (AI) is increasingly integrated into students’ everyday learning practices, yet little is known about how mathematics students actually use AI systems in their university studies. This study investigates patterns of AI use among 112 mathematics students in their first four semesters at two German universities. A quantitative survey comprising five theoretically derived scales was administered and analysed. The results indicate that AI plays a substantial role in clarifying mathematical content, while lecture notes remain a very important learning resource. Students show a high level of error awareness, but this awareness does not influence their usage patterns. Strong associations were found among productive and substitutive use, leaning-supportive use and optimizing AI-Interaction, forming a triangular structure of mutual implications. Ergas, O., Gilead, T., Eilam, E., Kyza, E., Akar, B., Buck, M. F., & Kumar, S. (2022). A curricular framework for flourishing in education. In The International Science and Evidence-based Education Assessment:ISEE Assessment Working Group 1 (S. 180–250). UNESCO. Fang, M., Wan, X., Lu, F., Xing, F., & Zou, K. (2024). MathOdyssey: Benchmarking Mathematical Problem-Solving Skills in Large Language Models Using Odyssey Math Data (No. arXiv:2406.18321). arXiv. https://doi.org/10.48550/arXiv.2406.18321 Fock, A., & Siller, H.-S. (2025b). Generative artificial intelligence in secondary STEM education in the light of Human Flourishing: A scoping literature review. International Journal of STEM Education, 12(1), 67. https://doi.org/10.1186/s40594-025-00589-5 Gerlich, M. (2025). AI Tools in Society: Impacts on Cognitive Offloading and the Future of Critical Thinking. Societies, 15(1), 6. https://doi.org/10.3390/soc15010006 Gras, R., Suzuki, E., Guillet, F., & Spagnolo, F. (Hrsg.). (2008). Statistical Implicative Analysis: Theory and Applications (Bd. 127). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-78983-3 Noster, N., Gerber, S., & Siller, H.-S. (2024). Pre-Service Teachers’ Approaches in Solving Mathematics Tasks with ChatGPT. Digital Experiences in Mathematics Education, 10(3), 543–567. https://doi.org/10.1007/s40751-024-00155-8 Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric theory (3. ed., [Nachdr.]). McGrawHill. OECD. (2023). Is Education Losing the Race with Technology?: AI’s Progress in Maths and Reading. OECD Publishing. https://doi.org/10.1787/73105f99-en Rosseel, Y. (2024). lavaan: Latent variable analysis (R package Version 0.6-20). https://CRAN.R-project.org/package=lavaan Russell, S. J., & Norvig, P. (with Chang, M., Devlin, J., Dragan, A., Forsyth, D., Goodfellow, I., Malik, J., Mansinghka, V., Pearl, J., & Wooldridge, M. J.). (2022). Artificial intelligence: A modern approach (Fourth edition, global edition). Pearson. Sun, Y., Sheng, D., Zhou, Z., & Wu, Y. (2024). AI hallucination: Towards a comprehensive classification of distorted information in artificial intelligence-generated content. Humanities Social Sciences Communications, 11(1), 1278. https://doi.org/10.1057/s41599-024-03811-x Taulli, T. (2023). Generative AI: How ChatGPT and Other AI Tools Will Revolutionize Business. Apress. https://doi.org/10.1007/978-1-4842-9367-6 Wardat, Y., Tashtoush, M. A., AlAli, R., & Jarrah, A. M. (2023). ChatGPT: A revolutionary tool for teaching and learning mathematics. Eurasia Journal of Mathematics, Science and Technology Education, 19(7), em2286. https://doi.org/10.29333/ejmste/13272 Zhuang, Y. (2025). Lessons from Using ChatGPT in Calculus: Insights from Two Contrasting Cases. Journal of Formative Design in Learning, 9(1), 25–35. https://doi.org/10.1007/s41686-025-00098-2

1499 1506 Zelha Tunc Pekkan Beyda Yilmaz University of Agder Norway Eötvös Loránd University Hungary Fractions are challenging, requiring teachers to interpret and respond effectively to students’ reasoning. This study examines how 12 experienced teachers (ET) and 10 pre-service teachers (PST) interpret and respond to problematic situations involving fractional knowledge on number lines. Using professional noticing as a framework, we analyzed two dimensions: interpreting and responding. Fraction sub-constructs and units coordination guided analysis. Participants viewed fraction-related video clips and answered open-ended questions probing noticing skills. Findings show ET more often demonstrated advanced interpretations and responses involving units coordination, while PST relied on quotient subconstruct. Results underscore the need for targeted teacher education on content-specific noticing. Amador, J. M., Glassmeyer, D., & Brakoniecki, A. (2025). Teachers’ noticing of proportional reasoning. Journal of Mathematics Teacher Education, 28, 879–907. Empson, S. B., & Jacobs, V. R. (2021). Exploring teachers’ responsiveness to children’s fraction thinking and relationships to fraction achievement. In D. Olanoff, K. Johnson, & S. Spitzer (Eds.), Proc. of the 43rd annual meeting of the NA Chapter of the International Group for the Psychology of Mathematics Education (pp. 1449–1456). PME-NA. Hackenberg, A. J., & Sevinc, S. (2024). Students’ units coordinations. In P. C. Dawkins et al. (Eds.), Piaget’s Genetic Epistemology for Mathematics Education Research (pp. 211– 230). Springer. https://doi.org/10.1007/978-3-031-47386-9_11 Jacobs, V. R., & Empson, S. B. (2021). Profiles of teachers’ expertise in professional noticing of children’s mathematical thinking. In D. Olanoff, K. Johnson, & S. Spitzer (Eds.), Proceedings of the forty-third annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 652–659). Philadelphia, PA: PME-NA. Jacobs, V. R., Lamb, L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49–84)., Lawrence Erlbaum Associates. König, J., Blömeke, S., & Kaiser, G. (2022). Video-based research on teacher education: A systematic review. Educational Research Review, 35, 100426. https://doi.org/10.1016/j.edurev.2021.100426 Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., ... & Chen, M. (2013). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101 Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York, NY: Springer.

1515 1522 Natsuki Uchikubo Graduate School of Comprehensive Human Sciences, University of Tsukuba Japan This study explores how mathematical giftedness is actualized through interactional dynamics in the mathematics classroom. Adopting a single-case study design, we analyzed a critical episode in which a mathematically gifted student’s new insight was introduced and adapted into the classroom. The analysis identified three distinctive phases, each involving diverse interactions among the gifted student, teacher, classmates, and shared mathematical context. As an initial attempt to elucidate the socio-dynamic nature of mathematical giftedness, this study provides insights into how each classroom element contributes to the actualization of mathematical giftedness. Beghetto, R. A. (2019). From static to dynamic: Toward a socio-dynamic perspective on creativity in classrooms. In I. Lebuda, V. P. Glăveanu (Eds.), The Palgrave Handbook of Social Creativity Research (pp. 473-485). Palgrave Macmillan. Brandl, M. (2011). A constructive approach to the concept of mathematical giftedness based on systems theory. In M. Avotiņa, D. Bonka, H. Meissner, L. Ramāna, L. Sheffield, & E. Velikova (Eds.), Proc. 6th International Conference on Creativity in Mathematics Education and the Education of Gifted Students (pp. 35-39). MCG. Corbin, J., & Strauss, A. (2014). Basics of qualitative research: Techniques and procedures for developing grounded theory. Sage publications. Csikszentmihalyi, M. (2014). Society, culture, and person: A systems view of creativity. In M. Csikszentmihalyi (Ed.), The Systems Model of Creativity (pp. 47-61). Springer. Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. The University of Chicago. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 2963. Lo, C. O., & Porath, M. (2017). Paradigm shifts in gifted education: An examination vis-àvis its historical situatedness and pedagogical sensibilities. Gifted Child Quarterly, 61(4), 343-360. Merriam, S. B., & Tisdell, E. J. (2016). Qualitative research: A guide to design and implementation (4th ed.). Jossey Bass. Mora, M., Jaime, A., & Gutierrez, A. (2022). A Framework of descriptors to characterize flexibility as a trait of mathematical giftedness. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proc. 45th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, p. 379). PME. Renzulli, J. S. (1978). What makes giftedness?: Reexamining a definition. The Phi Delta Kappan, 60(3), 180-184, 261. Singer, F. M., Sheffield, L. J. & Leikin, R. (2017). Advancements in research on creativity and giftedness in mathematics education: Introduction to the special issue. ZDM, 49, 5-12. Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students. The Journal of Secondary Gifted Education, 14(3), 151-165. Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41, 13-27. Sriraman, B. (2022). Uncertainty as a catalyst and condition for creativity: The case of mathematics. ZDM, 54, 19-33. Yin, R. K. (2018). Case Study Research and Applications: Design and Methods (6th ed.). Sage.

1523 1530 Nitchada Kamlue Laura Van Zoest Western Michigan University United States This exploratory qualitative study used 26 prospective elementary teachers’ (PTs’) mathematical justifications before and after they participated in a whole-class discussion of a high-leverage PT contribution to provide insight into who benefits from optimizing mathematically productive struggle. We used their own words to code and analyzed the 52 written responses, and identified five types of understanding: Robust, Substantial, Emerging, Preliminary, and No Evidence. We learned that, regardless of the PTs’ initial type of understanding and whether they expressed struggle, 85% demonstrated progress after the discussion, and 69% evidenced an increase in their type of understanding. We conclude that optimizing productive struggle may be a particularly productive approach for developing PTs’ flexible thinking. Baker, K., Clark, M., & Gallagher, D. M. (2024). Together in a productive struggle: Unpacking student and teacher productive struggle in mathematics. The Electronic Journal for Research in Science & Mathematics Education, 27(4), 107-117. Bishop, J. P. (2021). Responsiveness and intellectual work: Features of mathematics classroom discourse related to student achievement. Journal of the Learning Sciences 30(3), 466-508. https://doi.org/10.1080/10508406.2021.1922413 Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann. Clarke, D. M., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Ed. St. in Math., 72, 127-138. Creswell, J. W., & Poth, C. N. (2018). Qualitative inquiry and research design: Choosing among five approaches (4th ed.). Sage publications. Elsevier. (n.d.). Scopus [Database]. Elsevier. https://www.scopus.com Kalinec-Craig, C. A. (2017). The rights of the learner: A framework for promoting equity through formative assessment in mathematics education. Democracy and Education, 25(2), 1-11. Kamlue, N., & Van Zoest, L. R. (2025). An Exploratory Study of Mathematically Productive Struggle in Prospective Elementary Teachers’ Whole-Class Discussion of a HighLeverage Contribution. [Manuscript submitted for publication]. Department of Mathematics, Western Michigan University. National Council of Teachers of Mathematics [NCTM]. (2014). Principles to actions: Ensuring mathematical success for all. Author. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313-340. Stockero, S. L., Leatham, K. R., & Peterson, B. E., Van Zoest, L. R. (in press). A conceptualization of the teaching practice of building. Journal of Mathematics Teacher Education. Warshauer, H. K. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18, 375-400.

1531 1538 Hamsa Venkat Angel Mukuka CASTeL, Dublin City University Ireland University of the Witwatersrand South Africa In this paper, we share the process of developing an analytical framework for considering data comprised of teachers’ responses to an instructional explanation video offering help to a child who is stuck on how to solve: _ - 23 = 14. In the paper, we overview the relatively small literature base on instructional explanations and illustrate some of the ways in which the available (almost exclusively ‘global north’) frameworks failed to capture salient aspects of a South African dataset. This led us into a grounded, but literature-informed, development of an analytical framework that better represented the underlying data. In the body of the paper, we detail this analytical framework and the axial coding that was used to develop indicators linked to a hierarchy based on ‘nature/extent of reference to mathematics’. Bowie, L., Venkat, H., & Groenewald, C. (2025). Teachers’ understandings of early number models: Data from an online pilot study. In Proceedings of the 33rd Annual Conference of SAARMSTE, 52-57. Corbin, J. & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory (3rd ed.). Thousand Oaks. SAGE. https://doi.org/10.4135/9781452230153 Güneş, C., Kercher, A., & Zaskis, R. (2024). Teachers’ noticing mathematics within imprecise instructional explanations. Canadian Journal of Science, Mathematics and Technology Education, 24, 400-417. https://doi.org/10.1007/s42330-025-00355-3 Kinach, B.M. (2002). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: Toward a model of effective practice. Teaching and Teacher Education, 18, 51-71. https://doi.org/10.1016/S0742051X(01)00050-6 Leinhardt, G. (1993). Weaving instructional explanations in history. British Journal of Educational Psychology, 63, 46-74. Mukuka, A., & Venkat, H. (In press, 2026). Insights from primary school teachers’ instructional explanations in early grade mathematics. Proceedings of the 34th Annual Conference of the Southern African Association for Research in Mathematics, Science and Technology Education. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26. Spaull, N., Pampallis, I., Ardington, C., Sapire, I., & Isaac, P. (2022) Not adding it up: Grade 1 mathematics outcomes in the Eastern Cape and Limpopo. In H. Venkat & N. Roberts. Early Grade Mathematics in South Africa. Oxford University Press.

1539 1546 Aikaterini Vissariou Despina Desli School of Primary Education, Aristotle University of Thessaloniki Greece The purpose of the present study was to examine the changes in children’s declarative knowledge related to mathematical problem solving following metacognitive instruction. Participants were 38 Greek Year 3 children, half of which received metacognitive instruction (experimental group), while the others did not (control group). Results indicated that students who were assigned to metacognitive instruction outperformed their counterparts in declarative knowledge, whereas no significant differences between the two groups were observed prior to the intervention. The educational implications of these findings are discussed. Anderson, N. J. (2002). The role of metacognition in second language teaching and learning. ERIC Digest: ED 463659. Cornoldi, C. (1998). The impact of metacognitive reflection on cognitive control. In G. Mazzoni, & T.O. Nelson (Eds.), Metacognition and cognitive neuropsychology: Control and monitoring processes (pp. 139–160). LEA. Cornoldi, C., Carretti, B., Drusi, S., & Tencati, C. (2015). Improving problem solving in primary school students: The effect of a training programme focusing on metacognition and working memory. British Journal of Educational Psychology, 85(3), 424–439. De Corte, E., Verschaffel, L., & Masui, C. (2004). The CLIA-model: A framework for designing powerful learning environments for thinking and problem solving. European Journal of Psychology of Education, 19(4), 365–384. Desoete, A. (2008). Multi-method assessment of metacognitive skills in elementary school children: How you test is what you get. Metacognition and Learning, 3(3), 189–206. Ellis, A.K., Denton, D.W., & Bond, J.B. (2014). An analysis of research on metacognitive teaching strategies. Procedia-Social and Behavioral Sciences, 116(21), 4015–4024. Flavell, J.H. (1979). Metacognition and cognitive monitoring: A new era of cognitive development inquiry. American Psychologist, 34(10), 906–911. Gascoine, L., Higgins, S., & Wall, K. (2017). The assessment of metacognition in children aged 4–16 years: a systematic review. Review of Education, 5(1), 3–57. Hohn, R.L., & Frey, B. (2002). Heuristic training and performance in elementary mathematical problem solving. The Journal of Educational Research, 95(6), 374–380. Kim, Y.R., Park, M.S., Moore, T.J., & Varma, S. (2013). Multiple levels of metacognition and their elicitation through complex problem-solving tasks. The Journal of Mathematical Behavior, 32(3), 377–396. Lee, N.H., Yeo, D.J.S., & Hong, S.E. (2014). A metacognitive-based instruction for primary 4 students to approach non-routine mathematical word problems. ZDM, 46(3), 465–480. Mevarech, Z.R., & Kramarski, B. (1997). IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms. American Educational Research Journal, 34(2), 365–395. Schunk, D.H. (2008). Metacognition, self-regulation, and self-regulated learning: Research recommendations. Educational Psychology Review, 20(4), 463–467. Sperling, R.A., Howard, B.C., Miller, L.A., & Murphy, C. (2002). Measures of children's knowledge and regulation of cognition. Contemporary Ed. Psychology, 27(1), 51–79. Tarricone, P. (2011). The taxonomy of metacognition. Psychology Press. Vissariou, A., & Desli, D. (2019). Metacognition in non-routine problem solving process of Year 6 children. In U.T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the 11th Congress of the European Society for Research in Mathematics Education (pp. 519–526). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME. Wilson, J., & Clarke, D. (2004). Towards the modelling of mathematical metacognition. Mathematics Education Research Journal, 16(2), 25–48. Yildiz, E., Akpinar, E., Tatar, N., & Ergin, O. (2009). Exploratory and confirmatory factor analysis of the metacognition scale for primary school students. Educational Sciences: Theory and Practice, 9(3), 1591–1604. Yimer, A., & Ellerton, N.F. (2006). Cognitive and metacognitive aspects of mathematical problem solving: An emerging model. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures, and learning spaces (pp. 575–582). Mathematics Education Research Group of Australasia.

1547 1554 Robert von Hering Lena Jacobsen Henning Sievert University of Flensburg University of Hildesheim Germany Mathematical modelling involves translating real-world situations into mathematical problems, with real-world contexts playing a central role. The challenge of transferring modelling competencies across different contexts is insufficiently explored in primary education. This study examines how various real-world contexts influence primary students’ modelling performance and transfer ability. Using a mixed-methods design, 136 students solved problems constructed in parallel with private, school, and vocational contexts, followed by stimulated recall interviews with 10 students. Results reveal significant context effects and transfer issues despite students’ recognition of structural similarities between problems. Findings emphasize the need for diverse authentic contexts and targeted support in early modelling education. Almuna Salgado, F. J. (2017). The role of context and context familiarity on mathematics problems. Revista latinoamericana de investigación en matemática educativa, 20(3), 265– 292. Brown, J. P. (2019). Real-world task context: Meanings and roles. In G. A. Stillman & J. P. Brown (Eds.), Lines of inquiry in mathematical modelling research in education (pp. 53– 81). Cham: Springer International Publishing. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational researcher, 18(1), 32–42. Christensen, R. H. B. (2022). Cumulative link models for ordinal regression with the R package ordinal. Version 2022.6-16. Evans, J. (1999). Building bridges: Reflections on the problem of transfer of learning in mathematics. Educational Studies in Mathematics, 39(1–3), 23–44. von Hering, R., Grube, A., & Sievert, H. (2025). Vocational Orientation in Primary School Mathematics Textbooks. In B. Pepin, I. Kohanová, & M. B. Langfeldt (Eds.), Proceedings of the Fifth International Conference on Mathematics Textbook Research and Development (pp. 437–444). Norwegian University of Science and Technology. von Hering, R., Rietenberg, A., Heinze, A., & Lindmeier, A. (2018). The applied knowledge of trainees as industrial clerks solving problems with vocational and non-vocational context. Bergqvist, E., Österholm, M., Granberg, C., & Sumpter, L. (Eds.), Proc. 42nd Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 379– 386). PME. Krawitz, J., Schukajlow, S., Chang, Y.-P., & Yang, K.-L. (2017). Reading comprehension, enjoyment, and performance: how important is a deeper situation model? In K. Berinderjeet, H. Weng-Kin, T. Tin-Lam, & C. Ban-Heng (Eds.), Proc. 41st Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 97⁠–⁠104). PME. Mayring, P. (2022). Qualitative Content Analysis. A Step-by-Step Guide. Sage. Niss, M. & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. Niss, M. & Højgaard, T. (2011). Competencies and Mathematical Learning. Ideas and inspiration for the development of mathematics teaching and learning in Denmark. Roskilde University. OECD (2023). PISA 2022 assessment and analytical framework. OECD Publishing. Orón, J. V. & Lizasoain, I. (2023). Achieving transfer from mathematics learning. Education Sciences, 13(2), 161. Smith, C., & Morgan, C. (2016). Curricular orientations to real-world contexts in mathematics. The Curriculum Journal, 27(1), 24–45. Spiro, R. J., Coulson, R. L., Feltovich, P. J., & Anderson, D. K. (1988). Cognitive Flexibility theory: Advanced knowledge acquisition in ill-structured domains. University of Illinois at Urbana-Champaign. Verschaffel, L., Schukajlow, S., Star, J., & van Dooren, W. (2020). Word problems in mathematics education: a survey. ZDM – Mathematics Education, 52(1–16). Vessonen, T., Dahlberg, M., Hellstrand, H., Widlund, A., Korhonen, J., Aunio, P., & Laine, A. (2024). Task Characteristics Associated with Mathematical Word Problem-Solving Performance Among Elementary School-Aged Children: A Systematic Review and MetaAnalysis. Educational Psychology Review, 36(117).

1555 1562 Nils Völlinger Stanislaw Schukajlow Janina Krawitz University of Münster Germany University of Cologne Germany Solving open mathematical modelling problems is challenging for students. Students’ beliefs may be a reason for these difficulties. In this study, we analyzed (1) students’ beliefs about open mathematical modelling problems and (2) how their beliefs are related to their solution processes. Ten ninth graders solved open mathematical modelling problems and verbalized their solution processes. Inductive qualitative content analysis revealed that students held multiple inappropriate beliefs, for example, that applying a mathematical procedure is more important than critically engaging with the real-world situation. This belief was negatively associated with the correctness of students’ solutions to modelling problems. This study contributes to the theory of beliefs by revealing a belief system with respect to open modelling problems. Kaiser, G., & Maaß, K. (2007). Modelling in lower secondary mathematics classroom— Problems and opportunities. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and Applications in Mathematics Education (Vol.10, pp. 99–108). Springer US. Mayring, P. (2014). Qualitative content analysis—Theoretical foundation, basic procedures and software solution. Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. https://doi.org/10.4324/9781315189314 Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). National Council of Teachers of Mathematics. Reusser, K., & Stebler, R. (1997). Every word problem has a solution—The social rationality of mathematical modeling in schools. Learning and Instruction, 7(4), 309–327. https://doi.org/10.1016/S0959-4752(97)00014-5 Schukajlow, S., Krawitz, J., Kanefke, J., Blum, W., & Rakoczy, K. (2023). Open modelling problems: Cognitive barriers and instructional prompts. Educational Studies in Mathematics, 114(3), 417–438. https://doi.org/10.1007/s10649-023-10265-6 Schukajlow, S., Krawitz, J., Wiehe, K., & Rakoczy, K. (2025). Students’ preferences and beliefs about the openness of modelling problems—Does teaching students how to solve open problems matter? In T. Ikeda, A. Saeki, V. Geiger, & G. Kaiser (Eds.), International Horizons in Mathematics Modelling Education (1st ed., pp. 369–379). Springer Nature. Silver, E. A. (1995). The nature and use of open problems in mathematics education: Mathematical and pedagogical perspectives. ZDM - Mathematics Education, 95(2), 67– 72. Simon, H. A. (1973). The structure of ill structured problems. Artificial Intelligence, 4(3–4), 181–201. Skott, J. (2015). Towards a participatory approach to ‘beliefs’ in mathematics education. In B. Pepin & B. Roesken-Winter (Eds.), From beliefs to dynamic affect systems in mathematics education: Exploring a mosaic of relationships and interactions (pp. 3–23). Springer International Publishing. https://doi.org/10.1007/978-3-319-06808-4 Van Dooren, W., Lem, S., De Wortelaer, H., & Verschaffel, L. (2019). Improving realistic word problem solving by using humor. Journal of Mathematical Behavior, 53, 96–104.

1563 1570 Jennifer Way Katherin Cartwright University of Sydney Australia Neuropsychological research has established the value of finger-number representations in mathematical development, yet indecision remains in education. The responses of 31 children (age 6/7 years) when asked to show how they count to ten using their fingers, were videoed, and six months later the task was repeated. Micro-analysis of the Time 1 video clips revealed substantial variety in finger-counting methods, and Time 2 analysis showed that most children changed their method. We argue that more education-focused research is needed to resolve questions such as whether a stable finger-counting pattern is needed to maximise learning benefits Australian Curriculum, Assessment and Reporting Authority (ACARA). (2020). National numeracy learning progression, Version 3. https://www.ofai.edu.au/ Barrocas, R., Roesch, S., Gawrilow C., & Moeller, K. (2020). Putting a finger on numerical development – Reviewing the contributions of Kindergarten finger gnosis and fine motor skills to numerical abilities. Frontiers in Psychology, 11:1012. https//:doi.org/10.3389/fpsyg.2020.01012 Bender, A., & Beller, S. (2012). Nature and culture of finger counting: diversity and representational effects of an embodied cognitive tool. Cognition, 124(2), 156–182. https://doi.org/10.1016/j.cognition.2012.05.005. Chinello, A., Cattani, V., Bonfiglioli, C., Dehaene, S., Piazza, M. (2013). Objects, numbers, fingers, space: clustering of ventral and dorsal functions in young children and adults. Developmental Science, 16(3), 377–393. https://doi.org/ 10.1111/desc.12028 Di Luca, S., Granà, A., Semenza, C., Seron, X., & Pesenti, M. (2006). Finger–digit compatibility in Arabic numeral processing. The Quarterly Journal of Experimental Psychology, 59(9), 1648–1663. https://doi.org/10.1080/17470210500256839 Di Luca, S., & Pesenti, M. (2011). Finger numeral representations: more than just another symbolic code. Frontiers in Psychology, 2,272. https://doi.org/10.3389/fpsyg.2011.00272. Dupont-Boime, J., & Thevenot, C. (2018). High working memory capacity favours the use of finger counting in six-year-old children, Journal of Cognitive Psychology, 30(1), 35-42, https//:doi.org/10.1080/20445911.2017.1396990 Fischer, M. H. (2008). Finger counting habits modulate spatial-numerical associations. Cortex, 44(4), 386–392. https://doi.org/10.1016/j.cortex.2007.08.004 Gracia-Bafalluy, M., and Noël, M.-P. (2008). Does finger training increase young children’s numerical performance? Cortex 44, 368–375. https//:doi.org/10.1016/j.cortex.2007.08.020 Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2008). Development of number combination skill in the early school years: When do fingers help? Developmental Science, 11, 662–668. https//:doi.org/10.1111/j.1467-7687.2008.00715.x Krenger, M., & Thevenot, C. (2025). The role of children’s finger counting history on their addition skills. Developmental Psychology. Advance online publication. https://doi.org/10.1037/dev0002099 Moeller, K., Martignon, L., Wessolowski, S., Engel, J., & Nuerk, H. (2011). Effects of finger counting on numerical development - the opposing views of neurocognition and mathematics education. Frontiers in Psychology, 2, Article 328. https://doi.org/10.3389/fpsyg.2011.00328 Moeller, K., Fischer, U., Link, T. et al. (2012). Learning and development of embodied numerosity. Cognitive Process, 13 (Suppl 1), 271–274 https://doi.org/10.1007/s10339012-0457-9 Olliver, F., Noël, Y., Legrand, A., & Bonneton-Botté, N. (2020). A teacher-implemented intervention program to promote finger use in numerical tasks. European Journal of Psychology of Education, 35, 589–606. https://doi.org/10.1007/s10212-019-00441-9 Penner-Wilger, M., & Anderson, M. L. (2013). The relation between finger gnosis and mathematical ability: why redeployment of neural circuits best explains the finding. Frontiers in Psychology, 4, 877. https//:doi.org/10.3389/fpsyg.2013.00877 Poletti, C., Krenger, M., Létang, M., & Thevenot, C. (2023). French preschool and primary teachers’ attitude towards finger counting. Acta Psychologica, 241, Article 104079. https://doi.org/10.1016/j.actpsy.2023.104079 Roesch, S., and Moeller, K. (2015). Considering digits in a current model of numerical development. Frontiers in Human Neuroscience, 8, 1062. https://doi.org/10.3389/fnhum.2014.01062 Sato, M., & Lalain, M. (2008). On the relationship between handedness and hand-digit mapping in finger counting. Cortex, 44, 393–399. https://doi.org/10.1016/j.cortex.2007.08.005 Way, J., & Cartwright. (2024). Finger counting patterns in Year 1 children. Paper presented at the 15th International Congress on Mathematical Education. Sydney, 7-14 July, 2024.

1571 1578 Hang Wei Catalina Lomos Xiaoran Li Alik Palatnik Dor Abrahamson Utrecht University, The Netherlands Luxembourg Institute of Socio-Economic Research Luxembourg Beijing Language and Culture University China The Hebrew University of Jerusalem Israel University of California, Berkeley, USA United States Korea University South Korea ‘Heteroperception’ is a socio-cognitive phenomenon where participants in a joint activity perceive shared resources differently. Rather than viewing such divergence as disruptive, we consider advantages of heteroperception for problem solving by investigating how 3 prospective math teachers co-construct a body-scale icosahedron. Data include multi-person mobile eye-tracking (MET) and synchronized MET-overlaid point-of-view videos. Across 3 microgenetic episodes, we trace a recurring triadic progression of heteroperception: implicit, explicit, resolved. We show how divergent perceptual foci coalesce discursively, tacitly negotiated through situated operations, to formulate actionable construction solutions. Heteroperception is not a bug but a feature, which teachers should anticipate, foster, valorise, and mediate. Abrahamson, D., Lomos, C., & Palatnik, A. (2025). The icosahedron in the room: Revisiting pedagogical dilemmas of individual learning in collaborative activities. In M. Bosch, G. Bolondi, S. Carreira, C. Spagnolo, & M. Gaidoschik (Eds.), Proceedings of CERME14 (pp. 2569–2576). Free University of Bozen-Bolzano and ERME. Benally, J., Palatnik, A., Ryokai, K., & Abrahamson, D. (2022). Learning through negotiating conceptually generative perspectival complementarities: The case of geometry. For the Learning of Mathematics, 42(3), 34–41. Chow, J. Y., Davids, K., Button, C., Shuttleworth, R., Renshaw, I., & Araújo, D. (2007). The role of nonlinear pedagogy in physical education. Rev. of Ed. Research, 77(3), 251–278. Goodwin, C. (2013). The co-operative, transformative organization of human action and knowledge. Journal of Pragmatics, 46(1), 8–23. Hutto, D. D., & Sánchez-García, R. (2015). Choking RECtified: Embodied expertise beyond Dreyfus. Phenomenology and the Cognitive Sciences, 14(2), 309–331. Leontiev, A. N. (1981). Problems of the development of the mind (M. Kopylova, Trans.). Progress. Murphy, K. M. (2004). Imagination as joint activity: The case of architectural interaction. Mind, Culture, and Activity, 11(4), 267–278. Ohlinger, B. D., & Sasaki, T. (2026). How miscommunication can improve collective performance in social insects. Trends in Cognitive Sciences, 30(1), 10–12. Palatnik, A., & Abrahamson, D. (in press). Building knowledge: The ecological dynamics of co-operative action in collaborative construction of body-scale geometric structures. Journal of the Learning Sciences. Rączaszek-Leonardi, J., Dębska, A., & Sochanowicz, A. (2014). Pooling the ground: Understanding and coordination in collective sense making. Frontiers in Psychology, 5. Rosenbaum, L. F., Reimer, P., Palatnik, A., & Abrahamson, D. (2024). Building mathematical spaces through multi-faceted, body-scale geometry. Ed. Designer, 4(16). Schwarz, B. B., & Baker, M. J. (2017). Dialogue, argumentation, and education: History, theory, and practice. Cambridge University Press. Surowiecki, J. (2004). The wisdom of crowds. Random House, Doubleday. Varela, F. J. (1999). Ethical know-how: Action, wisdom, and cognition. SUP. Wigner, E. P. (1990). The unreasonable effectiveness of mathematics in the natural sciences. Mathematics and Science, 13, 1–14. Wittgenstein, L. (1953). Philosophical investigations. Basil Blackwell.

1579 1586 Ole Weigt Timo Leuders Katharina Loibl Lars Holzäpfel University of Education Freiburg Germany To date, there is only limited knowledge about the errors students make when solving differentiation tasks. This study presents an exploratory investigation of students’ errors in differentiation in high school–level tasks. 69 students produced a total of 2281 answers to differentiation tasks, resulting in 1231 errors. An inductive categorization of these errors revealed 26 distinct data-driven error categories. These 26 categories were assigned to four concept-driven error categories. A cluster analysis identified four groups of individuals with different error profiles based on these concept-driven categories. The findings provide insights into the heterogeneity of errors in differentiation and students’ general error behaviour. Moreover, the results show that the empirical data align well with the concept-based classification of errors. Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological Review, 89(4), 369–406. https://doi.org/10.1037/0033-295X.89.4.369 Bardy, T., Holzäpfel, L., Reinhold, F., & Leuders, T. (2024). Mathematics teachers’ multiple perspectives on adaptive tasks: Task evaluation and selection as core practices for teaching quality. ZDM – Mathematics Education, 56(5), 981–996. https://doi.org/10.1007/s11858024-01626-3 Ben-Zeev, T. (1998). Rational errors and the mathematical mind. Review of General Psychology, 2(4), 366–383. https://doi.org/10.1037/1089-2680.2.4.366 Ben-Zeev, T., & Ronald, J. (2002). Is procedure acquisition as unstable as it seems? Contemporary Educational Psychology, 27(4), 529–550. https://doi.org/10.1016/S0361476X(02)00005-X Brown, J. S., & Vanlehn, K. (1980). Repair Theory: A Generative Theory of Bugs in Procedural Skills. Dorner, C., Ableitinger, C., & Krammer, G. (2025). Revealing the nature of mathematical procedural knowledge by analysing students’ deficiencies and errors. International Journal of Mathematical Education in Science and Technology, 1–22. https://doi.org/10.1080/0020739X.2024.2445666 Greefrath, G., Oldenburg, R., Siller, H.-S., Ulm, V., & Weigand, H.-G. (2016). Aspects and “Grundvorstellungen” of the Concepts of Derivative and Integral: Subject Matter-related Didactical Perspectives of Concept Formation. Journal Für Mathematik-Didaktik, 37(S1), 99–129. https://doi.org/10.1007/s13138-016-0100-x Hennessy, S. (1994). The stability of children’s mathematical behavior: When is a bug really a bug? Learning and Instruction, 3(4), 315–338. https://doi.org/10.1016/09594752(93)90022-R Kuckartz, U. (2019). Qualitative Text Analysis: A Systematic Approach. In G. Kaiser & N. Presmeg (Hrsg.), Compendium for Early Career Researchers in Mathematics Education (S. 181–197). Springer International Publishing. https://doi.org/10.1007/978-3-03015636-7_8 Neidorf, T., Arora, A., Erberber, E., Tsokodayi, Y., & Mai, T. (2020). Student Misconceptions and Errors in Physics and Mathematics: Exploring Data from TIMSS and TIMSS Advanced (Bd. 9). Springer International Publishing. https://doi.org/10.1007/978-3-03030188-0 Norman, D. (1981). Categorization of Action Slips. Psychological Review, 88, 1–15. https://doi.org/10.1037//0033-295X.88.1.1 Ohlsson, S. (1996). Learning from performance errors. Psychological Review, 103(2), 241– 262. https://doi.org/10.1037/0033-295X.103.2.241 Radatz, H. (1979). Error Analysis in Mathematics Education. Journal for Research in Mathematics Education, 10(3), 163–172. JSTOR. https://doi.org/10.2307/748804 Rasmussen, C., Marrongelle, K., & Borba, M. C. (2014). Research on calculus: What do we know and where do we need to go? ZDM, 46(4), 507–515. https://doi.org/10.1007/s11858014-0615-x Rittle-Johnson, B., & Schneider, M. (2014). Developing Conceptual and Procedural Knowledge of Mathematics. In R. Cohen Kadosh & A. Dowker (Hrsg.), The Oxford Handbook of Numerical Cognition (1. Aufl., S. 1118–1134). Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199642342.013.014 Siyepu, S. W. (2013). An exploration of students’ errors in derivatives in a university of technology. The Journal of Mathematical Behavior, 32(3), 577–592. https://doi.org/10.1016/j.jmathb.2013.05.001 Stewart, S., Reeder, S., Raymond, K., & Troup, J. (2018). Could algebra be the root of problems in calculus courses. Proceedings of the 21st annual conference on research in undergraduate mathematics education, 1023–1030. VanLehn, K. (1982). Bugs are not enough: Empirical studies of bugs, impasses and repairs in procedural skills. The Journal of Mathematical Behavior, 3(2), 3–71.

1587 1594 Carl Wolfert Irene Neumann Daniel Sommerhoff IPN - Leibniz Institute for Science and Mathematics Education, Kiel Germany Kiel University Germany Online self-assessments (OSAs) are increasingly important, particularly during the transition from school to university. While feedback is integral to OSAs, it is unclear how to design feedback to be as useful as possible for learners. The empirical study compares the perceived usefulness and intended use of task-specific and aggregated feedback in mathematical OSAs, using data from N = 101 participants transitioning from school to university in a mathematics bridging course. The results show that it is not possible to favour one feedback aggregation level over the other in general. Instead, the results on perceived usefulness indicate that aggregated feedback is preferred by weaker learners, whereas task-specific feedback may better suit stronger learners. Individualization may thus be key for feedback design. Hattie, J., & Timperley, H. (2007). The Power of Feedback. Review of Ed. R., 77(1), 81-112. Henderson, M., Phillips, M., Ryan, T., Boud, D., Dawson, P., Molloy, E., & Mahoney, P. (2019). Conditions that enable effective feedback. Higher Education Research & Development, 38(7), 1401-1416. Janke, S., & Karst, K. (2025). Online Self Assessments as guidance systems for self-reflection and decision making in higher education – Broadly implemented yet poorly understood. Zeitschrift für Pädagogische Psychologie, 39(1–2), 1–9. Jordan, S. E. (2014). E-assessment for learning? Exploring the potential of computer-marked assessment and computer-generated feedback, from short-answer questions to assessment analytics [Doctoral dissertation, The Open University]. The Open University. Laaziz, Y., Chemsi, G., & Radid, M. (2024). The Influence of E-Assessment on Students’ Cognitive Engagement in Higher Education. International Journal of Engineering Pedagogy (iJEP), 14(4), 54-67. MBWK, IQSH, & IPN. (2022). MaLeMINT-I - Konkretisierung der mathematischen Lernvoraussetzungen der MaLeMINT-Studie durch Aufgaben. Leibniz Institute for Science and Mathematics Education. Rach, S., Sommerhoff, D., & Ufer, S. (2021). Technical Report - Knowledge for University Mathematics (KUM) and Mathematics Online Assessment System (MOAS) (MCLS Report No. 1.) Munich Center of the Learning Sciences, LMU. Rach, S., & Ufer, S. (2020). Which Prior Mathematical Knowledge Is Necessary for Study Success in the University Study Entrance Phase? Results on a New Model of Knowledge Levels Based on a Reanalysis of Data from Existing Studies. International Journal of Research in Undergraduate Mathematics Education, 6(3), 375-403. Rach, S., Ufer, S., & Sommerhoff, D. (2023). Mathematics online assessment: Do future mathematics students find assessment-based feedback useful? In Hanse-Kolloquium zur Hochschuldidaktik der Mathematik 2021 (pp. 3-14). WTM. Reich, B., Liebendörfer, M., Eichler, A., Schürmann, M., Stroh-Maraun, N., & Müller, R. (2025). How Students Evaluate Different Task Formats: An Exploratory Study in a Large Mathematics Course for Business Students. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proc. 48th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 211-218). PME. Small, F., & Attree, K. (2016). Undergraduate student responses to feedback: expectations and experiences. Studies in Higher Education, 41(11), 2078-2094. Tang, F., & Zhan, P. (2021). Does Diagnostic Feedback Promote Learning? Evidence From a Longitudinal Cognitive Diagnostic Assessment. AERA Open, 7. Van der Kleij, F. M., & Lipnevich, A. A. (2021). Student perceptions of assessment feedback: a critical scoping review and call for research. Educational Assessment, Evaluation and Accountability, 33(2), 345-373. Wagner, S., Sibley, L., Weiler, D., Burde, J.-P., Scheiter, K., & Lachner, A. (2024). The more, the better? Learning with feedback and instruction. Learning and Instruction, 89, 101844. Wiliam, D. (2011). What is assessment for learning? Studies in Educational Evaluation, 37(1), 3-14.

1595 1602 Mira Hykkelbjerg Wulff Marcel Funk Femke Sporn Aiso Heinze Leibniz Institute for Science and Mathematics Education Germany Christian-Albrechts-Universität zu Kiel This paper examines how students attribute relevance to mathematically equivalent tasks in different contexts. Drawing on expectancy-value theory, relevance is considered across four perspectives (subjective/objective × present/future). A teaching unit on circle calculations and radian measure in the context of wind energy was implemented in grade 10 (N = 26). An analysis of group interviews shows that the authentic context mainly elicited objective relevance, while subjective relevance attributions remained rare. Quantitative data indicate higher relevance beliefs about mathematics in general than for the specific content. Consequently, contextualization can highlight societal and curricular relevance, but additional metacognitive reflection appears necessary to foster students’ subjective relevance attributions. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53, 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eilks, I., Stuckey, M., Hofstein, A., & Mamlok-Naaman, R. (2013). The meaning of ‘relevance’ in science education and its implications for the science curriculum. Studies in Science Education, 34, 1-34. https://doi.org/10.1080/03057267.2013.802463. Gaspard, H., Parrisius, C., Piesch, H., Kleinhansl, M., Wille, E., Nagengast, B., Trautwein, U., & Hulleman, C. (2021). The Potential of Relevance Interventions for Scaling Up: A Cluster-Randomized Trial Testing the Effectiveness of a Relevance Intervention in Math Classrooms. Journal of Educational Psychology, 113, 1507-1528. https://doi.org/10.1037/edu0000663. Geisler, S. & Rach, S. (2023). Who benefits from modelling tasks combined with experiments. Quadrante 3(2), 130-152. https://doi.org/10.48489/quadrante.31452 Mayring, P. (2022). Qualitative Content Analysis. A Step-by-Step Guide. Sage. Niss, M. (1994). Mathematics in Society. In R. Biehler, R. W. Scholz, R. Strässer, & B.Winkelmann (Eds.), Didactics of Mathematics as a Scientific Discipline (pp. 367–378). Kluwer Academic Publishers. OECD (2023). PISA 2022 Results (Volume I): The State of Learning and Equity in Education, PISA, OECD Publishing. Paris, https://doi.org/10.1787/53f23881-en. Rohenroth, D., Neumann I. & Heinze, A. (2023b) Mathematical prerequisites for non-STEM programs. Frontiers in Education, 8:1089509. https://doi.org/10.3389/feduc.2023.1089509 Walton, G. M., & Yeager, D. S. (2020). Seed and soil: Psychological affordances in contexts help to explain where wise interventions succeed or fail. Current Directions in Psychological Science, 29(3), 219–226. https://doi.org/10.1177/0963721420904453 Wulff, M. H., Radkowitsch, A., & Heinze, A. (2023). Learning about digital technologies of the working world in regular math classes? Teaching composite bodies with 3D print as a learning context. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel., & M. Tabach (Eds.), Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4., pp. 355-362). PME 46

1603 1610 Antonia Wunsch Robert von Hering Henning Sievert University of Hildesheim Germany University of Flensburg Stochastic competencies are crucial in both daily lives of primary school children and their social participation. Hence, developing stochastic competencies is a mandatory goal in mathematics education. However, challenges are apparent in the three subdomains of data, combinatorics, and chance. Despite their growing societal importance, the state of research on primary school stochastic competencies and their development remains insufficient. Accordingly, in this study, we conducted a longitudinal analysis of the development of students’ stochastic competencies in Grade 4 by collecting data from 586 students across 42 classes in 18 schools in Germany. The results show developments in all three subdomains as well as significant differences both between schools and classes. Álvarez-Arroyo, R., Batanero, C., & Gea, M. M. (2024). Probabilistic literacy and reasoning of prospective secondary school teachers when interpreting media news. ZDM Mathematics Education, 56(6), 1045–1058. Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181–199. von Davier, M., Kennedy, A., Reynolds, K., Fishbein, B., Khorramdel, L., Aldrich, C., Bookbinder, A., Bezirhan, U., & Yin, L. (2024). TIMSS 2023 International Results in Mathematics and Science. Boston College, TIMSS & PIRLS International Study Center. English, L. D. (2014). Promoting Statistical Literacy Through Data Modelling in the Early School Years. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking. Presenting Plural Perspectives (pp. 441–457). Springer Science+Business Media. Falk, R., Yudilevich-Assoulione, P., & Elstein, A. (2012). Children’s concept of probability as inferred from their binary choices – revisited. Educational Studies in Mathematics, 81(2), 207–233. Fischbein, E., Nello, M. S., & Marino, M. S. (1991). Factors affecting probabilistic judgements in children and adolescents. Educational Studies in Mathematics, 22(6), 523– 549. Frischemeier, D. (2020). Building statisticians at an early age: Statistical projects exploring meaningful data in primary school. Statistics Education Research Journal, 19(1), 39–56. Frischemeier, D. & Birk, L. (2024). Different lenses on distributions and the integration of context knowledge in data‐ing processes of primary school students. ZDM - Mathematics Education, 57(1), 1–15. Frischemeier, D. & Schnell, S. (2023). Statistical investigations in primary school – the role of contextual expectations for data analysis. Math.Ed. Research Journal, 35, 217–242. Jaeger, L. (2024). Zur Erfassung des intuitiven Begriffsverständnisses junger Kinder im Kontext des frühen probabilistischen Denkens. [To assess young children's intuitive understanding of concepts in the context of early probabilistic reasoning.] Stochastik in der Schule 44(3), 2–7. Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (2015). Data seen through different lenses. Educational Studies in Mathematics, 88(3), 305–325. Kultusministerkonferenz [KMK] (2022). Bildungsstandards für das Fach Mathematik Primarbereich. Beschluss der Kultusministerkonferenz vom 15.10.2004. [National educational standards in mathematics at primary level]. KMK. Leavy, A. & Hourigan, M. (2021). Data Modelling and Informal Inferential Reasoning: Instances of Early Mathematical Modelling. In J. M. Suh, M. H. Wickstrom, & L. D. English (Eds.), Exploring Mathematical Modeling with Young Learners (pp. 67–93), Springer Nature. Lookwood, E. (2014). A set-oriented perspective on solving counting problems, Educational Studies in Mathematics, 34(2), 31–37. Martignon, L. (2014). Fostering Children’s Probabilistic Reasoning and First Elements of Risk Evaluation. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking. Presenting Plural Perspectives (pp. 149–160), Springer Science + Business Media. National Council of Teachers of Mathematics [NCTM] (2000). Principles and Standards for School Mathematics. NCTM Nikiforidou Z. (2018). Probabilistic Thinking and Young Children: Theory and Pedagogy. In A. Leavy, M. Meletiou-Mavrotheris, & E. Paparistodemou (Eds.), Statistics in Early Childhood and Primary Education. Supporting Early Statistical and Probabilistic Thinking (pp. 21–34). Springer Nature. Palmér, H. & van Bommel, J. (2023). Young Students’ Choice of Representation When Solving a Problem-Solving Task on Combinatorics. In H. Palmér, C. Björklund, E. Reikerås, & J. Elofsson (Eds.), Teaching Mathematics as to be Meaningful – Foregrounding Play and Children’s Perspectives. Results from the POEM5 Conference, 2022 (pp. 119–129). Springer Nature. Paparistodemou, E. (2014). Children’s Constructions of a Sample Space with Respect to the Law of Large Numbers. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking. Presenting Plural Perspectives (pp. 599–612). Springer Science+Business Media. Ridsdale, C., Rothwell, J., Smit, M., Ali-Hassan, H., Bliemel, M., Irvine, D., Kelley, D., Matwin, S., & Wuetherick, B. (2015). Strategies and Best Practices for Data Literacy Education: Knowledge Synthesis Report. Thomas, C. & Pöhler, B. (2025). Analysis of students' self-written word problems related to arrangement without repetition. In C. Cornejo, P. Felmer, D. M. Gómez, P. Dartnell, P. Araya, A. Peri, & V. Randolph (Eds.), Proc. of the 48th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 2, pp. 331–338). PME. Wild, C. 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1611 1618 Huiyan Ye Oi-Lam Ng Zhihao Cui The Education University of Hong Kong The Chinese University of Hong Kong AI literacy has gained growing significance in K-12 education, with an urgent need to explore how to integrate AI literacy into mathematics classrooms. Underpinned by constructionism, this study designed a three-phase AI lesson themed around machine learning from a regression perspective. Three themes emerged in students’ conceptualizations of AI after the lesson: the learning process of AI, AI’s capabilities and the role of data in developing AI models. Furthermore, students’ understanding of data’s role shifted markedly from a general focus on data as input or data quantity to a heightened emphasis on data quality. The findings contribute to a deeper understanding of how data literacy develops within an AI context and will inform both theory and practice for integrating AI literacy into K-12 mathematics education. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa Lin, Z., Dai, Y., & Ng, O. (2025). Constructionism in K-12 AI literacy education: A systematic review of pedagogical designs, student outcomes, and learning mechanisms. Journal of Educational Computing Research, 63(7–8), 1748–1781. Long, D., & Magerko, B. (2020). What is AI literacy? Competencies and design considerations. Proc. of the 2020 CHI Conf. on Human Factors in Comp. Systems, 1–16. Ng, O., & Cui, Z. (2021). Examining primary students’ mathematical problem-solving in a programming context: Towards computationally enhanced mathematics education. ZDM – Mathematics Education, 53(4), 847–860. https://doi.org/10.1007/ s11858-020-01200-7 Ng, O., Ye, H., Cui, Z., Lin, Z., Dai, Y. (Minor Revision). Computational thinking and AI literacy integration in mathematics education: Design principles for curricular and instructional innovations. ZDM-Mathematics Education. OECD (2025). Empowering learners for the age of AI: An AI literacy framework for primary and secondary education (Review draft). OECD. Paris. https://ailiteracyframework.org Papert, S., & Harel, I. (1991). Situating constructionism. In S. Papert & I. Harel (Eds.), Constructionism (pp. 1–11). New York, NY: Ablex Publishing.

1619 1626 Linxuan Yi Di Liu College of Natural Science, Michigan State University School of Mathematical Sciences, East China Normal University United States As generative AI enters mathematics education, understanding how students use these tools becomes critical. We examine instrumental genesis and cognitive offloading in student-GenAI interaction through case analysis of one eleventh-grade student solving a modelling problem, with help-seeking behaviors served as observable indicators. Analysis revealed that the student developed a learning-oriented utilization scheme within a four-stage cycle of instrumentalization and instrumentation, demonstrating low cognitive offloading: delegating computational tasks while maintaining critical authority. This framework distinguishes productive appropriation (scaffolded engagement) from problematic patterns (complete delegation), offering analytical tools for supporting productive GenAI use in mathematics learning. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245-274. Fan, Y., Tang, L., Le, H., Shen, K., Tan, S., Zhao, Y., Shen, Y., Li, X., & Gašević, D. (2024). Beware of metacognitive laziness: Effects of generative artificial intelligence on learning motivation, processes, and performance. British J. of Ed. Technology, 56(2), 489-530. Hong, H., Vate-U-Lan, P., & Viriyavejakul, C. (2025). Cognitive Offload Instruction with Generative AI: A Quasi-Experimental Study on Critical Thinking Gains in English Writing. Forum for Linguistic Studies. Jose, B., Cherian, J., Verghis, A. M., Varghise, S. M., S, M., & Joseph, S. (2025). The cognitive paradox of AI in education: Between enhancement and erosion. Frontiers in Psychology, 16. Kasneci, E., Seßler, K., Küchemann, S., Bannert, M., Dementieva, D., Fischer, F., Gasser, U., Groh, G.L., Günnemann, S., Hüllermeier, E., Krusche, S., Kutyniok, G., Michaeli, T., Nerdel, C., Pfeffer, J., Poquet, O., Sailer, M., Schmidt, A., Seidel, T., Stadler, M., Weller, J., Kuhn, J., & Kasneci, G. (2023). ChatGPT for good? On opportunities and challenges of large language models for education. Learning and Individual Differences. 103, 1-9 Newman, R. S. (2000). Social influences on the development of children’s adaptive help seeking: The role of parents, teachers, and peers. Developmental Review, 20(3), 350–404. Trouche, L. (2004). Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students’ Command Process through Instrumental Orchestrations. International Journal of Computers for Mathematical Learning. 9, 281–307. Rabardel, P. (2002). People and technology : A cognitive approach to contemporary instruments. Université Paris 8. Risko, E. F., & Gilbert, S. J. (2016). Cognitive offloading. Trends in Cognitive Sciences, 20(9), 676-688. Verillon, P., Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of though in relation to instrumented activity. European Journal of Psychology of Education, 10(77).

1627 1634 Bilge Yilmaz Aslan Begüm Özmusul Mehmet Nuri Öğüt Gaziantep University Turkey Middle East Technical University Turkey Manisa Celal Bayar University This study examines preservice primary teachers’ epistemological beliefs about geometry and their implications for pedagogical orientations. Using a phenomenological qualitative design, data were collected through a structured reflective form focusing on definitions of geometry, interpretations of Poincaré’s statement, conceptual meanings of basic geometric objects, and beliefs about the source of knowledge. Thematic analysis revealed two dominant epistemological orientations: a school-oriented/instrumental view and a life-oriented/ontological view of geometry. Based on these findings, the study proposes the Dual Epistemological Filter Model, offering a theoretical lens for understanding epistemologically grounded differences in preservice teachers’ approaches to geometry teaching. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554 Battista, M. T. (2007). The development of geometric and spatial thinking. Second handbook of research on mathematics teaching and learning. National Council of Teachers of Mathematics. Boakes, N. J. (2009). Origami instruction in the middle school mathematics classroom: Its impact on spatial visualization and geometry knowledge of students. Rmle Online, 32(7), 1-12. https://doi.org/10.1080/19404476.2009.11462060 Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processing. In Exploiting mental imagery with computers in mathematics education (pp. 142-157). Springer. Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of education for teaching, 15(1), 13-33. https://doi.org/10.1080/0260747890150102 Furinghetti, F., & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In Beliefs: A hidden variable in mathematics education? (pp. 39-57). Springer. Hofer, B. K., & Pintrich, P. R. (1997). The development of epistemological theories: Beliefs about knowledge and knowing and their relation to learning. Review of educational research, 67(1), 88-140. Musser, G. L., Peterson, B. E., & Burger, W. F. (2013). Mathematics for elementary teachers: A contemporary approach. John Wiley & Sons. Moustakas, C. (1994). Phenomenological research methods. Sage. Raymond, A. M. (1997). Inconsistency between a beginning elementary school teacher's mathematics beliefs and teaching practice. Journal for research in mathematics education, 28(5), 550-576. Schoenfeld, A. H. (1985). Making sense of “out loud” problem-solving protocols. The Journal of Mathematical Behavior, 4(2), 171-191 Schommer-Aikins, M. (2004). Explaining the epistemological belief system: Introducing the embedded systemic model and coordinated research approach. Educational psychologist, 39(1), 19-29. https://doi.org/10.1207/s15326985ep3901_3 Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational researcher, 15(2), 4-14. Sinclair, N., & Bruce, C. D. (2015). New opportunities in geometry education at the primary school. ZDM, 47(3), 319-329. https://doi.org/10.1007/s11858-015-0693-4 Van Hiele, P. M (1986). Structure and insight: A theory of mathematics education. Academic Pres.

1635 1642 Kübra Nur Yüce Betül Fatma Yıldırım Muhammet Sıtkı Özger Canan Şahin Serkan Özel Boğaziçi University Turkey Middle East Technical University Turkey Kilis Aralık University Turkey This systematic review synthesizes 92 empirical studies published between 2000 and 2025 to examine the trajectory of research on patterning in mathematics education. Trend analysis reveals a shift from foundational descriptions toward cognitive mechanisms and mathematical outcomes, yet a critical disconnect persists: preschool research is saturated with repeating patterns, while growing patterns are significantly underrepresented until primary school. We synthesize findings on mathematics achievement to argue that mixed intervention results likely stem from failing to distinguish the specific cognitive roles of repeating versus growing patterns. We propose integrating growing patterns into early childhood curricula to bridge this "algebraic gap." Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global survey of curricula and research (pp. 5–23). Springer. Borriello, G. A., Grenell, A., Vest, N. A., Moore, K., & Fyfe, E. R. (2023). Links between repeating and growing pattern knowledge and math outcomes in children and adults. Child Development, 94(2), e103–e118. De Keyser, L., Bakker, M., Rathé, S., Wijns, N., Torbeyns, J., Verschaffel, L., & De Smedt, B. (2020). No association between the home math environment and numerical and patterning skills in a large and diverse sample of 5- to 6-year-olds. Frontiers in Psychology, 11, 547626. Fox, J. (2006). Connecting algebraic development to mathematical patterning in early childhood. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 89–96). Charles University. Kidd, J. K., Carlson, A. G., Gadzichowski, K. M., Boyer, C. E., Gallington, D. A., & Pasnak, R. (2013). Effects of patterning instruction on the academic achievement of 1st-grade children. Journal of Research in Childhood Education, 27(2), 224–238. Mayring, P. (2014). Qualitative content analysis: Theoretical background and procedures. In Advances in mathematics education (pp. 365–380). Springer. McKnight, P. E., Kidd, J. K., Gallington, D. A., Strauss, L. I., Lyu, H., Gadzichowski, K. M., & Pasnak, R. (2021). The efficacy of teaching advanced forms of patterning to kindergartners. Creative Education, 12(3), 557–572. Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49. Page, M. J., McKenzie, J. E., Bossuyt, P. M., Boutron, I., Hoffmann, T. C., Mulrow, C. D., Shamseer, L., Tetzlaff, J. M., Akl, E. A., Brennan, S. E., Chou, R., Glanville, J., Grimshaw, J. M., Hróbjartsson, A., Lalu, M. M., Li, T., Loder, E. W., Mayo-Wilson, E., McDonald, S., . . . Moher, D. (2021). The PRISMA 2020 statement: An updated guideline for reporting systematic reviews. BMJ, 372, n71. Rittle-Johnson, B., Fyfe, E. R., Hofer, K. G., & Farran, D. C. (2017). Early math trajectories: Low-income children’s mathematics knowledge from ages 4 to 11. Child Development, 88(5), 1727–1742. Steen, L. A. (1988). The science of patterns. Science, 240(4852), 611–616. Strauss, L. I., Peterson, M. S., Kidd, J. K., Choe, J., Lauritzen, H. C., Patterson, A. B., Holmberg, C. A., Gallington, D. A., & Pasnak, R. (2020). Evaluation of patterning instruction for kindergartners. The Journal of Educational Research, 113(4), 292–302. Vanluydt, E., Wijns, N., Torbeyns, J., & Van Dooren, W. (2021). Early childhood mathematical development: the association between patterning and proportional reasoning. Educational Studies in Mathematics, 107(1), 93–110. Wijns, N., Torbeyns, J., De Smedt, B., & Verschaffel, L. (2019). Young children’s patterning competencies and mathematical development: A review. In Mathematical Learning and Cognition in Early Childhood (pp. 139–161). Springer International Publishing. Zippert, E. L., Douglas, A., & Rittle-Johnson, B. (2020). Finding patterns in objects and numbers: Repeating patterning in pre-K predicts kindergarten mathematics knowledge. Journal of Experimental Child Psychology, 200, 104965.

1643 1650 Maya Zastrow Bianca Fink Leander Kempen Florian Schacht Lars Holzäpfel University of Education Freiburg Germany University of Greifswald University of Duisburg-Essen Germany This study examines the extent to which teachers’ conceptions of good mathematics teaching correspond to facilitators’ anticipations. Open-ended responses from 333 facilitators and 120 secondary school teachers were analysed using qualitative content analysis with inductive and deductive coding. Results show that both facilitators and teachers refer early on to deep-structures of instruction, indicating a shared research-based orientation toward instructional quality. However, facilitators systematically underestimated teachers’ emphasis on these aspects. Overall, facilitators’ and teachers’ conceptions were more closely aligned than facilitators had anticipated, highlighting implications for participant-oriented PD and facilitator expertise. Barzel, B., & Selter, C. (2015). Die DZLM-Gestaltungsprinzipien für Fortbildungen. Journal für Mathematik-Didaktik, 36(2), 259–284. Brewer, M. B. (1991). The Social Self: On Being the Same and Different at the Same Time. Personality and Social Psychology Bulletin, 17(5), 475–482. Decristan, J., Hess, M., Holzberger, D., & Praetorius, A.-K. (2020). Oberflächen- und Tiefenmerkmale. Eine Reflexion zweier prominenter Begriffe der Unterrichtsforschung. Zeitschrift für Pädagogik, 66, 102–116. Klieme, E., Pauli, C., & Reusser, K. (2009). The Pythagoras Study. Investigating effects of teaching and learning in Swiss and German mathematics classrooms. In T. Janík & T. Seidel (Eds.), The Power of Video Studies in Investigating Teaching and Learning in the Classroom (pp. 137–160). Waxmann. Landis, J. R., & Koch, G. G. (1977). The Measurement of Observer Agreement for Categorical Data. Biometrics, 33(1), 159. https://doi.org/10.2307/2529310 Marques, J. M., Yzerbyt, V. Y., & Leyens, J. (1988). The “Black Sheep Effect”: Extremity of judgments towards ingroup members as a function of group identification. European Journal of Social Psychology, 18(1), 1–16. Murray, J., & Male, T. (2005). Becoming a teacher educator: Evidence from the field. Teaching and Teacher Education, 21(2), 125–142. Perry, E., & Boylan, M. (2018). Developing the developers: Supporting and researching the learning of professional development facilitators. Professional Development in Education, 44(2), 254–271. Prediger, S., Roesken-Winter, B., Stahnke, R., & Pöhler, B. (2021). Conceptualizing contentrelated PD facilitator expertise. Journal of Mathematics Teacher Education, 25(4), 403– 428. Prediger, S., Götze, D., Holzäpfel, L., Rösken-Winter, B., & Selter, C. (2022). Five principles for high-quality mathematics teaching: Combining normative, epistemological, empirical, and pragmatic perspectives for specifying the content of professional development. Frontiers in Education, 7, 1–15. Prediger, S., & Selter, C. (2024). Establish Shared Visions and Support Productive Adaptations on All Levels: Aims, Strategies, and Architecture of a Nationwide Implementation Program. Implementation and Replication Studies in Mathematics Education, 1–35. Tajfel, H., & Turner, J. C. (1979). An integrative theory of intergroup conflict. In W. G. Austin, & S. Worchel (Eds.), The social psychology of intergroup relations (pp. 33-37). Monterey, CA: Brooks/Cole. Zastrow, M., Bianca, F., Schacht, F., Kempen, L., Barzel, B., & Holzäpfel, L. (2025, February). Teaching quality in mathematics lessons from the perspective of facilitators and how they perceive the teachers’ perspective. Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14), Free University of Bozen-Bolzano.

1651 1658 Veysel Akcakin Ali Zengin Usak University Turkey This study adopted a person-centred approach to identify latent profiles of mathematical resilience among 815 middle school students (Grades 5-8) and examine differences in academic achievement across profiles. Latent profile analysis revealed three profiles: Fully Resilient, Persistent but Growth-Deficient, and Moderate Resilience. Students in the Fully Resilient profile demonstrated significantly higher achievement than the other profiles, with no significant difference between Persistent but Growth-Deficient and Moderate Resilience profiles. These findings highlight the multidimensional nature of mathematical resilience and underscore the critical role of growth-oriented beliefs in translating persistence into success. Educational interventions should be tailored to resilience profiles, emphasizing growth beliefs. Agtarap, R., & Miranda, A. T. (2022). The mediating effect of students’ resiliency on the relationship of self-concept and mathematics performance. Asian Journal of Education and Social Studies, 36(2), 1-10. Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition analysis: With applications in the social, behavioral, and health sciences. Wiley. Demir, B. (2023). The mediating effect of academic resilience on the relationship between psychological resilience and academic achievement. International Journal of Educational Research, 14(3), 52-67. Dweck, C. S. (2006). Mindset: The new psychology of success. Random House. Gürefe, N., & Eryılmaz, E. (2025). The Mediating Role of Metacognitive Awareness in the Relationship between Middle School Preservice Mathematics Teachers’ Beliefs About the Nature of Mathematics and Mathematical Resilience. Bartın University Journal of Faculty of Education, 14(2), 455-466. https://doi.org/10.14686/buefad.1452974 Kooken, J., Welsh, M. E., McCoach, D. B., Johnston-Wilder, S., & Lee, C. (2016). Development and validation of the mathematical resilience scale. Measurement and Evaluation in Counseling Development, 49(3), 217-242. https://doi.org/10.1177/0748175615596782 Lee, C., & Johnston-Wilder, S. (2014). Mathematical resilience: what is it and why is it important? In S. Chinn (Ed.), The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties (pp. 337-345). Routledge. Lee, C., & Johnston-Wilder, S. (2017). The Construct of Mathematical Resilience. In U. Xolocotzin Eligio (Ed.), Understanding Emotions in Mathematical Thinking and Learning (pp. 269-291). Elsevier. https://doi.org/10.1016/B978-0-12-802218-4.00010-8 Lee, C., & Ward-Penny, R. (2022). Agency and fidelity in primary teachers’ efforts to develop mathematical resilience. Teacher Development, 26(1), 75-93. https://doi.org/10.1080/13664530.2021.2006768 Seligman, M. E. P. (2011). Flourish : A visionary new understanding of happiness and wellbeing. William Heinamann. Turgut, S., & Uğurlu, M. (2024). A moderated mediation analysis on the relationship between metacognition and mathematical resilience of gifted students. Psychology in the Schools, 61(5), 1848-1867. https://doi.org/10.1002/pits.23141 Vera Gil, S. (2024). The Influence of Gender on Academic Performance and Psychological Resilience, and the Relationship Between Both: Understanding the Differences Through Gender Stereotypes. Trends in Psychology. https://doi.org/10.1007/s43076-024-00370-7 Wong, P. T. P. (2011). Positive psychology 2.0: Towards a balanced interactive model of the good life. Canadian Psychology/Psychologie canadienne, 52(2), 69-81. https://doi.org/10.1037/a0022511 Zengin, A., & Akcakin, V. (2025). Mathematical resilience and mathematical modeling competency: Exploring the mediation effect of subjective academic well-being. Journal of Pedagogical Research, 9(4), 281-295. https://doi.org/10.33902/JPR.202536864.