Using reasoning-and-proving Cycles to Foster Teacher Development in Argumentation and Proof: An Illustration of Practice
Keywords:
argumentation and proof, circles, consecutive numbers, reasoning-and-proving, tasks for teacher developmentAbstract
This illustration of practice features two reasoning-and-proving tasks that fostered pre-service teachers’ development in argumentation and proof. Argumentation and proof are significant to the learning of mathematics. In a design-based study, a teacher development programme was developed and tested through multiple research cycles. The programme was shown to be effective in promoting pre-service teachers’ positive beliefs and knowledge about argumentation and proof. In the programme, the two reasoning-and-proving tasks linked argumentation and proof to the work of teaching. Each task featured one of the two key topics in school mathematics—properties of integers, and circles. The tasks modelled the process of reasoning-and-proving and invited learners to explore the featured mathematical ideas through multiple methods, representations, and cycles. The tasks provided pre-service teachers with alternative experiences in argumentation and proof, arousing positive emotions and motivation. This illustration of practice also discusses the key role that teacher educators could play in using the tasks to support teacher development.
References
Australian Curriculum, Assessment and Reporting Authority. (2022). Australian curriculum: Mathematics (Version 9). https://v9.australiancurriculum.edu.au/
Buchbinder, O., & McCrone, S. (2020). Preservice teachers learning to teach proof through classroom implementation: Successes and challenges. Journal of Mathematical Behavior, 58, Article 100779. https://doi.org/10.1016/j.jmathb.2020.100779
Buchbinder, O., & McCrone, S. (2023). Preparing prospective secondary teachers to teach mathematical reasoning and proof: The case of the role of examples in proving. ZDM–Mathematics Education, 55, 779–792. https://doi.org/10.1007/s11858-023-01493-4
Csikszentmihalyi, M. (1990). Flow: The psychology of optimal experience. Harper & Row.
Dreyfus, T. (1987). Approximation of area under a curve: A conceptual approach. The Mathematics Teacher, 80(7), 538–543.
Education Bureau. (2020a). Explanatory notes to junior secondary mathematics curriculum. Hong Kong. https://www.edb.gov.hk/attachment/en/curriculum-development/kla/ma/curr/EN_KS3_e.pdf
Education Bureau. (2020b). Explanatory notes to primary mathematics curriculum (Key Stage 2). Hong Kong. https://www.edb.gov.hk/attachment/en/curriculum-development/kla/ma/curr/EN_KS2_e.pdf
Fredrickson, B. L. (2001). The role of positive emotions in positive psychology: The broaden-and-build theory of positive emotions. American Psychologist, 56(3), 218–226. https://doi.org/10.1037/0003-066X.56.3.218
Goos, M., & Geiger, V. (2010). Theoretical perspectives on mathematics teacher change. Journal of Mathematics Teacher Education, 13(6), 499–507. https://doi.org/10.1007/s10857-010-9166-4
Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM Mathematics Education, 40(3), 345–353. https://doi.org/10.1007/s11858-008-0080-5
Lee, G. C.-Y. (2026). Conditions for change in pre-service teachers’ beliefs about proof and proof teaching. Mathematics Education Research Journal, 38(1), 187–218. https://doi.org/10.1007/s13394-025-00525-4
Lehmann, T. H. (2025). Learning to measure the area of circles. Research in Mathematics Education, 27(3), 410-436. https://doi.org/10.1080/14794802.2024.2304329
Leung, K.-C. I. (2014). Prospective teachers’ understanding of the constant pi and their knowledge of how to prove its constant nature through the concept of linearity., Mathematics Education Research and Practice, 18(1), 1–29. https://www.kci.go.kr/kciportal/landing/article.kci?arti_id=ART001866681
Mathematics Teaching and Learning to Teach. (2010a). BetsyProof-Start. University of Michigan. http://hdl.handle.net/2027.42/65012
Mathematics Teaching and Learning to Teach. (2010b). SeanNumbers-Ofala. University of Michigan. http://hdl.handle.net/2027.42/65013
Ozgur, Z., Ellis, A., Vinsonhaler, R., Dogan, M. F., & Knuth, E. J. (2019). From examples to proof: Purposes, strategies, and affordances of example use. Journal of Mathematical Behavior, 53, 284–303. https://doi.org/10.1016/j.jmathb.2017.03.004
Shilling-Traina, L. N., & Stylianides, G. J. (2013). Impacting prospective teachers’ beliefs about mathematics. ZDM Mathematics Education, 45(3), 393–407. https://doi.org/10.1007/s11858-012-0461-7
Skovsmose, O., & Säljö, R. (2008). Learning mathematics through inquiry. Nordic Studies in Mathematics Education, 13(3), 31–52. https://doi.org/10.7146/nomad.v13i3.148123
Stacey, K., & Vincent, J. (2009). Finding the area of a circle: Didactic explanations in school mathematics. The Australian Mathematics Teacher, 65(3), 6–9.
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In Á. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315–351). Brill.
Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314–352.
Stylianides, G. J., Stylianides, A. J., & Moutsios-Rentzos, A. (2024). Proof and proving in school and university mathematics education research: A systematic review. ZDM Mathematics Education, 56(1), 47–59 https://doi.org/10.1007/s11858-023-01518-y
Stylianides, G. J., Stylianides, A. J., & Shilling-Traina, L. N. (2013). Prospective teachers’ challenges in teaching reasoning-and-proving. International Journal of Science and Mathematics Education, 11(6), 1463–1490. https://doi.org/10.1007/s10763-013-9409-9
Stylianides, G. J., Stylianides, A. J., & 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.
Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Winicki-Landman, G. (2012). The need for proof and proving: Mathematical and pedagogical perspectives. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 215–229). Springer. https://doi.org/10.1007/978-94-007-2129-6_9
