Theory and Practice in the Curriculum

Theory and Practice in the Curriculum

From Intuition to Deduction: A Qualitative Analysis of the Structure of Students’ Geometric Argumentation

Document Type : Original Article

Authors
Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran
Abstract
This study aimed to analyze the structure of students’ argumentation while solving a geometry problem. As a qualitative inquiry, the study examined the structural and cognitive components involved in the development of students’ argumentation throughout problem solving. Participants were three groups of three eleventh-grade students from a county in Tehran Province. The groups were organized from among 15 students based on similarities in their reasoning patterns, and each group then solved the same geometry problem. Data were collected through video recordings, transcripts of group discussions, field notes, and students’ written work, enabling reconstruction of their reasoning processes and interactions. Data analysis was conducted using an integrated analytical framework combining Toulmin’s model of argumentation and Balacheff’s cK¢ model. The findings revealed that students’ argumentation was dynamic and multilayered, emerging through the interaction of components such as data, claims, warrants, backing, control, rebuttals, operators, and representational systems. Four forms of reasoning were identified: visual reasoning, inductive reasoning, abductive reasoning, and informal deductive reasoning. Among these, abductive and informal deductive reasoning played the most influential role in supporting students’ movement toward formal deductive argumentation. Furthermore, the coordination of multiple representations—particularly the transformation of visual and numerical representations into algebraic relationships—was essential for constructing coherent argumentation. The activation of control and rebuttal components, together with collaborative interactions, enabled students to refine their reasoning, revise their arguments, and strengthen the validity of their conclusions. The findings may contribute to the revision of mathematics curricula, the design of instructional activities, and inform future research.
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Ahmadpour, F., Fadaei, M. R., Rafi'pour, A. (2017). The necessity of reconsidering the content of seventh- and eighth-grade mathematics textbooks from the perspective of reasoning and proof. Curriculum Studies, 12(46), 59–.84 [In Persian]
Almpani, S., & Stefaneas, P. (2023). Bridging informal reasoning and formal proving: The role of argumentation in proof-events. Foundations of Science, 1–25.
Arab Yousefabadi, F. (2025). Rhetorical and argumentative analysis of persuasive language in Majāles-e Sabʿeh of Rumi: A study based on Toulmin’s theory. Literary Arts, 17(4), 19–32. [In Persian]
Balacheff, N. (2013). Cognitive vs. situational perspectives in mathematics education research: The case of conceptions. Educational Studies in Mathematics, 82(3), 249–265.
Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education. Springer International Publishing.
Bock, C., & Dimmel, J. (2021). Dynamic Spatial Diagrams and Solid Geometric Figures. North American Chapter of the International Group for the Psychology of Mathematics Education.
Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (1999). Argumentation and proof in mathematics education. In Proceedings of CERME 1 (pp. 125–139).
Cramer, J., & Kempen, L. (2022, February). Toulmin and beyond: Structuring and analyzing argumentation. In Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) (No. 07).
Creswell, J. W. (2014). Research design: Qualitative, quantitative, and mixed methods approaches (4th ed.). Thousand Oaks, CA: Sage Publications.
Duval, J.-P. (1998). Périodes locales et propagation de périodes dans un mot. Theoretical Computer Science, 204(1–2), 87–98.
Educational Research and Planning Organization. (2025). Geometry 1: Grade 10, Second Secondary Education. Printing and Publishing Company of Textbooks of Iran.
Ferrari, P. L. (2024). Toulmin’s model of argument and mathematics education: A critical view. For the Learning of Mathematics, 44(2), 15–20.
Fiallo, J., & Gutiérrez, A. (2017). Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove in a Grade 10 trigonometry course. Educational Studies in Mathematics, 96(2), 145–167.
Garuti, R., Boero, P., & Lemut, E. (1996). Cognitive unity of the theorem: A tool for analyzing proving processes. Proceedings of PME, Valencia.
Gholamazad, S., Liljedahl, P., & Zazkis, R. (2003). One Line Proof: What Can Go Wrong?. International Group for the Psychology of Mathematics Education, 2, 437-444.
Ghousseini, H., & Herbst, P. (2016). Pedagogies of practice and opportunities to learn about classroom mathematics discussions. Journal of Mathematics Teacher Education, 19(1), 79–103.
Gooya, Z. (2012). Editor’s note. In F. Kalahdooz, E. Reyhani, & F. Hamidi (Eds.) , Reasoning and proof in mathematics education (Roshd Mathematics Education, No. 108). Educational Publishing and Educational Technology Office. [In Persian]
Hanna, G., & Knipping, C. (2020). Proof in mathematics education, 1980–2020: An overview. Journal of Educational Research in Mathematics, 30(Special Issue), 1–13.
Jafari Tabar, H. (2019). Applying Toulmin’s model of argumentation in law. [In Persian]
Jonsson, B., Granberg, C., & Lithner, J. (2020). Gaining mathematical understanding: The effects of creative mathematical reasoning and cognitive proficiency. Frontiers in Psychology, 11, Article 574366.
Koichu, B., & Zazkis, R. (2013). A dialogic method of presenting proofs: Focus on Fermat’s little theorem. In Proceedings of the Conference for Research in Undergraduate Mathematics Education.
Krippendorff, K. (2004). Measuring the reliability of qualitative text analysis data. Quality and quantity, 38(6), 787-800.
Lepore, M. (2024). A holistic framework to model students’ cognitive process in mathematics education through fuzzy cognitive maps. Heliyon, 10(16), Article e35961.
Nardi, E. (2025). Bridging informal reasoning and formal proof through interactive learning activities and digital technologies. Educational Studies in Mathematics, 98(1), 1–20.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM.
Pedemonte, B. (2001). Relation between argumentation and proof in mathematics: cognitive unity or break. In Proceedings of the 2nd Conference of the European Society for Research in Mathematics Education (pp. 70-80).
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.
Pedemonte, B., & Balacheff, N. (2016). Establishing links between conceptions, argumentation and proof through the cK¢-enriched Toulmin model. The Journal of Mathematical Behavior, 41, 104–122.
Pino-Fan, L. R., Guzmán, I., Font, V., & Duval, R. (2024). Analysis of the underlying cognitive activity in the resolution of a task on derivability of the absolute-value function: Two theoretical perspectives. Educational Studies in Mathematics, 105(3), 327–348.
Rapanta, C. (2021). Can teachers implement a student-centered dialogical argumentation method across the curriculum? Teaching and Teacher Education, 105, Article 103404.
Reyhani, E., Hamidi, F., & Kalahdooz, F. (2012). Investigating second-year secondary school students’ understanding of mathematical reasoning and proof. Iranian Journal of Curriculum Studies, 7(24), 157–182. [In Persian]
Reynolds, K. A., Aldrich, C. E. A., von Davier, M., Fishbein, B., Khorramdel, L., Bookbinder, A., Bezirhan, U., & Yin, L. (2024). TIMSS 2023 Encyclopedia: Education policy and curriculum in mathematics and science. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.
Rodríguez-Nieto, C. A., & Moll, V. F. (2025). Mathematical connections promoted in multivariable calculus classes and in problem solving about vectors, partial and directional derivatives, and applications. Eurasia Journal of Mathematics, Science and Technology Education, 21(4), Article em2619.
Sabzali, N., & Tavanaei, N. (2021). Investigating the effect of geometry instruction based on the Van Hiele theory on students’ learning. Research in Mathematics Education, 1(2), 1–11. [In Persian]
Safabakhsh Chakosari, A., & Yaftian, N. (2018). Van Hiele levels of geometric thinking among eighth-grade students. Roshd Mathematics Education, 37(3), 8–13.
Sasanipour, M., & Rashidi, N. (2025). A reasoning-based instructional approach for teaching academic texts to Iranian students of English language teaching. Modern Research in Education, 11(42), 59–76.
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.
Sowder, L., & Harel, G. (1998). Types of students’ justifications. Mathematics teacher, 91(8), 670-675.
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.
Sukirwan, S., Muhtadi, D., Saleh, H., & Warsito, W. (2020). Profile of students’ justifications of mathematical argumentation. Infinity Journal, 9(2), 197–212.
Supreme Council of Education. (2012). National curriculum of the Islamic Republic of Iran. Ministry of Education.
Syahar, P. H., Lukman, L., & Prabawanto, S. (2026). Utilizing Praxeology and the cKȼ Model to Developing Didactical Design in Derivatives: An Indonesian Mathematics Textbook Study.
Tall, D. (2006). The nature of mathematical proof. Roshd Mathematics Education, (83), 3–12.
Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.
Toulmin, S. E. (2003). The uses of argument (2nd ed.). Cambridge University Press.
van Hiele, P. M. (1957). De probleemstelling voor het inzicht van meetkunde-leerlingen en een daarop afgestemd leerplan [Doctoral dissertation, Universiteit van Amsterdam].
van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Academic Press.
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.
Weber, K., & Mejía-Ramos, J. P. (2014). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 85(1), 79–95.
Wilson, D. (2009). An integrated approach to teaching and learning geometry. In T. V. Craine (Ed.), Understanding geometry for a changing world (71st ed.). National Council of Teachers of Mathematics.
Yaftian, N., & Ansari, M. R. (2020). Eleventh-grade students’ ability to evaluate mathematical arguments. Educational Technology (Technology and Education), 14(3, Serial No. 55), 643–656. [In Persian]
Yaftian, N., & Pazoki, L. (2022). Evaluating tenth-grade male and female mathematics and physics students’ understanding of geometric figures based on Duval’s theory. New Approaches in Educational Research, 16(2), 153–170. [In Persian]