Constructing Indicators of Mathematical Proof Reasoning for the Topic of Sytems of Linear Equations in Two Variables Based on Hanna-Raman Theory
Abstract
This study aims to build a framework of indicators for mathematical proof reasoning grounded in Hanna-Raman theory, applied to the topic of Systems of Linear Equations in Two Variables (SLETV) at the senior high school level. Hanna-Raman theory draws a distinction between proof that proves, which functions to verify the truth of a statement, and proof that explains, which reveals why the statement holds. Using a Content Analysis approach combined with an Analytical-Conceptual Design, this article works through the epistemological structure of SLETV, brings together the defining features of proof within school algebra, and proposes three levels of indicators that together form a continuum running from procedural toward structural thinking. The analysis shows that Level 1 (Procedural Verification) corresponds to proof that proves, whereas Level 2 (Conceptual Justification) and Level 3 (Structural Generalization) correspond to proof that explains. The construct validity of the framework is examined through three sample problems, each accompanied by a deconstruction analysis, a mapping of relevant indicators, a comparison between proves-type and explains-type responses, and an account of likely misconceptions. The article argues that SLETV assessment, which has so far tended to focus on procedural end results, needs to shift toward valuing the process by which students construct mathematical arguments. The framework is proposed as a diagnostic tool for qualitative studies aiming to identify the proof-reasoning profiles of senior high school students.
References
Anderson, L. W., & Krathwohl, D. R. (Eds.). (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom's educational objectives. Longman.
Agusantia, D., & Juandi, D. (2022). Kemampuan Penalaran Analogi Matematis di Indonesia: Systematic Literature Review. Symmetry: Pasundan Journal of Research in Mathematics Learning and Education, 7(2), 222–231. https://doi.org/10.23969/symmetry.v7i2.6436
Arjmand, R. (2018). Introduction to Part I: Islamic education: Historical perspective, origin, and foundation. In H. Daun & R. Arjmand (Eds.), Handbook of Islamic education (pp. 27–52). Springer. https://doi.org/10.1007/978-3-319-64683-1_3
Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). Hodder and Stoughton.
Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII: Some mental processes underlying producing and proving conjectures, and conditions suitable to enhance them. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 249–264). Sense Publishers.
De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
Dreyfus, T. (1999). Why Johnny can't prove. Educational Studies in Mathematics, 38(1–3), 85–109. https://doi.org/10.1023/A:1003660018579
Dreyfus, T., & Hadas, N. (1996). Proof as answer to the question why. Zentralblatt für Didaktik der Mathematik, 28(1), 1–5.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13. https://doi.org/10.1007/BF01809605
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1–2), 5–23. https://doi.org/10.1023/A:1012737223465
Hanna, G., & Raman, M. (2004). Proof and proving in the classroom. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 40–41). Bergen University College.
Kemendikbudristek. (2022). Capaian pembelajaran mata pelajaran matematika pada kurikulum merdeka. Badan Standar, Kurikulum, dan Asesmen Pendidikan, Kemdikbudristek.
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Information Age Publishing.
Krippendorff, K. (2018). Content analysis: An introduction to its methodology (4th ed.). SAGE Publications.
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 (pp. 173–204). Sense Publishers.
Nasir, T. M., Purnama, W., Haryanti, E., Komara, D., & Sain, Z. H. (2025). Integrating reason, logic, and Islamic epistemology for holistic science in modern education. Atthulab: Islamic Religion Teaching and Learning Journal, 10(1), 34–51. https://doi.org/10.15575/atthulab.v10i1.44031
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. NCTM.
Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319–325. https://doi.org/10.1023/A:1024360204239
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
Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). SAGE Publications.
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
Tall, D. (2002). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 3–21). Kluwer Academic Publishers.
Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2–3), 209–234. https://doi.org/10.1023/B:EDUC.0000040410.57253.a1
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