There is a longstanding conversation in the mathematics education literature about proofs that explain versus proofs that only convince. In this essay, we offer a characterization of explanatory proofs with three goals in mind. We first propose a theory of explanatory proofs for mathematics education in terms of representation systems. Then, we illustrate these ideas in terms of combinatorial proofs, focusing on binomial identities. Finally, we leverage our theory to explain audience-dependent and audience-invariant aspects of explanatory proof. Throughout, we use the context of combinatorics to emphasize points and to offer examples of proofs that can be explanatory or only convincing, depending on how one understands the claim being made.
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Providing such a definition was not necessarily their intent.
Balacheff (2009) also defined explanation as a psychological construct, proof as a social construct, and both characterized in terms of representation systems. However, at least in terms of student-generated proofs, he viewed a proof as a socially accepted explanation (i.e., all proofs are explanations). Our characterization says a student-generated proof might not be explanatory if it was generated with non-natural inferences or in an RS that a student did not personally value.
The theorem extends to other, non-integer values, but we focus on this version.
Balacheff, N. (2009). Bridging knowing and proving in mathematics: A didactical perspective. In Pulte, H., Hanna, G., & Jahnke, H. J. (Eds.). (2009). Explanation and proof in mathematics: Philosophical and educational perspectives. New York, NY: Springer.
Bartlo, J. R. (2013). Why ask why: An exploration of the role of proof in the mathematics classroom. (Unpublished doctoral dissertation). Portland State University. Portland, OR.
Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181–199.
Brown, S. A. (2014). On skepticism and its role in the development of proof in the classroom. Educational Studies in Mathematics, 86(3), 311–335.
De Villiers, M. (1999). The role and function of proof. In M. de Villiers (Ed.), Rethinking proof with the Geometer’s Sketchpad (pp. 3–10). Emeryville, CA: Key Curriculum Press.
Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulty of proof. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 345–352). Stellenbosch, South Africa.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zaskis (Eds.), Learning and teaching number theory (pp. 185–212). Norwood, NJ: Ablex.
Harel, G., & Sowder, L. (1998). Students' proof schemes. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on Collegiate Mathematics Education (vol. III, pp. 234–283). Providence, RI: American Mathematical Society.
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.
Lange, M. (2009). Why proofs by mathematical induction are generally not explanatory. Analysis, 69(2), 203–211.
Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32, 251–265. https://doi.org/10.1016/j.jmathb.2013.02.008
Mejia-Ramos, J. P., & Inglis, M. (2017). ‘Explanatory’ talk in mathematics research papers. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 20thConference for Research in Undergraduate Mathematics Education (pp. 373–382). San Diego, California.
Nelsen, R. (1993). Proofs without words: Exercises in visual thinking. Washington, DC: MAA.
Nunokawa, K. (2009). Proof, mathematical problem-solving, and explanation in mathematical teaching. In H. Pulte, G. Hanna, & H. J. Jahnke (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives. New York, NY: Springer.
Paseau, A. (2010). Proofs of the compactness theorem. History and Philosophy of Logic, 31(1), 73–98.
Pulte, H., Hanna, G., & Jahnke, H. J. (Eds.). (2009). Explanation and proof in mathematics: Philosophical and educational perspectives. New York: Springer.
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.
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, 1–36.
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151.
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
Stylianides, G. J., Sandefur, J., & Watson, A. (2016). Conditions for proving by mathematical induction to be explanatory. Journal of Mathematical Behavior, 43, 20–34.
Weber, K. (2010a). Proofs that develop insight: Proofs that reconceive mathematical domains and proofs that introduce new methods. For the Learning of Mathematics, 30(1), 32–37.
Weber, K. (2010b). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12, 306–336.
Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.
Weber, K. & Alcock, L. (2009). Semantic and syntactic reasoning and proving in advanced mathematics classrooms. Invited chapter on research on proof at the undergraduate level for M. Blanton, D. Stylinaiou, and E. Knuth (Eds.), The teaching and learning of proof across the K-16 curriculum (pp. 323–338). New York, NY: Routledge.
Weber, K., & Mejia-Ramos, J. P. (2019). An empirical study on the admissibility of graphical inferences in mathematical proofs. In A. Aberdein & M. Inglis (Eds.), Advances in experimental philosophy of logic and mathematics (pp. 123–144). London, UK: Bloomsbury.
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Lockwood, E., Caughman, J.S. & Weber, K. An essay on proof, conviction, and explanation: multiple representation systems in combinatorics. Educ Stud Math 103, 173–189 (2020). https://doi.org/10.1007/s10649-020-09933-8
- Proofs that explain and convince
- Binomial identities