A framework for proofs and refutations in school mathematics: Increasing content by deductive guessing
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The process of proofs and refutations described by Lakatos is essential in school mathematics to provide students with an opportunity to experience how mathematical knowledge develops dynamically within the discipline of mathematics. In this paper, a framework for describing student processes of proofs and refutations is constructed using a set of heuristic rules formulated by Lakatos. A salient feature of this framework resides in the notion of increasing content by deductive guessing, which Lakatos considered to be one of the cores of proofs and refutations, where deductive reasoning is used to create a general conjecture that is true even for counterexamples of an earlier conjecture. Two case studies involving a pair of fifth graders and a pair of ninth graders are presented to illustrate that the notion of increasing content by deductive guessing is useful in examining student processes of generalisation of conjectures. The framework shown in this paper contributes to the current knowledge of mathematics education researchers about proof and proving by providing a tool to investigate how a proof can be used not only to establish the truth of a given statement but also to generate new mathematical knowledge.
KeywordsProof Refutation Counterexample Lakatos Heuristic rule Increasing content by deductive guessing
I am grateful to Taro Fujita (University of Exeter, UK), the editor, and the anonymous reviewers for their valuable suggestions on earlier drafts of this paper. This study is supported by the Japan Society for the Promotion of Science (nos. 23330255 and 15H05402).
- Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics (D. Pimm, Trans.). In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London: Hodder and Stoughton.Google Scholar
- Boero, P., Garuti, R., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 113–120). Valencia, Spain.Google Scholar
- Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.Google Scholar
- Herbst, P. (2005). Knowing about “equal area” while proving a claim about equal areas. Recherches en Didactique des Mathématiques, 25(1), 11–56.Google Scholar
- Komatsu, K. (2012). Lakatos’ heuristic rules as a framework for proofs and refutations in mathematical learning: Local-counterexample and modification of proof. In Pre-proceedings of the 12th International Congress on Mathematical Education (pp. 2838–2847). Seoul, Korea.Google Scholar
- Komatsu, K., & Sakamaki, A. (2014). Invention of new statements for counterexamples. In P. Liljedahl, S. Oesterle, C. Nicol, & D. Allan (Eds.), Proceedings of Joint Meeting of PME 38 and PME-NA 36 (Vol. 4, pp. 17–24). Vancouver, Canada.Google Scholar
- Polya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton: Princeton University Press.Google Scholar
- Semadeni, Z. (1984). Action proofs in primary mathematics teaching and in teacher training. For the Learning of Mathematics, 4(1), 32–34.Google Scholar
- Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.Google Scholar
- Weber, K., & Alcock, L. (2009). Proof in advanced mathematics classes: Semantic and syntactic reasoning in the representation system of proof. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 323–338). New York: Routledge.Google Scholar
- Yim, J., Song, S., & Kim, J. (2008). The mathematically gifted elementary students’ revisiting of Euler’s polyhedron theorem. The Montana Mathematics Enthusiast, 5(1), 125–142.Google Scholar