Applying Lakatos’s Theory of Generating Mathematical Knowledge to School Mathematics:Focusing on Refinement from Zero-definitions to Proof-generated definitions

Abstract

The purpose of this study is to reveal Lakatos’s theory of generating mathematical knowledge and consider its application to school mathematics. The logic of mathematical discovery (Lakatos, 1976) has been referred to in mathematics education research to achieve dynamic learning of proof and proving. While previous studies focused on the aspects of conjecturing, proving, refuting, and generating mathematical knowledge, this study especially focuses on the aspects of constructing and refining mathematical definitions. I analyzed Lakatos’s Ph.D. dissertation (Lakatos, 1961), focusing on heuristic rules. Five phases of generating mathematical knowledge through proofs and refutations were specified. These phases include conjecturing, proving, refuting, refining the conjecture using the method of lemma-incorporation or deductive guessing, and formulating proof-generated definitions. Based on these phases, I presented an imaginary student’s activity to refine a zero-definition of a polygon to proof-generated definitions in secondary school mathematics. The activity has a two-fold educational significance: realization of mathematics learning, in which mathematical definitions play a crucial role; and a change of students’ absolute and solid views of mathematical definitions. Lastly, I discuss some matters in order to realize the activity in secondary school mathematics classrooms.