Kotaro Komatsu

Rethinking the discovery function of proof within the context of proofs and refutations

Proof and proving are important components of school mathematics and have multiple functions in mathematical practice. Among these functions of proof, this paper focuses on the discovery function that refers to invention of a new statement or conjecture by reflecting on or utilizing a constructed proof. Based on two cases in which eighth and ninth graders engaged in proofs and refutations, we demonstrate that facing a counterexample of a primitive statement can become a starting point of students’ activity for discovery, and that a proof of the primitive statement can function as a useful tool for inventing a new conjecture that holds for the counterexample. An implication for developing tasks by which students can experience this discovery function is mentioned.

Proof Validation and Modification by Example Generation: A Classroom-Based Intervention in Secondary School Geometry

Recent curriculum reforms underline mathematical activity related to proof validation , but few studies have explicitly addressed proof validation at the secondary school level. This chapter reports on our study of this issue. We suggest a specific kind of task for introducing proof validation in secondary school geometry and define the meanings of proof validation and proof modification in terms of Lakatos’s notion of the local counterexample . We briefly report on a classroom-based intervention implemented using such tasks in a lower secondary school in Japan. We then analyze the results of a task-based questionnaire conducted after the intervention to investigate how well the students did in proof validation and modification. The analysis shows that student failure in proof validation arose mainly from their difficulty with producing diagrams that satisfied the condition of the proof problem.