A Study on the Viewpoints for Analyzing Interactions using Various Proofs in Mathematics Lessons

Abstract

　　This paper aims to clarify some viewpoints for analyzing the interactions using various proofs in mathematics lessons. Students who have already learnt basic mathematical proving can give various verifications to a theorem. In this paper, we propose a framework which has two viewpoints, which were invoked by Voigt (1995), “interaction patterns” and “thematic patterns of interactions.” Furthermore we divide the former viewpoint into “comparison pattern” and “discussion pattern,” and the latter into two thematic classifications, “ways of proof” and “structures of proof.”

　　Using the framework, we analyzed the lessons where 31 third graders at a junior high school tackled a proof problem. As a result, we get the following two findings in the interactions using various proofs in the mathematics lesson. i) To distinguish between the comparison and discussion patterns is available for us to analyze the complicated interactions. ii) To analyze the thematic patterns of interactions by distinguishing between ways and structures of proof is useful in describing the students’ processes of proving a theorem and generating another proof through their interactions. Especially it can be described the students’ interactions in generating various proofs by referring to the thematic patterns, integrating the thematic patterns of various proofs, and comparing themes with another theme of new proof.