数学的問題設定における証明の活用の様相～「数と式」の授業における生徒の問題設定の過程～

抄録

This study clarifies aspects of utilization of proofs within the context of mathematical problem posing, by analyzing individual students’ processes of problem posing in a mathematics classroom. First, we establish a framework to analyze students’ processes of utilizing proofs in problem posing by focusing on functions of proofs. Second, we describe the methods of data collection and the outline of the whole class activities in the lesson on “numbers and algebraic expressions” for Japanese 8th graders. Third, we analyze two students’ processes of utilizing proofs in problem posing, which resulted in gaining the students’ insight into why the statement of the starting point is true. As a result, we clarify two aspects of utilizing proofs within the context of problem posing: (1) utilizing others’ proofs concerning different functions of the proofs properly based on their own conviction (or non-conviction) of the functions by reviewing the proofs. (2) Repeatedly utilizing the explanatory function of proofs while gradually expanding the extent of their consideration. Finally, we discuss educational implications, significance, and limitations of the analysis.