事柄の証明と論駁における定義活動を促進するための教材開発研究―定義の洗練に焦点を当てて―

抄録

Definitions play a significant role in mathematical activities such as proving a conjecture, extending a concept, and so forth. However, there is an issue that definitions are presented with authority in school mathematics textbooks. We address this issue by focusing on a mathematical activity, defining that students construct and revise definitions by themselves. The purpose of this study is to design a task that fosters students’ defining in proofs and refutations. The research was conducted based on the design research methodology. We first explained a normative model of generation of mathematical knowledge through proofs and refutations. We used this model to design a task that considers a definition of polygon in proving a statement. To investigate the effect of this task, we analyzed classroom lessons involving public secondary school students. Our analysis showed that students noticed the existence of counterexamples which invalid the given statement and modified their proof, statement, and definition. Even if students revised their zero-definition to a monster-barring definition at first, they could refine it to a proof-generated definition by receiving the teacher’s support. This paper highlights the importance of how to revise definitions based on ideas of proof, and a continuous task design to encourage students’ learning through mathematical activities.