Prescriptive Framework of Activities Necessary for Constructing Mathematical Knowledge-How: Theoretical Consideration with Examples of Indirect Proof

Abstract

　　The purpose of this paper is to elaborate a prescriptive framework for designing mathematical activities necessary for shifting students’ indirect argumentations from naïve to proof-like.  The framework was derived from the following four theoretical resources: 1) Features and structures of indirect proofs; 2) Critiques of contents-general proof research; 3) Reflection on mathematical methods; 4) Mathematical literacy focusing on knowledge-how.  As a result, we proposed a general prescriptive framework for designing mathematical activities necessary not only for shifting students’ indirect argumentations from naïve to proof-like, but also for constructing mathematical knowledge-how.  More concretely, we made a conclusion that four questions should emerge from students’ mathematical activities in the following order: 1) How can we solve a particular problem? (The construction of an indirect argumentation); 2) Why can we solve in that way? (The construction of the method of indirect proof); 3) When can we solve in that way? (The construction of a situation where the method of indirect proof is applied); and 4) Why can we solve in that way at that time? (The construction of a list of heuristics to apply indirect proof methods).  Although our discussion starts from the particular topic of indirect proof, this paper succeeded in extending the general theory of mathematical literacy.