辻山 洋介

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.

It has become gradually accepted that proof and proving are essential at all grades of mathematical learning. Among the various aspects of proof and proving, this study addresses proofs and refutations described by Lakatos, in particular a part of increasing content by deductive guessing, to introduce an authentic process into mathematics classrooms. This paper analyzes proof problems with diagrams as an appropriate task for this mathematical action, and illustrates, with a sequence of three lessons in the eighth grade, that a certain type of such problems can provide students with an opportunity to engage in this action.

In constructing a proof, students need to examine the probable arguments they found when devising a plan of proof. However, many students do not seem to acquire effective methods of making such an examination. In order to provide ideas for improving the learning and teaching of proving, this paper aims at deliberating how to examine probable arguments in constructing a proof from an argumentative perspective. Towards this aim, this paper elaborates the nature of argumentation based on Toulmin (1958/2003), who focused on the probability of practical arguments and the essential aspects of using such arguments. Based on the nature of argumentation and the relationship between constructing a proof and devising a plan of proof, this paper theoretically characterizes how to examine probable arguments as the following: a) clarifying the probability of arguments by changing their types; b) improving arguments by identifying and justifying the condition under which the arguments hold; c) exploring alternative arguments by utilizing dismissed arguments; and d) modifying arguments by justifying the truth of their warrants. Finally, this paper discusses the validity of the characterization and derives an implication for teachers' means to facilitate students' examination of probable arguments.

The purpose of this paper is to theoretically examine the role of looking back at proof-planning processes in school mathematics from the perspective of argumentation. Various research studies have elaborated the significance of proof and proving in school mathematics. A particular focus of this paper is on what students can gain by looking back at proving processes that do not appear explicitly in the presentation of proof. Examples of hypothetical proof-planning processes are chosen to illustrate that two kinds of ideas can occur by looking back at the processes. In particular, the ideas are related to how to identify why the statement is true and how to prove the initial statement and another related statement.