# 数学的探究におけるaction proofの活用の促進：事例研究を通して

## Bibliographic Information

Other Title
• スウガクテキ タンキュウ ニ オケル action proof ノ カツヨウ ノ ソクシン ジレイ ケンキュウ オ トオシテ
• Facilitating Pupils' Utilization of "Action Proof" in Mathematical Inquiry : A Case Study(Part I Research Articles)

## Abstract

"Action proof" is, in short, proof in which one performs certain operations with manipulative objects in a generic example. The previous studies on action proof have discussed its utilization in the context of verifying statements or learning formal proof. However, we also need to focus on mathematical inquiry in which one makes a conjecture, proves it and refines the conjecture and its proof, because such inquiry reflects mathematical research and may lead to productive learning of formal proof. Thus, this study explores how we can facilitate pupils' utilization of action proof in mathematical inquiry. Firstly, the author considered the meaning of utilizing action proof in mathematical inquiry from two viewpoints: grasping why statements are true and inventing new statements. Then the methods for facilitating pupils' grasp and invention were theoretically examined. After that, the author planned and conducted a case study in which a pair of fifth graders utilized action proof in mathematical inquiry and the author intended to facilitate their activities according to the methods that had been theoretically examined before. The methods were then discussed and elaborated, based on the findings of the case study. The conclusions of this study are as follows. -In mathematical inquiry, we have to provide pupils with opportunities to make conjectures themselves. It is then important to promote them to describe the subjects and predicates of their conjectures clearly, such as "S becomes (is) P". If a conjecture includes several matters and therefore becomes complicated, we need require pupils to divide the conjecture into several statements. -When pupils try to grasp why a statement is true through action proof, we should facilitate them to grasp the original character of actions on manipulative objects with making reference to the conditions of the statement, for example questioning "why the conditions make the statement always true?". Then, it seems that, through selecting examples which are troublesome for being represented by manipulative objects, we can promote pupils to represent exactly the conditions of the statement by manipulative objects. -After grasping why a statement is true, pupils pursue actively new statements if we raise counter-examples to the statement. Furthermore, facilitating pupils to seize the following two points is important for them to modify the original statement and to invent more general statement: conditions in which the original statement becomes false, and part of the previous examination which is applicable to the conditions. -When pupils try to invent new statement through the previous action proof, it is important that after we provide them with opportunities to become aware of their previous views of manipulative objects, we promote them to change their views and to invent new actions on manipulative objects. After that, we need require them to interpret the meaning of new actions on manipulative objects and to find the conditions of new statement. Then we should facilitate them to compare previous and new statements in terms of generality through examining differences of action proof of the two statements.

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