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学校数学におけるaction proofの機能に関する研究:発見に焦点をあてて

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  • ガッコウ スウガク ニ オケル action proof ノ キノウ ニ カンスル ケンキュウ ハッケン ニ ショウテン オ アテテ
  • A Study on Function of "Action Proof" in School Mathematics : Focusing on Discovery

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Abstract

The previous studies on "action proof" have discussed its function in the context of verifying statements or learning formal proof. However, there are any other functions of proof in mathematics than verification: explanation, systematization, discovery and communication. The purpose of this study is, however, to clarify discovery function of action proof in school mathematics among all. The reason why this study selects its discovery function is that we could achieve many productive learning activities in even primary school mathematics. Firstly, this study reviews and summarizes the previous studies on action proof and then points out that they have not discussed discovery function of action proof. Next, this study defines the concept of action proof in terms of actions on manipulative objects, the original character of the actions and a representative special case. According to this definition and to comparison between action proof and formal proof, this study points out "embodiment", "movability" and "genericity" as the characteristic properties of action proof. After that, through examining discovery function of formal proof in terms of the characteristic properties of action proof, this study clarifies two kinds of discovery function of action proof, as follows. First, in order to clarify relation between the conditions of the statement and a truth of the statement, one has to grasp not only the original character of actions on manipulative objects but also the reason why the actions can be applied to all cases of the statement; after grasping the reason, one must notice and eliminate unnecessary parts for the relation or examine the process of actions on manipulative objects with consciousness of the relation; then by interpreting facts on manipulative objects which one finds out, one can produce new statements. Second, one has to view manipulative objects from new viewpoints which are not limited to the given statement and then, if necessary, organize the actions on manipulative objects; then by interpreting facts on manipulative objects which one finds out, one can produce new statements.

Article

日本数学教育学会誌. 数学教育学論究. 92:33-45 (2010)

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