数学教育における概念変容の特徴づけに関する一考察 : 離散量から連続量への展開を例として

抄録

The purpose of this paper is to address the following question: what kinds of viewpoints should be taken into account in characterizing conceptual change in mathematics education? For attaining this purpose, at first some theoretical problems of conceptual change research in mathematics education are summarized. Secondly "the global perspective" and "the local perspective" are introduced as methodological points of view for characterizing conceptual change, and the needs for these perspectives are discussed. Furthermore from these perspectives we make illustrations in the case of development from discrete to continuous quantity. And in the final place, some issues as didactical implications in developing from discrete to continuous quantity can be shown. On above considerations we have come to the conclusion that "multiplication with decimal numbers" and "irrational numbers (incommensurable magnitudes)" can be identified as problematic situations for conceptual change. As a result of the preliminary analysis, the following issues as didactical implications in developing from discrete to continuous quantity can be shown: ・In the problematic situation "multiplication with decimal numbers", learners need to become aware of inconsistency in terms of existing meaning of multiplication. ・In the problematic situation "multiplication with decimal numbers", the proportional meaning of multiplication that has been implicit status can become more explicit. ・In the problematic situation "irrational numbers (incommensurable magnitudes)", learners need to become aware that the familiar notation (i.e., place value system of decimal notation) cannot represent the quantity (incommensurable magnitude) in question precisely. ・In the problematic situation "irrational numbers (incommensurable magnitudes)", the awareness of incommensurability can lead to advance the degree of rigor on the infinity. As the future tasks, we need to build some theoretical framework/model in order to describe and/or assess learner's conception (status of knowing) about above issues, and to design the mathematics classrooms of "multiplication with decimal numbers" and "irrational numbers (incommensurable magnitudes)".