図形を定義する活動の位置づけに関する基礎的考察 : 図形の相互関係の理解に関する調査と関連して

抄録

Geometrical definitions play important roles in learning geometry. Especially, in demonstrative geometry students need to function well since geometrical logic is constructed based on the definition. In junior high school mathematics, the inclusion relation between geometrical figures is one of relatively simple teaching materials to be understood by using definitions. In this study, firstly we analyze students' understanding of inclusion relations between geometrical figures, as an indication of whether the definitions are functional for junior high school students. The findings are follows; ・Inclusion relations between parallelogram and rhombus, and between rhombus and square are already conceived by more than half of 1st graders. On the contrary, the scores of inclusion relations between parallelogram and rectangle, and between rectangle and square are less than 50 % in 3rd graders. ・On the whole, the degree of achievement of the contents is low. Therefore, we cannot say that learning inclusion relations in demonstrative geometry is effective. Secondly, we discussed activities in which students construct geometrical definitions, from the viewpoints of transition to demonstrative geometry. Geometrical definition is initially used by young children to classify or construct geometrical shapes, and therefore it shows the characteristics of the shape. On the other, the definitions in demonstrative geometry are necessary and sufficient conditions, and they are used as bases and premises for constituting geometrical logic. We suggested that in order to fill the gap, it is important for students to construct definitions as the formalization of geometrical concepts in the level of action. The activities have the following two aims; ・To understand the definition as the condition for determining the geometrical figure ・To understand the definition as a starting point of ordered relations among geometrical properties The former is recognized in the activity of classification among geometrical figures by using operational sheets (Nakahara, 1995), and the latter will be recognized in the activity of geometrical construction of perpendicular bisector, perpendicular line, and bisector of angle based on the geometrical figure of kite or rhombus. Through these activities, student will be able to conceive the geometrical figure as a set of geometrical properties and control the figure using the definition.