メタ知識としての「限界(Grenze)」の意味とその役割 : 新しい数学的内容と学習者との間の関係の問題

抄録

If we regard students as decision-makers who decide to participate in teaching situations and they not decide to, then it is a very important task for mathemtics education how to help the students relate themselves to any new mathematical ideas or ways of thinking. From this perspective, clarifing the meanings of the new mathematical ideas we intend to teach or to make clear what are these for us becomes more important than simplifing them. In this paper, the concept of "Grenze" will be considered. It have been introduced by German mathematics educators in IDM. They regard it as the best characterizing of metaknowledge. It could be regarded as the kernel of what are any new mathematical ideas or ways of thinking. The goals of this paper are as follows: (1) To make clear the meanings of the concept of Grenze and it's epistemological roles. (2) To show that a Grenze was realized in one of the mathematics classrooms which engendered eager participations and interests. (3) To identify the facts that need for realizing the Grenze by analysing the teacher-students interactions in the classroom. For the goal (1), the theory of proportion, especially the definition 5 in THE ELEMENTS BOOK V was considered. The main results of the consideration are summerized as follows: The meanings of Grenze can be characterized as a developmental relationships between unknown new ideas and known old or more familiar ideas. We need a context for constructing the developmental relationships. If we can construct the context and recognize the Grenze of our familiar ideas, then we can have a good perspective about the new ideas' novelty without pre-understanding of the new ideas detailed. So Grenze could play an important role in an introductry phase of didactical situations. Especially, if we regard the students as decision-makers, the role must become crucial. For goals (2) and (3), microethnographical case studies were utilized. About 10 minits introduction episodes including teacher-students interactions were taken from an 8th grade mathematics class, where the students' eager participations and interests were observed. The content which the teacher tried to introduce was basic triangle congruence theorems. Analysis of the episodes reveals that the Grenze was realized through teacher-students interactions in the classroom. The main facts that need for realizing the Grenze were as follows: (i) In the mathematics classroom, we observed a classroom culture which gave students a standard of valuable argumentations. The students seemed to find that it is reasonable to explain why it (geometric relations) is true in terms of other well-known geometric relations or results, that is to say, they were in the classroom micro-culture of mathematical argumentations. (ii) The students were permited to make free use of their familiar ideas of triangle congruence and encouraged to argument based on these ideas.