図形概念の現実的表現に関する認識論的研究

抄録

The main purpose of this research is to reflect upon the epistemological meanings of realistic representation of the concept of geometry and to clarify the crucial roles of it in the teaching of geometry. There are five styles of representation on the mathematical concept of geometry that are imformed by teachers in the teaching of geometry: realistic, operational, figural, linguistic and symbolic representation. Realistic representation is to represent real objects by use of pictures, photos, illustrations or the objects themselves. After observation of elementary mathematics textbooks, it came out that there are two types of situations to show realistic representation in the teaching of geometry: situation for introduction and situation for application. This difference between two types of showing situations is interpreted from an epistemological view point of realistic representation, that is to say from an interaction between real world and mathematical world. ・Situation for introduction: developing concept of geometry from real world to mathematical world (a) From object to shape "Shape" is identified from realistic representation by the operartion of mathematization of real objects. In the process of mathematization we must change our consciousness to reflect a view of objects. And a new view of "shape" that is acquired as "mathematical tool" is put to practical use in the latter teaching of geometry. (b) From object to diagram Realistic representation (object) and figural representation (diagram) are not rerated originally. But according to the recognition of shape, realistic representation and figural representation are connected and "image" of shape is constructed. (c) From real world to mathematical world Recognition is sifted from real world to mathematical world. The connection of object and diagram is to sift a consciousness from real world to mathematical world and to construct individual concept of geometry. ・Situation for apprication: mathematical recognition from matehmatical world to real world (d) Ontological meaning According to a confirmation of mathematical entity by use of realistic representation, intuitionismic recognition that there exist figure in the real world is constructed. (e) Epistemological meaning in a narrow sense According to an observation of realistic representation, an external experience about real object is internalized to an internal experience and an implicit operative tool is translated to an explicit geometrical object. So mathematical concept of geometry in elementary mathematics is recognized as an empirical concept. (f) Pragmatismic meaning Realistic representation can be viewed as playing the role of object itself with shape as well as playing the role of realistic model of figure. So realistic representation has a dual nature. The pragmatismic meaning of realistic representation is a role of realistic model. What is significant in the teaching and learning of geometry in the light of realistic representation is to make not only students but also teachers become aware of the meanings of interaction between real world and mathematical world.