図形概念の不整合に関する認識論的研究

抄録

There have been a little epistemological interests in the phenomenon of inconsistencies in the concept of geometry. So the purpose of this research is to clarify the characteristics of inconsistencies in the concept of geometry. For this purpose I have to consider here three main issues. 1. What are inconsistencies in the concept of geometry? 2. What are the factors of the inconsistencies? 3. What are the functions of the inconsistencies in the teaching and learning of geometry? The term "inconsistence" is used as the situation that if p is a proposition, then both p and〜p hold simultaneously. And inconsistencies in the concept of geometry can be classified into two kinds of types. One kind of type is from a viewpoint of obuject of inconsistencies. External inconsistencies: inconsistencies between the student's concept of geometry and mathematical concept of geometry. Internal inconsistencies: inconsistencies within the student's concept of geometry. The other kind of type is from a viewpoint of student's awareness. Explicit (cognitive) inconsistencies: student is aware of inconsistencies. Inplicit inconsistencies: student is not aware of inconsistencies. External inconsistencies have been called "conflict" or "misconception" and explicit inconsistencies have been called "cognitive conflict" or "disequillibrium". External inconsistencies are easily recognized by observers because mathematical varidity of the student's concept of geometry is judged on the basis of abstract mathematical concept of geometry. According to the representational model of concept of geometry, the inconsistencies can be interpreted. For example the definition of rectangle can not be written because of the lack of meaning of linguistic representation about rectangle and rectangle is recognized as long shape because of the effect of figural representation about rectangle. On the other hand internal inconsistencies are not always recognized by both observers and students. According to the representational model, internal inconsistencies are considered as the difference between linguistic representation and imaginary representation. Furthermore according to the aspect model of understanding on individual concept of geometry, internal inconsistencies are described as the inconsistencies between the aspects of understanding. For example definition of rectangle is written by use of linguistic representation but rectangle shape is identified by use of imaginary representation. There are many factors that interpret inconsistencies: natural language, meaning of definition, linguistic interpretation, prototype judgement and generalization of figure for external inconsistencies and the difference of cognitive operation, the awareness of definition and the context-bound nature for internal inconsistencies. What is significant in the teaching and learning of geometry in the light of inconsistencies is to make student become aware of the inconsistencies, that is to exchange from implicit inconsistencies to explicit inconsistencies. Then on the basis of the facters of inconsistencies they can dissolve the inconsistencies. It is the conscious of teachers on the student's individual concept of geometry that needs to be reformed.