An Aspect of the Development of the Functional Concept in Mathematics： Based on the Viewpoint of Levels of Thinking and Symbolization

Abstract

　　The purpose of this research is to clarify the aspect of symbolization of tables, formulas,and graphs,and signalization in development of a functional concept on the basis of the theory of levels of thinking by van Hiele (1986).　In development of the functional concept, I think that symbolization is that it is to recognize the situation in tables, formulas, and graphs, and when the situation that a symbol works on other expression appeared, I consider that a symbol functions as a signal.

　　In order to clarify the aspect of symbolization and signalization, I observed and analyzed the lessons of a fourth grader “how to change” and the fifth grader “formula using ○ and △”, and　investigated the aspect of development of a children’s functional concept.

　　As a result, the following knowledge was acquired :

・ In order to symbolize graphs, it is important that children consider motions of hands (gestures) as the concept of the function and the change.

・ In order to symbolize formulas, it is important that children focus on the structure of the situation by unitizing and norming (Lamon, 1994).

・ By following a change of the phases of thinking (van Hiele, 1986), children found that it was difficult to exceed “free orientation (phase 4)”.