Research of Mutual Relationships between Addition and Subtraction: From a Viewpoint of Algebraic Reasoning

Abstract

　　 This research is to approach the problem of connection of arithmetic and algebra from a viewpoint of algebraic reasoning.  Therefore, in this paper, we focus on the transition from pragmatic recognition to semantic recognition from a point of view of linguistics research.  The purpose of this paper is to practically examine that we constituted classes of mutual relationships between addition and subtraction from a viewpoint of algebraic reasoning and to clarify aspects of the transition based on aspects of algebraic reasoning.  

　　 As a result, the following things become clear. 

(1) The classes are effective to students’ understanding of mutual relationships between addition and subtraction.

(2)  Aspects of algebraic reasoning in the classes are firstly cognition of antonym by agreement of rules in diagrams and objectification of them, then cognition of synonym by interaction between inverse operation of  diagram, inverse calculation, and part-whole diagrams.

(3) Aspects of the transition from pragmatic recognition to semantic recognition are firstly cognition of antonym, secondly ambiguity, finally synonym.  The cognition of ambiguity, however, means not only meanings of  expression but also ideas of expression.