Abstract
The main aim of this paper is to analyze the process of problem-solving by changing an equation with letters from point of Analogy, Mathematical Generalization and Abstraction Theory and to clarify how the process contains difficulties in understanding. In order to attain this aim, at first I divide the process into 4 parts as follows. the process(1); constructing the equation. the process(2); changing the equation. the process(3); reading the equation. the process(4); verifying that the values don't change or some values add. By the analysis of this process, I could obtain the following results. In the process(1) that the students construct the equation from the word-problem given, the their mind functions toward so called Primitive Abstraction and Removal Generalization, while the values remain to be invariable or some values add. In the process(2) of changing the equation, the values of the equation don't decrease so long as we follow the valid deductive rules. In the process(3), the values are specialized or materialized, and they are checked whether the values are fit for word-problem or not. In process(4), it is very difficult that teacher make students to understand logically the values remain to be invariable or some values add, then I propose that there can be the teaching method using the Valance-Model based on Analogical Understanding as a kind of understanding modes.
