Abstract
There are crucial issues in the transition from the primary school to the secondary school mathematics curriculum. One of the main issue of the transition is concerning with the notion of variable. The purpose of this paper is to address the following question: how we should interpret the evolution of students' conceptions of mathematical expression with variable as a whole?
For attaining this purpose, firstly, the conceptual change approach and the theory of reification are discussed. Secondly, as a preliminary analysis, the notion of variable can be considered in terms of the theory of mathematical symbolism; for example, some semiotic viewpoints for mathematical expression, the syntactic definition of mathematical expression, and the difference between constant and variable and quasi-variable.
As a result of such preliminary considerations, the evolution of conception of mathematical expression can be characterized as both "horizontal development" and "vertical change". The "horizontal development" can be explained as transition from the interiorization to the condensation in the same semiotic level. The "vertical change" can be explained as transition from the condensation to the reification among the different semiotic level. In the final place, the horizontal and vertical evolution of conceptions can be identified in two different didactic situations with the help of the following scheme.
Fig. 1: An interpretative framework for conceptual change on the notion of variable
