Symbolic writings and heterogeneous expression substitution operations: the conditions for understanding in elementary algebra

Abstract

This article is a translation of a chapter written in French by Raymond Duval, who sought to analyze different types of substitution operations that can be carried out with symbolic scripts in order to think about comprehension in algebra. These analyses concern the awareness of semiocognitive operations that will allow us to understand how to work with algebraic scripts and recognize when to apply them. Duval presents four questions related to the semiocognitive conditions that condition the understanding and acquisition of knowledge in algebra, as well as its spontaneous use in problem-solving situations outside the mathematical sphere: designating objects in natural language, using letters or symbols; visualizing the mathematical structure of the formulation of a problem, in a text that articulates several sentences; formulating problems whose solutions require the functional designation of a second unknown quantity to write two incomplete expressions, starting from the problem statement and forming the two members of an equation. The conclusion of the linguistic analysis of symbolic algebraic writings presented by Duval in the chapter points to the need for Frege's semantic distinction between the meaning of an expression and what it denotes. With the chapter Duval points out that a semiocognitive analysis of symbolic writings means analyzing the needs and difficulties of learning algebra before organizing its teaching whose aim is, on the one hand, to raise awareness of both the specific discursive operations of natural language and symbolic writings, and on the other, to break the glass wall that separates them.