How to analyze the crucial problem of understanding mathematics?

Abstract

The major challenge of mathematics teaching is to get students into the way of thinking and working that is specific to mathematics since that is the condition that precedes all acquisition of concepts in mathematics. But for this we must analyze the mathematical way of thinking and working on what is radically different from the more spontaneous ways of thinking and working in other domains of knowledge. The theory of registers of semiotic representation is essentially an instrument that was designed to analyze the way of thinking and working in mathematics whatever the concepts and domains (geometry, algebra, analysis ...) treated. In a way, mathematical activity has two sides: the side that appears when one considers the mathematical point of view and another side that reveals itself when one considers the cognitive point of view. In this article, we discuss the importance and necessity of getting students to insert themselves in the way of thinking and working that is specific to mathematics. To this end, the following issues will be addressed: - How to describe the way of thinking and working in mathematics? - Is the conversion of representations the first obstacle to understanding mathematics? - What does "understanding math" mean? - Are both sides of mathematical activity considered in teaching and research in mathematics education?