History of mathematical analysis and cognitive development

Abstract

This research discusses the historical construction of mathematical analysis concepts in the light of genetic epistemology, to obtain conclusions about the learning in this area. According to the research problem "How is structured knowledge related to mathematical analysis?", we aim to understand the process of constructing real knowledge analysis, in history and comparing the cognitive subject development, according to the genetic epistemology perspective. For this, we studied the history of mathematics from its main references, and Piaget's theory. As a result, it was observed that the course that culminated in the current formalization of the analysis occurred in four stages: development of differential and integral calculus; organization of this calculus; analysis of this calculus; arithmetization of this analysis. Each of these stades parallels the notion of developmental stage, and the main difference between these stages would be the level of conceptualization. It is concluded that the learning of analysis takes place through successive moments of taking the awareness process that culminate in conceptualization. This implies that to present axioms and establish properties and theorems from them does not consist of a potential teaching activity, except for those students who already have reached the cognitive level necessary to carry out these assimilations and, therefore, the answer was already in the subject before. For those who have not reached that level, it is necessary to act on mathematical objects so that, from this action, the process of taking awareness of the concerned actions may occur. The aim was to contribute to broadening the debate about mathematics education in higher education, favoring a democratization of learning mathematics at all levels, including the most formal ones.