Invariantes Operatórios e a Compreensão da Definição Formal de Limite de Função: Uma Análise à luz da Teoria dos Campos Conceituais

Abstract

This research aims to investigate the operative invariants mobilized by Mathematics students at the Federal University of Mato Grosso do Sul when facing situations designed to foster the understanding of the formal definition of the limit of a function at a point. Grounded in Gérard Vergnaud's Theory of Conceptual Fields, the study analyzes the theorems-in-action expressed through the students' representations and strategies. The adopted methodology is Didactical Engineering, encompassing the stages of preliminary analysis, design and a priori analysis of the didactical sequence, experimentation, and a posteriori analysis with validation. The proposed didactical sequence was implemented with two pairs of students—one from the teaching degree program and the other from the bachelor's program—and consisted of eleven activities organized into three sessions. These activities focused on the progressive construction of the notion of distance and its algebraic and geometric representations, culminating in the characteristic notation of the formal definition of limit. The results showed that students mobilized different theorems-in-action throughout the activities, revealing affiliations with concepts previously encountered, especially in their use of inequalities, intervals, graphical representations, and operations involving absolute value. The a posteriori analysis also revealed that some of these theorems-in-action, which did not contribute to the understanding of elements related to the formal definition of limit, were abandoned—ruptured—and replaced by more pertinent ones, such as one involving the notion of distance. This shift was made possible through the challenges posed by activities requiring the algebraic generalization of the idea of approximation, by epsilon and delta, as developed with the students during the didactical sequence.