Teoria das situações didáticas: uma proposta para o ensino da função quadrática em suas formas canônica e fatorada

Abstract

Considering the studies analyzed during the development of the bibliographic review of this work, involving the solution of quadratic equations, which are associated with the search for the roots of quadratic functions and the importance highlighted by official and guiding documents regarding the resolution and elaboration of problems that can be represented by polynomial equations/functions of the second degree, as well as related competencies and skills, we understand that these topics become essential for the development of algebraic thinking. Thus, there are several studies focused on the teaching and learning processes of polynomial equations/functions of the second degree, in search of a deeper understanding of the influential factors in the effective learning of these concepts. Thus, this research aims to analyze the learning of the Quadratic Function observing the canonical and factored form having as theoretical assumptions the Theory of Didactic Situations, since this theory has been used by researchers as a way of organizing teaching situations and evaluating the teaching and learning of Mathematics at different levels of education. The study was developed with academics, from the first semester, of the Mathematics courses – Degree and Bachelor’s degree of a federal institution of higher education in the state of Rio Grande do Sul, because we understand that the understanding of the basic concepts related to the functions and the application of these concepts in the solution of problems practical, constitutes a knowledge base both in the scope of personal/cognitive development and for professional development, as they will be future teachers. For that, we selected eight activities that approach the canonical form and the factored form of the quadratic function and that contemplate the phases of the didactic and didactic situations foreseen in the Theory of Didactic Situations. In addition, these activities addressed the resolution of tasks using the GeoGebra dynamic geometry software and in pencil and paper, with the twenty-three research participants. In view of the data analysis, it was possible to perceive that when going through the situations of action, formulation and validation, phases foreseen in the Theory of Didactic Situations, the students acted actively in the construction of their knowledge when they tested conjectures, formulated hypotheses, proved, built models, concepts and theories. Thus, in the institutionalization phase, in which the teacher methodically organizes the knowledge involved in the activities in mathematical language, the students already understood and/or were able to relate the hypotheses formulated, during the development of the activities, with the concepts presented by the teacher.