Do Pensamento Funcional ao Campo Conceitual de Função: o desenvolvimento de um conceito

Abstract

The concept of function is essential in Mathematics, as it is a constituent part of a large number of mathematical operations. For this reason, it forms an integral part of the Mathematics curricula of virtually all countries in the world. However, research has shown that students have not correctly appropriated this concept. Several reasons are given, such as: teaching approach based on abstract representations and a lack of clarity about what are the previous concepts that students need to know to conceptualize the concept of function. The practical aspect left aside, together with the lack of structuring of the situations necessary for the teaching of function, have led the Group of Studies and Research in Didactics of Mathematics (GePeDiMa) to establish the possible Conceptual Field of Functions and Affine Function. In this scenario, this thesis aimed to verify the existence of the Conceptual Field of Affine Function and, consequently, of Functions; beyond the Fields of Additive and Multiplicative Structures, already established by Vergnaud. To prove this existence, the organizing concepts, the basic ideas, the representations, and situations that make up this Conceptual Field were mapped. To this end, a bibliographic investigation was carried out, guided by the Theory of Conceptual Fields regarding functions, considering three perspectives: historical, cognitive, and didactic. The investigation was qualitative, with methodological referrals assured in the assumptions of a State-of-the-Art research and analyzes based on the Theory of Conceptual Fields. From a historical perspective, nine stages in the evolution from the functional thinking to the concept of function have been identified. The cognitive perspective provided support to identify that the cognitive structures necessary for the subject to be able to conceive the concept of function start with notions of relations between magnitudes and are established in formalized regularities. Finally, from the didactic perspective, the algebraic focus on function teaching and the difficulties encountered by students with regard to different representations were detected. From these results and associated with the quartet (situation, organizing concept, basic idea, representation) the existence of the Conceptual Field of Functions was confirmed, as well as its basic ideas were identified, namely: dependence, generalization, regularity, and variable.