A mobilização das ideias-base de função por estudantes do ensino fundamental e ensino médio em situações associadas à função afim

Abstract

The concept of function stands out as one of the most important to be considered during the schooling of students in the processes of Mathematics teaching and learning. Research in the field of Education, especially of Mathematics Education, has highlighted the need for students to use a large number of concepts, such as the basic concepts of functions (dependence, variable, regularity, and generalization), in this conceptual construction. Based on this, the objective of this research was to investigate the level of competence of elementary and high school students in mobilizing the basic concepts of functions. With theoretical support from Gérard Vergnaud's Theory of Conceptual Fields, an instrument was developed considering nine situations associated with the concept of affine function in order to collect data. With the intention of observing students activity as they attempted to solve the situations, the instrument was implemented with 21 students from the 6th grade of Elementary School to the 3rd year of High School at a State school in the city of Quedas do Iguaçu – PR through semi-structured individual interviews developed following the theory of the Piagetian Clinical-Critical Method. In data analysis (written protocols and interview recordings), based on the Theory of Conceptual Fields, it was aimed to identify students' competencies and levels of understanding of the concept of function (intuitive understanding, initial mathematization, and abstraction) when solving the situations, observing the mobilization of the basic concepts of functions and other operational invariants. It was inferred that students at different educational levels may have acquired practically equivalent competencies when solving situations associated with the concept of affine function. It was identified that the mobilization of the basic concepts of functions by the students during the resolution of the situations occurred in association with the manifestation of eight true action theorems and two false action theorems. The students who, at the moment of solving a situation, showed mastery of the necessary competencies to address it, mobilized all the basic concepts of functions associated with true action theorems. Regarding the level of understanding of the concept of affine function, at the moment of solving the situations, it was inferred that the most of the students were at the level of initial mathematics. The results of this investigation highlight the need to offer situations that provide, during their resolution by students, the mobilization of basic concepts of functions so that they develop competencies related to the concept of affine functions so they can achieve the level of abstraction in understanding the concept of functions during their schooling in Basic Education.