Conceitualização da função afim mobilizada por estudantes de Licenciatura em Matemática: uma análise a partir da resolução de situações articuladas a gráficos cartesianos

Abstract

This research is grounded in the premise that a concept is understood by learners through exposure to a variety of problem situations. In this context, the main objective of the study is to design, implement, and evaluate a didactic sequence for first-year undergraduate Mathematics students, aimed at fostering the conceptualization of the linear function. The methodology adopted was Didactical Engineering, developed in four phases: preliminary analyses; design and a priori analysis; experimentation; and a posteriori analysis and validation. The preliminary analyses involved the development of an Epistemological Reference Model (ERM), as well as the examination of curricular documents, textbooks, and studies on errors and obstacles related to the linear function. This process resulted in the construction of a Current Epistemological Model (CEM) and the proposition of an Alternative Epistemological Model (AEM). The didactic sequence proposed to students comprised ten problem situations involving the linear function, encompassing variations of the simple proportion class and measurement transformation, articulated with corresponding graphical representations. The analysis of students’ solutions, grounded primarily in the Theory of Conceptual Fields, enabled the identification of the mobilized schemes, the theorems-in-action, and the levels of conceptualization demonstrated by the students. The results indicate that, throughout the resolution of the problem situations, students showed significant progress in interpreting and constructing Cartesian graphs of linear functions, particularly in the correct identification of axes, the appropriate choice of scales, and the coherent plotting of points based on the relationship between the variables involved. A greater articulation among different representations was also observed, suggesting that students became more confident in moving between them as forms of expressing the concept of the linear function, which contributed to a reduction in the use of false theorems. These advances were supported by the diversity of problem-situation classes explored, their articulation with graphical records, and the institutionalization moments conducted throughout the didactic sequence. Three levels of conceptualization were identified among students – elementary, intermediate, and advanced – revealing a progressive movement of expansion and refinement of the schemes mobilized. The study concludes that the proposed approach contributes to strengthening these schemes, indicating that the diversity of situations, the systematic analysis of errors, and the planned institutionalizations enhance the process of conceptualizing the linear function in Higher Education.