Tarefas de Derivadas em Livros de Cálculo na Perspectiva da Educação Matemática Realística

Abstract

The teaching and learning of Differential and Integral Calculus are a frequent topic of discussion among Brazilian and foreign researchers, mainly due to the students' difficulties and the high failure rates in Calculus subjects. This research intends to reconcile three pillars: the analysis of statements of mathematical tasks, the tasks of derivatives in Calculus books and Realistic Mathematics Education. The investigation is guided by the following question: How are the tasks of derivatives, present in Calculus books, presented from the perspective of Realistic Mathematics Education? The approach to teaching Mathematics that underlies this analysis, Realistic Mathematics Education, infers that the protagonism in the process concerns the student, who is encouraged by his guide, the teacher, to reinvent Mathematics; to enjoy a feeling of discovery similar to that of a mathematician. Therefore, the purpose is made possible by investigative practices, mathematical organization and problematization. In addition to the behavior of those involved, the theoretical perspective argues that it is essential to propose good context problems, a basic item of this research. Two Calculus books were chosen for the analysis, namely, the one by Louis Leithold and the one by Hamilton Guidorizzi. Based on the characteristics and objectives of good context problems, according to the approach, incongruence between the tasks of the books and the realistic conceptions was found. From the beginning, the disagreement, to a certain extent, was assumed, given that there is no evidence of concern on the part of the authors with regard to Realist Mathematics Education; however, this was not an objection to the research. The purpose of this investigation was based on the analysis, from the perspective of Realistic Mathematics Education, of statements of derivatives arranged in books considered classic and commonly used in Brazil. The research infers that the use of these books, at least in the study of derivatives, can be fruitless from the perspective of Realistic Mathematics Education, given that, at first, the actions stimulated by the tasks arranged in the books are incompatible with the realistic conception of approach to Mathematics, whose activity is supported by the organization and problematization of reality. While it is feasible to use books in a classroom setting based on a realistic approach, it was found that this use will require the teacher's readiness to establish a link between the books and the approach. As a result, it was also intended to subvert some statements, in order to bring them closer to the precepts disseminated by Realistic Mathematics Education, as well as to provide examples of statements that converge to the principles of the realistic approach to the scientific community and teachers. The subversion of utterances proved to be a laborious task, in particular, due to the lack of a precise profile of the student to whom the subversions are intended. Sometimes, however, it was found that subtle changes in the statements or in the way tasks are handled in the classroom are sufficient adjustments for the books to be used in class. Therefore, although this research is dedicated to a study of task statements, it is inconceivable to disassociate it from the practical dimension.