Provas e demonstrações em livros didáticos do Ensino Médio: análise de organizações matemáticas e didáticas relativas à semelhança de triângulos

Abstract

The objective of this research was to analyze the mathematical and didactic organizations of textbooks for the 1st year of high school regarding proofs and demonstrations involving the similarity of triangles. To achieve this, we analyzed the approach to the concept of similarity of triangles in textbooks for the 1st year of high school approved in the National Book and Teaching Material Program (NBTMP), 2021, as well as their tasks, and respective techniques, involving tests and demonstrations relating to this concept. This analysis consisted of three main stages: choice of the textbooks analyzed, ecological analysis, and praxeological analysis based on the praxeological structure constituted by the notions of type of task, technique, technology and theory, and was based on the Anthropological Theory of Didactics (ATD) as a tool for analyzing teaching materials. From this perspective, we use, as a theoretical and methodological framework, constructs from the ATD developed by Yves Chevallard and collaborators, and similar research by authors such as Saddo Ag Almouloud; in addition to theoretical constructs about proofs and demonstrations from the perspective of Nicolas Balacheff, especially the distinction between explanation, proof and mathematical proof (demonstration) and its importance for the introduction of teaching these concepts in Basic Education, and the typography of proofs, classified as pragmatic and intellectual, which in turn encompass four other important types, namely: Naive Empiricism, Crucial Experiment, Generic Example and Mental Experiment. By analyzing the tasks and establishing the techniques necessary for their resolution, we seek to outline how the authors of the selected textbooks approach and explore the concepts of proof and demonstration. The predominance of a traditional approach to the concept of similarity of triangles stands out, that is, definition, followed by examples, solved and proposed exercises, although in some cases there was a certain contextualization by associating the similarity with the enlargement and reduction of figures, for example. In most of the textbooks analyzed, the authors did not explore proofs and demonstrations. Many authors did not demonstrate cases of similarity, of which only one was demonstrated in one of the textbooks. We also note that in most textbooks, the authors chose to demonstrate only the fundamental similarity theorem. In general, in the analysis of organizations, we infer that there is an approximation between mathematical and didactic choices, that is, based on what was presented in the introduction of the notion of similarity of triangles, it was possible to develop in a justifiable way the tasks chosen to mobilize knowledge relating to the respective notion. Furthermore, although in approaching the notion of similarity of triangles there is no exploration of proofs and demonstrations, the teacher can develop strategies to mobilize these concepts through the tasks that were presented, since the textbook is a tool that must be used to assist you, subject to the objectives relating to what you intend to teach.