O recurso da demonstração em livros didáticos de diferentes níveis do ensino de matemática

Abstract

This present research, of qualitative nature, was guided by the question: what characterizes and what are the functions of school demonstrations in different level textbooks for teaching mathematics? In order to answer this question, three high school and three Junior high school textbook collections –which are assessed and approved by the National Textbook Program (PNLD, in Portuguese), the National Curricular Parameters (PCN, in Portuguese) and the PNLD guides for the same presented school levels –were considered as documents. Inspired by the Depth Hermeneutics methodological referential, these demonstrations were considered symbolical. Firstly, the way how research in academic mathematics and Mathematical Education discuss the demonstrations was pointed out. Afterwards, the demonstrations, in different historical periods, were exposed –such as in the literary work “The Elements” by Euclid and in textbooks which were published during reforms in the teaching of mathematics in Brazil. Through a referential of the sociology of the science, the symbolic value of these school demonstrations was discussed. The development of this study has pointed to a discussion about the naturalization processes of the uniqueness and verity of logical values. In addition, the demonstration was presented as a belief linked to symbols of stringency, precision, mathematical scientificity, proof, subdual, respectability and authority. Upon performing content analysis on the selected books, four categories of school demonstrations were set: (1) through experiments and particular cases; (2) through deductive reasoning with an exploratory character; (3) through formal elements of classical reasoning; (4) through particular cases, generalization and explanation. The analysis has shown changes regarding: the methodology for the development of a demonstration; the type of language applied; the way a procedure was introduced and concluded; the use of pictures and inductive, intuitive and visual procedures. In order to complement interpretations and understand the different demonstration fashions expressed in these categories, official documents were used, what made it possible to identify that the school demonstrations in the textbooks fulfill their role in preparing students for the understanding and future development of a formal demonstration, besides being aligned with the goal of developing deductive reasoning and approximation in forming a professional mathematician. Upon that, it can be understood that the formal demonstration is appreciated and motivated in curricular propositions which, despite guiding the use of different demonstration forms adapted to each teaching degree, seek to build a unique idea of demonstration.