Indução matemática na prova de fórmulas combinatórias: dos livros didáticos à sala de aula

Abstract

Mathematical induction is an extremely important and useful proof technique in establishing results in various areas of mathematics, such as number theory, combinatorics, geometry, and others, However, currently, this technique is not widely taught or explored in the context of high school education. Given this, the central objective of this work is to propose the use of mathematical induction at this level of education, specifically in the study of combinatorial analysis (Newton's binomial theorem and Pascal's triangle). To achieve this objective, we initially selected four collections of high school textbooks and conducted a critical analysis of how these books address the mentioned concepts. We deliberately chose collections from different decades to highlight the change in approach over a thirty-year period. We concluded the work by presenting a didactic sequence that allows teachers to use mathematical induction in the classroom to prove combinatorial formulas (Newton's binomial theorem and Pascal's triangle theorems), aiming to facilitate the understanding of the mentioned combinatorial content for secund year high school students. The approach adopted in our didactic sequence contrasts with the approaches traditionally found in high the analyzed school textbooks. Our work is intended for elementary school teachers and other interested individuals.