Análise combinatória: fundamentos teóricos e propostas didáticas para o ensino das técnicas de contagem na Educação Básica

Abstract

The transition from direct counting to structured combinatorial reasoning represents a milestone in the development of mathematical thinking. However, the traditional didactic approach in Basic Education often reduces Combinatorics to the mechanical application of formulas, leading to learning difficulties and recurring student errors. Given this scenario, the main objective of this dissertation is to present the theoretical foundations of Combinatorics and to structure didactic proposals applicable to Basic Education, aiming to improve teaching practices and to strengthen student learning. The theoretical framework is developed across two chapters: the first addresses the Fundamental Methods of Counting (Additive and Multiplicative Principles, simple groupings, circular permutations, permutations with repetition, and combinations with repetition); the second explores Advanced Counting Approaches (Principle of Inclusion-Exclusion, Derangements, Kaplansky’s Lemmas, Reflection Principle, Pascal’s Triangle, Binomial Theorem, and Leibniz’s Multinomial Theorem). Finally, the third chapter presents the Educational Product: an educational booklet featuring inquiry-based learning sequences aligned with the BNCC, progressively covering the early and final years of Elementary School through High School, employing resources ranging from concrete manipulatives to path counting on rectangular grids (lattice paths).