Introducing Counting Concepts through Problem Solving:

A Teaching Experience with 8th Grade

Abstract

The Fundamental Principle of Counting (FPC) is an essential concept in Discrete Mathematics, with broad applications in everyday life — from organizing menu combinations and arranging clothing to calculating statistical probabilities. However, many students encounter difficulties in understanding this principle, which can impact their performance in more advanced subjects. Thus, this experience report aims to discuss the initial potential for constructing the FPC concept through the Problem-Solving Methodology, going beyond the mere mechanical and uncritical application of algorithms. To achieve this goal, a qualitative, participatory research approach was adopted, developed with a class of 32 8th-grade students at a public elementary school in Brejo da Madre de Deus, Pernambuco, Brazil. Data analysis focused on the solutions provided to a generating problem involving the creation of name tags with specific restrictions regarding numbers and letters. The results revealed a variety of strategies, ranging from systematic enumeration to intuitive generalization of the FPC, as well as difficulties in communicating reasoning and maintaining logical organization. The findings suggest that this approach fostered meaningful conceptual understanding, going beyond the mechanical use of formulas. Thus, problem solving proved to be an effective pathway for developing intellectual autonomy and combinatorial reasoning, although it requires teacher mediation that is attentive to errors as learning opportunities.