Mobilização de conceitos de análise combinatória à luz da Teoria das Situações Didáticas

Abstract

Combinatorial Analysis plays a significant role in mathematics and, despite being full of problems capable of motivating students, it is considered a complicated subject, in which students have difficulty finding the correct way to solve each problem. Students have difficulty correctly identifying the count of possibilities in a combinatorics problem. This can result in incorrect answers and frustration on the part of students. The Theory of Didactic Situations (TSD) can be a valuable strategy for teaching Combinatorial Analysis concepts and for developing students' Combinatorial Reasoning. In this sense, the use of Didactic Sequences, based on TSD, can be an effective strategy to improve learning and overcome the difficulties faced by students. We defined the following guiding question: how do Didactic Sequences mobilize Combinatorial Analysis concepts in higher education undergraduate students? To achieve this, our general objective is to analyze the mobilization of Combinatorial Analysis concepts through didactic sequences in higher education students. Our theoretical framework is based on TSD and the methodological framework used is Michelè Artigue's Didactic Engineering. Taking a qualitative approach, the study was conducted in a discipline of a Science Degree course: Mathematics and Physics at the Federal University of Amazonas (UFAM), where a priori and a posteriori analyzes were compared, in addition to identifying and describing levels of combinatorial reasoning of a sample of six students. This approach has proven to be an effective pedagogical strategy for student learning. The collaborative construction of knowledge among the participants was observed, as well as the enrichment of the researcher, who simultaneously explores educational objectives and acquires learning through his practice. During the application of the didactic sequence, one can observe the mobilization of Permutation concepts, in addition to concepts such as the Fundamental Principle of Counting and Factorial. A common difficulty was noted, which was the difficulty in correctly applying permutation concepts with repetition. The practical application of the didactic sequences proved to be effective, as it allowed students, even those with limited prior knowledge in Combinatorial Analysis, to reach the correct solutions or adjust their reasoning after the teaching intervention. Furthermore, it promoted the socialization and collaborative development of combinatorial reasoning, proving to be a valuable pedagogical strategy not only for Higher Education, but also with potential for adaptation for Secondary Education.