Resolução de problemas e história da matemática no ensino de equações polinomiais de 1° grau: uma sequência didática investigativa aplicada no ensino médio

Abstract

The teaching of Algebra, particularly first-degree polynomial equations, still faces challenges associated with instructional practices centered on the memorization of rules and the mechanical application of procedures, which hinder students’ conceptual understanding and their ability to attribute meaning to mathematical content. In light of this reality, this study was guided by the following research question: How does the articulation between the History of Mathematics and Problem Solving, developed through an Investigative Didactic Sequence (IDS), contribute to the teaching and learning process of first-degree polynomial equations? The general objective of this study was to analyze the contributions of this articulation to the teaching and learning of first-degree polynomial equations through an IDS. This objective was further developed into the following specific objectives: to investigate the contributions of the History of Mathematics to mathematics teaching; to understand the Problem-Solving methodology in the teaching and learning process; to design an IDS that integrated these two theoretical frameworks; and to analyze the contributions of this IDS to the teaching and learning process. This research is characterized as a qualitative study of an interpretative nature, grounded in a bibliographic review on the History of Mathematics, Problem Solving, and Investigative Didactic Sequences. The IDS developed, entitled From History to Problem Solving: Exploring First-Degree Polynomial Equations, was organized into six lessons and implemented in a third-year high school class at a public state school in São Luís, Maranhão, Brazil, linked to the Institutional Scholarship Program for Teaching Initiation (PIBID). Data were produced through participant observation, field notes, students’ written productions, and photographic records, which were analyzed qualitatively in light of the adopted theoretical framework. The results indicated that the introduction of elements from the History of Mathematics, such as rhetorical algebra, the method of false position, and Al-Khwarizmi’s concepts of al-jabr and al-muqabala, contributed to demystifying Algebra, fostering its understanding as a human construct developed throughout history. Simultaneously, the incorporation of George Pólya’s problem-solving stages promoted the development of students’ intellectual autonomy, as they began to employ different solution strategies and validate their answers according to the conditions presented in the proposed problems. However, difficulties related to reading and interpreting problem statements were observed, as well as limited spontaneous use of the retrospective stage of Problem Solving, revealing constraints of the proposal due to the limited time available for its implementation. It is concluded that the articulation between the History of Mathematics and Problem Solving contributed to the construction of meaning, the development of mathematical reasoning, and the active participation of students in the teaching and learning process of first-degree polynomial equations.