Função polinomial de primeiro grau: uma abordagem através de resolução de problemas

Abstract

This study aimed to investigate the implications of Problem Solving, articulated with the Theory of Registers of Semiotic Representation (TRSR), for the learning of the concept of First-Degree Polynomial Function by 10th-grade students from a public high school in the countryside of São Paulo state, Brazil. The investigation was guided by the following research question: What are the implications of using Problem Solving, supported by the Theory of Registers of Semiotic Representation, for the learning of the concept of First-Degree Polynomial Function? The study followed the assumptions of a qualitative approach and was structured according to the four phases of Didactic Engineering: preliminary analysis, a priori analysis, experimentation, and a posteriori analysis. The didactic sequence was organized into six interdependent sessions, grounded in the ten steps of Problem Solving, and composed of problem situations designed to mobilize and coordinate different registers of representation, such as natural language, algebraic, tabular, and graphical registers. This organization fostered the development of cognitive activities of formation, treatment, and conversion, enabling the analysis of transitions between registers and of phenomena of congruence and non-congruence. Sixteen students participated in the study, and the written productions generated throughout the didactic sequence constituted the corpus of analysis. The results indicated that, initially, students mobilized the registers of representation in a fragmented way; however, over the course of the didactic sequence, they showed progress in the coordination between different representations. Data analysis showed that the articulation between Problem Solving and the Theory of Registers of Semiotic Representation constitutes a consistent theoretical-methodological framework for understanding the processes of construction of mathematical knowledge. It is concluded that this articulation fostered the understanding of the First-Degree Polynomial Function as a model of variation and as an instrument for analyzing contextualized situations, promoting the development of more elaborated forms of mathematical thinking.