Investigando o pensamento algébrico à luz da teoria dos campos conceituais

Abstract

This master's thesis was developed within the scope of the Programa de Pós Graduação em Educação Matemática (PPGEMAT) at the Universidade Federal de Pelotas (UFPel). This qualitative research sought to investigate algebraic thinking from the perspective of the Theory of Conceptual Fields (TCF). The question that guided the research was: how does the manifestation of algebraic thinking occur in solving problem situations that that explore relationships and comparisons between patterns delimiting part of the Algebraic Conceptual Field (ACF)? Based on the delimitation of part of the ACF by a set of situations that aimed at establishing relationships and comparisons between numerical expressions and geometric patterns, the objective was to identify the manifestations of operational invariants and representations of the ACF in the solutions of a set of situations explored by eleven students from the 8th year of elementary school in the city of Canguçu/Rio Grande do Sul. It also sought to identify whether and how the processes of identification and expression of regularities and invariance and the expression of generalization would happen in these solutions. The students were invited to develop strategies to solve nine exploratory investigative situations. It was possible to identify the representative solutions of the class through the analysis of the representations of the solutions. The discussion about the expected and divergent representative solutions allowed to identify the components of the schemes used by the students. Thus, it was possible to establish relationship between the TCF triplet (sets of situations, invariants and representations) with the manifestation of algebraic thinking and the development of algebraic language. The manifestation of students' algebraic thinking was identified from the representations in the solutions of the activities in which operative invariants were found. Analyzing the solutions of the activities, it was identified that the students understand algebraic notions, however, they still have difficulties to represent them in formal mathematical language. Only in the last activity, students used algebraic writing as a representation and only one solution was in accordance with the expected response. Finally, perspectives of continuity of the research are presented, such as expanding the sets of situations to other parts of the CCA.