Impactos da combinatória no contexto da formação inicial de professores de matemática na prática profissional docente

Abstract

Motivated by my personal-academic trajectory, the small number of studies on combinatorics over the last decade, especially with regard to the initial and ongoing teacher education, and the concern with emphasizing a combinatorics education that prioritizes combinatorial reasoning, this work aims to understand how the initial training of mathematics teachers can impact them, specifically regarding the teaching of combinatorics in the context of basic education. To achieve this, through a qualitative investigation, we conducted semi-structured interviews with teachers working in basic education, exploring the validity (or lack thereof) of non-standard solutions to counting problems and aspects of their initial teacher education, based on a few specific criteria for selecting the research subjects. These criteria were derived from a pilot study conducted with one teacher, with the main criterion being the length of time since graduation – focusing on those who had not been in the field for too long. The methodological approach of this study was divided into three stages: selecting the teachers, administering a questionnaire to the teachers, and conducting the semi-structured interviews. For data analysis, we applied Content Analysis (BARDIN, 2004), identifying three main thematic axes: University, Teachers, and School. Each axis contains categories that we built from the units of registration found in the full transcripts of the interviews. As a result, we concluded that there is a direct (and negative) impact of initial teacher training on the knowledge of combinatorics and on the professional practice of graduates from the respective universities. The teachers did not reference their undergraduate training when planning combinatorics lessons; instead, their own teaching practice was the primary reference, as well as textbooks, teaching manuals, and the exchange of knowledge with fellow teachers. We point to shared teaching as a way to strengthen the link between university and school, and the need to establish a compulsory combinatorics course that values the development of combinatorial reasoning at the university level. From the school perspective, we indicate the need to use problematized approaches in teaching, meaning mathematics teaching that does not rely on "formula-application" or rote learning, but rather on the construction of combinatorial properties and concepts, which should begin in elementary education. We also recommend the use of Error Analysis methodology as a potential tool for implementation in basic education, which could contribute to the teaching learning process. Additionally, we identified the need to record solutions to combinatorics counting problems in a way that allows for the recognition of the intrinsic combinatorial reasoning within the problem, rather than focusing solely on numerical solutions. It is hoped that this work will contribute to mathematics educators and teachers, connecting the mathematical content of combinatorics with the professional development of teachers who teach mathematics, two central axes of this research.