Conhecimento matemático especializado de futuros professores de matemática em um contexto formativo no âmbito de proporcionalidade

Abstract

Proportionality is a central concept in Mathematics due to its wide presence across different fields of the discipline, its applications in other sciences, and its recurrence in everyday situations. Despite its relevance, research has identified persistent difficulties in the teaching and learning of this topic, spanning from Basic Education to initial teacher education, particularly regarding the understanding of the different classes of problems that mobilize proportional reasoning. Considering the decisive role of the teacher in this process, this study aimed to investigate the specialized mathematical knowledge revealed and developed by prospective mathematics teachers from a public university in Amazonas, Brazil, within a teacher education context focused on proportionality. The investigation was guided by the theoretical-analytical model Mathematics Teachers’ Specialized Knowledge (MTSK), created by Carrillo and collaborators, with emphasis on the KoT subdomain, and the educational context was structured around tasks involving distinct classes of proportional reasoning problems, developed according to the phases of action, formulation, validation, and institutionalization proposed by the Theory of Didactical Situations, by Brousseau. The study adopted a qualitative approach, designed as an instrumental case study, and the data were analyzed through Content Analysis in light of the MTSK categories. The results indicate that the prospective teachers mobilized relevant knowledge related to proportionality; however, this knowledge was uneven and exhibited varying levels of formalization, articulation, and generalization. An initial predominance of procedural strategies, such as the rule of three, was observed, which gradually evolved into approaches requiring greater conceptual understanding, including scalar and functional strategies and the search for the unit rate, as well as more substantiated justifications concerning the existence of proportional relationships. Nevertheless, difficulties persisted in using algebraic expressions as problem-solving tools and in appropriating classes of problems as organizing conceptual categories of proportional reasoning. Although the institutionalization phase supported the recognition of structural characteristics of the proposed situations, its limitations became evident with regard to conceptual generalization and the autonomous production of new problems. It is concluded that the development of specialized mathematical knowledge on proportionality in initial teacher education does not result merely from the use of diversified tasks, but requires educational contexts structured by an explicit intentionality to reveal, problematize, and develop such knowledge. This process demands systematic institutionalization that makes explicit the underlying mathematical structures and fosters awareness of the categories that organize proportional reasoning.