Panorama das situações relacionadas à função afim em teses e dissertações brasileiras

Abstract

Knowing the typology of situations related to the affine function, the contexts in which they are located, the main didactic variables present, the several ways of presenting their statements and the different methodological approaches indicated, can favor the determination of a set of situations that enable the construction of this mathematical concept by students. Therefore, the overall objective of this research was to analyze situations related to the affine function present in Brazilian master's dissertations and doctoral theses, using Gérard Vergnaud's Theory of Conceptual Fields as a theoretical framework. The investigation, of theoretical and documentary nature, evaluated all the available documents in the Brazilian Digital Library of Theses and Dissertations and in the Catalog of Theses and Dissertations, of which 479 dissertations and 109 theses met the defined search criteria. The selection criterion required that the situations belonged to didactic sequences developed in the classroom, reflecting everyday contexts beyond Mathematics. The final corpus consisted of sixty-six documents, sixty-three dissertations and three theses, from national postgraduate programs defended between 2007 and 2022. The research was categorized into thematic focuses, predominating the researchers' interest in the investigation of methodological proposals to favor the construction and consolidation of the concept of affine function and its applications. The regional distribution shows a greater concentration of studies in the southern and southeastern regions of Brazil. There were analyzed and categorized 1,140 situations considering the Theory of Conceptual Fields and, regarding the typology of situations, the presence of twelve distinct classes was found, with absolute primacy of classes of simple proportion and composition of measures (mixed structure) and simple proportion (multiplicative structure). It was also identified the presence of situations in which a structure is duplicated, such as: double simple proportion and double composition of measures; simple proportion and double composition of measures; simple proportion and double transformation of measures (mixed structure); double simple proportion; simple proportion and multiplicative comparison (multiplicative structure). It was shown that situations belonging to the classes of multiplicative structures product of measures, bilinear function and multiple proportion are incompatible with affine and linear functions. The following didactic variables of the situations were identified: their typology, the presentation of the statements and the numerical set used in the data. The predominant value of the statement presentation variable is natural language, although graphs and tables were also observed. For the numerical set variable, the natural numbers are the most common, and for the context variable, the cost of products and services is the most frequent. These results offer contributions to educators and researchers, highlighting the diversity of relational calculation structures, didactic variables, and methodological approaches to contribute to teaching action in the construction and consolidation of this concept by students.