Uma matemática para o ensino do conceito de derivada

Abstract

In this study, we did not set a general objective. Instead, we proposed a macro research project based on independent objectives, which resulted in three distinct but interconnected studies that sought to capture the discursive variability emerging from the different repertoires mobilized in the communication of the concept in question. The research was developed based on the qualitative approach and, as a methodological procedure for the first study, we used the ideas of the Concept Study to understand the different uses of the concept of derivative. This approach, combined with Anna Sfard's theoretical definitions, was used as a tool for data analysis and production. The first study aimed to structure and synthesize the different ways of communicating the concept of derivative, identified in the scientific articles that make up the corpus, with the intention of constructing a Theoretical Model of Mathematics for Teaching this concept. To compose the research corpus, we selected articles published in journals in the area of Mathematics Education, in which the identified achievements were categorized into six panoramas: Formal definition, Graphics, Tangent line to a curve, Derivative function, Derivation rules and Applications of derivatives. Data production for the second and third studies occurred during fourteen hours of class time in a class composed of twelve students enrolled in the first Calculus course, and was recorded through filming, photographs and a field diary. For data analysis, we used the Theoretical Model of Mathematics for Teaching the Concept of Derivative. The second study aimed to identify and characterize the achievements mobilized by students in the transition between the panoramas of the aforementioned Theoretical Model. After analyzing the material produced during the execution of the tasks, we concluded that the transitions occurred for the purposes of Confirmation, Verification and Instrumental. The third study was conducted with the objective of identifying and categorizing the cognitive conflicts observed in the way students transition between the different panoramas of the Theoretical Model. After analysis, we identified five types of conflicts: Definition, Representational, Strategy, Linguistic and Contextual. The results obtained provide a detailed view of the concept of derivative, and can contribute as a reference for new research in the field of Mathematics Education, in addition to assisting in teacher training and the production of educational materials related to teaching.