Progressões aritméticas e geométricas : praxeologias em livros didáticos de matemática - Almeida, Eliane Aparecida Martins de (2012)
This research aims to investigate how textbooks propose the study of arithmetic and geometric progressions in the first year of High School. The textbooks that aggregate to this research belong to the first year of High School, which have been selected because they are part of the catalogue of the Textbook Nacional Program for High School, and also because they are used in State schools of Cuiabá. A qualitative research has been used as the methodology of choice, with emphasis on document analysis. The studies of documents of textbooks are based primarily on the Anthropological Theory of Didactics (TAD), more specifically on the praxeologies, proposed by Chevallard (1999) and on the Square Games Theory of Douady (1992). In order to conduct the studies, praxeologies found in textbooks were identified, as well as the tasks, techniques, theoretical-technological discussions and numerical tables, both geometric and algebraic, with attention to the historical aspects concerning progressions, the conceptual part, the tasks addressed by the authors, and the ones proposed to the students. From this, such praxeologies were verified to see if they match the Official Documents proposals and the intentions of authors shown in the manuals. Among the recommendations of the Official Documents, the study highlights the articulation of arithmetic and geometric progressions with the content functions, as well as tasks proposals that prompt the general standards for the development of algebraic thinking. The results of this investigation, associated with the relevant theory, made the diagnosis of textbooks possible, pointing that praxeological organizations of two of the books analyzed did not explain the close relationship between progressions and functions. This articulation is contemplated in both of the books investigated in the conceptual part, although such relationship has not been mentioned in the intentions shown in the manuals. It has also been noticed of the books selected for this study do not frequently propose tasks to stimulate the generalization of patterns, even if this intention is explicit by the authors of four books selected for the study. There is still the predominance of tasks of the type calculate, determine, constituting imitation tasks, which can lead to routinization of the technique. Among other aspects, a limited number of tasks has been identified, which can facilitate the articulation of different frames (numerical, algebraic and geometric) as proposed by Douady (1992). In most cases priority to the algebraic framework. Amongst the books selected for the studies, it has been noticed that none of them is “complete”, with authors that propose a study that contributes effectively to the development of algebraic thinking in the study of progressions. They do not contemplate the recommendations of the Official Documents, and even if they did, it would not be complete. Thus, it is for the teachers to select the book that they consider most appropriate, according to their opinion and according to the reality of their students.