Condições e restrições sobre uso de atividades de estudo e pesquisa relacionadas a frações no 6º ano do ensino fundamental

Abstract

The objective of this dissertation was to analyze conditions and constraints on the use of Study and Research Activities (SRA) involving fractions associated with the measurements of perimeter and area in a 6th-grade class of Elementary School. By taking the Anthropological Theory of the Didactic (ATD) as a reference, we sought to elaborate a sequence of five activities based on studies conducted by Silva focused on the object of fractions, using models respective to the three dimensions of the didactic problem. In these studies, it is stated that classroom approaches to this object focus on the part-whole conception idea, using the technique of double counting of parts, mainly on the aspect of "know-how" to the detriment of developing an understanding that justifies the use of such techniques. This results in a lack of depth by students to understand this mathematical object. To this end, we framed the work as qualitative research, whose data collection process involved bibliographic and documentary study on fractions, as well as including elements of exploratory research, with pedagogical intervention. For this data collection process, we followed the methodology used by ATD, which has four phases. The first phase referred to a preliminary analysis from an epistemological study of the mathematical object in question, aiming to identify the conditions and constraints on this same object, through the three dimensions of the didactic problem: epistemological dimension (Epistemological Reference Model – ERM – referred to the study of the reason for being and existence of the mathematical object in play; economic dimension (Dominant Epistemological Model – DEM, focused on the study of curriculum documents) and ecological dimension (Praxological Reference Model – PRM, also including an analysis of 6th-grade mathematics textbooks approved by the National Plan of Textbooks and Teaching Materials – PNLD). The second phase corresponded to an a priori analysis for the pedagogical intervention, at which point we developed the Study and Research Activities (SRA) based on the studies conducted in the first phase and the development of the chronogenesis, topogenesis, and mesogenesis dialectics, which were also part of studies by ATD. The third phase focused on in vivo experimentation and analysis of the SRA, being applied in a federal school in the state of Sergipe – the UFS Application School, whose data collection involved video recordings and student notes. The final phase emphasized the a posteriori analysis, in comparison to the a priori analysis. Through the aforementioned dialectics, predicted issues were compared with those evoked during experimentation. Thus, when applying the SRA, we aimed to promote a study of fractions related to conceptions of measurement, distribution, and comparison, which can be constructed through five relations: part-whole, measurement, quotient, ratio, and operator. In this application, we identified conditions and constraints affecting the use of the SRA related to the participating students' difficulty in formulating questions, due to not being accustomed to an investigation-based teaching model, and the difficulties in transferring responsibilities between these students and the applying teacher