Praxeologias e modelos praxeológicos institucionais: o caso da álgebra linear

Abstract

This research deals with our concern to teach the Linear Algebra discipline in a Mathematics Degree Course, due to the students' difficulty in understanding the objects studied. The objects linear systems, matrices, vector spaces, vector subspaces, linear combinations, base and dimension were studied from the qualitative study of linear systems, since our idea was to make content less abstraction. National and international studies have already approached the subject using theories, others proposing ways that demonstrate the actions and functionalities of their approaches. Thus, this paper answers the following questions: What characteristics do mathematical and didactic organizations assume as institutional praxeologies, referring to the teaching of Linear Algebra and What conditions can we establish in a higher institution to make certain mathematical and didactic organizations with specific characteristics? In response, we propose an epistemological model of reference, composed of a task system constituted from a historical epistemological study in original works, which we had access, besides the constitution of a course of study and research, that was used as teaching methodology In a course of Mathematics undergraduates, lasting for 5 months between the years 2014 and 2015. The theoretical basis was based on the Anthropological Theory of Didactics. Starting from a concrete problem, we compared our model proposal with textbook adopted to teach the discipline under analysis and use the categories of analysis supported in the Theory, in addition to comparing the model with the text of the knowledge of a teacher who minister this Discipline in the course of Mathematics. In our analyzes we verified that the mathematical object linear systems comes as an appendix at the end of the book, and the tasks are not articulated with arrays, vector spaces and sub spaces, but that linear combinations is a technology that justifies the study of linear dependencies and independence , Base and size of space. The dominant epistemological model of the textbook analyzed and the teacher was to present the definition, with direct applications in the tasks. In order to develop the study, we created three didactic systems: the first was the creation of the mathematical and didactic organization of the reference model, the second was the study course with 14 undergraduate students and the last one was the mathematical and didactic organization presented through Of the groups on Linear Algebra objects studied. The study course proved that studying linear systems is to study Linear Algebra itself. From the study, our thesis was based on a proposal for an Epistemological Model of Reference on Linear Algebra, focused on basic education, with a direct impact on teacher training, becoming an alternative epistemological model for the undergraduate course In Mathematics of the Federal Institute of Pará.