The present study has the theme of Functional Teaching and as an object of investigation the teaching and learning difficulties of the subject, having as a perspective the indication of possibilities of action from the looks of scholars, authors of books, teachers and students , From the perspective of the researcher and her experience teaching mathematics. My intention is to analyze the theme, seeking to understand its historical process of construction, analyze books produced and adopted in schools, listen to teachers who work with role teaching and students of these teachers, so that their experiences, wishes, doubts, difficulties and expectations Can also be considered, all aiming to identify difficulties, possibilities and trends, in terms of teaching function to, from there, define principles and guidelines for Teaching Role and this be translated in the form of a teaching product to be presented As a teaching proposal of the theme Function, more specifically, Function Afim. To this end, after a brief description of my training trajectory, I propose a reflexive study on the historical construction of the theme Function, on teaching proposals of Function present in textbooks, results of studies on the subject and on practices of teachers from their Considering the problem of research that concerns the teaching and learning difficulties in the middle level related to the topic Function and I finish by establishing principles for the Teaching of Function and materializing these principles in Cadernos for the Teaching of Function, destined to teachers who work with the theme. The qualitative research is based on Garnica (2004) and was defined in two stages, with the first one, with a bibliographical character, being configured as a diagnostic, descriptive character and performed in three moments, being finalized through a study of the Type of state, based on Fiorentini and Lorenzato (2006). In the second stage, the research is configured in the narrative mode, using the comprehensive interview, from Kaufmann's (1996) perspective, in describing them, according to Moraes & Galiazzi (2007) and in their analysis, from emerging categories Denominated by Moraes (2003). The two stages allowed the identification of principles for the teaching of Function, which are: The teaching and learning process of the topic Function should have as a first reference the previous knowledge of the students in relation to the subject; Understanding must precede formalization; The proposition of introductory activities to the Teaching of Function should have as reference the reality of the students; Students' participation in their learning is a necessary condition in order to make it a personal process in which learning becomes mathematical; The approach to activities must take place in a broad way and allow the connection with other areas of knowledge and diversity in the adoption of Methodological Trends should be a characteristic of the Pedagogical Practice. The principles were then taken as the main references for the elaboration of a proposal of teaching of Function called Cadernos for the Teaching of Function, destined to teachers that act with the teaching of the subject, in which the conceptual understanding of Function , Based on daily activities, on the gradual formalization of the theme and on the consolidation of learning.
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