Conocimiento especializado del profesor de matemáticas acerca del infinitoun estudio de caso

Abstract

This research tackles the special kind of knowledge that a secondary mathematics teacher can use, in order to achieve its identification, characterization and categorization. For this purpose, we rely on the �Mathematics Teachers� Specialized Knowledge'-MTSK- model, developed together with this research. Thus, we endow content to each one of the subdomains proposed by the model, particularizing in the case of the knowledge that teachers� posses about infinity to teach mathematics. The methodology used is of qualitative kind, following the methodological approach of the Grounded Theory, being framed in a interpretative paradigm, with a methodology of case study. The methodological tools developed and used to obtain the data were: - An exploratory questionnaire, that tacked beliefs about teaching and learning mathematics, beliefs about the nature of mathematics, and a first approach to the mathematical idea of infinity. An interview, carried out in several sessions, that allowed us to engage questions about infinity as a mathematical concept, as epistemológica! base of other mathematical topics, and as element about which the teacher was instated to develop both mathematical reflections, as linked to the teaching and learning of mathematics where it could be involved. We attended the classes of the Teacher object of the case study to add some elements of discussions to the interview. The data gathered were analyzed several times, from different approaches to get a higher rate of triangularon. This analysis allowed the appearance of different categories: Localization of course; Thinking and actions of the students; Language; Explication in the classroom and teaching examples; Phenomenology; Cognitive development of infinity; Conflicts-Cognitive Dissonance; Conceptual mistakes; infinity meanings. These categories were exhibited in the data analysis, showing fragments of the data that showed their content. In the chapter of result discussion and conclusions, we approach to the different subdomains of the teacher knowledge model MTSK, developing a discussion about the content of each one in relation with infinity, establishing the relationship between subdomains and the arising categories, and also between the the subdomains themselves, due to the transversal nature of the arising categories to them. With this relations established, we deepened in the nature of the knowledge of infinity showed by the teacher, proposing fuzzy generalizations (in coherence with the methodological approach of Grounded Theory) about the knowledge that a teacher could use. The conclusions of this research show the complex nature of teachers� knowledge, particularized on the case of infinity. This conclusions are based in the assumption of the �knowledge1 concept in the sense of Schoenfeld (2010), being knowledge the 'information available to use', which allows us to do a flexible discussion about the meaning of �knowing infinity as a teacher�, going beyond the purely formal mathematics considerations. Also, the theoretical sensitivity developed in this research allowed us to establish some reflections about infinity and teacher knowledge, as both independent parts of the object of study.