El conocimiento del formador de maestros en la etapa de formación inicial, en relación con la enseñanza de la Didáctica de las Matemáticas. Un estudio de caso

Abstract

The studies on the knowledge necessary to teach mathematics teachers constitute an underdeveloped line of research, although an emerging one. Our research objective, to collaborate in the advances in this line, is to understand and characterize the mathematical and pedagogical knowledge mobilized by a teacher educator during the teaching of the content of initial training and during his reflections on teaching. We are facing a research problem that is necessary, due to its limited presence in the research literature; relevant, due to the importance of the trainer in the teacher training processes; and, applicable, from the point of view of training and selection of teacher educators. The study of the teacher educator's knowledge becomes complex if we take into account the variety of professional profiles and the heterogeneity of initial teacher training programs. We have selected a case in which the teacher educator, also a researcher in Mathematics Didactics, develops a subject of a mostly cognitive nature in which content of a different nature is combined, related to: the specialized knowledge of the mathematics teacher, the professional practices of teaching and the construction of professional identity. The research background on the knowledge of mathematics teacher educators (MTE) is based on the differentiation between content knowledge and pedagogical content knowledge applied to the study of the teacher's knowledge. The relationships between both knowledge, mean that in some investigations the teacher's knowledge has been interpreted as part of the teacher educator's knowledge. However, the idiosyncratic characteristics of teacher training mean that the translation of professional teacher knowledge models is not straightforward. In initial training, prospective teachers (PT) have to reconstruct a mathematics already learned and understand its foundations to turn it into a teaching object in their future teaching practice. The professionalizing nature of the stage also involves considering the learning of teaching practices such as the design of tasks, the interpretation of the thinking of Primary students and the teacher's decision-making in this regard; and the development of the teacher's professional identity and the feeling of belonging to the teaching community of practice. Thinking about the knowledge of the MTE to develop these knowledge, skills and attitudes in their PTs, is the starting point for the conceptualization of a model of knowledge of the trainer and constitutes the first indication of the differences in its structure with respect to the knowledge of the teacher. We have followed a methodological approach according to the top-down, bottom-up cycle with special attention to the emergence of data, from the Grounded Theory budgets. To obtain the information, non-participant observation and interview techniques have been used, and for data analysis, we have transcribed the observed episodes to analyse them from the point of view of content analysis, validating the evidence using a multi methodological approach and through triangulation between experts. The analysis has been structured in two stages, chronological and transversal to understand the evidence of knowledge within the organized whole that is the knowledge of MTE. The results of this research have allowed us to understand in depth the mathematical knowledge of the trainer, which is well-founded, related and decisive in the construction of the training sequence. We have also found evidence of knowledge that allows us to understand the multidimensional and integrated nature of the MTE's knowledge, which attends to each of the pillars of the content of the training that we have recorded in the analysis. Finally, we have managed to identify categories of knowledge of the trainer related to how to teach the content of initial training that include: (i) knowledge of the trainer about teaching tasks - reconstruction of mathematical content, didactic isomorphism and analysis of difficulties in Primary Education -, (ii) knowledge of the trainer about the learning characteristics of their EPM -about learning difficulties based on the understanding of the mathematical content- and, (iii) knowledge of the trainer about the design and organization of initial training programs.