El conocimiento especializado de un profesor de matemáticas, cuando enseña sucesiones numéricasrelaciones entre subdominios en la selección y uso de ejemplos

Abstract

This paper describes the process we undertook to understand how the different knowledge mobilised by a teacher when selecting and using different types of examples to teach numerical sequences in a third year secondary school class was related. Our research is framed within the interpretative paradigm and corresponds to an instrumental case study. The pillars that support it are (i) the teacher's professional knowledge, the MTSK as an instrument for analysing specialised knowledge, (ii) examples as a didactic tool for teaching and learning mathematics and (iii) numerical sequences and their approach at secondary school level. Two techniques were used to obtain the information: classroom observation and interview. Through classroom observation we entered a secondary education classroom, without generating any interaction with the students or the teacher, only to record, by means of audio and video recordings, and observe the work carried out by the teacher in each of the classes. The interview was used to clarify the different doubts that emerged when analysing what was observed in class. The filming of the classes and the interview were transcribed verbatim. The analysis of the information was carried out on the transcripts of the different moments in which the use of an example was identified and, where appropriate, the relevant extract from the interview transcript was considered. By analysing these transcripts, we sought to obtain evidence of the knowledge mobilised by the teacher, and then to determine the relationships generated between this knowledge during the selection and use of the examples. In presenting the results, we begin with a description of the knowledge mobilised by the teacher, by knowledge subdomain of the MTSK model, when approaching the teaching of sequences by means of examples. This is followed by a description of the relationships we observed during the selection and use of examples, distinguishing between passive and active examples and teaching- and practice-focused examples. In conclusion, while exemplification itself is recognised as a favourable setting for the study of MTSK, there are times and types of examples that are more favourable than others. Studying the MTSK that determines the selection of an example allows us to identify a greater number and variety of relationships between subdomains, compared to what occurs when analysing the development of an example. And in the case of the different types of examples, we observe that active examples and those focused on teaching are the ones that allow us to identify a greater number and variety of relationships between subdomains. For their part, beliefs influence both the selection and use of different types of examples, determining the subdomains of knowledge that are mobilised in each of these instances and the way in which they are related.