Análisis didáctico de lecciones de libros de texto sobre proporcionalidad basado en el enfoque ontosemiótico. Implicaciones para la formación de profesores

Abstract

This doctoral thesis is structured around three focuses of interest in Mathematics Education: textbook analysis, teacher training and proportionality. After a review of the background in these areas of research (Chapter 1), two problems are clearly identified. On the one hand, the need for tools to guide the systematic analysis of lessons in primary and secondary mathematics textbooks, particularly those focusing on the study of proportionality. These tools must employ specific and objective criteria based on mathematical and didactil knowledge derived from research in mathematics education. On the other hand, the need to include in teacher training programmes the development of the competence for critical and reflective analysis of mathematics textbooks. The theoretical framework that underpins this research is the Ontosemiotic approach to knowledge and mathematical instruction. Within this framework, theoretical and methodological notions have been developed to address both problems, in particular, the theory of didactical suitability and the model of didactic-mathematical knowledge and competences of the teacher, which are explained in detail in Chapter 2 of this report. In relation to the first problem presented, Chapter 3 shows the process of developing two textbook analysis guides, one applicable to any mathematical content (GALTmathematics) and the other focused about proportionality (GALT-proportionality). Subsequently, Chapters 4 to 7 describe the design, implementation, and evaluation of several training cycles with future teachers (two aimed at future secondary school teachers and two aimed at future primary school teachers), which seek to promote their competence in the analysis of the didactical suitability of textbook lessons, using the instruments previously developed. The results show that, even when tools are available to guide them, the professional analysis of textbook lessons is a task that is not without difficulties for prospective teachers in both primary and secondary education. These difficulties appear to be connected to deficiencies in the didactic-mathematical knowledge of the participants, which is then reinforced by the training. However, it is observed that participants can overcome these limitations with appropriate instruction and that GALT-proportionality supports the development of the participants' reflective competence. It is also found that the future teachers have reached different levels of achievement of this competence, with better results in terms of learning in the second cycles of intervention. In this respect, it is worth mentioning that the retrospective analysis of the first cycles of intervention implemented allowed us to improve the instruments for the analysis of the textbook lessons (the GALT-proportionality) used in the following cycles. Specific training strategies are derived from our study, which, it is hoped, will serve to guide teacher educators in the complex task of characterising and promoting this competence in future teachers. Other contributions of this research are presented in detail in Chapter 8, together with a summary of the limitations encountered and future lines of research.