Niveles de algebrización en el razonamiento proporcional desde las perspectivas institucional y personal. Implicaciones para la formación de profesores de matemáticas

Abstract

In this thesis, after a systematic study of previous research on proportional and algebraic reasoning, we first, address the characterization of proportional reasoning and its relation to algebraic reasoning from an epistemic perspective. We analyze the various meanings of proportionality by applying theoretical tools of the Onto-semiotic Approach (OSA) of mathematical knowledge and instruction, in particular, the interpretation of meaning in terms of systems of operative and discursive practices related to solving types of problems, and the model of algebrization levels of mathematical activity. These meanings should be taken into account in the planning and management of the teaching and learning processes of proportionality in primary and secondary education. Basing on the OSA epistemological, ontological, semiotic and instructional assumptions, we justify the importance of a dialogical-collaborative didactic model for the situations of first encounter with mathematical objects in which the teacher and the students work together in the resolution of the problem situations. With these premises, in the second place, we carry out a case study on the design, implementation and evaluation of training interventions to develop proportional reasoning in a group of 23 primary school students. We analyze the results, in terms of the emergence of algebraic reasoning forms, obtained with the application of a didactic model of mixed dialogical-collaborative type. Although it is a case study that does not allow to generalize the results, the evaluation of the learning achieved, served to formulate hypotheses about the influence of the mixed instructional model on students’ learning, which should be checked in new research cycles on this topic and in similar contexts. Thirdly, in the specific area of mathematics teachers’ education, and applying the OSA model of Didactic-Mathematical Knowledge and Competencies, we develop some training cycles for prospective primary and secondary mathematics teachers, with the aim of developing the didactic-mathematical competences and knowledge about proportional reasoning and its connection with algebraic reasoning. In particular, we take into account the competencies of epistemic, cognitive and didactical suitability analysis. The results show the need to deepen in developing the specialized content knowledge of prospective teachers and recognize the complexity involved in developing these skills.