Generalización y mediación en primeros cursos de educación primaria en un contexto de pensamiento funcional como aproximación al pensamiento algebraico

Résumé

This document constitutes the author's Doctoral Thesis, with the aim of obtaining the degree of Doctor with International Mention in the Education Sciences Program of the University of Granada, academic year 2023/2024. This research was initiated during the academic year 2020/2021 after finishing the Master in Didactics of Mathematics and is framed in two R&D research projects, funded by the Spanish State Research Agency. In this work we address algebraic thinking, which has been a relevant research topic for Didactics of Mathematics in the last decades. Specifically, we focus on functional thinking, a key component of algebraic thinking. Functional thinking is considered an appropriate option to address algebraic thinking due to its relevance in the application of fundamental practices of the algebraic domain (Blanton et al., 2015). Functional thinking focuses on the relationship between two variables, being fundamental the study of regularities and, in particular, generalization (Blanton, 2008). However, working with this algebraic component can be challenging, especially when students have not previously worked on generalization. Teacher or researcher-teacher mediation has proven to be an effective strategy to address these difficulties and facilitate the generalization process. The general research objective of this Doctoral Thesis is to analyze and describe the generalization process in elementary school students and to examine the mediations employed during their work in this context. In order to fulfill this objective, we set out four specific objectives: (a) to characterize the generalization process of students in the first years of elementary school; (b) to describe the relationships between students' generalizations and the mediations performed by a researcher-teacher; (c) to characterize students' justifications when solving a generalization task; and (d) to describe students' reasoning when solving a generalization task. Our doctoral dissertation is composed of four research studies. The first one is a bibliometric review on algebraic thinking, allowing us to characterize the scientific production on this topic. The other three studies focused on answering the specific research objectives, using data obtained from a teaching experiment in a school in Granada, with students in the second and fourth grades of primary education. The results obtained in relation to our specific objectives were significant. In the first specific objective, when characterizing the generalization process, we observed that regardless of the level, students showed generalization for indeterminate quantities and general cases, even without previous knowledge of algebraic symbology or concepts of indeterminacy (infinity, many, etc.). These finding highlights students innate ability to establish relationships between quantities and discover patterns beyond concrete cases, extending our understanding of their generalization abilities. In the second objective, we identified specific mediations that were effective in supporting this process. Evidencing how the researcher-teacher interaction influenced the quality of the generalizations. Regarding the third objective, we explored students' justifications during generalization, highlighting their importance in the formulation of arguments and the relationship between students' justifications and the generalization process. Finally, in the fourth objective, by describing students' reasoning in generalization tasks, focusing on the abduction, induction, and generalization phases, we identify the importance of collective responses in formulating and confirming the task structure, facilitating the generalization process. In addition to common patterns in students' reasoning, in particular the formulation and maintenance of initial conjectures during abduction and subsequent refinement through induction. In our conclusions, we highlight the generalization ability of elementary students, where the mediation of the researcher-teacher had a remarkable role within the generalization process. In addition, we highlight the importance of justifications in functional thinking, as well as the relevance of the phases of abduction, induction and generalization in the generalization process.