metic to algebraic thinking in individuals. This, by itself, can be recognized as a field of research in which it is feasible to draw lines of evolution from one thought to another, based on locating on the epistemological level the presence of ruptures (in the sense developed by Bachelard) that they correspond, at the level of didactics, with didactic obstacles of epistemological origin (in the sense of
Brousseau).

 

 

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The semiotic approach

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Due to the need to give plausible theoretical explanations to the results of empirical studies carried out for more than two decades, the knowledge, not only about the construction of meanings, but also about sense production, began to gain relevance. The strings of signs that constitute algebraic texts were the focus of new studies. Thus, the notions of the mathematical system of signs, the production of meaning and intertextuality became the theoretical support of my research. This was possible thanks to the work of my colleagues Eugenio Filloy and Luis Puig, who adapted to the field of mathematics education ideas and concepts from the field of semiotics. The book we wrote together Educational Algebra: A Theoretical and Empirical Approach, synthesizes this process.

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The sense of structure

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Inspired by the works of Hoch & Dreyfus, Novotna & Hoch, and D. Kirshner, I am at the beginning of a new stage of research, in which I have embarked on studying the internal structure and order of algebraic expressions and equalities and the awareness of subjects of that structure. On the other hand, along with a team of young colleagues, we have laid the theoretical and technological foundations for the design, development and testing of a web application that helps individuals, with different backgrounds and experience in mathematics, to develop their algebra structure sense. The ongoing frontier science project Development of the Sense of Structure in Algebra (funded by Conacyt) reports on the progress of these works.