### Interests

Algebraic Thought

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The study of symbolic algebra -understood as a system of signs- and the interaction of individuals with this system, are at the center of my research interests.

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From arithmetic to algebra

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In the 80s, the conjugation of the historical-epistemological analysis of certain features of pre-symbolic algebra (13th-15th centuries) and the clinical interview with students between 12 and 15 years of age, constituted a powerful methodology that allowed me to deepen in the transition processes from arith-

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.

Technology Learning Environments

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Spreadsheets and algebra

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Due to the possibility of the systematic variation of numerical values â€‹â€‹in functional relationships and of displaying said variation in numerical tables, Spreadsheets allow proposing methods of solving problems and equations that are reconciled with children’s non-formal intuitive methods (e.g. that of trial and error). The above, together with the generality of the formulas with which the relationships between variables are represented on the Spreadsheets environment, provide the student with a way to access general notions and methods such as the algebraic ones. The set of papers that Rosamund Sutherland and I wrote in the 90s report results that confirm such hypothesis.

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Parameterized modeling, Spreadsheets and smart dialogs

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Through a longitudinal study of cases carried out in Mexico and England, the feasibility of modifying mathematical practices in the science classroom by modeling with spreadsheets was shown, which allow the student to create an artificial world to explore phenomena of the physical world. The incorporation of theoretical elements introduced by J. Lave allowed us to analyze the experimental data in this type of study, revealing aspects of mathematical modeling that are not evident in more conventional analyzes. The digital version with an artificial intelligence layer of the parameterized modeling activities, allowed to provide students with accurate and timely feedback at critical moments of modeling, such as understanding the phenomenon, building the model and predicting. The articles written with José Luis Abreu and Montserrat García-Campos report results from this stage.