Pensamiento Aritmético-Algebraico a través de un Espacio de Trabajo Matemático en un Ambiente de Papel, Lápiz y Tecnología en la Escuela Secundaria

In this paper, we want to address the problem of teaching algebra from the point of view of the theoretical framework of Mathematical Working Spaces (MWS). In our study, we carried out two experiments, one in Quebec and one in Mexico. In this research, we consider the visualization has started, once the individual communicates his ideas to others (or himself) and builds relationships between the elements involved, including articulations among representations. The evolution of these cognitive structures within a perspective of the processes of objectification, requires shared experiences in which we have designated as a teaching and collaborative learning, self-reflection and scientific debate (ACODESA) under the point of view of activity theory. Specifically, it investigates polygonal numbers (especially triangular numbers) in an environment of paper, pencil, and technology. Our interpretation of students’ (14-15 year old) cognitive processes of a population at secondary school in Mexico, following the theoretical framework of the ETM. This interpretation focuses on a step towards a cognitive epistemological plane, where mathematical activity enables articulation among visualization processes, production processes, and arithmetico-algebraic validation providing a support precisely to an arithmetico-algebraic thinking.