Our study began with the notion that multiplication is a complex mathematical object, in both its epistemological and cognitive dimensions. The fact that geometric representations can make a mathematical object's meanings more obvious led us to structure our research around the geometrization of multiplication for different sets of numbers. To study the relationship between this complex mathematical object -- multiplication -- and the construction of meaning by students we designed experimental lessons that were put in place in French high school and junior high school classrooms. This experimental study allowed us to closely analyze students' understanding of the topic, or, on the other hand, the obstacles they encountered in a mathematics assignment requiring frame changes and changes in registers of semiotic representation. Our experimental data were analyzed using a combination of several theoretical approaches. The notion of the Mathematical Work Space and its geneses allows us to account for the complexity of students' mathematical work. In order to study collaborative work between students, as well as the teacher's role in this process of cultural mediation, we also applied theories of semiotic mediation and the social construction of knowledge. Our resulting theoretical framework allows us to give a detailed description of the relationships between the epistemological and cognitive levels of the MWS. We conclude with the identification and analysis of individual students' chosen paths, resulting from interactions within a Mathematical Work Space.
