Growing mathematical objects in the classroom - The case of function

This article is devoted to some of the educational quandaries stemming from the fact that mathematics is a discourse that creates its own objects. More specifically, we ask how the participants of classroom learning-teaching processes cope with the seemingly paradoxical situation in which they are supposed to talk about objects, of the existence or nature of which they are not yet sufficiently aware (it is through participation in a conversation about them that these objects are being brought into being). To answer this question, we watched videos of 7th grade students as they were making their first steps in the discourse on functions. The learning-teaching processes were followed for nearly two months. Curricular materials and the teachers' discourse were documented and analyzed as well. We found out that the students were able to participate in the discourse on function without ever dealing directly with this as-yet nonexistent object. They managed to cope with problems by associating them with solution routines through all kinds of discursive clues, to which they were sensitized through their former experience. This said, we were also able to conclude that although the learners were still in the early phase of the project of objectification, their participation in the project laid a solid foundation for their future discourse on functions.