Abstract

Previous research has found that within elementary university courses in single variable and multivariable Calculus, the activities proposed to students may enable and encourage the development of non-mathematical practices. More specifically, the research has shown that students can obtain good passing grades by learning highly routinized techniques for a restricted set of task types, with little to no understanding of the mathematical theories that justify the choice and validity of the techniques. We were interested in knowing what happens as students progress to more advanced courses in Analysis. The study presented in this thesis focussed on a first Real Analysis course at a large urban North American university. To frame our study, we turned to the Anthropological Theory of the Didactic, which offers theoretical tools for modelling practices as they exist within and across institutions. We analyzed various course materials to develop models of practices students may have been expected to learn in the course. These were then used to inform our construction of a task-based interview that would allow us to elicit and model practices students had actually learned. Interviews were conducted with fifteen students shortly after they passed the course. In our qualitative analyses of the resulting data, we found that students’ practices were (non-)mathematical in different ways and to varying degrees. Moreover, this seemed to be linked not only to the kinds of activities students had been offered in the course, but also to the characteristically different ways in which students may have interacted with those activities. As theoretical tools for thinking about these links, we introduce the notion of a path to a practice and a framework of five positions that students may adopt in a university mathematics course institution: the Student, the Skeptic, the Mathematician in Training, the Enthusiast, and the Learner. We discuss the possibility of designing paths of activities that might perturb students’ positioning and encourage the development of practices that are more mathematical in nature.