Task Design for Students’ Work with Basic Theory in Analysis: the Cases of Multidimensional Differentiability and Curve Integrals

Abstract

We investigate the challenges students face in the transition from calculus courses, focusing on methods related to the analysis of real valued functions given in closed form, to more advanced courses on analysis where focus is on theoretical structure, including proof. We do so based on task design aiming for a number of generic potentials for student learning, developed from and within the theory of didactic situations: "adidactic potential," "linkage potential," "deepening potential" and "research potential." The context of investigation is a first year course on analysis in which the tasks thus constructed were considered relevant to solve a number of operational problems. The experimental method involves careful a priori analysis of each task in terms of the potentials, specifically related to the knowledge at stake; this analysis in confronted with a posteriori analyses of observations of student work before and in class sessions. Two cases are analyzed in detail. While some of the potentials were partly realized, we also identified clear limitations resulting from a variety of factors, including teaching assistants' management of the class sessions and students' perception of the importance, difficulty and meaning of the tasks.