Students’ Understanding of Riemann Sums and Double Integrals: The Case of Task Design in APOS Theory

In this study, we investigate students’ understanding of the relation between a double integral of a continuous function over a rectangle and the corresponding Riemann sums. To do so, we explore the relation between (1) a proposed model (genetic decomposition) of mental constructions that students may do to understand the relation between Riemann sums and double integrals, (2) tasks designed to help students make these constructions, and (3) the results of semi-structured interviews with eleven students who completed the tasks. We focus on the construction differences between students who engaged in tasks designed according to the genetic decomposition and those in a previously studied lecture-based course. The study aimed to underscore the task's effect on students' learning in order to refine the genetic decomposition if needed. This study contributes a set of tasks that enable students to relate Riemann sums and double integrals. The results showed that students using the proposed materials in class and a collaborative didactical strategy provided evidence of constructing the structures proposed in the genetic decomposition. The tasks are based on a genetic decomposition, so the study also contributes by showing that it is an effective model to guide instruction. The constructions inferred from students’ work are discussed in detail and compared to those proposed in the genetic decomposition and those resulting from previous research.