Students' understanding of Riemann sums for integrals of functions of two variables

In this paper we report on a study of how students understand some of the most fundamental ideas of Riemann integrals of functions of two variables. We apply Action-Process-Object-Schema (APOS) Theory to pose a preliminary genetic decomposition (GD), conjecturing mental constructions that students would need to relate Riemann sums to integrals of functions of two variables over rectangles. The genetic decomposition is informed by the researchers' classroom experience, findings of a previous study that applied semiotic representation theory, and by a study on integrals of functions of one variable. We pay particular attention to the case of an integral of a continuous function over a rectangle and the simplest partition possible, that consisting only of the rectangle itself. We then explore students' geometrical understanding of the relation between the single term f(a, b)Delta x Delta y, where (a, b) is a point on the rectangle, and the double integral over the rectangle. We tested the GD by performing student interviews with 10 students who had just finished taking a lecture-based multivariable calculus course. The findings underscore the importance of each of the mental constructions described in the genetic decomposition and suggests that students have difficulty in some mental constructions that may commonly be assumed to be obvious during instruction.