Students' Obstacles to Using Riemann Sum Interpretations of the Definite Integral

Students use a variety of resources to make sense of integration, and interpreting the definite integral as a sum of infinitesimal products (rooted in the concept of a Riemann sum) is particularly useful in many physical contexts. This study of beginning and upper-level undergraduate physics students examines some obstacles students encounter when trying to make sense of integration, as well as some discomfort and skepticism some students maintain even after constructing useful conceptions of the integral. In particular, many students attempt to explain what integration does by trying to use algebraic sense-making to interpret the symbolic manipulations involved in using the Fundamental Theorem of Calculus. Consequently, students demonstrate a reluctance to use their understanding of "what a Riemann sum does" to interpret "what an integral does." This research suggests an absence of instructional attention to subtle differences between sense-making in algebra and sense-making in calculus, perhaps inhibiting efforts to promote Riemann sum interpretations of the integral during calculus instruction.