RUME 28
2026
Alexandria, Virginia
Connecting Multiplicative Structures Across Contexts: Students’ “Product Layer” Reasoning in Water Pressure and Electricity Tasks
Page: 170
Students often struggle to interpret the multiplicative structure within integrals––what we call the “product layer” (f(x)∙dx)––when reasoning about applied problems. In this study, we analyze the reasoning of Melinda, an undergraduate in a calculus-based physics course, as she works on two tasks: estimating the total force on a tank wall (water pressure) and the total voltage across a capacitor (electricity). Using conceptual blending theory, we examine how Melinda connects symbolic, mathematical, and physical ideas when setting up integrals. Melinda frequently interprets integration as summation of small pieces or as area under a curve but struggles to coordinate multiplicative relationships among varying quantities (e.g., pressure x area, current x time). These findings suggest that challenges with product layer reasoning persist across contexts, highlighting the need for instruction that explicitly supports blended multiplicative reasoning.