Student reasoning about the signs of backward definite integrals in mathematics and physics*

Researchers have documented student difficulties with the signs of definite integrals in both physics and calculus, particularly negative integrals. Less studied are how students reason about the sign of a “backward definite integral”, i.e., taken right to left, which is important in several physical contexts. We report on two related studies that explore student reasoning about backward integrals in graphical and/or symbolic representations. Our analysis uses the concept image framework and a recent categorical framework for mathematical sense making. The concept images of dx and ?x affected student reasoning. We found four prevalent ways students made sense of the integral signs: invoking the Fundamental Theorem of Calculus, using macroscopic area under the curve (spatial or graphical), considering the Riemann sum (microscopic area, both types), and using a physical context. Students were better able to interpret backward integrals when they linked the mathematical concepts of integrals to some physical context beyond spatial area, whether self-generated or prompted. The context seemed to provide meaning to the difference represented by ?x or dx and thus to the sign of that difference and the definite integral.