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## What a mathematical representation tells us about reasoning about integrals

Michael C. Wittmann, University of Maine,

### Submitted Abstract to PERC 2011

When solving two integrals arising from the separation of variables in a first order linear differential equation, students have multiple correct choices for how to proceed. They might set limits on both integrals, use integration constants on both, or on only one equation. In each case, the physical meaning of the mathematics is equivalent. But, how students choose to represent the mathematics can tell us much about what they are thinking. Typically, limits indicate more physical thinking, constants more mathematical thinking. Furthermore, evidence for the way they interpret the physical situation is found in choices of limits or constants. At the University of Maine, using a resources (knowledge-in-pieces) framework, we organize these results to show the many skills needed in solving seemingly simple integrals.