Calculus as taught by mathematicians typically involves a large toolbox of algebraic manipulations. Almost all computations are done using rectangular coordinates and, later, the associated standard basis of unit vectors. Vector calculus as used by physicists, on the other hand, typically involves geometric reasoning, and the frequent use of coordinates and basis vectors adapted to the symmetries that are present. Thermodynamics goes even further, fundamentally altering the notion of “standard coordinates.” These treatments are sufficiently different from each other that they constitute different languages; students are often unable to translate.

Our research group at Oregon State University has been working to bridge this gap between mathematics and physics for more than two decades, primarily by restructuring upper-division physics courses, but also by developing materials for second-year calculus that emphasize geometric reasoning. The Paradigms in Physics project, continuously supported since 1997 by the NSF, has evolved from “merely” designing novel curricula to studying student learning of mathematical concepts such as partial derivatives.

This talk describes several examples of these language differences, the curricular materials we have developed to help students bridge this gap (including an online textbook and a website featuring more than 300 classroom activities), and some of the education research in which our materials are grounded.

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