Students' Concept Images of Derivatives from Contour Graphs

We interviewed upper-level physics students in an effort to explore their understanding of multivariable derivatives in the context of contour graphs, a common and powerful representation frequently used in electrostatics and thermodyanmics. We identified three overarching actions in students' approach to the derivative tasks: (1) orienting to the graph, (2) finding one or more derivatives, and (3) reflecting. Students identified and relied on many different representational features, many of which highlighted what is held constant while finding a derivative. These results have helped us to generalize the existing concept image framework for derivatives to multivariable derivatives. Primarily, we have added a new layer (Change) to the framework about attending to which variables are changing (or remaining constant). This new layer has been instrumental in our development of classroom activities for physics majors and has the potential to support further research into students' ideas about multivariable derivatives.