  Partial derivatives are used in a variety of different ways within physics.
Most notably, thermodynamics uses partial derivatives in ways that students
often find confusing. As part of a collaboration with mathematics faculty, we
are at the beginning of a study of the teaching of partial derivatives, a goal
of better aligning the teaching of multivariable calculus with the needs of
students in STEM disciplines. As a part of this project, we have performed a
pilot study of expert understanding of partial derivatives across three
disciplines: physics, engineering and mathematics. Our interviews made use of
the Partial Derivative Machine (PDM), which is a mechanical system featuring
four observable and controllable properties, of which any two are independent.
Using the PDM, we probed expert understanding of partial derivatives in an
experimental context in which there is not a known functional form. Through
these three interviews, we found that the mathematicians exhibited a striking
difference in their understanding of derivatives relative to the other groups.
The physicists and engineers were quick to use measurements to find a numeric
approximation for a derivative. In contrast, the mathematicians repeatedly
returned to speculation as to the functional form, and although they were
comfortable drawing qualitative conclusions about the system from measurements,
were reluctant to approximate the derivative through measurement. This pilot
study led us to further questions. How do fields differ in their experts'
concept image of partial derivatives? What representations of partial
derivatives are preferred by experts? We plan to address these questions by
means of further interviews with a wider range of disciplinary experts.
