Experts' understanding of partial derivatives using the Partial Derivative Machine

TL;DR

This study investigates how experts from physics, engineering, and mathematics understand partial derivatives using the Partial Derivative Machine, revealing disciplinary differences in approaches to measurement and functional form speculation.

Contribution

It provides initial insights into disciplinary differences in expert understanding of partial derivatives through innovative use of the PDM in interviews.

Findings

Physicists and engineers quickly used measurements for derivatives.

Mathematicians focused on functional form and were hesitant to measure derivatives.

Disciplinary differences influence how experts conceptualize partial derivatives.

Abstract

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.