Experts’ understanding of partial derivatives using the partial derivative machine

Abstract

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching of partial derivatives, with a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. In this paper, we report on an initial study of expert understanding of partial derivatives across three disciplines: physics, engineering, and mathematics. We report on the central research question of how disciplinary experts understand partial derivatives, and how their concept images of partial derivatives differ, with a focus on experimentally measured quantities. Using the partial derivative machine (PDM), we probed expert understanding of partial derivatives in an experimental context without a known functional form. In particular, we investigated which representations were cued by the experts' interactions with the PDM. Whereas the physicists and engineers were quick to use measurements to find a numeric approximation for a derivative, the mathematicians repeatedly returned to speculation as to the functional form; although they were comfortable drawing qualitative conclusions about the system from measurements, they were reluctant to approximate the derivative through measurement. On a theoretical front, we found ways in which existing frameworks for the concept of derivative could be expanded to include numerical approximation.

Student Interpretations of Partial Derivatives

“…One important extra step that must be taken when finding a partial derivative is identifying what is held constant. This step is not always a straightforward one, particularly in physics contexts [9][10]12]. We focus on questions 2 and 3, since very few responses to question 1 were relevant.…”

Students’ strategies for solving a multirepresentational partial derivative problem in thermodynamics

“…Furthermore, the importance of these different representations for understanding derivatives has been highlighted by Zandieh's concept image framework [23]. We have extended this framework to include the numerical representation that has particular relevance in physics [7], and we are also in the process of extending it to encompass partial derivatives as well as ordinary derivatives [11].…”

“…OSU has used it for content such as numerical integration, numerical differentiation, exploration of partial derivative concepts, and development of analogies to cyclic processes in thermodynamics. A description of the partial derivative machine has previously been published [7,19].…”

“…We find that the students in this study seem to have more difficulties with the representational aspect than with the other aspects. In this article, we report results from part of a long-term ongoing research project focused on the teaching and learning of partial derivatives [5][6][7][8][9][10][11][12][13][14]. One result of this project has been the development of several activities and pedagogical approaches aimed at improving the teaching and learning of various mathematical concepts in thermodynamics, including partial derivatives [13][14][15][16][17][18][19].…”

Editorial: Focused Collection: PER in Upper-Division Physics Courses

“…The two articles in this group reveal new insights into both faculty and student thinking in the area of mathphysics connections. The first article, by Roundy and colleagues [40] at Oregon State University, describes an exploratory study of the way in which experts in various STEM fields understand and interpret partial derivatives. Their study revolves around a novel mechanical device of their own design, used for both research and instruction.…”