Interpreting Derivatives

Abstract

Calculus, as commonly taught, describes certain operations on explicit functions, but science relies on experimental data, which is inherently discrete. In the face of this disparity, how can we help students transition from lower-division mathematics courses to upper-division coursework in other STEM disciplines? We discuss here our efforts to address this issue for upper-division physics majors by introducing a new representation for derivatives in terms of experiments to go along with the traditional symbolic, graphical, verbal, and numerical representations, and by emphasizing infinitesimal reasoning through the use of differentials. These ideas culminate in the concept of thick derivatives. By providing examples of “physics” reasoning about both ordinary and partial derivatives, and methods for incorporating such reasoning into the classroom, we hope to give instructors of calculus new insight into the needs of many of their students.

Keywords:

• Derivatives
• differentials
• experiment
• measurement
• physics

AcknowledgEments

Much of this work was done under the auspices of three overlapping projects. The Vector Calculus Bridge project [3, 9] seeks to bridge the gap between the way mathematicians teach vector calculus and the way physicists use it. The Paradigms in Physics project [26, 27, 33] has redesigned the entire upper-division physics curriculum at OSU, incorporating modern pedagogy and deep conceptual connections across traditional disciplinary boundaries; its website documents both the 18 new courses that resulted, and the more than 300 group activities that were developed. The Raising Calculus to the Surface [50] project uses plastic surfaces and accompanying contour maps, all dry erasable, to convey a geometric understanding of multivariable calculus. Figure 2 first appeared in [37]; Figure 1 is taken from the Surfaces project website [50] and is used with permission.

Figure 1. One of the transparent plastic surface models developed by Aaron Wangberg at Winona State University as part of the Surfaces project. Each of the six color-coded surfaces is dry-erasable, as are the matching contour maps, one of which is visible underneath the surface. For further details, see [49, 50].

Figure 2. The Partial Derivative Machine designed by David Roundy at Oregon State University. In this mechanical analog of a thermodynamic system, the variables are the two string positions (the flags) and the tensions in the strings (the weights). However, it is not obvious how many independent variables there are, and which variables are considered independent depends on the context. For further details, see [37].

Figure 3. An extended framework for the concept of the derivative [35].

Figure 4. Shown on the left is the standard chain rule diagram for converting the temperature T in rectangular coordinates (x, y) to polar coordinates $(r,\varphi ).$ The diagram on the right shows the same situation, expressed in terms of differentials.

Figure 5. The chain rule diagram when changing only one of two variables, in this case expressing the magnetization M in terms of the magnetic field B and either the temperature T or the entropy S.

Notes

¹Thompson and collaborators [43–45] have recently given a new interpretation of differentials which describes known smooth functions (e.g., $\text{sin}(\theta )$) as piecewise linear on a “small enough” scale.

Additional information

Funding

The Bridge and Paradigms projects have been supported by the NSF through grants DUE–9653250, 0088901, 0231032, 0231194, 0618877, 0837829, 1023120, and 1323800; the Surfaces project is supported by the NSF through grants DUE–1246094, 1611946, 1611970, and 1612480.

Notes on contributors

Tevian Dray

Tevian Dray, professor of mathematics at Oregon State University, is a geometer with a longstanding interest in the interface between mathematics and physics, as well as in mathematics education. He has been the director of the Vector Calculus Bridge project since 2001, a co-PI on the Paradigms project since its inception in 1996, and has also participated in several projects aimed at improving the content knowledge of mathematics teachers. The underlying theme in both his traditional research in mathematical physics and his work in education has been the importance of geometric reasoning.

Elizabeth Gire

Elizabeth Gire does research in physics education, with emphasis on problem-solving and the development of epistemological beliefs and metacognitive skills of undergraduate physics majors. She worked on the Paradigms project as a postdoctoral Research Associate from 2007–2009, where she taught and conducted research in the context of the Paradigms courses. She has recently returned to Oregon State University as an assistant professor of physics at Oregon State University, and is a co-PI on the Paradigms project.

Mary Bridget Kustusch

Mary Bridget Kustusch, assistant professor of physics at DePaul University, specializes in physics education research. As a postdoctoral scholar with the Paradigms project from 2011–2013, she conducted some of the initial research on expert use of partial derivatives in thermodynamics. Her current research continues to explore the development of scientific expertise and identity, although now with more of a focus on the impact of group interactions in the classroom. In addition to implementing many of the pedagogical approaches from the Bridge and Paradigms projects in her own classes, she has conducted local workshops on embodied learning activities and the use of whiteboards for promoting student discourse.

Corinne A. Manogue

Corinne A. Manogue, professor of physics at Oregon State University, has directed the Paradigms project from the beginning. She has 20 years experience not only developing and teaching multiple courses in the Paradigms program, but also working with the entire Paradigms team to create a coherent curriculum characterized by active learning. Her special interest, investigating the role that active-engagement experiences play in helping students transition from lower-division mathematics to upper-division physics, led to her involvement as co-PI in the Vector Calculus Bridge project. Her traditional research in theoretical quantum gravity uses the octonions to describe the symmetries of high energy particle physics.

David Roundy

David Roundy, associate professor of physics at Oregon State University, has been co-PI on the Paradigms project since 2010. He does research in computational condensed matter physics, and his curriculum development has focused on improving the teaching of computational and thermal physics.