*by Tevian Dray, Professor, Department of Mathematics, Oregon State University*

One of the iconic messages of the calculus reforms that took place in the 1990s is the “Rule of Four,” emphasizing the use of multiple representations: algebraic, geometric, numeric, and verbal. But what is a numerical representation of the derivative?

In a recent study [1], we asked faculty in mathematics, physics, and engineering to determine a derivative based on experimental data they had to collect themselves, using the apparatus shown in Figure 1. The physicists and engineers had no trouble doing so—but the mathematicians refused to acknowledge a computed *average* rate of change, however accurate, as a derivative. The physicists and engineers knew full well that their computation was an approximation, but they also knew how to ensure that it was a good one.

*Figure 1: The Partial Derivatives 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 which variables are independent, nor even how many independent variables there are. For further details, see [1].
*

Context is everything in applications. Ask a physicist how small an infinitesimal distance is, and she will surely ask, “With respect to what?” Furthermore, even when working with quantities \(x\ll L\) for some scale \(L\), she might well add, “but not so small that atomic structure matters!” Physicists know that derivatives do not in fact describe the real world; they are a (very useful) idealization. This awareness of the allowed regime is second nature to scientists and engineers, even if often left unstated.

We recently argued [2] that mathematicians’ “bright line” distinction between average and instantaneous rates of change is therefore misplaced. How does one talk about instantaneous rates of change numerically, anyway? It is not only atomic structure that imposes a lower bound: Roundoff error becomes a problem for “infinitesimal” numerical computations—and experimental error plays the same role when measurements are involved. In both cases, the very notion of numerical derivative *requires* a lower bound on the step size; it is simply not possible to compute actual limits numerically, nor from experimental data.

Should we therefore reject numerical or experimental representations of derivatives? Of course not. Rather, we should move the line; what matters is not whether a rate of change is average or instantaneous, but whether it is “good enough.” Do we need to teach students complicated techniques of data analysis to determine what “good enough” means? Not necessarily, although it wouldn’t hurt to acknowledge that such techniques exist.

Our group has coined the name *thick derivative* for the resulting notion of “good enough approximation to the instantaneous rate of change.”

The lesson here goes well beyond a discussion of how best to teach students what a derivative is. The mathematics community is well aware that negative experiences with calculus are the single biggest factor causing students to switch out of STEM majors [3, 4, 5]. There is clearly a mismatch between what we mathematicians believe such a course should teach and the needs and abilities of our students. Perhaps we are focusing too much on dotting the i’s, and not enough on the underlying concepts.

I once asked my physicist wife whether physicists cared about the difference between the functions \(\frac{x^2-1}{x-1}\) and \(x+1\). Her straight-faced response was, “What difference?” This was not an instance of “sloppy math,” but rather a very deliberate attempt to point out that there are no physical situations where such removable singularities matter. So why do we start our calculus courses with them?

Similarly, mathematicians delight in constructing examples (and, with the advent of 3-d printing, models) of functions with direction-dependent limits, or of critical points that are *not* local extrema. Shouldn’t we be emphasizing the examples that *are* well behaved?

One of my favorite books as a student had the marvelous (and accurate) title, *Counterexamples in Topology* [6]. One thing I learned from this book is that some other mathematician is always going to be smarter than I am. As a successful mathematician, I have learned how to clarify my assumptions. But a calculus student should be learning how calculus works, not the largely unphysical mathematical contexts in which it doesn’t.

Is there a better way? I would argue that calculus is the study of infinitesimal reasoning, not limits. Calculus had been used successfully for 150 years before limits were invented—and the real numbers on which such limits depend were not properly defined until even later. Another 100 years would pass before nonstandard analysis would justify infinitesimal reasoning without limits, but by then it was too late; limits were here to stay.

So what do I suggest? Skip the fine print. Emphasize examples, not counterexamples. Use numerical data, and discuss the implications. Ask students to determine derivatives experimentally. No fancy apparatus is necessary; just measure rise over run! But be sure to include some examples that are not based on graphical data.

Emphasize the need to be fluent with multiple representations, not merely the ability to perform symbolic manipulations.

Much of our own work has emphasized geometric reasoning as the key to conceptual understanding. The dot product is fundamentally a *projection*; the cross product is fundamentally a *directed area*; the divergence is fundamentally about *flux*. In each case, the formulas follow from these conceptual underpinnings, rather than the other way around.

Use and encourage infinitesimal reasoning, the art of working with quantities that are “small enough”. As we have argued in a series of papers [7, 8, 9] and an online multivariable calculus text [10], differentials provide a robust, geometric, conceptual framework for working with such quantities; there are also others, such as power series.

All of these suggestions align well with the recommendations of the Curriculum Foundation Project of the MAA [11], after seeking extensive input from partner disciplines: Emphasize conceptual understanding, problem solving skills, communication skills, and a balance between perspectives.

Small group activities supporting many of these ideas are available through the project websites described below, which include indexes of activities suitable for vector calculus and multivariable calculus.

Each activity is documented separately, in hopes of allowing instructors to use as many or as few activities as they wish. Although our own work has focused on second-year calculus, many of the ideas—and some of the activities—could be easily restricted to single variable calculus. The Partial Derivatives Machine in Figure 1 becomes a *Derivatives Machine *if one string is locked down. Similarly, use just one edge of the surfaces in Figure 2.

Finally, tell a story. After all, there are really only two ideas in calculus: ratios of small quantities, and chopping and adding. Let’s not lose sight of the coherence of that underlying message.

*Figure 2: One of the plastic surface models developed by Aaron Wangberg at Winona State University as part of the Surfaces project. Each of the color-coded surfaces is dry-erasable, as are the matching contour maps, one of which is visible underneath the surface. For further details, see the Surfaces project website.*

**Acknowledgements**

Most of the ideas presented here grew out of more than 20 years of collaboration with my wife, Corinne Manogue, as well as many colleagues too numerous to name. David Roundy deserves the credit for introducing “experiment” as a representation of the derivative, leading directly to the concept of *thick derivatives*. Much of this work was done under the auspices of three overlapping projects.

The Vector Calculus Bridge project seeks to bridge the gap between the way mathematicians teach vector calculus and the way physicists use it.

The Paradigms in Physics project 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 project uses plastic surfaces and accompanying contour maps, all writable, to convey a geometric understanding of multivariable calculus.

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

Figure 1 first appeared in [1]; Figure 2 is taken from the Surfaces project website, and is used by permission.

**Bibliography**

[1] David Roundy, Eric Weber, Tevian Dray, Rabindra R. Bajaracharya, Allison Dorko, Emily M. Smith, and Corinne A. Manogue, *Experts’ understanding of partial derivatives using the PartialDerivative Machine*, Phys. Rev. ST Phys. Educ. Re.s **11**, 020126 (2015).

[2] David Roundy, Tevian Dray, Corinne A. Manogue, Joseph F. Wagner, and Eric Weber, *An Extended Theoretical Framework for the Concept of Derivative*, in Proceedings of the **18th Annual Conference on Research in Undergraduate Mathematics Education**, (Pittsburgh, 2015), eds. Tim Fukawa-Connelly, Nicole Engelke Infante, Karen Keene, Michelle Zandieh, MAA, pp. 838–843.

[3] *Engage to Excel: Producing One Million Additional College Graduateswith Degrees in Science, Technology, Engineering, and Mathematics*, President’s Council of Advisors on Science and Technology, The White House, Washington, DC, 2012.

[4] Chris Rasmussen and Jessica Ellis, *Who is Switching out of Calculus and Why?*, In:

Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, eds. Anke M. Lindmeier and Also Heinze, PME, Kiel, Germany, 2013, pp. 73–80.

[5] David Bressoud and Chris Rasmussen, *Seven Characteristics of Successful Calculus Programs*, Notices of the AMS **62**, 144–146 (2015).

[6] Lynn Arthur Steen and J. Arthur Seebach, Jr., **Counterexamples in Topology**, 2nd edition, Springer Verlag, New York, 1978.

[7] Tevian Dray and Corinne A. Manogue, *Using Differentials to Bridge the Vector Calculus Gap*, College Math. J. **34**, 283–290 (2003).

[8] Tevian Dray and Corinne A. Manogue, *Putting differentials back into calculus*, College Math. J. **41**, 90–100 (2010).

[9] Tevian Dray, *Using differentials to determine the derivatives of trigonometric and exponential functions*, College Math. J. **44**, 17–23 (2013).

[10] Tevian Dray and Corinne A. Manogue, *The Geometry of Vector Calculus*, (online only).

[11] Susan Ganter and William Barker, eds., *Curriculum Foundations Project: Voices of the Partner Disciplines*, MAA, 2004.

Your ideas about teaching are similar to those of the late Soviet mathematician V. I. Arnol’d. 😀

Excellent idea I was looking for. Thank you very much for sharing it.

Very nice article, thank you.