By David Bressoud @dbressoud

This month, Princeton University Press will publish my new book, Calculus Reordered: A History of the Big Ideas (Figure 1). At one level, this is a history of calculus, a successor to such works as Boyer’s The History of Calculus and its Conceptual Development, Toeplitz’s The Calculus: A Genetic Approach, and Edwards’ The Historical Development of the Calculus. But this book was also written to encourage those who teach calculus to rethink how we approach this pivotal subject.

Those who regularly read my columns and articles will know that I have long railed against the way we teach integration. Whatever we may say, whatever we may intend, the message that students retain from the way we teach it is that integration is all about reversing differentiation. If it has applications, then those rely on obscure procedures that must be memorized individually. They miss the fundamental fact that integration is all about problems of accumulation. The connection to differentiation via the Fundamental Theorem of Integral Calculus (to give it its original and proper name) is not the essence of integration but merely a tool that can be applied in the last step of solving an accumulation problem.

To make this point, I started to write a book on the history of this fundamental theorem. In the process, I realized that the historical development of calculus illuminates more than how to teach integration. It also has important lessons for how we should approach differentiation, series, and limits. I have long embraced the belief that every course should be built around a story, a quest to answer certain burning questions. In writing this book, I sought to unearth the questions that drove the historical development of calculus. I found that this historical lens supported many of the pedagogical innovations promoted by researchers in undergraduate mathematics education: Pat Thompson’s decision to begin the study of calculus with problems of accumulation, Mike Oehrtman’s explanations of limits in terms of bounds on the output, and Marilyn Carlson’s emphasis on the co-variation of variables that actually vary.

So this is a history of calculus, but embedded within it and called out explicitly in the appendix is a cry to recast how we teach this subject. The book is built around the four big topics of single variable calculus, taken in historical order: integration as accumulation, differentiation as ratios of change, series as limits of sequences, and limits as the algebra of inequalities.

I believe that it is a serious mistake to start calculus with a discussion of limits, commonly expressed in phrases such as “the limit of f as x approaches c is L.” Implicit in the words “as x approaches c” is the assumption that that x cannot equal c. Mathematicians know that we do not literally mean “approaching,” but students assume that when someone says, “the dog approaches the door,” they have ruled out the possibility that the dog has been sitting at the door. More than this, the phrase suggests that the focus should be on the behavior of x and its effect on f. In fact, the mathematical definition of the limit begins with the problem of bounding f and asks what has to be done to x to accomplish the desired bounds. The solution is not to start with talk of limits, but of bounds, and to save a serious discussion of limits to the end of the course, in line with the historical order in which the work on limits came long after mathematicians had wrestled with integration, differentiation, and series.

I also do not like the way we treat differentiation, as a means of determining the slope of a tangent line. Slope is a slippery concept for undergraduates. As I’ve written, for most students it is simply a numerical value that indicates steepness (see The Derivative is Not the Slope of the Tangent Line, Launchings, November 2018). Few undergraduates recognize it as a ratio of changes. A better approach is to start with a historically rooted discussion of ratios of changes. I explain in my book that the first function to be differentiated was the sine, it occurred around the year 500 CE in India, and it was done to aid in the interpolation of tables of sine values. Aryabhatta needed to know how small changes in arc length were reflected in small changes in the sine. Shortly after 1600, Napier was led to the discovery of the derivative of the natural logarithm while exploring his invented function that turned multiplication into addition.

Integration is not about area under curves but rather accumulation problems. The paradigmatic example of accumulation is the determination of distance traveled from knowledge of velocity, a problem that arguably was first tackled by the Babylonians almost 2500 years ago (Ossendrijver, 2017), but which was a major topic of scholarly investigation in the 14th century. Starting calculus with problems of accumulation is the approach taken by Thompson, Milnor, and Ashbrook in Project DIRACC and about which I have written in Re-imagining the Calculus Curriculum I, Launchings, May 2017 and Re-imagining the Calculus Curriculum II, Launchings, June 2017. In my book, I strongly recommend it as a better way to introduce calculus.

Infinite series are too often viewed as sums with a LOT of terms, a topic tacked onto the end of the first year of calculus and dominated by rules for determining convergence. For most students, these rules possess little cohesion and present nothing more than another set of procedures to be memorized. It is unclear to me what this accomplishes. How much better to focus on Taylor polynomials. Rather than convergence tests, I would like that time spent explaining how the Lagrange error bound arises as an elegant application of the Mean Value Theorem. After all, convergence as such as less useful than the ability to bound the error when the Taylor polynomial is substituted for the actual function.

I hope that others will enjoy reading this book as much as I enjoyed writing it and find it a source of insights and explanations that can be carried into the classroom.

Read Bressoud’s Launchings archive.

References
Boyer, C.B. (1959). The History of Calculus and Its Conceptual Development. Mineola, NY: Dover.

Edwards, C.H., Jr. (1979). The Historical Development of the Calculus. New York, NY: Springer-Verlag.

Ossendrijver, M.  (2018). Bisecting the trapezoid: tracing the origins of a Babylonian computation of Jupiter’s motion. Arch. Hist. Exact Sci. (2018) 72:145–189 https://doi.org/10.1007/s00407-018-0204-4

Toeplitz, O. (2007). The Calculus: A Genetic Approach. Chicago, IL: University of Chicago Press.