Is Calculus Necessary?

Question

That title is a quote from Fred Roberts:

Fred Roberts. "Is Calculus Necessary?" Proceedings of the Fourth International Congress on Mathematical Education. 1980. p.52ff. "Calculus is not necessary. That will be my premise in this debate."

There was much discussion in the 1980's around the issue of whether or not calculus might be displaced with some form of discrete mathematics, e.g.:

Lynn Arthur Steen. "Developing Mathematical Maturity." In: Ralston, Anthony, and Gail S. Young, eds. The Future of College Mathematics. New York: Springer-Verlag, 1983. p.104.

Steen says, "The gradient from elementary to advanced mathematics is far easier along the road of discrete mathematics than it is along the highway of calculus."

But I am not finding a sustained literature on this debate since those early days. A more recent advocacy is Art Benjamin's TED talk:

Art Benjamin, "Teach Statistics before Calculus!" TED video. 2009. Video link.

He says,

"The world has moved from analog to digital, and it is time for our mathematics curriculum to change from analog to digital—from the more classical continuous mathematics to the more modern discrete mathematics."

He believes students should still learn calculus, but calculus should not be the pinnacle high-school course.

My question is:

Q. Where does this debate stand today? Have some schools implemented Benjamin's suggestion? Is there current, thoughtful literature on the topic?

Answer 1

Society always benefits when new mathematical tools are developed to get insights into the world around us. There is little doubt that Calculus has helped change the quality of the lives we can lead over the hundreds of years since it was born and Calculus has evolved in many unexpected ways.

The real question should be to what extent it is worth distorting the mathematical education of all students for the benefit of those who eventually need and use a particular a mathematical tool whether it be Calculus, queuing theory, sorting algorithms, or what have you? Engineering and many other professions need Calculus but many careers don't "need" Calculus. However, knowing what Calculus can do is also valuable.

My own position is that K-12 mathematics education should be much broader than the curiously narrow CCSS-M (now to some extent collapsed and replaced by each of 50 states' curriculum often using the CCSS-M in modified form) or even, from my vantage point, the somewhat better NCTM Standards. Why emphasize depth over breadth for mathematics in K-12 as many in the mathematics community did when they cut back on breadth to put together the CCSS?

Currently there are technical topics involving algebra and trigonometry that students are taught, which prepare them for Calculus but have relatively little value for a majority of students. Current curriculum emphasizes techniques (roots of quadratic equations, trig functions etc.) over more thematic topics such as optimization, growth and change, codes, information, risk, etc. This emphasis on mastering highly technical technique skills is part of what turns so many otherwise intellectually curious and accomplished students into mathophobes.

The mathematics that leads to Calculus is much more hierarchical than what constitutes many of the nifty topics one can study that belong to discrete mathematics (graph theory, recursion and difference equations, social choice theory, elections, fairness models, etc.). And, one could treat Calculus in K-12 as a nifty topic (not covering what you need to know to compute the integral of sec x) but where students could get the idea of the difference between instantaneous speed and average speed, as well as understand what marginal revenue and costs mean, etc. and how sometimes one needs to go beyond the Bolyai-Gerwien-Wallace Theorem. The fact that discrete mathematics builds on important ideas that are not as hierarchical as topics that lead up to Calculus is another big advantage to doing more with discrete topics in K-12. We certainly need more discrete mathematics in K-12.

Answer 2

I don't believe that there is much of debate about Calculus today. Major universities continue to push the overly rigorous vision embodied in texts like Stewart or Hughes Hallet. The AB/BC Calculus curricula have seen little to no change in decades.

It seems lost on many arguing for Stats over Calc, that they are one and the same. Statistics developed from Physics and Astronomy using many of the typical ideas from first semester calculus. Despite this architecture, too often we find people arguing for statistics or calculus, or even stats over calculus.

I am not suggesting that when you are introducing measures of center and spread in 9th grade algebra class you need to fully develop centers of mass in the language of calculus. I am acknowledging that historically, statistics was developed using the tools of calculus. All this is to say that the stats vs. calc division is an imaginary one that is reinforced rather than remedied by the arbitrary battle lines so often drawn in teaching and learning mathematics that wish to favor a certain stylistic application of the tools of calculus.

There are many examples of groups who have tried to take the stats over calc path. A large example would be the Statway work from Carnegie.

I would argue that the discrete and continuous should have a presence in all mathematics coursework; particularly the undergraduate and high school curriculum. I don't fully agree that the CCSS is a step backwards here, as there is an explicit effort to include functions with discrete domains throughout the high school work with functions. To improve this wedding, the CCSS should have included the matrix and vector concepts throughout rather than as optional additions.

In a calculus class, if we are discussing integration we should include centers of mass and understand how this plays a role in Pearson's r. These ideas have also emerged as important in the field of machine learning and classification algorithms like k-means. Students can understand remedial examples of these topics at much earlier levels.

When it comes to differentiation we should use first difference to motivate a conceptual understanding. Optimization lends itself to such an interpretation, move to the limiting case to discuss continuous situations. Now we can engage with Nash equilibrium and with an understanding of differentiation with respect to a given variable we can derive least squares formulas for linear, polynomial, and exponential situations using our knowledge of minimizing functions.

I suppose what I'm arguing is that it should not be a question of one or the other but our efforts should target unification rather than division. Discrete mathematics deserves its place in the curriculum but I don't believe this needs to be at the expense of the continuous. Instead, these domains reinforce one another and would support a broader learning experience for students.

My favorite writing on this subject would be Felix Klein's Elementary Mathematics from an Advanced Standpoint books I and II or any of Tom Apostol's writing on Calculus including his textbooks. Klein argues beautifully for a larger role of the difference calculus in the curriculum. Also, Leibniz's History and Origins is a great read where he describes the motivation of his work through sums and differences from the familiar Pascal's triangle.

Answer 3

The debate seems to be where it was 20 years ago. The reason is that most universities have a "generic" mathematics curriculum, so if the engineers need calculus as it is now, everyone will also get it. That said, the quotation rings true to me: "The world has moved from analog to digital, and it is time for our mathematics curriculum to change from analog to digital—from the more classical continuous mathematics to the more modern discrete mathematics." I just think that the latter can't and shouldn't exclude calculus. What might make this possible is discrete calculus. Discrete calculus is nothing but the familiar calculus before we take that fateful limit. This limit makes the algebra simpler and easier to handle manually but when the problem falls outside the familiar scope, one must use approximations, i.e., circle back to discrete calculus. Maybe we can emphasize the discrete at the expense of the continuous? $$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ discrete }\\ \text{ calculus }\end{array} \right)= \text{ calculus }$$ An attempt to develop the latter and the former in parallel is in a draft of a book of mine, Calculus Illustrated.

Answer 4

I actually have some sympathy for this point of view in that there is an awful lot of useful insight that can be gathered from algebra, exponential curves, trig, stats, investment calculations, nomographs, maneuvering boards, etc.

However, anyone heading into engineering, physics, or chemistry still needs standard (Granville) calculus.