The fundamental theorem of calculus has not only made calculus one of the most powerful intellectual tools known to man, but it has also created a dichotomy that makes calculus very difficult to teach. Should calculus be presented as the taproot of geometry? Or should it be presented as the tip of the analysis iceberg? Is calculus the first step toward an understanding of the topology of the real line? Or is calculus the first step in the exploration of manifolds and the geometry of mechanics?
Of course, the answer is “both,” and therein lies the crisis. This idea was explored in detail by Halmos in a 1974 essay, and likewise, many generations of mathematicians have declared in their own words that “all calculus books are bad.” In fact, it was once accepted that traditional approaches are flawed, as evidenced by so many of us saying we did not know calculus until graduate school. At one time nearly everyone talking about “Calculus: a Pump, not a Filter” and the need for a “Lean and Lively Calculus.” Reform textbooks are also widely viewed as flawed products, so much so that they are driving calculus instructors back to the traditional approaches they once condemned. After centuries of discussing how calculus should be taught, we still today find that most mathematicians do not learn calculus until they are in graduate school.
So is there anything wrong with calculus? If so, what is it? What should calculus be about? How can calculus be presented to the general population in a meaningful way? These were the questions we began to address when we decided to write our own calculus book. This paper presents the answers we formulated in the course of writing that calculus book. Hopefully, even those who do not accept our calculus book as part of the solution will find this discussion helpful in identifying the problem and how it might be approached.
It is unlikely that there will ever be a means of teaching calculus that allows every theorem to be proven rigorously, every concept to be developed completely, and every meaningful application to be explored. This is true in many introductory math courses and is not evidence of any crisis in calculus, in our opinion.
However, there is a great deal of evidence of a crisis in calculus that has nothing to do with its being an introductory course. We focus on 3 categories of such evidence. We admit up front that the analysis below is solely our opinion and is in all likelihood biased by our desire to have compelling reasons for writing our own calculus textbook.
How Mathematicians Discuss Calculus Among Themselves
The crisis is evident in how we as mathematicians discuss calculus with each other. In fact, nearly all discussions of calculus I have been involved in are stilted and disjoint. It is as if our knowledge of calculus is rote rather than logical, or as if we are using a different part of our brain when we begin discussing ideas from a first year calculus course. Seemingly, even the most proficient mathematicians struggle with calculus and fail in much the same way that our students do.
For example, I read a test question written by an accomplished researcher that asked for the “tangent line to a function at a given point.” Although mixing function terminology with geometric terminology is admittedly a minor misstatement at worst, it is only the tip of the iceberg. I have heard calculus instructors make statements like “locally a tangent line intersects a curve at only one point” and then almost immediately make the contradictory statement “the tangent line to a line is the line itself.” And this is still rather tame compared to statements like “Riemann sums converge to the function, so the integral converges to the area under the function,” and the following mind twister I once overheard from a hallway outside of a classroom: “if a sequence converges conditionally, then so does its series—or not at all, unless its sequence converges to 0.”
Also, Calculus books are full of errors even though they are written and reviewed by mathematicians. There are exercise instructions imploring the student to “Let F(x) be the antiderivative of f(x) in which C=0.” (Calculus, Stewart, 4th edition, page 522) (to see why this does not make sense, consider that F(x)=sin2(x)+C and G(x)=–cos2(x)+C are both antiderivatives of p(x)=sin(2x) for any value of C). There are also nonsensical definitions of limits of powers (Calculus, Thomas/Finney, 9th edition, page 61), and flawed chain rule proofs (Calculus, Larson/Hostetler/Edwards, 6th edition). And I am picking on these three because they are arguably among the best calculus textbooks available. Space does not permit the number of errors in calculus books we actually uncovered, including large numbers of errors in reformed textbooks.
In contrast, mathematicians and textbooks rarely make nonsensical statements when discussing trigonometry, or linear algebra, or even measure theory. My suspicion is that many of us could not understand calculus we were being taught at the time, so we relied primarily on memorization in our first calculus course. The result is that when we try to fall back on our calculus background, it comes out more like a memorized poem than a well-understood collection of concepts (more on this idea later).
How Calculus Students use Calculus
If our own stilted conversations about calculus are not enough, then consider the evidence all around us that even our best students do not learn calculus in a calculus course. Check any computer algebra system to see how well our students picked up on the necessity of the “+C” when computing antiderivatives.
Consider also that we make a great many seemingly absurd statements in a calculus course, yet even the most inquisitive students display no intellectual curiosity of any kind. Have you ever had a student ask how an average can be equal to the ratio of two differences? Do any of them ever snicker when they first hear the oxymoronic statement “C is an arbitrary constant?” Research has established that even our best students reduce limits to a set of rules to be memorized. Not surprisingly, many instructors and many students view Calculus as a course which reinforces algebra and trigonometry and does little else.
We need not belabor this point, because the evidence over the past decade has been overwhelming in showing that student's are not learning much calculus in our calculus courses, including studies on retention of the material, ability to adapt their calculus experience to new settings, and so on. Suffice it to say that student performance is sufficiently low to support a decade of annual calls to new reform ideas.
How Relevant Calculus Courses are to the Other Sciences
In spite of a calculus course that is saturated with “applications,” laden with references to physics and chemistry, and packed with numerical techniques, most of our colleagues in other disciplines see very little relationship between their fields and the calculus course. In fact, except for possibly in colleges of engineering, our colleagues tend to think of our calculus sequence as quaint and curious, important but irrelevant. And most engineers will tell you that they have to "undo" much of what the calculus course does to their students.
Of course, this is due in part to the fact that much of the calculus taught in those courses is irrelevant. Simpson’s rule is no longer used for numerical integration (except by mathematicians). The “applications of the integral” do not even resemble the ways in which those concepts are examined in their respective fields. That is not to say that applications of the integral are not important, but rather that that mathematicians do not know how calculus is used outside of a calculus textbook and thus skew all applications toward mathematical contexts and away from their natural settings.
Moreover, the calculus that our colleagues receive makes them struggle mightily when calculus does occur in their area. They teach their own statistics courses (usually quite poorly) as if calculus and statistics were not inextricably intertwined. ( Can we actually expect to give students a meaningful introduction to the Central Limit Theorem without any concept of a limit?) And even when instructors in other disciplines do discuss small quantities and local approximation, they tend to hand-wave through any real use of calculus and ignore all but the basic concepts of tangents and areas.
What is surprising is that many of these instructors will say that they rarely use calculus, if ever, or that all they need are a few simple derivatives. What is not surprising, however, is that the most “mysterious” topics in science are often those that rely heavily on calculus. The study of fields in physics is essentially an exploration of the definition of the integral and the fundamental theorem of calculus, and yet it is the rare student who has any grasp of Maxwell’s equations, how they are derived, and what they imply.
Clearly, our calculus course does not prepare scientists in other fields to recognize, understand, and utilize the calculus that many of their fields are based upon. Thus, when it comes to calculus, we don’t get it the first time around, our colleagues don’t get it, and our students are still not getting it. It’s no wonder that one of the most common occurrences in higher education is that of a non-mathematics faculty member discovering that something they were doing is calculus. And at the very least, we feel justified in asserting that there still is a crisis in calculus instruction.
How to address that crisis it the topic in the next paper, "Facing the Crisis."