This post from Friday, in which I asked for reasons why business majors should be required to take calculus, has gotten a lot of comments, most of which seem to be wondering the same thing and adding disciplines like biology into the question. That post did actually come from some specific thoughts I’ve been having about making calculus a required course.

I can think of just two legitimate reasons for making *any* course a requirement within a major or minor field of study:

1. The course is fundamental to getting a coherent, well-rounded view of the discipline it comes from; or

2. The course has demonstrable, significant applications to that field to such an extent that teaching the material outside a traditional course is impractical.

You could also add that in some cases, a major contains a course of its own that requires prerequisite material from some other course — so much material that it can’t just be built in. That’s pretty much an alternate take on #2.

Note that I don’t include “liberal arts” or general education courses in this, because some schools care about well-roundedness while others would prefer to give a deep and not-so-broad approach to students’ education. These are two different approaches to higher ed, and the merits and demerits of each are not the issue here. We’re talking about courses within a major, thought of as integral and inextricable parts of what it means to be trained in a discipline.

If you grant me those two criteria, and only those two, then the big question becomes: Which majors can justify requiring calculus? And what of the other majors which do require calculus but can’t justify it according to our criteria?

Well, it’s pretty clear that math, physics, engineering, chemistry, (probably) computer science and (I would add) economics are definitely “yes” for requiring calculus. These are all disciplines that are either deeply rooted in the mathematical methods that arise out of calculus — so leaving calculus out would create a foundational misunderstanding of the major subject — or involve heavy amounts of calculus-based applications. Note that I am drawing a distinction between Computer Science and “information systems”; the former is more theoretical and the latter mostly applied.

It’s also clear that most of the majors that traditionally don’t require calculus — English, art history, etc. — are probably justified in not doing so.

But then we have these majors that do require calculus, but I’m not so sure — based on my knowledge of the disciplines and the people who practice them — that you can justify it. The major programs on our campus, for instance, that require calculus apart from those listed above are Accounting, Biology, Business (all tracks), and Computer Information Systems (business-oriented CS type major). What’s the rationale for requiring calculus of these folks?

Let’s take accounting and business. As I mentioned in the comments to the first post, there are actually no courses required of the Accounting or Business majors that have calculus as a prerequisite. So criterion #2 appears not really to be in effect. If calculus were that important, you would see it in courses that are required of these majors to an unavoidable extent. As for criterion #1, is it really true to say that a person simply can’t be an accountant or a businessperson without a semester of calculus — in the same sense that a person can’t really be a mathematician if they haven’t seen abstract algebra? Is calculus really that central to an understanding of accounting or business? If it is, where is it in the prerequisite list for the rest of the curriculum?

It’s certainly the case that there are concepts in calculus that are of great use to people studying accounting or business — for example, the notion of concavity and inflection points. But are there so many concepts that a semester of limits, derivatives, and integration are necessary to get them? This is like saying that everybody needs to get a little protein in his diet each day, and on the basis of that fact, requiring people to eat a five-course steak dinner every night. It seems to me that the concepts can easily be taught in the context of accounting, business, or economics courses without requiring an entire course — most of which consists of something other than the concepts students need to get. I don’t think you require an entire course of somebody, only because there are half a dozen or so important concepts embedded in the course somewhere.

Bottom line here? I think we require calculus way too much, and usually for the wrong reason or for no reason at all. The result is not more students with a calculus background. The result is more students who enter calculus with an inadequate background, struggle through the course with insufficient motivation (and an inadequate background), and very often drop out of their majors — or out of college altogether — because they can’t get past the calculus course. These are not loser students. They are bright and perfectly capable of doing well in accounting, business, biology, or what have you — and in other kinds of math, like stats or “discrete math” — but calculus drives them away, and not for anything like the right reasons.

My own thoughts echo virusdoc’s — what we ought to be doing instead of just requiring calculus uncritically is looking at what kind of quantitative equipping students in a major need, and then providing it, in a way that is clearly relevant and has a low signal-to-noise ratio. In fact, one could argue that calculus as the primal first-semester math course in college is on its way out in terms of usefulness, and can (should?) be replaced by some combination of probability, statistics, linear algebra, and discipline-specific quantitative methods.

Please don’t misunderstand me — I love calculus, and it has a special place in my life. It was the course in high school that won me over to math, the first course I ever taught, and the one course more than all the others that I teach and blog about. But I want students to have a better experience both in calculus (for the ones who truly need it) and in math outside of calculus (for the ones who don’t).

Technorati Tags: Calculus, Curriculum, Education reform, Higher education, Math, Teaching

I find it interesting that you list computer science as one of the majors that “probably” requires calculus. I have a masters in C.S., and, though I’m glad that I took calculus, I can think of exactly one instance where I’ve actually

usedit. Even then, I used calculus in a non-traditional way: I was working on a program to display information and came up with a finite sum that computed the screen position of the output based on the input. After a few failed attempts to find a closed form for the sum, I used calculus tointegratethe formula, thus finding the general form of the solution. Once I had the general form, it was relatively simple to specialize it and use mathematical induction to prove that it was correct. This amused me even at the time because usually finite sums are thought of as approximations of integral sums, while I used an integral to get an approximation of a finite sum!đź™‚Amusingly, at a college meeting some time ago, someone proposed that a semester of calculus (real calculus, not watered-down fake calculus) be required for all B.S. degrees in the college of Arts & Sciences. The person who made that motion was a historian. Prettymuch everyone in the math department voted against the motion, mostly because we were justifiably concerned about either a) not getting the extra people we’d need to teach these new sections of calculus, or b) having to condense our calculus courses into a large lecture format.

If that wasn’t frightening enough, a colleague in the Philosophy department, bless his heart, asked me if it would be a good idea to require Abstract Algebra for all B.S. degrees in the college. If I recall correctly, my response was along the lines of “That would be an excellent idea…if you wanted to ensure single digit percentage graduation rates.”

In regards to the calculus for computer scientists discussions (hopefully this won’t derail the more general discussion), I supect the calculus requirement is more about demonstrating mathematical maturity rather than specific mathematical knowledge.

Most graduate CS programs require a 1-year sequence of calculus (along with discrete math, though this is often taught in CS departments as “dicrete structures”). Yet, as Justin indicated, actual use of calculus in CS is rare. Probability, linear algebra, number theory (and other areas of discrete math) are far more common. Graph algorithms come up more often integrals or polar coordinates.

I suspect part of the reason for requiring calculus is due to the standardization of calculus curriculum. You know what you are getting with someone who has passed Calc I and Calc II. There may be minor difference in the curriculum (did they cover polar coordinates or not?), but for the most part, you know they got through the main ideas behind derivatives and integrals. There is far less standardization in stats courses or discrete math courses. Some stats courses barely get beyond standard deviations. Does a specific discrete math course cover spanning trees? Recurrence relationships? Proofs by induction? There is too much wiggle-room to completely trust the results. Calculus, on the other hand, is set, assuming it’s not “business calc” or some other bastardized form.

Coderprof, a couple of years ago our CS guys redid the CS and IS curricula here based on some updated recommendations for the undergrad CS curriculum, which were put out by (I think) the ACM. They used this list of content and skills that, according to the ACM, were considered canonical for anybody getting a CS or IS degree. This list included not only math but also computing courses. For us, that’s what currently drives the inclusion of Calculus I in both majors (and Calculus II for the CS majors). It’s not exactly like an accreditation requirement, but we’re trying to deliver a curriculum that’s in line with what the ACM thinks is good. If ACM ever changed that list, I suspect we’d change our curricula to match.

So it seems like the people who care about what undergrad CS majors actually study, like grad programs, can just insist on a curriculum that is demonstrably built around these standards.

And I’m not sure what good the standardization of calculus does, if so little of the course is of actual value to the student. I’m also not sure just how standard calculus is these days; there’s a lot more variation in a first-year calc course than there used to be.

“thereâ€™s a lot more variation in a first-year calc course than there used to be.”

Really? I’m in my department’s calculus textbook committee and I’m in the middle of looking through a *lot* of calculus books right now. And I gotta tell you that the tables of contents are nearly isomorphic. The only major differences are slight rearrangements of topics (the most hairbrained of which, in my opinion, is talking about parametric equations at the very beginning).

That’s what we do — after the students have taken the math requirement (calculus or finite math). This has come up at the business school, and they looked into requiring finite (the math requirement is a university requirement for all graduates), but because the university has finite as an option for the math requirement, the business school cannot require it.

Too bad. Finite is far more useful to them because it covers probability.