Bernie on calculus

First off, I've become very wary of Amazon's reader reviews ever since I realized that they remove negative comments in order to boost the ratings of the books. That's not kosher. [Catherine speaking: I posted 2 5-star reviews on Amazon that have disappeared, so I'm not sure Amazon has a systematic policy against negative reviews....]

Ok, what's the big deal about Calculus? Why are there thousands of Calculus books and none of them any good?

The reason is that the subject is simultaneously too big and too deep. And there's really no good way to split it up into manageable digestible pieces.

If you want to understand a computer, say, you can split it into pieces (power, case, motherboard, plug-in cards) which are you can then study and understand separately. But with Calculus, learning the subject is more like approaching a huge ship in the fog. At first you don't have any idea what is there. Then a few points become clear, but they are disconnected and make no sense. Then a few structures show themselves, and gradually, very gradually, the whole thing starts to come together. It takes much more energy and much more determination to carry through with such a program than with simpler subjects. So most people don't carry through with it, and it becomes a filter, a flunk-out class.

Linear algebra is a much more useful subject which is amenable to being broken into manageable chunks, and perhaps for this reason it doesn't carry the same mystique as Calculus.

Let's lay out what Calculus is in order to make this clear. It consists of two new operations called "differentiation" and "integration"--roughly analogous to subtracting and adding--both of which are based on a totally new view of the world, called "limits". Limits are a pretty deep concept, much deeper than is generally supposed or understood by most people taking Calculus. In fact, I would venture to say that most people taking Calculus never really grasp limits and, as a result, end up more confused and resentful about mathematics than when they started. Moreover, limits cannot be tackled until one has already achieved a certain mastery of both algebra and geometry, for they entail a melding of these two subjects. Both subjects must have been learned down to the "have it at my fingertips" level before limits will start to make sense.

To be perfectly honest, the problem is even worse than that, because I think it's fair to say that in some sense the human race doesn't really understand Calculus yet. This is because, although there is complete agreement on what basic Calculus is and how to use it, there is still sharp disagreement on what the logical underpinnings of it should be. It's really kind of like Quantum Mechanics in this regard, and that makes it quite unlike all the other kinds of mathematics young students have ever seen, which is all cut and dried.

So, to take the larger view once more, Calculus has three aspects which the student must master more or less simultaneously: 1) the mechanics of integration and differentiation and limits, 2) a philosophical understanding of limits, 3) the thing we discussed yesterday--an understanding of the underlying meaning of the formalism of Calculus in terms of real-world problems. Because there is so much interconnected stuff to learn, the connection between formalism and real-world meaning is even more tenuous, and must be held in even greater abeyance, than is the case with standard school mathematics. The student must suspend disbelief for a much longer period than ever before. Which means that there are inevitably many more Calculus students who get left by the wayside than occurs in elementary mathematics.

It is generally accepted among mathematicians that the hardest part of learning Calculus is 2), the philosophical part, and therefore the teaching of Calculus is usually broken into two subjects, taught to two different groups. "Mechanical Calculus" (high-school Calculus) is taught to students who are deemed too hopeless to ever really learn it deeply. Almost all standard Calculus taught to freshmen college students is of this kind. The students are only taught the basic formulas for differentiation and integration and some of the applications are shoved down their throat. Limits are hand-waved and never really explained, and most students don't realize there's a problem. They're just left with a vague feeling of uneasiness. If they're engineering students, then they are drilled on the applications for another 3 or 4 years, so that they become quite good at them, without worrying too much about what it all means. It works, why worry about it?

For students believed to be budding mathematicians, the whole subject is taught, with an emphasis on the meaning of limits and being able to deeply understand the logical underpinnings of the whole enterprise, i.e., to do proofs. Applications are only lightly touched upon. That's the audience Apostol's book is written for. That's a completely inappropriate book for almost all people.

The mechanics of Calculus, i.e., the basic formulas for integrating and differentiating, aren't really that big a deal except for one fly in the ointment. They are operations applied to functions rather than operations applied to numbers, which is all that the students have ever seen before. So even here there is a philosophical hurdle, because it's hard for people to think of functions as objects. We are used to thinking of functions as the "verbs" of mathematics, not the "nouns", so operating on them seems very strange and most young students probably never really grok it. It's yet another philosophical nut to chew on before one can really understand what one is doing with Calculus. It takes time for that fact to sink in.

The single most important obstacle precluding most students from mastery of Calculus is that they don't really have any idea what functions are when they start Calculus. And that's usually because they don't have a firm grasp of algebra. This, however, is a solvable problem. I personally would reorganize the curriculum so that a year is spent just messing with functions before Calculus is tackled.

But of course that runs headlong into the problem that people in high school and college--unlike students in elementary school--have very little desire to suspend disbelief: if they can't see an immediate payoff for what they are learning right now, they don't want to learn it. This leads to a quandary for the teachers/professors, namely, in order to motivate the students they have to tell them the applications. But in order to do the applications, the students need the full machinery of differentiation and integration. This leads inexorably to the continual cycle of Calculus "reform" which changes textbooks every couple of years, seeking to do the undoable by squeezing in years of difficult philosophical struggle and mechanical practice into far too short a time period.

There's also the problem that many of today's soccer mothers and fathers want to push their children into Calculus as quickly as possible in order to put another feather in their own cap, so they have no tolerance for an extra year "wasted" on learning functions. But that's a subject for a different thread.

Back to main page.