Friday was the last day of classes for the fall semester. We have finals this week. Then, a big pile of grading. Yuck! But then, winter break. Yay!
This semester I taught three sections of calculus. More specifically, I taught our first semester calculus course intended for students with weaker mathematical backgrounds. Essentially, we stretch the standard one-semester calculus syllabus over two semesters and use the extra time to review algebra and trigonometry.
I have taught this course a number of times, and I have yet to find a fully satisfactory way of doing it. The textbook we use presents the material in the standard and logically correct way. It begins with a chapter reviewing various ideas from algebra: solving inequalities, finding equations of straight lines, that sort of thing. Then we come to a chapter on limits. Then derivatives. Then various applications of the derivative, such as curve sketching, related rate problems, and optimization problems. And then we go on from there.
In the past I have always done what most people do: Just plod through the textbook in order. Sadly, there are some serious problems with this approach, the main one being that it’s extremely boring. You begin with a big algebra dump, in which symbols are manipulated apparently just for the sheer joy of manipulating symbols. You then discourse at length about limits, which is a formidable and highly abstract topic for beginning students. You can imagine how cheated they feel when, after slogging through all sorts of gibberish about deltas and epsilons, they then discover that they can pretty much always just plug in the point when evaluating a limit.
And it is only after all of this tedium that you finally get to derivatives, which is, after all, the main concept of the course. After dutifully summoning forth all of the necessary rules, your reward is to rush through the word problems, which is both the most difficult and most interesting material you do all semester. By this time, alas, it is already so late in the term that you hardly have time to do it properly.
So, this term I tried something different. The first problem calculus seeks to solve is that of measuring the slopes of arbitrary curves. So let’s start with that! I began by talking about straight lines, which are especially simple sorts of functions in which measuring slope is a straightforward matter. Then I jumped straight to derivatives.
No algebra dump. No endless nattering about the nuances of limits. Why should I teach them how to solve an inequality until we are confronted with a concrete problem whose solution depends on solving inequalities? (Such problems arise naturally in calculus, for example when you are trying to determine the intervals on which a function is increasing or decreasing.) And why should I go on about limits when a glancing familiarity with the concept is sufficient for most of what they will be asked to do? If you’re whole world is continuous functions, as it is for most of the course, then limits are not an especially interesting or important idea. (Not with respect to the material in first semester calculus, at any rate.)
But don’t you need limits to define derivatives? Indeed you do. So my approach does require a bit of hand-waving when you get to that part. Some rigor must be sacrificed to do what I am suggesting. I have no problem with that. For beginning students, rigor is often the enemy of clarity.
Mind you, we did go back to the chapter on limits. After spending quite a few weeks luxuriating in the promised land of smooth functions, delighting in how civilized and well-behaved they are, we then started wondering what can be said about functions that are not so easily tamed. The formalism of limits is just what you need to characterize the different ways in which a function might be discontinuous, and we introduced it for that purpose. But I made only the briefest mention of deltas and epsilons, a fact for which I make no apology. In my view, it is just a flat-out pedagogical blunder to discuss that little topic in an introductory course. Deltas and epsilons are not “calculus.” They are “real analysis,” and that is a different course altogether. Isaac Newton knew nothing about deltas and epsilons, but he seemed to have a pretty good grasp of calculus.
In this post from last week we discussed some of the problems inherent in an over-emphasis on rigor. Sadly, dissing rigor is seen as something of a faux-pas in polite mathematical circles. Most textbook authors seem to think their job is done when they have presented their definitions, lemmas, theorems and proofs in a pristine logical sequence. That is why most of them are unreadably dull and largely incomprehensible to students. I hated slogging through those books when I was a student, and I loved math. Why should I expect my students, most of whom are lukewarm at best toward math, to find them any more engaging?
We mathematicians are constantly gushing about the beauty of our subject. We talk about the thrill of solving problems, and of finding unexpected beauty and order in the objects we study. So why is that attitude almost entirely absent from our textbooks? When did the emphasis switch from solving problems to strict logical rigor?
Of course, I am not at all suggesting that we should present no rigor at all. Strict rigor at one extreme, or rote memorization at the other, are not our only options. Rather, I am suggesting that the emphasis should be on presenting convincing arguments, not logically rigorous arguments. You want to present enough of the proofs so that they understand where the rules come from (while emphasizing that the rules have reasons, which is not something they necessarily realize at the start of the semester), but you don’t want to drown them in details they do not understand. Presenting too much rigor too soon is like presenting abstract set theory to eight-year-olds. They’re not ready for it.
So how did things go this term? Well, I didn’t work any miracles. I’m not saying I took a group of discouraged students and turned them into math majors. But I do think it went a good deal better than my past experiences with this course. I was much better able to focus them in on what was important, while avoiding extraneous details they will never see again in any subsequent course. I didn’t worry so much if there were nuggets here and there in the textbook that I failed to cover, and focused instead on telling them a story. I emphasized how each idea led naturally to the next, without sweating it if I didn’t dot every i and cross every t.
In short, I may not have persuaded them to love calculus or even to like it. But I feel quite certain that they have seen the real thing, and I did not bury the beautiful and clever ideas beneath a deluge of pointless rigor.