Teaching Calculus

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.

1. #1 Neil Rickert
December 9, 2013

Thanks.

That’s the way I have tried to teach calculus. I want students to see the derivative as a slope, or as a rate of change, not as something fitting a fussy rigorous definition.

Most, but not all, students have some sort of intuition about limits that’s good enough.

Rigor is fine in its place. But, if you want to give a rigorous definition of limit, then you owe it to your students to present counter examples to show what goes wrong with our intuition. It’s such examples that motivate the rigorous definitions, and our students deserve to see that motivation.

2. #2 KeithB
December 9, 2013

Have you tried professor e mc squared?
http://www.math.sjsu.edu/~swann/mcsqrd.html

3. #3 Michael Kelsey
SLAC National Accelerator Laboratory
December 9, 2013

Your approach sounds a lot like the “calculus for physics and engineering” I took at UCLA a lifetime or two ago (back when the lobby of the math building still had a pair of operational punch-card readers for submitting batch jobs).

At least for students who have already memorized things like ballistic motion equations (y = y0 + v_y t + (1/2)at^2, x = x0 + v_x t), learning about _why_ those equations work is much more effective than building up to them abstractly.

And for the non-physics students, using practical applications (slopes of lines or curves from high-school algebra, or practical things like car performance curves) also gives them hooks on which to hang their new knowledge.

As you imply, rigor is important, but getting the ideas across in an applicable way is often more important, especially when this is likely to be the students’ only set of math courses.

4. #4 MNb
December 9, 2013

“focused instead on telling them a story”
That’s always a good idea. And every beginning of the story on limits should contain the Ancient hare and turtle problem. The two have a running contest; the turtle start 10 m ahead. The hare can’t catch up, because when he has run 5m, ie half the turtle has moved forward a bit. Then the hare runs the half of 5 m, when the turtles has walked a bit more. Etc. etc.

5. #5 oldebabe
December 9, 2013

Ah, calculus. Apparently, tho, it’s just as hard to teach it (for Profs), as it is to learn it (for students)… as it is to retain an understanding of it (elderly)..

6. #6 rehana
December 9, 2013

I’d add one thing (and maybe you did)–that a slope isn’t necessarily just a geometric description but a rate of change of something you want to measure.

7. #7 Ned Rosen
December 9, 2013

So why is that attitude almost entirely absent from our textbooks? When did the emphasis switch from solving problems to strict logical rigor?
————————————
My guess is: because the big textbook orders come from large universities making a committee decision about which text to use, and what happens is that the texts that have fewest vociferous objections get chosen, in order to keep peace. So any “radically” different text, even with a number of strong supporters, is likely to be beaten by a book everyone can live with. Which means you get traditional topics, traditional order, traditional definitions, and mostly, traditional attitude that it is more important that a text book be a good (and mathematically precise) reference book than that it be readable for beginners.

You might look at the old Hughes-Hallett “reformed” calculus text. It may be out of fashion now (late 90’s was it’s golden age) but it approaches the subject a little bit more along the lines you have described, at least in the first few editions from which I taught ten to fifteen years ago. For example, there are lots of problems where you do calculus things on functions given only by a graph, a table of data, or a story, for which (* shudder *) no formula is given or derivable.

8. #8 Jim Harrison
December 9, 2013

I’d like somebody to do an updated version of Sylvanus Thompson’s approach in Calculus Made Easy, i.e. an explanation of calculus entirely based on applications. Unless you’re a mathematician, after all, you’re never going to fool around with epsilons and deltas.

9. #9 Poincare
Maryland
December 9, 2013

I would like to see the derivative developed by looking for approximations.

10. #10 Poincare
December 10, 2013

In a similar vein, on Dec 8, the NY Times published an editorial titled “Who Says Math Has to Be Boring?” (It was published online Dec 7)

The NYT says,
“One of the biggest reasons for that lack of interest [in STEM, i.e., Science, Technology, Engineering, and Math related fields] is that students have been turned off to the subjects as they move from kindergarten to high school. Many are being taught by teachers who have no particular expertise in the subjects. They are following outdated curriculums and textbooks. They become convinced they’re ‘no good at math,’ that math and science are only for nerds, and fall behind.”

“That’s because the American system of teaching these subjects is broken. For all the reform campaigns over the years, most schools continue to teach math and science in an off-putting way that appeals only to the most fervent students. The mathematical sequence has changed little since the Sputnik era: arithmetic, pre-algebra, algebra, geometry, trigonometry and, for only 17 percent of students, calculus. Science is generally limited to the familiar trinity of biology, chemistry, physics and, occasionally, earth science.”

1) A More Flexible Curriculum
2) Very Early Exposure to Numbers
3) Better Teacher Preparation
4) Experience in the Real World
There are over 900 reader comments on-line split into the the 4 listed categories and a general category.

11. #11 Another Matt
December 10, 2013

I have often taught music courses for non-musicians, and computer-music courses for musicians and non-musicians both. It’s exactly the same story — it’s so easy to get caught in the technical weeds that the students can lose focus, and by the time you get to the practical stuff after the technical preliminaries they’ve forgotten all the technical stuff anyway. One of the problems I have noticed in my syllabi, and which I think is a temptation for teachers in general is to try to build everything up logically from what seems like a secure “foundation” — we try to make a “rational reconstruction” of our own process of learning the material, trying to cull all the dead ends, and trying to find an order that moves through the prerequisites “in logical order,” so that when we get to something in the middle of the course, the student can learn everything about it using all the things it depends on logically. But that isn’t how we learn most of the time! Nobody ever starts with ZFC and builds up from that.

I think your idea of starting in the middle where the concepts are easiest to retain and apply is a good way to go.

12. #12 Jim
Texas
December 10, 2013

What you have identified is a second tier of student – one that needs to know calculus enroute to the understanding of something else. Engineers and some physical sciences need to know that there is a connection between the drama of mathematics and the world in which we work. Mathematicians – who will advance the field to new levels – need to know the inside understanding of how we arrived at this point. Your course is clearly targeted to the former, and will encourage them later in life to explore calculus more deeply. Keep up the good work!

13. #13 Niall
December 11, 2013

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.

I’m guessing it was Adams. I wasn’t enamoured of the topic order either, I just dived right into limits and the derivative.

In retrospect, I would have done limits differently. I would have removed piecewise functions entirely from the beginning of the course and just dealt with the limits of polynomials (alway continuous), rational functions (only a problem when denominator is zero, and algebraic functions (only a problem when argument of radical is negative). That would enable you to get to the definition of the derivative in week 1 vs week 2.

I also did not bother with epsilon-delta limit definitions. In fact, I don’t really “believe” in epsilontics as the be all end all of calculus. One concession I did make to the canonical style was to avoid all use of the concept and term “infinitesimal”, “infinitesimally small”, etc. I have since come to regret this, and wouldn’t do the same when teachign a calculus course again.

Along the same lines, when teaching integration, I plan to focus fully on anti-derivatives, area, etc, and barely discuss Riemann sums at all. As you say, you’re teaching calculus, not real analysis.

Personally, I think that analysis tends to be over-rated. When you go further in mathematics and discover even further techniques like Lebeque integration, divergent series, etc, you begin to realise that the supposed rigor of real-analysis either a) keep you in a box or b) is really just its own more particular “calculus” anyway, with requisite deus ex machina and exceptions to boot.

I’m going to teach the students how to integrate 1/(x^2+1)^(5/2) instead.

14. […] identify overarching laws and properties.  And in a third post, Rosenhouse explores strategies for making introductory calculus less boring for undergrads.  He writes, “Some rigor must be sacrificed to do what I am suggesting. I […]

15. #15 rork
December 12, 2013

#10: “most schools continue to teach math and science in an off-putting way that appeals only to the most fervent students” – that’s about K-12, not college.
I’m appalled at highschool math for nearly the opposite reason – withholding most of what is beautiful, boring any good students. Experience with my daughter’s texts:
1) sqrt(2) is irrational, no proof. The proof is enormously interesting. The fact is nearly irrelevant.
2) Your calculator computes “the best fitting line” – but book never said what “best” means. Why even bother?
3) volume of sphere is a formula to remember. In my mind I see a double-cone shaped like an hour-glass (height 2, radius 1) next to a sphere (radius 1) and that any (horizontal) slice though both created two areas that sum to the same area no matter where you slice. Only dementia could ever erase that picture from my mind. For kids to not see that demonstration is robbery. Archimedes spins.

In each of these examples (and I have more), if teacher claims the fact, I want every student to immediately demand “how do you know that” – instead not a single student asks that question. They’ve been trained not to think that way.

Tactic of build intuition first, and later ask “now how do we go about convincing ourselves” is OK, but what I see in High School is never asking the second question.

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December 17, 2013

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