# Four things I used to think about calculus, and what I’ve replaced them with

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I’ve been teaching calculus since 1993, when I first stepped into a Calculus for Engineers classroom at Vanderbilt as a second-year graduate student. It hardly seems possible that this was 16 years ago. I can’t say whether calculus itself has changed that much in that span of time, but it’s definitely the case that my own understanding of how calculus is used by professionals in the real world has developed, from having absolutely no idea how it’s used to learning from contacts and former students doing quantitative work in business amd government; and  as a result, the way I conceive of teaching calculus, and the ways I implement my conceptions, have changed.

When I was first teaching calculus, at a rate of roughly three sections a year as a graduate student and then 3-4 sections a year as a newbie professor:

• I thought that competency in calculus consisted in the ability to think through difficult mechanical calculations. For example, calculating $\displaystyle{\lim_{x \to 9} \frac{9-x}{3-\sqrt{x}}}$ using multiplication by the conjugate was an essential component of learning limits.
• There were certain kinds of problems which I felt were inseparable from a proper understanding of calculus itself: related rates, trigonometric integrals, and a few others.
• I thought nothing of calculus that didn’t involve algebra. I’m not saying I held a low opinion of numerical or graphical calculus problems or concepts; I’m saying I didn’t even have them on my radar screen. I spent no time on them, because I didn’t know they were there.
• Mechanical mastery was the main, and in some cases the sole, criterion for student learning.

Since then, I’ve replaced those criteria/priorities with these:

• I care a lot less about mechanical fluency in algebra and trig, and I care a lot more about whether a student can read a problem for comprehension and then get an optimal solution for it in a reasonable amount of time and using a reasonable method.
• I don’t think twice about jettisoning any of the following topics from a calculus course if they impede the students’ attainment of the previous bullet point: epsilon-delta proofs of limits*, algebraic limits that involve sophisticated algebra tricks that students saw five times three years ago, formal definitions of continuity, related rates problems, calculation of integrals using limits of Riemann sums, and so on. I always want to include these, and I do it if I can afford to do so from the standpoint of managing class time and maximizing student learning. But if they get in the way, out they go.
• I care very much about whether students can do calculus on functions of all shapes and sizes — not only formulas but also tables of data and graphs — and whether students can convert one kind of function to the other, and whether students can judge the relative pros and cons of doing calculus on one kind of function versus another. The vast majority of functions real people encounter are not formulas — they are mostly evenly split between tables and graphs — and it makes no sense to spend 90% of our time in calculus working with formulas if they are so rarely the only option.
• I don’t get bent out of shape if a student struggles with u-substitution and the like; but it drives me up the wall if a student gets the units of a derivative wrong, or doesn’t grasp that a derivative is a rate of change, or doesn’t realize that the primary purpose of calculus is to quantify what we mean by “rate of change”. I guess that means my priorities for student learning are much more about the big picture and the main ideas than they are the minute, party-trick algebra/trig calculations.

Perhaps the story would have been different if I’d remained tasked with teaching calculus to an all-engineer audience. But here, my classes are usually 50% business majors, about 25% biology or chemistry majors, and 15% undecided with only a fraction of the remaining 10% being declared majors in mathematics (which includes students in our 3:2 engineering program). But that’s the story as it is, and I’m sticking to it.

* Technically I never have to omit these, because we don’t do them in our intro Calculus class here.

Filed under Calculus, Life in academia, Math, Teaching

### 10 responses to “Four things I used to think about calculus, and what I’ve replaced them with”

1. samjshah

Ummm… I want to have all your teaching materials! I am currently in a middle state between those two. I like my calculus students to learn some algebra along the way, but the way I am doing it doesn’t keep the “big picture” in mind. I feel like I lose the concepts for the minutae and my goal for next year is to keep the forest for the trees. (My current idea is to teach 2-3 days of algebra skills needed for a unit and THEN teach the unit.) In any case, if you have created any resources (e.g. your spreadsheets post, worksheets, projects, concept questions, activities, PPT/Smartboard), throw ’em online! We’ll all be grateful!

Sam

• Will Farris

Now, Sam, aren’t you teaching high school calculus such that there is considerably more time available for working through problems than is available in the college setting? I have found that the success students generating the correct answer to a problem through manipulative algebra builds necessary confidence. That confidence is required for a student to know that she understands at least something about the subject. From there you can build a bigger picture, perhaps. With added time in class comes that added advantage or looking at more of the minutae.

I think some things are better learned bottom up and others top down. Do you learn the trees and then mensuration or vice versa? Is this a learning styles issue or a general question of pedagogical philosophy?

Alas, since the great majority of students will never really use the minutae past formal education better to teach the ontology of calculus than the clever tricks of solving problems.

Best case: there needs to be 3 calculus tracks, one for the math majors, one for the quantitative technical majors, and one for the general case. Surely this is a viable way to straighten out the mess.

2. Thanks for sharing this very interesting and useful summary of your shifts in thinking about teaching calculus. I think these are very appropriate shifts, and your comment about basing them on hearing from contacts and former students about “real world” uses of calculus lends a lot of credibility to these shifts.

I’m particularly intrigued by your point about doing calculus on functions represented as tables and graphs. That’s a learning goal I haven’t that much about, but it sounds entirely appropriate.

One question: If the primary purpose of calculus is to understand rates of change (as you note in your final point), then why not include related rates as a key topic? I find that related rates problems give students a useful way to understand the idea that more than two variables and their rates of change can be related, which is useful for “getting” the idea of rates of change and sets up differential equations nicely.

Caveat #1: I often teach engineers (at Vanderbilt, no less), so it’s important to prep my students for DE.

Caveat #2: Students can easily be distracted by algebraic nastiness in related rates problems, so the problems need to be well-chosen to surface concepts not bog students down in computations.

Thoughts?

3. sumidiot

Thanks for this post! You’re saying the things I’ve been thinking about a little, and giving those thoughts some credibility – and encouraging me to persue it. And I always wonder how calc (or anything close) is used in the real world, but I know so few people there 🙂 I asked my sister, doing bio research at John’s Hopkins, and she said she didn’t see any calc in what she did. I don’t want to believe her.

I was also a little curious about @Derek’s question: why no related rates?

4. I’d rather have all of it… but I guess there are real world constraints. Is it common to have such a variety of majors in one class?

When I went to a big University, we had separate calc for different majors.

When I went to a littler engineering school, it was 70% engineers, 10% math, 20% science.

But nothing as mixed up as what you describe.

Jonathan

5. @jd2718: At a small liberal arts college, the kind of mix of students, majors, high school backgrounds, etc. that I describe is pretty typical. It’s challenging but it’s fun to have to work up examples and problems from a variety of different areas.

@sumidiot and @derek: I was planning on making a quick post today or tomorrow (now that classes are over) on related rates problems. The short version is, I do always include *some* treatment of related rates, but how extensive I make that treatment I usually leave up to the situation. I have a “budget” plan for doing related rates which basically fits within what I normally teach for the chain rule and doesn’t bring in a lot of algebra (for example, under that plan I do not teach implicit differentiation as a separate topic) but does do IMHO a good job of teaching the core idea behind using the derivative to calculate one rate in terms of another. I use the budget plan when I am short on time, the students are short on algebra skill, or both. Like I said, hopefully more to come on that soon.

6. Barry Garelick

Algebra can get in the way, as you describe, for students whose mastery of algebraic techniques and concepts is weak. While it is true that calculus is not about algebraic “tricks”, reading literature is not about phonetics either. Picking problems that minimize algebra so as to bring out the main concepts of calculus seems to be accomodating the bad math backgrounds many students bring with them.

A glance at a calculus book from 1942 is instructional in this regard. It was written by Harold Bacon, a math professor at Stanford who taught the first year calculus class (for which the text was written) from the 40’s until the 70’s when he retired, and was a revered and loved professor. His book is interesting in the level of algebra and trigonometry that he assumes students know coming into the course. The book doesn’t skimp on concepts at the expense of mechanics.

I agree that there are different levels of calculus that should be taught according to the students taking it. Having a class of mixed ability students would hamper teaching, just as the “full inclusion/mixed ability” practice in K-12 is hampering the teaching of mathematics in general.

7. i not understand well, couse i from indonesia…..
i just take your post to may home work…
thanks..

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