Four things I used to think about calculus, and what I’ve replaced them with
Notes: This post first appeared on my blog all the way back in 2009... 12 years ago?! It resurfaced this morning through a Twitter mention and as I re-read it now, having taught Calculus for 12 more years on top of the 16 years that was mentioned in the article, I can see how my own conception of Calculus and how it's taught has been an exploration of the trajectory that I laid out here. Read for yourself; I'm adding some updates at the end along with some edits in the middle.
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. Originally, despite having a math degree, I had absolutely no idea how it’s used; now, I'm learning from contacts and former students doing quantitative work in business and government. 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
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.
Some updates on this for 2021:
- Maybe the biggest change in the way I teach Calculus, and everything else, since 2009 is that back in 2009 I don't think I had thought much about the value of clear, measurable learning objectives and building one's courses around them. If I recall, this concept didn't hit me until 2010 when I started following my now-colleague John Golden on Twitter and he was writing about learning objectives – specifically, the idea that when we give an assessment, we should explicitly label it with the learning objectives that are being assessed. The moment I started being intentional about my learning objectives and aligning activities and assessments with them, I started seeing much more clearly that I was spending too much energy on topics that didn't matter. In fact I stopped thinking about my courses in terms of "topics" altogether – that's probably the biggest paradigm shift of my teaching career.
- My personal valuation of algebra skills in Calculus has steadily declined since 2009. Today I explicitly tell my students, over and over, that algebraic methods in calculus are all fine and good, but let's not put 90% of our energy into something that only is useful 10% of the time. This is a letdown for a lot of students who see Calculus as Algebra III, but it's the truth. Formulas are rare; tables and graphs aren't.
- I got a laugh out of re-reading the paragraph about topics that I jettisoned, and thinking back in 2009 I was going to get a lot of angry commenters saying How can you call yourself a mathematician if you don't teach _____ in calculus? Today, I'm, like: Dude, epsilon-delta proofs? Finding where a function is continuous by actually evaluating a limit using algebra? Are you serious? Save it for real analysis. Or graduate school. Otherwise, basically nobody cares and there are a lot of other concepts that actually signal understanding of calculus than these, that could use the time and space to breathe.
- I think the biggest shift in my thinking about calculus since 2009 has been a clearer vision of just how overrated calculus is, in the mathematics undergraduate curriculum. Don't get me wrong – I like calculus, I teach it at least 1-2 times a year, and it's what got me into math in the first place. But it has a pride of place in undergraduate math studies that it used to deserve but no longer does, a halo of relevance that's more of an artifact of pre-Information Age times than anything earned from actual use today. We still need to teach calculus because it is useful and relevant, also beautiful in its own way. But it no longer really makes sense to set up calculus as the backbone, front door, and main corridor to the undergraduate math experience. We need to be giving students mathematics that makes sense for the entire world they inhabit — including calculus in proper amounts, but also and especially linear algebra, discrete mathematics, and statistics among other things. That's why my department revamped the linear algebra course to make it a freshman-level introductory sequence that can be taken before calculus – to reclaim that front/center position for a subject that makes more sense for the 2020's.
Now cue the angry commenters.