2019 Spring Class Reflections: Calculus

Now that the term is over, I want to reflect a bit on the courses I taught, what worked well, and what I might want to do differently next time. (Honestly, it probably would have been more useful to write this sooner after finishing the courses, when they were fresher in my mind. But I don’t have a time machine, so I can’t do much about that now.) In this post I’ll talk about my calculus class; I’ll try to write about the others soon.

My previous course design had limited success

Math 114 at Occidental is intended for students, usually freshmen, who have seen calculus before but haven’t mastered the material sufficiently to be ready for calculus 2. This has the advantage that everyone in the course is familiar with the basic ideas, and that I can sometimes reference ideas we haven’t talked about yet to help justify what we do in the early parts of the course. It also has the disadvantage that my students arrive with a lot of preconceptions and confusions about the subject.¹

It also means that we have extra time available to learn about extra topics that are interesting or useful or just help explain the ideas of calculus better, even if those topics aren’t really necessary to prepare for calculus 2.

In past years I had used this extra time to do the epsilon-delta definition of limits. I’m still proud of having successfully taught many freshmen to write clean epsilon-delta proofs. But over time I came to the conclusion that this wasn’t the best use of class time.

I had wanted the epsilon-delta proofs section to accomplish two things: help my students learn to write and reason more clearly, and give them a taste of what higher math was like. Neither of these goals were complete failures, but neither was really a success either.

• My students got better at writing proofs, but I don’t think they learned this in a way that transferred skills to their other writing and communication. Beginner proofs tend to be written in a very restrictive, formal organization, effectively following a template. This template looks like it does for a reason, and is useful as a baseline for people to grow from. But in practice my students were just repeating the template to me instead of growing beyond it, so I don’t think they were gaining much.

• And my students got a taste of higher math, but I’m pretty sure it was an unfortunately bitter taste. Epsilon-delta proofs are actually pretty complicated things and especially hard for novice proof-writers to execute successfully, so they don’t make a great first experience in proofs.

• Making things worse, it tends to be really unclear why we need to prove any of these things. Most of the limit facts that come up in a first calculus course are “obviously true,” and so the effort we’re putting in often doesn’t feel like it’s actually accomplishing anything.² Proofs often come across as a particularly obnoxious hoop that I’m making my students jump through to satisfy some perverse math–professor urge. Ben Orlin makes this case pretty clearly: calculus 1 students haven’t run into any of the problems that epsilon-delta proofs were invented to solve, and so they seem like an unnecessary runaround.

• Most of all, it actually took quite a lot of time to do this well! Getting freshmen with no proof experience to the point where they could mostly write epsilon-delta proofs took a good three weeks out of a thirteen-week course. That’s a huge chunk of the course, and needs to be accomplishing a lot to justify itself. An epsilon-delta approach to limits just wasn’t worth the time and effort we were putting into it.

An approximate approach

Over time I realized that my course had gotten less focused on using the formal limits ideas anyway. I had drifted more and more to talking about two big ideas once we got out of the limits section: models and approximation.

Models are the big idea I’ve been thinking about lately.³ On its own terms, math is a purely abstract enterprise; to use math to understand the world we need to have some model of how the world can be described mathematically. This modeling is a really important skill for any field where you’re expect to apply math to solve problems—and the same skills can help reason about situations with no explicit mathematical model.

Approximation is the big idea of calculus. This is true on a surface level, where we can think of limits as taking an “infinitely good” approximation of the value of a function at a point, and derivatives are an approximation of the rate of change. But it’s also the case that many of the applications of calculus and especially of derivatives have to do with notions of approximation.

After some wrestling with both ideas, I decided to take the latter approach in this term’s course. It meshed well with the way I tend to think about the ideas in calculus 1, and the way I had been explaining them to students. So I reorganized my course into five sections.

1. Zero-order approximations: Continuity and limits. We can think of a continuous function as one where $f(a)$ is a good approximation of $f(x)$ when $x$ is close to $a$. A lot of the facts about limits we need to learn are answers to questions that arise naturally when we want to approximate various functions. And “discontinuities” make sense as “points where approximation is hard for some reason”.
2. First-order approximations: Derivatives. We started with the linear approximation formula $f(x) = f(a) + m(x-a)$ and asked what value of $m$ would make this the best possible approximation. A little rearrangement gives the definition of derivative, but now that definition is the answer to a question, not a definition just dropped on our heads from the sky. We want to be able to compute derivatives so that we can approximate functions easily, and as a bonus we can reinterpret all of this geometrically, in terms of the tangent line.
3. Modeling: Word problems and differential equations. We reinterpret the derivative a third time as an answer to the problem of average versus “instantaneous” speed, and then as the answer to all sorts of concrete “rate of change” problems. We can talk about the idea of differential equations, and practice turning descriptions of situations into toy mathematical models with derivatives. We can’t solve these equations explicitly without integrals, but we can approximate solutions using Euler’s method, and get a good definition of the function $e^x$ in the bargain. Implicit derivatives and related rates also show up here, using derivatives in a different type of model.
4. Inverse Problems: Inverse functions and antiderivatives. We take all the questions we’ve asked and turn them around. We define inverse functions, especially the logarithm and inverse trig functions, and use the inverse function theorem to find their derivatives. We can use the intermediate value theorem and Newton’s method to approximate the solutions to equations. We finish by defining the antiderivative as the (not-quite) inverse of the derivative.
5. Second-Order approximations: The second derivative allows us to find the best quadratic approximation to a given function. This is a natural setting for thinking about extreme value problems, so we cover all the optimization topics, along with Rolle’s theorem and the mean value theorem, and then put all this information together to sketch graphs of functions. We finished up with brief explanations of Taylor series and of imaginary numbers.

Most of it worked pretty well.

This course was basically successful, but there are lots of ways to improve it. I think my students both had a more comfortable experience and gained a much better understanding of some of the core ideas of calculus, especially the basic idea of linear approximation.

The first section, on limits, was okay. It’s still a little awkward, and I’m tempted to Ben’s approach of starting with derivatives entirely. But I really liked the way it started, with making the point that $\sqrt{5}$ is “about 2”. This simplest-possible-approximation made a good anchor for the course, and helps reinforce the sort of basic numeracy that helps us understand basically any numerical information we learn. I still need to do a bit more work on the logical flow and transitions, and the idea of limits at infinity is important but doesn’t sit in here entirely comfortably.

The section on derivatives and first-order approximations worked wonderfully. This is the section that contains many of the ideas driving this course approach, and I’ve used many of them before, so it makes sense that this worked well.

The section on inverse functions again worked pretty well. It’s pretty easy to justify “solving equations” to students in a math class, and “this equation is too hard so let’s find a way to avoid solving it” is pretty compelling.

And finally the section on the second-order stuff felt pretty strong as well, but could still be improved. While in my head I have a clear picture relating “approximation with a parabola” to the maxima and minima of a function, I don’t know that it came across clearly in the class. And I was feeling a little time pressure by this point; I really wish I had had an extra couple of days of class time.

Modeling is hard

But the section on modeling needs a lot of work. A lot of the ideas that I wanted to include in here aren’t things I’ve ever taught before, so the material is still a little rough. I also got really sick right when this section was starting, so my preparation probably wasn’t as good as it could have been.

In particular, I wasn’t very satisfied with the section on describing real-world situations in terms of models, and coming up with differential equations. I showed a bunch of examples but don’t know that we really got a clear grasp on the underlying principles as a class. And my homework questions on this modeling process probably contained a bit too much “right answer and wrong answer” for a topic that’s as inherently fuzzy as modeling.

I’m toying with the idea of assigning some problems where I ask students to argue for some modeling choices they make—handle it less like there’s one correct model, and more like there are a bunch of defensible choices. But I don’t know how well I can get that to fit in to the calculus class and the framework of first- or second-order ODEs.⁴ (Maybe I should do some modeling that doesn’t involve derivatives since understanding modeling is a goal on its own.)

I also wish I could fit the mean value theorem into the discussion of speed, but proving it really requires a lot of ideas I wanted to hold off on until later. Maybe I should state and explain it here, but then prove it later when the proof comes up for other reasons.

One thing I did really like in this section is the way I introduced the exponential $e^x$ as the solution to the initial value problem $y’ = y, y(0)=1$. This makes $e$ seem less like a number we made up to torture math students, and more like the answer to a question people would reasonably ask again.

Final thoughts

Overall, I feel pretty good about this redesign. I’m definitely not going back to the epsilon-delta definitions for this course any time soon, and I think this course will be really strong with a bit of work.

But there are a lot of ideas in the modeling topic that are important but that I don’t quite feel like I’m doing justice to yet. I need to go over that section carefully and figure out how to improve it.

I’m also thinking about moving some of my homework to an online portal. If we take all the “compute these eight derivatives” questions and have them automatically graded, I can use scarce human-grading time to give thorough comments on some more interesting conceptual questions.

To anyone who’s read this entire post, I’d love your feedback—on the course design as a whole, and on how to fix some of the problems I ran into. And if anyone is curious how I handled things, I’d be happy to share my course materials. You can find most of them on the course page but I’m happy to talk or share more if you’re interested!

Have ideas about this course plan? Have questions about why I did things? Tweet me @ProfJayDaigle or leave a comment below, and let me know!