The semester that was: A 3x3x3 reflection on Calculus
The 2020-2021 academic year is now over. It was certainly a year. Although I'm not scheduled to teach Calculus again until 2022-2023 at the earliest, I've been reflecting in "3x3x3" fashion on how my Calculus class went this semester, and here's what I've got.
Three things I learned
- Calculus is better when you decouple it from algebra. A few weeks ago I wrote about my open-technology policy in Calculus. I put that policy in place out of expedience (any other policy about technology would be impossible to enforce). But also, I wanted students to keep their focus on concepts, on crafting good solutions to problems, and on correct and meaningful explanations, and I felt they would be much more likely to do so if they didn't have to re-learn algebra on top of learning (or unlearning) Calculus. So for example, on this assignment, students explored the behavior of logistic functions, mostly using technology. Apart from taking the first derivative by hand, all computations were to be done on Wolfram|Alpha. This freed up brain space to focus on interesting questions: How come your logistic function doesn't have any critical numbers? When you look at the graph, why does this make sense? Does this mean it also has no inflection points? What's the y-coordinate of that inflection point, and does it have anything to do with the constant in the numerator of the function? The message I wanted to send to students was Professionals use tools to help them think and things are just more interesting and fun when you're not mired in algebra. (But, see below about questions I have.)
- Calculus does actually work in an online setting, in fact better in some ways than in a F2F setup. In the past, teaching Calculus face-t0-face, the use of computer tools to help students think was a battle. Students are used to math being all about hand computations and I always had this struggle with many students about there being "too many websites" (i.e. too much technology) being used, and why can't I just lecture and give them notes and worksheets? Online, there are no such arguments. I use only as much tech as needed to help master the learning objectives but otherwise it just goes unstated that we are doing what professionals do, i.e. using tools appropriately to help us think. It's actually hard to see how I could replicate the good parts of this course in a face-to-face setting where computers and the internet aren't ubiquitous.
- Calculus needs a diet. I have said that over the last year of teaching Calculus, I have removed a lot of previously-untouchable topics from the course without telling anyone, to cut things down to a defensible core so we can focus on a simple, single core narrative. The number of comments or complaints I've received about having done so — from students, my math colleagues, or our client disciplines — is zero. In fact nobody has said anything about my reverse pilot program at all. This makes me think that university Calculus courses, as currently constituted in most places, contain a lot of stuff that are at best niche topics — a personal favorite of some guy in the past who wrote a textbook — and at worst a dramatic waste of time and effort that distracts students from the real purpose of the course. We should probably be running reverse pilots like this on every course we teach.
Three things that surprised me
- Students were reluctant to use technology, even with no limitations. Maybe it's learned helplessness, maybe it's a lingering sense that using technology is against the rules, I'm not sure — but despite the open policy we had on technology use, I had to badger students to use it to check their work on take-home assessments (i.e., every assessment). If you do a problem asking to find and classify the critical numbers of a function, and you are allowed to check this with Desmos by throwing up a quick graph to see whether the local extreme values are where you say they are, to me it seems like manna from heaven — a pathway to mistake-free work. But not all students saw it this way, and there were many retakes of assessments done that could have been avoided by investing 90 seconds to examine a graph or check a calculation.
- Students resonate with integration a lot more than they do with differentiation. Our calculus course covers the usual gamut of limit and derivative content (modulo the stuff I removed) and then the basics of integration at the very end — just Riemann sums and the Fundamental Theorem. (Calculus 2 starts with u-substitution.) I'm not sure why, but students in both the fall section and in the one just completed love the stuff on integration — and they're good at it, flying through the learning targets and application problems. Maybe they're just sick and tired of derivatives after 10 weeks straight of it and it's just recency bias. Or maybe we have given differentiation a bigger role in Calculus than it deserves?
- Academic dishonesty was scarce. I've been teaching for 25 years, and I like to think I know academic dishonesty when I see it. And I really didn't see much of it, despite having no restrictions on technology. This kind of surprises me, but then again maybe not: Because of the tech use, the focus was on crafting good solutions and explaining the meaning of concepts and results. This kind of thing is highly Chegg-resistant and hard to fake. And because of the mastery grading system that's installed, if I sensed that a solution was not totally the product of a student, I would just mark it as "P" (Progressing; please revise and resubmit) and ask the student to explain what they were doing in more detail. Having an open tech policy, an emphasis on conceptual understanding, and a grading system that allows for revisions is much, much more effective at stemming academic dishonesty than proctoring software.
Three questions I still have
- Are we teaching the right things in Calculus, and in the right order? My Calculus courses have dramatically improved by removing things that appear to be inessential to the core narrative of the subject. So the question is, what should we be teaching in Calculus? What is that defensible core that we need to create and defend? And based on what I mentioned above about integration, is it possible that we should teach integration before differentiation? My colleague John Golden did exactly this in a Calculus class a few years ago, and it was fascinating. There is no ironclad law that says we have to teach Calculus in the same way, same stuff, same order as it's been done for 100 years or more.
- What is the narrative? Cutting down the size and scope of Calculus in order to focus on a single, simple narrative is a good thing. But what's the narrative? What is Calculus about, exactly? If you had to give a one-sentence description that is understandable to a first-year student, and which could be repeated over and over again throughout the course to explain why we are studying topic "X", what would it be? In my syllabus, I state that Calculus is "about modeling and understanding change". This isn't wrong but it also seems unhelpful and not very meaningful to the ordinary student. What's the right message?
- Will Calculus ever outgrow its connection to algebra and manual computations? Calculus has a branding problem. Many good students have come to view Calculus, because of prior experience, as Algebra III — a collection of algorithms and tricks, with no underlying meaning. They experienced Calculus in prior coursework as hand computations, got good at those computations, and now by focusing on concepts, problem solving, and so on we are messing up that good thing. I've found many students are more than happy to think more deeply about Calculus and relegate computation to computers. But many aren't so happy, and it makes me wonder what if anything can be done. At one point in the past I proposed renaming the Calculus sequence from "Calculus" 1, 2 and 3 to something like "Mathematical Analysis" 1, 2, and 3. That's not the right phrase to use because analysis already exists and it's not what I want first-year Calculus to become. But the idea is the same — maybe the only way to place the right focus on the course, and get students to stop thinking of it as a souped-up version of their AP class in high school, is to completely rebrand it.
Next up, a similar reflection on my other course, Modern Algebra.