Re-envisioning Calculus


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I had a really good couple of calculus sections last semester and another really good section last summer. Part of that has to do with the quality of the students I had — hard working and smart, even if not all of them had very deep mathematics backgrounds, and most importantly they all had a sense of humor about themselves which kept them from getting bogged down in despairing self-reflection when the course got difficult. I also think part of it has to do with a re-envisioning of how, exactly, I am framing the subject of calculus to these students and how I am approaching the way I teach it. I did some things differently, and I wanted to take this post to try to put into words what those things were.

One to approach teaching calculus is to make it a kind of mini-analysis class, something that would be quite “pure” in the classical mathematical sense, along the lines of Hardy’s A Course of Pure Mathematics. My Calculus I-II courses in college were like this, and it was the rigor and purity of that approach that lured me from my unsure position as a psychology major to being a mathematics major and then into being a mathematician. I didn’t fully understand the epsilon-delta proofs we were doing at the time, but I did understand that we were solidly against the idea of taking things for granted and were instead trying to establish absolute truths through logic and reason. This is an approach that still resonates with me today, not just in math, and that experience is one of the many reasons that my love of mathematics has grown stronger in a nonlinear way over the years.

But — it’s not exactly the approach that I want to take with my own students. My students are coming to calculus with a very wide variety of interests, backgrounds, and sets of assumptions about mathematics and college work in general. The most mathematically pure approach will not lead them to the greatest knowledge or appreciation of mathematics. So as much as that approach is pure and works for me, I don’t take it with them.

Another way is to go to the other end of the spectrum and make calculus about applications only and teach the rules of computation in a mechanistic sort of way. This so-called “brief calculus” approach does have its merits — you don’t miss the interesting array of applications — but also its demerits, particularly the fact that you’re not learning why things work. It represents an artificial simplification of the subject, simplification by sweeping stuff under the rug. That approach might “work” with my students in the sense that they could perform calculations correctly and maybe even apply what they learn in the book to new situations, but I don’t think that would represent a real grasp of mathematical knowledge.

Instead, I want to take a third way of teaching calculus that does not take much for granted — only the stuff that might be better left for later work once students get the main ideas — but still highlights the utility of the subject and gives students to tools to extrapolate what they learn to new situations.

Above all, I decided back before the summer that I wanted students to grasp the idea that, underneath the calculations and the epsilon-deltas, calculus is really based on some very simple questions and is itself a pretty simple subject when you peel away some of the layers of algebra and symbology. I think a course on differential calculus could, and perhaps should, be ordered by considering the following questions, one after the other:

  1. We know that quite often, two quantities are related in a cause-and-effect way. How do we represent such a relationship in a precise way and then use this representation to say things about the relationship? (Motivates the idea of a function and multiple representations and families of functions.)
  2. Having established a function relationship between two quantities, we can see through experience that knowledge of the amount of the dependent variable is often insufficient information for real problems. (Example: The value of a stock price; the location of a storm front.) Instead, we need to know the rate at which the amount is changing. What is the best way to measure the rate at which a function is changing? (Motivates the ideas of average rate of change and ultimately the derivative via the slope of the tangent line.)
  3. Having established that the slope of a tangent line is the best way to measure instantaneous rate of change, how can we measure it with as little error as possible? (Motivates the idea of the limit and leads to the algebraic derivative rules.)
  4. How can we create a function which will compute the derivative quickly and with as little error as possible? (Establishes the idea of the derivative function, as opposed to the derivative as a quantity — a very important difference that most calculus books do not stress nearly enough.)
  5. Having established that the derivative can be calculated by a function, what relationship is there between f(x) and f'(x)? And f”(x)? (Leads to the ideas of the Increasing/Decreasing Test, and then to relative and absolute extrema and concavity.)

These questions then fan out into the various applications of the derivative which can be covered pretty much whenever the class wants to.

The main thing about this approach is that it’s based on sensible questions that are easy to pose and straightforward to answer. Students seem a lot more ready to absorb new ideas if there is some sensible question to motivate the new ideas, rather than ideas appearing seemingly out of the blue. Structuring the course around questions (rather than “topics”) also helps students remember the big picture and provides a unifying framework to the course. That’s helpful, even comforting, when the algebra is flying hot and heavy and students just need to remind themselves what exactly they are doing here.

This approach is also agnostic when it comes to the level of rigor in the course. You could teach a course this way using epsilon-delta proofs to address the third question, or you could use well-designed diagrams or computer software, or whatever. You wouldn’t really even need to touch algebra in a course like this because none of these questions presuppose that we are using algebraic representations of functions. I’ve often thought that the best way to teach a calculus course to the kind of audience I serve is to address all of these questions once using only graphical and tabular representations of functions; and then do it all over again a second time using algebra.

That’s one major way I’ve re-envisioned teaching my calculus courses, and I’ll talk about some more ways later. In the meanwhile, your comments and brickbats, please?

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5 responses to “Re-envisioning Calculus

  1. Zac

    Thanks for the post, Robert.

    There is always a struggle to find the best way to introduce the concepts of calculus. If you don’t do the “first principles” stuff, some students complain that they don’t know where everything is coming from, but it usually leaves the rest of them (because of their poor algebraic manipulation skills) very anxious along the lines of “If this is the introduction to calculus, I sure don’t want to see when it starts getting hard!”

    I like the approach you are talking about. I refer to What is the best way to measure the rate at which a function is changing?. I’m not sure if you already do this, but it is valuable to get students to measure, then determine average velocity for an accelerating object for (say) the first second, then the second second, then the fifth. Then get them to think about how to determine the velocity at a specific time (say 2.8 sec).

    I used to find that students don’t have a good conceptual sense of “velocity”, “acceleration” and “average acceleration”, and if that is the case, they will have a nightmare trying to figure calculus.

  2. I indeed do the kind of thing you’re talking about re: measuring average rates of change. In fact I think every calculus book I’ve ever seen does something like this. But I have not often seen — in fact I can’t remember ever seeing — a calculus book that stresses this topic (average velocity) as anything but a thing that we do in calculus, rather than as an initial step towards answering an important question (What is the best way to measure rate of change?)

    Students can usually do the math on an average velocity calculation but very frequently cannot articulate the purpose of doing this and just plain miss the connection between average and instantaneous velocity. Or, they use average velocities over an interval size of their own choosing to calculate derivatives without wondering about minimizing error. The end is to calculate instantaneous velocity; the means of doing so is average velocity; but there’s no articulation that one thing is the means and the other the end. They’re just two topics among dozens.

  3. samjshah

    Interesting post! I’m teaching a non-AP calc class and I too am grappling with how to structure the course and teach it.

    I often get trapped in the muck and mire of algebra. A lot of times, I know my students are “getting” the larger conceptual bits, but struggle with the algebra. Or they get so consumed with the algebra that they lose sight of the big picture. It seems like that’s a common problem from most calculus teachers I’ve talked to (e.g. the “Why can’t my kids deal with fractions?!” syndrome).

    What I have to think about this summer (for next year) is rework my course to teach algebra skills AND calculus concepts at the same time — but divorces them initially and then merges them together. Like at the beginning of each chapter, we spend a few days going over the algebra skills needed in that chapter — totally NOT with any calculus ideas. Just algebra skills. Then we switch to “calculus mode” and then dive into the concepts.

    Right now I’ve been teaching them the algebra skills that they should know by now (but don’t) while I’m teaching them calculus. Let’s just put it this way: teaching how to find the derivative of \sqrt{x} using the formal definition was a nightmare.

  4. @samjshah: I hear you re: algebra. Actually I’m wondering if they way to change things is to do the opposite of what you’re suggesting — instead of reviewing algebra skills first and then teaching calculus ideas, teach calculus ideas first independently of any algebra at all and then introduce the algebra later once the students understand the main ideas. That’s the gist behind my comment about teaching calculus all the way through once using only graphs and tables, and then again with algebra.

    That won’t miraculously make students better at pre-calculus algebra, but it at least might provide some basis for believing the algebra is going to be worth it. As it is, students have to wade through all this algebra — all of which is ultimately necessary — just to get to the main ideas, which aren’t really about algebra at all.

  5. Al Gorgon

    I like the approach you are talking about. I refer to What is the best way to measure the rate at which a function is changing?. I’m not sure if you already do this, but it is valuable to get students to measure, then determine average velocity for an accelerating object for (say) the first second, then the second second, then the fifth. Then get them to think about how to determine the velocity at a specific time (say 2.8 sec).