Peer instruction and Calculus 2

May 4, 2012, 4:00 pm

Sorry for the boring title and lack of catchy image, but since my first post about the upcoming six-week Calculus 2 course, I’ve expended all my creativity getting the course put together and getting ready for Monday. In the earlier post I laid down some design ground rules for the course. Here, I’m going to say a little more in detail about what we’ll be doing.

It’s especially important on a highly compressed schedule like ours to use the class meetings themselves to jumpstart the assimilation process and then train students on how to carry that process forward as they go to work on the day’s material in the afternoon and evening. This is always an important goal of class meetings in any course — I’d go as far as to say that this is why we have class meetings at all. But when you cram a 14-week course into 6 weeks, it doesn’t take long for one incorrectly-assimilated concept to corrupt large portions of your knowledge, and it’s very hard to fix after the fact.

So for me, the natural choice for designing the class was to use peer instruction. Readers of this blog know that I am a fan of PI and that I’ve used it to some degree in every class I’ve taught for the last 3-4 years. I used PI in my Calculus 2 class in the Fall and I thought it was really successful in keeping students engaged and helping them learn the concepts of Calculus 2 — no small thing, since Calculus 2 tends to be heavy on calculation techniques and recipes. I’m making the decision to commit to using PI as the central organizing principle behind the class meetings, moreso than perhaps any course I’ve done up to this point.

If you’re unfamiliar with peer instruction, here’s a good basic explanation. Or if you have 80 minutes to spare, Eric Mazur’s “Confessions of a Converted Lecturer” video is well worth your time.

While I still plan on including lots of modeling of working through calculations and problems with students, the balance between using conceptually-oriented PI and mechanical calculations is going to tilt more in the direction of conceptual this time. This is because I think — and Mazur’s research bears this out — that students who have a stronger conceptual baseline knowledge in a subject are better at mechanical problem-solving than students who only see mechanical calculations in a lecture format.

Common sense bears this out too. Why do students have such a hard time doing volumes of revolution, for instance? Skill at calculating integrals by hand doesn’t seem to be the main culprit, because the issues start a long time before the integration happens. Students struggle with this particular problem, in my experience, because they lack a coherent principle from which to begin. Students think — I thought, when I was taking Calculus 2 back in 1989 — that there are about half a dozen very loosely-connected methods for finding the volume of something through integration, when really there is only one concept behind all of these: the concept of working with cross-sectional slices. If you really “get” the idea of slicing, everything in that chapter about applications of the integral suddenly makes sense because the separate instances fit into a coherent framework. A major job of the class meetings is to help construct that framework, and PI is the tool I will use to do that.

By way of example, here’s a PI question I will use on Tuesday. (Here’s a full-sized version.) This shows up in the section on the Fundamental Theorem of Calculus and is keyed to the essential concept that a definite integral calculates a signed area. It’s not necessarily a hard question — this is not really a hard concept. But it’s a pretty standard PI-type of question: Short, simple to grasp, and not overly focused on calculation. It really assesses whether the students understand some important points: that distance travelled is the integral of velocity, that integrals calculate area, and that area under the horizontal axis is considered negative. (In fact this might be considered a somewhat crowded question because it’s not just one concept being assessed here.)

PI implements more of my design principles than just not lecturing much and having students work on assimilation in class. PI gives me lets me assess “small, early, and often” and come away with actionable formative assessment data in real time. If a whole bunch of students get the wrong answer after two votes on that question above, then it means I need to change what I was about to do next. By the time students leave the meeting, they know and I know what they know — and we also know what they don’t know. PI also stresses collaboration and community through the rich and lively group work that results during the discussion phase. And finally, PI is perfect for stressing connections and concepts, which are hard to capture when they’re wrapped in a heavy-duty calculation.

Finally, a technical note. You might have noticed that I’m using Learning Catalytics for the clicker question. I’ve used TurningPoint clickers for some time now, and while I like TP’s products a lot, I felt like it was time to investigate the “bring-your-own-device” approach to classroom response systems due to try to keep expenses down for students. Learning Catalytics is web-based, so students use any web-enabled device that has a web browser on it to access a web site that serves the clicker questions and registers their responses. Yes, there’s a risk that not every student will have a web-enabled device. I sent an email out to the students this week asking them to let me know ASAP if this was an issue, and I got no responses. We do meet two hours a week in a computer lab, so every student can use the system then. If it comes to it, I can use Learning Catalytics to display clicker questions and then we can use a spare set of Turning Point clickers I have in a desk drawer. I hope we can use Learning Catalytics, though, because it really opens up some intriguing possibilities for audience responses.

We start the class on Monday, but I’m not done describing the setup. Next I’ll start going through the technical infrastructure of the class — our class blog (, online homework, grade book, and all that.

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