Teaching experiments in Calculus

I’ve started to teach a calculus class for our International Business Engineers. This is the fourth time around. I’ve done it differently each time, but never really been satisfied with what I have achieved (or the students for that matter). The problems are as I’ve written before: too weak mathematics background and two short time (7 weeks). How do you fit in: functions, limits & continuity, derivatives, integrals, optimization, approximation, curve sketching and differential equations in seven weeks. Well it can’t be done of course, but this is the situation we’re stuck with because someone did not think, but copied from memory, twenty years ago.

So in the spring, being inspired by a seminar I listened to, I decided to focus on 7 themes, one theme for each available week:

  1. Introduction and algebra
  2. Functions
  3. Derivatives
  4. Integrals
  5. Optimization
  6. Approximation (Taylor)
  7. Differential equations.

I cut away limits, only doing them

  • as they naturally appear doing the derivative. For integrals, I will rely on the fundamental theorem as taken for granted but motivated by area considerations and defining the integral in terms of the anti-derivative.
  • as they naturally appear when discussing continuity.

I drop curve sketching, only mentioning it in connection to max & min problems.

Still it looks impossible, so some careful design is needed. I decided to (as an experiment) have 8 hours of classes a week according to the scheme:

Mondays 3h Briefing and Introduction to the weeks’ theme and the weeks’ guiding problem (1h)
Examples (exercises 1h)
Theory (1h)
Tuesdays 2h Examples (exercises 0.5h)
Theory (0.5h)
Work on guiding problem in small groups (1h)
Thursdays 2h Examples (exercises 0.5h)
Theory (0.5h)
Work on guiding problem in small groups (1h)
Fridays 3h Presentations and discussion of work on guiding problem (1h)
Examples (exercises 0.5 h)
Theory (0.5h)
Debriefing and Outlook: What next? Philosophical questions about the nature of mathematics (1h)

This is indeed an onslaught of time in class. There is also half class tutorials (2h) and Matlab sessions (2h). This is not sustainable, so I regard it as an experiment. In any long-time version of this set-up, the course needs to be extended to 14 weeks and the number of hours cut down. But it will give time to experiment with different approaches. My idea is to “try every trick in the box”, and keep the ones that work.

“What is a “guiding problem”? I got the idea from a seminar (at the “Mattebron” conference in Goteborg during the spring. Mattebron is regular series of meetings between gymnasium and university mathematics teachers.) I listened to Dag Wedelin from Chalmers University of Technology describing his course “Mathematical modelling and problem solving”. I got the idea of devoting each week to a focused area of calculus (a theme) and designing a problem for each week that could force the students to think about that theme. The students should work on the problem in small groups during the week. I also got one concrete problem from the seminar: a problem suitable for the function theme. The rest I could think up myself (or copy from other sources).

I described this approach in some detail to Ranjan Roy at Beloit College and I thought about it now and then during my college odyssey. After talking to Leo Livshits at Colby College, I began to think in terms of “provoking a crisis” in the minds of the students. By presenting the students with an “easy to state but hard to solve problem” I could lure them to think – something that is sometimes painful. But in the comfortable environment of working with a few fellow students and without the grading threat hanging over them. Working on the problem for the simple purpose of learning some mathematics. This is my hope. I will describe the first guiding problem in the next post.

So what happened the first week? Well, I won’t report everything here but only the most interesting things. Things that happened that surprised me because I had not foreseen them happening.

My colleague Rustan Halldin, who have the tutorial groups each week, held exercise classes in algebra on Monday and Tuesday (while I hung out in Paris on my way home – thanks!). I met the students on Thursday and Friday and spent some time introducing calculus and the course. But since I want them to work in groups on the guiding problems I thought it might be a good idea to practice that with a “toy” problem just to see if it would work. So I had to device such a toy guiding problem.

Hmm …

At Bennington College, Andrew McIntyre had mentioned that he had given the students the problem to calculate the weight of a sphere with inhomogeneous density. I put that away in memory as a possible guiding problem. So now I thought about area problems.

Why not pose the problem to “Calculate the area of Australia”? I did that, writing it up as a real guiding problem would look. I gave it to the students and they did break up in groups of 3 to 5 students and they started to work. One group moved over to a neighboring empty class-room. Everyone seemed engaged. They worked for well over an hour and I passed around talking to them. I had some expectations but got more.

  • Many groups boxed in the continent within a rectangle. Of course they understood that this would overestimate the area. One group also put a smaller rectangle inside the coastline of the continent. Then they calculated these two areas and took the average. It came out too big. Then they drew a rectangle “eyeballing” the continent trying to balance sea inside with land outside. Then they got a quite good estimate.
  • Some groups worked with piecing together Australia from squares, rectangles, triangle and sections of circles.
  • Many groups ended up with a square grid in the obvious way. They all understood that taking smaller squares (finer grid) would give a better estimate.
  • One group thought about triangulating the area!

So their intuition was basically good. I think many of them realized that they would need a lot of data to do the calculation. Some realized that Australia is a curved surface on the globe. The somewhat arbitrary coastline came up in discussions as well questions about high and low tides. Et cetera.

We did this on Thursday. Friday morning on the bus to Borås I was thinking about what to do with all this. How could I connect this to calculus and in particular to the integral?

Obviously the area of Australia cannot be calculated by setting up a very complicated formula for the coastline and then trying to find the primitive function. Only a complete idiot would suggest that in class. But I had to come up with a connection that made sense. And the answer was all in the ideas that the students had.

There is a list of simple geometric patterns for which we can compute the area with a minimum of data: squares, rectangles, triangles, circles, sectors of circles, … Not very many. Irregular regions like continents need a lot of data to compute the area. The natural mathematical question is therefor:

How can we extend the list of geometrical shapes for which we can compute areas with a minimum of data?

This is one motivation for doing integrals.

It worked in class. I did other things as well during these two first days, but this is what I hadn’t expected and what surprised me. My teaching these two days turned humanistic. Less formulas – more ideas.

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2 Responses to Teaching experiments in Calculus

  1. Jeff Paules says:


    I stumbled upon your blog while I was looking for some materials on a class at Bennington College that was called Why Math? I have been a high school math and science teacher and currently work at the University of Oregon helping to connect the university science and math departments to high schools. One of the key issues I have been looking at is the motivation of students in math and science. I found this post a really compelling example of an authentic problem that could be used even for high school students, although the focus would be on geometry rather than calculus.

    Thanks for your blog.

    Jeff Paules

    (By the way, my daughter currently attends Skidmore College.)

    • Hi Jeff,
      It’s nice to see that I still have readers, although it’s quite some time since I wrote any new posts. I’m splitting my time between this project, research, teaching and book writing so there has not been much time. But I’m giving a talk on the subject of humanistic mathematics at a conference on engineering educational programs in Umeå, Sweden, next week. The text to that talk is in Swedish though. My report from the US trip will appear in the Journal of Humanistic Mathematics next year.

      I guess I should write something soon about how I try to teach humanistically in standard calculus courses.

      Skidmore is a great school to study at and Saratoga Springs a really nice small town!


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