After some weeks of intensive calculus teaching (I teach one class in English and one in Swedish) to the backdrop of my discussions in the US, some ideas are beginning to jell. The English class is experimental, the Swedish is more traditional, following a textbook, although I do emphasize “ideas captured by formalism”.

Last week, in the English class, was spent on integration. The guiding problem was to compute the weight (mass really) for a sphere where the density varies with the distance from the center. Many students worked quite hard at it, but no-one came close to the standard solution of splitting up the sphere in concentric spherical shells. The data provided in the problem text was the weight of a sphere with constant density, the radius of the sphere: R = 4, and a formula for a linearly varying density ρ(r) = 18 – 2r. Units were provided in decimetres and kilograms per cubic decimeter.

Many things were tried by the groups, like integrating the formula for the density, putting the formula for the density equal to zero, just multiplying the volume by the varying density. Many students were juggling with the formulas (nothing wrong in that) but without any clear idea behind. I guided towards the “correct” solution, but no-one came near.

Now, this is of course a standard problem that in a normal class is treated in perhaps a few minutes. I bet almost no-one gets it. Mind you, I had talked at length about the paradigmatic example of the integral as the area under a curve, splitting up in rectangular strips et cetera. But that apparently does not transfer well.

I thought about this experience (I had expected it of course) on Monday morning, and added in two other things: (1) discussions and reading about how much of integration techniques should be taught and, (2) the need from applied courses that the students be able to set up integrals. I was tempted to conclude that too much emphasis is put on techniques and too less on:

- What kinds of problems are the integral the solution to?
- And how to use it in such cases!

It is nothing dramatically new about this, I suspect, but it came into sharper focus for me. Then another thought struck me, having taught differentiation in the Swedish class.

How useful is it, really, to spend a lot of time training to compute derivatives of complicated functions? I mean, really. Why not move emphasis to:

- What kinds of problems are the derivative the solution to?
- And how to use it in such cases!

Enthusiastic about this old, but at the same time new insight, I presented it to my friend and colleague Mats. In a traditional calculus class almost all the time is spent on calculational techniques, I said. In my mind, the first question:

- What kinds of problems …

is a humanistic question, it’s about the concepts and classical problems, the history and the philosophy, …

whereas the second question:

- And how to use it …

is a practical, applied question about skills, …

I said that this is what a calculus class should concern itself with. Down with the calculational techniques, I proclaimed!

Mats wasn’t that impressed and had a good answer or two. We split up, the bell was ringing, classes about to start.

Ok, what does this has to do with language? I’ve written before that I think mathematics should be taught as a language because it is a language. Now I saw this from another perspective. A good course in French must consist of three things (I believe)

- The culture and history of France
- The grammar of French
- How to use French in practice is various circumstances, reading, speaking and writing.

You see the analogy? (BTW, the classes my daughter is taking in Paris at the Sorbonne is precisely of that kind.)

The first question: What kinds of problems are French the solution to? The third question: How to use it in practice in such cases.

You don’t teach and learn French just by teaching and learning the grammar (item 2 above). Without item 1 and 3 it’s more or less pointless and useless. But isn’t this the way we often teach calculus? We teach the grammar and only occasionally mention the culture and history of mathematics and the practical use of mathematics.

I have started to play with the idea of completely breaking lose from the calculus teaching tradition and start out with a blank piece of paper, setting down a syllabus for “a perfect calculus class” taught humanistically as a language and with humanistic content.

The funny thing about this is – and I haven’t understood that until now – that it does not preclude applications. Quite to the contrary!

… and mathematical software will have a natural place (taking over much of the “grammar” part).

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I think the leap in dimension is too much for many students to make, since the problem is no longer readily visualized. I might suggest adding an intermediate problem such as determining the volume of a solid of revolution or determining the mass of a non-uniform disk. Both raise the order of the integrand but are still visualizable. (I have a similar background to you, having first studied chemical engineering before moving into education. My experience in engineering is that many people struggled when the Navier-Stokes equations were introduced.)

I have really enjoyed reading your posts, although I differ slightly on the value of teaching colleges.