Past 2000 views – the blog is living!

Dear all,

I’ve been away from the typewriter for a week or two due to much teaching and end-of-term work and negotiations over a theoretical physics paper and a few other things. But the blog has anyway gone past 2000 views!

I hope soon to be back with new postings – and feel free to comment also!

All the best

Anders

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Calculus and the language metaphor revisited

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)

  1. The culture and history of France
  2. The grammar of French
  3. 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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In sharper focus: Ideas

I went to America because I was interested in humanistic approaches to teaching mathematics – not just talking and writing about it, but practicing it – in practice (so to speak). What I found I have written about in other places on this blog. But here it is again.

I found mathematics taught in a humanistic way, sometimes with humanistic content, but certainly in a context conducive to “bildung” (the Liberal Arts environment itself). I had no idea that it was this that I should find. So I certainly know much more now as compared to before the odyssey.

But this also points to an embarrassing fact. Now knowing quite a lot about American mathematics teaching – I realize I know very little about mathematics teaching at the university level in Sweden! I say this because what has come into sharper focus for me might be obvious to everyone else in Sweden. I wouldn’t know.

Here is what:

  • Focus on ideas and concepts – not formalism and manipulation. That is, make formalism dependent on ideas. The ideas and concepts of mathematics can be understood by closing your eyes and thinking with pictures and words. The symbols and the formalism then tries to capture this in a way that allows us to reason and calculate quantitatively in a reliable way.I’ve known about this for many years. But now it has moved into the center of my attention. And I realize that for almost every student I meet it is a completely new, alien and unbelievable thought.

Hey you out there! In particular in Sweden. Is this completely obvious to you? Have you been teaching like this all along? Why didn’t you tell me (yes, I’m a slow thinker)?

Now I’m so curious that I’ve started to think about travelling in Sweden to study university mathematics teaching!

  • Next thing to put into sharper focus will be the transfer problem. Hang on.
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Third week of teaching calculus

The third week was devoted to differentiation. I motivated the derivative briefly with it being a concept that captures the ideas of

  1. rates of changes
  2. linearization (tangent lines)
  3. finding local max and min

Then we dug into the definition and the derivatives of the elementary functions and the rules of differentiation. All this is of course standard stuff, very technical and mechanical, a kind of drudgery that has to be done. I proved (at a level that ought to be convincing to the audience) some of the rules:

  • Derivatives of small natural number powers of x. The students then did “square-root of  x” and “1 over x” as part of the weeks guiding problem(s). Hand-waving took care of general power functions.
  • I differentiated exponential functions b^x, doing the crucial limit in Matlab on the OH-projector. It is nice to see the limit approach 1 as you chose b = 3, 2.7, 2.71, 2,718 et cetera. Real nice – this went down well I think.
  • I also did the natural logarithm.
  • Once again I learned the bitter lesson that arguments for the Leibniz rule, an in particular the chain rule, are hard to get across. Suddenly there is an explosion of formalism, and this even though I shaved off a lot of hair. Well, hard to get across, but not impossible. I think I see how to do it in a clear way next time around.

So it was a week of quietly working along. More importantly though, I realize that I’m actually teaching in a different way now, after my trip to the US. It was in the making before – I mean, I did go there for a reason – but quite a few things that I’ve been thinking off for the last few years or so, came into sharper focus. I will devote the next post to that.

BTW, working with a naive theory (common sense) of limits works well for a course like this. It saves a lot of time. The cases where it breaks down has to be dealt with as they occur.

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Thanks to everyone

It’s time to start to write some kind of report on my college odyssey. So I thought I should start by thanking everyone in the US that made this trip of mine such a pleasant and exciting experience. And why not at the same time put together some statistics.

My warmest thanks (for interesting discussions or sitting in on classes, in many cases both) to

Paul Campbell, Ranjan Roy, Bruce Atwood, David Ellis, and Tatiana Dmitrieva at Beloit College, Beloit, WI.

Deanna Haunsperger, Stephen Kennedy, Sam Patterson, Gail Nelson, Eric Egge, Andrew Gainer-Dewar and Brian Shea at Carleton College, Northfield, MN.

Karen Saxe, David Bressoud, Chad Topaz and Daniel Kaplan at Macalester College, St. Paul, MN.

Paul Zorn and Ted Vessey at St. Olaf College, Northfield, MN.

Susan Colley, Robert Young, Michael Raney, Jim Walsh, Michael Henle and Kevin Woods at Oberlin College, Oberlin, OH.

Paul Melvin at Bryn Mawr College, Bryn Mawr, PA.

Thomas Hunter at Swarthmore College, Swarthmore, PA.

Joshua Sabloff at Haverford College, Haverford, PA.

Mark Huibregste, Gove Effinger and Michael Eckmann at Skidmore College, Saratoga Springs, NY.

Andrew McIntyre, Michael Reardon and Andrew Cencini at Bennington College, Bennington, VT.

Jan Holly, Richard Fuller, Scott Taylor, Fernando Gouvêa, Ben Mathes, Otto Bretscher, Justin Sukiennik, Andreas Malmendier, Leo Livshits and Scott Lambert at Colby College, Waterville, ME.

Meredith Greer, Bonnie Shulman, Catherine Buell, Pallavi Jayawant, Peter Wong and Benjamin Weiss at Bates College, Lewiston, ME.

Robert Benedetto and David Cox at Amherst College, Amherst, MA.

Stanley Chang, Alexander Diesl, Oscar Fernandez, Karen Lange, Andrew Schultz, Jonathan Tannenhauser and Ismar Volic at Wellesley College, Wellesley, MA.

And thanks to the students I talked to at Macalester, Oberlin and Bennington. And all the other people I just met briefly.

Also thanks to my friends Jean Capellos, Sarah Goodwin, Steve Goodwin and Bob DeSieno. Special thanks to Sheldon Rothblatt who encouraged me to go ahead with the project and who gave me the hint to contact Lynn Steen who also gave valuable encouragement.

For financial support: Thanks to my institute, the School of Engineering, and in particular to Anders Mattsson and Hans Björk for supporting the project. Thanks to Längmanska Kulturfonden for additional support.

Statistics

I visited 11 colleges (Beloit, Carleton, Macalester, Oberlin, Bryn Mawr, Skidmore, Bennington, Colby, Bates, Amherst and Wellesley) and met people from 14 colleges (add to the list: St. Olaf, Swarthmore and Haverford).

I sat in on 25 classes.

I gave 3 seminars along the way (at Beloit, Bryn Mawr and Skidmore).

I saw a lot of north-east US, many small towns and beautiful countryside, and ate a lot of good food. And the beer wasn’t bad either. What a wonderful country!

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Second week of teaching calculus

The week passed roughly according to my plan. The guiding problem, which is central to my approach, worked almost as intended. The students worked on it in groups on Monday and Tuesday. On Wednesday I had planned an extra algebra session. Since there were so many good questions on Tuesday about power functions, I had to use some of the Wednesday time for polynomials to keep up the pace. Teaching in the way I do invites questions and these questions reveal the depths of possible misunderstandings. Not tackling them is no way forward.

I also talked (on Wednesday as planned) about the axioms for the real numbers as the basis for algebra. And I had time to talk about irrational numbers and the fact that you need an algorithm in order to calculate the square-root of 2, because √2 is just a symbol satisfying √2·√2=2. I used an algorithm to compute successive approximations. So it all became very concrete. I think it worked well.

On Thursday, I decided to use all the time for a thorough discussion about exponential functions and logarithms, rather than having group time on the guiding problem.

A difference between Swedish university and American college is the fact that it is not compulsory to attend class (except laboratory classes and such). The students may come and go as they please (which they do). This means that if you start out with say 40 students in a class, the number actually attending drops down to say 25 after about two weeks. The half-life for group work of the kind I’m trying to have is also on the order of a few days. This is something you have to live with in Sweden. The teaching environment of college is more like the Swedish gymnasium. But I will continue with the guiding problems and try to convince them about to usefulness of them.

On Friday I ended with the trigonometric functions and we discussed the guiding problem. So we had a detailed run through of the elementary functions. And we had an outlook towards the next week which will be about the derivative.

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Proofs in a teaching context

What is a proof?

Or what constitutes a proof in a teaching context?

What’s the value of proof pedagogically?

Do proofs explain anything to students?

These are questions that I’ve been thinking about, and I was reminded about them today during a short discussion in the corridor. But it is too late to write about it tonight. What do you think? I’ll be back.

Update

Being a theoretical physicist, proof does not come natural to me. We derive things. We work in models. We live in big frameworks of theory such as Quantum Mechanics and General Covariance and we try to construct new interesting models. We use mathematics and we trust mathematics.

But, as I have written elsewhere, I find mathematics deeper and basically more intriguing, than theoretical physics. And I do teach mathematics, and I enjoy its history and philosophy. So I struggle with proof.

In my teaching I explain mathematics. Lately I started to experiment with explicitly proving things. So I’ve had to come to grips with what constitutes a proof in a teaching context. This is my answer:

A proof is a logical argument that’s convincing to the audience.

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