Chameleons

The old man stared at us and spoke thus: “Those were the days, in Mod Island… We lived surrounded by beautiful colorful chameleons… I remember that, the day when I reached, 17 of them were blue, 15 were red and 13 were yellow. But, whenever two chameleons of different colour met, they would start a funny dance after which both would change to the colour which neither of them had. For example, if a blue and a red chameleon met, after the dance they both would become yellow. These dances took place for a long time, until all the chameleons became the same colour, and we knew it was time to leave…”.

“You’re lying”, said Alice. The old man grinned… “How are you so sure?” And Alice replied back, “Because, with your numbers,  it’s impossible that all chameleons become the same colour”.

Is Alice right?

 

Three chameleons for the math-kings under the sky...

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Teaching fresh(wo)men calculus

I have just finished, for the second time, the calculus fall term for engineering freshwomen (and freshmen) ;) at UC3M. The classes were in English, split into two for practical sessions, around 70 students in total. It was a nice group (yes, some of my students will read this and no, I do not say this because of that… the teacher evaluation polls are already over) ;) I have been thinking about what do we teach, what is its purpose and how we should do it… and I have reached a few conclusions.

• There are two reasons to teach maths to non-mathematicians: (a) because they will need some tools which are standard in their trade or (b) because they should learn to think, they should learn real problem solving techniques. The contents of the calculus term (derivatives and integration in one variable, basically) is already covered in high school, only a few new things are taught here (Taylor, polar coords…) So, the real reason must be the second one.
• That’s why I have introduced two novelties: first of all, problems in “real life format”. With this I mean that they’re formulated vaguely, with no data. For example: “I want to leave my can of beer on the ground, but it is irregular and I’m afraid it might fall down. I should drink a little bit of beer so that it becomes more stable. How much?”
• Another point that was important for me was the ability to give numerical solutions, approximations… I mean: to obtain numbers even when an analytical solution is not available. We also introduced numerical calculation techniques via octave, but it had to be out of the class hours.

My only complaint: such a course, if it has to be taught correctly, requires a rather low number of students per class. When I teach linear algebra, it’s ok for me to talk about eigenvalues to 120 students. That’s because the idea is fully different. I don’t know how to teach problem solving from the blackboard. There are always a few lucky cases in which you have to teach nothing: they already get your point, almost before you’ve finished stating it. With the rest, we normal mortals, it has to be done one by one…

Another important point. I would like to change the “blocks”. There should be four of them

1. Visualization: sketching functions, curves in polar coordinates or parametric, surfaces… And the reverse: see data and “guess” an analytical expression. Fitting experimental data.
2. Computing: approximation schemes, estimation skills. Tayor, mean value theorem… And numerical programming skills.
3. Optimization: all sorts of problems where some target function has to be maximized or minimized. There are few “real life” problems which can not be re-cast in this form…
4. Cutting into pieces and pasting back: (for want of a better name) with this I mean all kinds of problems which “reduce” to integration: areas, volumes, lengths, work of a force, average of a function, etc.

Calculus at this level can be seen by the students as a bunch of tricks. And they’re right. All of us making a life as “applied mathematicians”, we have a bag of ideas that come to our mind when we see a new problem. Applied mathematics is just that: the ability to tackle a new problem, to make the “right metaphor” with another problem that you solved years back.

Just a finishing remark: why do we have so few girls??????? I want a convincing answer, or I’ll move to nursery school next year! And I’m serious about that!

Is this the reason?