Thursday, November 24, 2011



\displaystyle\int{\sqrt{1 + \sqrt{x}}}

The Calc I problem ends with:

Comment: The latest version of the big and powerful symbolic manipulation program Maple, available on most Rutgers systems, is unable to find this antiderivative (with the default settings for the program). Maple’s major competitor, Mathematica, considers the integral and returns

Mathematica could not find a formula for your integral. Most likely this means that no formula exists.

which is amusing.

Yes, hilarious.

The author has a common problem: he wants his students to sit down and grind through a few integrals but the students are getting bored and reaching for the computer. How do you convince them that learning to integrate by hand is worth the tedium?

Well, one thing to consider is that the students may be right. I'm not saying practice isn't worth something, I'm saying you're never going to run out of things to work on even if you delegate the monotonous tasks to the machines. There's always something more to do and your students might even discover something new. If you don't have the creativity to write problems that can survive the onslaught of technology, ask someone next door.

The thing is, even the author's attempted solution -- a tricky integral -- is only halfway to actually motivating the subject. Even if Maple can't do it (it can -- I checked, and so can Maxima, but that's besides the point), it's still just one random integral nobody cares about -- it's going to vanish by tomorrow.

Students are not used to being asked to create things of permanent value. They are asked to create temporary things that are sufficient to pass the grader's eye but will not serve them later. No matter how "real worldy" you make the problem, if the solution isn't generally useful, it's just going to vanish along with all the rest.

Let me emphasise that last point again because it's usually missed: it doesn't matter at all how realistic the problem seems, if the student isn't going to use their own answer again later, it's not something permanent. And the important part isn't that it will vanish, it's that when you know it will vanish, you approach it differently.

Do you ever notice that the people around you (and you too, though I was hoping not to rely on your own introspection) seem hesitant, nervous even to really finish something? This is because we are not used to permanence. We want the problem to have finite scope from the outset and vanish when it is complete. Do you notice that people have trouble grounding their thoughts in reality? If you're not producing something that lasts, why make it complete?

There was a chimpanzee named Nim Chimpsky whom researchers tried to teach language by positive and negative reinforcement. At first Nim seemed to be learning a lot of new words, then they found out he'd just been learning to the spec, imitating enough to pass the test, and had in fact learned nothing.

This is what we have been trained to do. Hey, if people are writing software to solve integrals, that's something of permanent value... how about having the students do that? Writing a program to solve a problem is better than solving it by hand 100 times because to make your code work you have to understand the steps completely. The compiler is unforgiving. I think this is often what scares teachers away from programming as part of their class -- they think they know, and want to believe, that the students understand the material, but the compiler is too revealing.

This is why you should use the compiler, though, because it breaks your illusions.

We are transient, our memories are transient, we forget often and dislike remembering, the best we can do is produce something outside ourselves that lasts.

1 comment:

  1. I think there's a misconception here. What you learn in school isn't the solution, it's the way to reach the solution. While the problem vanishes, the algorithm stays. That's why the fact that this integral is dumb doesn't matter.