TITLE: Learning Through Crisis AUTHOR: Eugene Wallingford DATE: November 20, 2009 3:35 PM DESC: ----- BODY:
... an author never does more damage to his readers
than when he hides a difficulty.
-- Évariste Galois
The impulse toward rigorous proof comes about when your intuition fails you. If your intuition is never given a chance to fail you, it's hard to see the point of proof.This is just as true for us as we learn to create programs as it is when we learn to create proofs. If our intuition and our current toolbox never fail us, it's hard to see the point of learning a new tool -- especially one that is difficult to learn. Blum-Smith then quotes Lockhart:
Rigorous formal proof only becomes important when there is a crisis -- when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind.This quote doesn't inspire cool thoughts in me the way so many other passages in Lockhart's paper do, but one word stands way out on this reading: crisis. It inspires Blum-Smith as well:
... what happens is that when kids reach a point in their mathematical education where they are asked to prove things, they findWe need to give our programming students problems in which the obvious solution, the solution that flows naturally from their fingers onto the keyboards, doesn't feel right, or maybe even doesn't work at all. There is more to the story; there is reason to learn something new. Teachers who know a lot and can present useful knowledge to students can be quite successful, and every teacher really needs to be able to play this role sometime. But that is not enough, especially in a world where increasingly knowledge is a plentiful commodity. Great teachers have to know how to create in the minds of their students a crisis: a circumstance in which they doubt what they know just enough to spur the hard work needed to learn. A good writer can do this in print, but I think that this is a competitive advantage available to classroom teachers: they operate in a more visceral environment, in which one can create safe and reliably effective crises in their students minds. If face-to-face university courses with domain experts are to thrive in the new, connected world, it will be because they are able to exploit this advantage. ~~~~ Postscript: Galois, the mathematician quoted at the top of this article, was born on October 25. That was the date of one of my latest confrontations with difficulty. Let me assure you: You can run, but you cannot hide! -----
The way out of this is to give them a crisis. We need to give them problems where the obvious pattern is not the real pattern. What you see is not the whole story! Then, there is a reason to prove something.
- that they have no idea how to accomplish what is being asked of them, and
- that they don't really get why they're being asked to do it in the first place.