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

Like many of the aphorisms we quote for guidance, this one is true, but not quite true if taken with the wrong sense of its words or at the wrong scale. First, there are different senses of the word "difficulty". Some difficulties are incidental, and some are essential. An author should indeed hide incidental difficulties; they only get in the way. However, the author must not hide essential difficulty. Part of the author's job is to help the readers overcome the difficulty. Second, we need to consider the scale of revelation and hiding. Authors who expose difficulties too soon only confuse their readers. Part of the author's job is to prepare the reader, to explain, inspire, and lead readers from their initial state into a state where they are ready to face the difficulty. At that moment, the author is ready to bring the difficulty out into the open. The readers are ready. What if the reader has already uncovered the difficulty before meeting the author? In that time, the author must not try to hide it, to fool his readers. He must attack it head on -- perhaps with the same deliberation in explaining, inspiring, and leading, but without artifice. It is this sense in which Galois has nailed a universal truth. If we replace "author" with "teacher" in this discussion we still have truths. The teacher's job is to eliminate incidental difficulties while exposing essential ones. Yet the teacher must be deliberate, too, and prepare the reader, the student, to overcome the difficulty. Indeed, a large part of the teacher's craft is the judicious use of simplification and unfolding, leading students to a deeper understanding. Sometimes, we teachers can use difficulty to our advantage. As I discussed recently, the brain often learns best when it it encounters its own limitations. Some say that is the only way we learn, but I don't think I believe the notion when taken to this extreme. But I think that difficulty is often the teacher's best source of leverage. Confront students with difficulty, and then help them to find resolution. Ben Blum-Smith expresses a similar viewpoint in his recent nugget on teaching students to do proofs in mathematics. He launches his essay with remarks by Paul Lockhart, whose essay I discussed last summer. Blum-Smith's teaching nugget is this:
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 find 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.
We 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! -----