Some examples of the type of problems we confronted indicate the abilities in the kids that were being cultivated. The problems require a strong algebraic and geometric intuition for solution but they are not from a pattern of like problems. They each have some unique aspect and hence their solution requires coming up with an approach that fits the problem rather than applying some pre-developed approach. As I've written in an earlier post, by the time I got to college I was somewhat vexed by where the approach that works comes from. But I do know I have some aptitude for it. I gave myself a smile by working on the problems on this sheet. I was able to do those, pretty quickly.

The fundamental skill is asking the question, "what do I need to do to solve this problem?" and then trying varying approaches that might produce a result. This is definitely not trial by error. That will never get you to the answer. This is trial by intuition. There is a "smart guess" about what might be a promising direction. And then there are the various math facts that we can apply to see if that guess yields an answer.

Consider this particular problem. Two independent random variables are uniformly distributed on the interval between 0 and 2. What is the probability that the absolute value of the their difference is less than 1?

My first guess was to write down the double integral and then perform the necessary integration. After about 30 seconds, I scratched that. The bounds of integration were the key and they seemed a bit tricky. But that was a clue to the next guess. Graph it. Draw the 2 by 2 square. Draw the region where that difference is less than 1. That turns out to be the main intuition. The next one is to draw the main diagonal of the square. To the right of the diagonal x is bigger than y. That happens half the time. Figure out the area in that case, then double the answer. After that it's a little algebra to plot x - y = 1 and then to realize that instead of computing the area of trapezoid that is a little hard to do one can compute the area of a complementary right triangle that is very easy.

At each step in the process one looks for --- is that it? do I see the answer? If not, then one looks for --- is there something simple that I know how to do that will help me make progress? And if not then one usually bails out and tries something else. The process is not foolproof. Sparks of intuition are required. The intuition is what makes it fun. It is discovery.

When I write, I go through something similar. What topic should I choose for the Blog today? Is that interesting? Why? What do I have to say about it? This old book I've mentioned before,

*Learning by Teaching*by Donald M. Murray, makes essentially the same point about writing. Perhaps the one big difference with writing is that the problem wasn't posed by someone else. So there is the choice of topic question with the writing. But really, especially if you are writing a brief essay like my blog posts rather than writing an entire book, there is not that much difference in the choice of topic than in the choice of method of approach to a unique problem formulated by someone else. My own view is that studying math in high school is a very good way to learn how to write.

In a recent Op-Ed piece in the Times, Stanley Fish articulated quite a different view about learning how to write. But taking sides in the form versus content "controversy" doesn't seem to be getting at the root of my process. As part of my process, after a sentence is written I do think you have to ask, "Does the sentence say what I had meant it to say?" Answering that, at least for me, is quite similar to answering whether I've figured out the math problem or if I have to work on it more. I don't think about subject, verb, object. To me, that is paralyzing.

In my view of the world both the math problem solving and the writing represent what we want of someone who "thinks critically," though that expression might include other modes of thought as well. One key way that they are unified is that these are solitary activities and that we associate such critical thinking with being off on your own, scratching your head, staring at wall, looking inward for the answer. Sure one can work through the math problems with others, parent with kid, teacher with student, student with other student, but the idea that each contributes to helping the other "see" the answer and in that way the solution is "negotiated" is contrary to my experience.

The current practices we seem to be promoting with learning technology - even inquiry based approaches - seem to give short shrift to introspection. I wonder what Mr. Conrad would think about the "constructivist" approaches that so many are advocating for online instruction. I'm afraid he'd not be in favor. Is there a way to reconcile the Mr. Conrad view with constructivism or should we be fearful about the negotiation in group approach in that we might actually be hurting students by not pushing them hard enough to see where their own intuitions will take them?

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