How to Solve It in review

[January 1, 2003] - How does a teacher go about teaching? It wasn’t something that interested me particularly when I started an aborted minor in education a number of years ago. But it is certainly a question that has become more pressing on my mind as I have labored [and sometimes labored to teach others] in the computer trade press vineyard for 20 years; as I taught briefly at a northeastern university; and as I struggled anew with math, English and the like with my son. The trick seems to be to open the door slightly, turn on a light, and then point the learner’s attention at the favorable pathway.

It is a hard trick. That alone makes it interesting. But having an understanding of the process of learning is needed in science and art in order to freshly view the often-viewed, in order to see what has been there but which hasn’t been seen, or at least not identified and described.

All of which led me to pick up “How to Solve it” by G. Polya of Stanford University. Written and published in the ‘40s, and then again subsequently Polya’s “How to Solve It” is an attempt to describe the general paths to Eureka! moments. How does a teacher teach? Certainly not by doing the problems for the student. It is rather done by leading a student to the path to discovery. By developing strategies. That and the tactics that might lead to accomplishing that overall strategy.

Polya describes the walkway to problem solving:

• The First step is to understand the problem. What is the unknown? What is the data? What are the conditions?
• The Second step is to devise a plan to solve the problem. Often this means using an analogy that derives from already known solved problems.
• Step three is to carry out the plan, checking each step.
• Finally, in step 4, one looks back – and checks the result.

Although the step-by-step progression described here is pure Polya, none of it conveys the music in his pursuit of mathematical mysteries. I can only hope to convey it in side glances.

Here’s a brief quote that hopes to stand to describe his larger story: “There is a grain of discovery in the solution of any problem.”

Polya’s consideration of the Various Approaches to problem solving hangs on several key structural bands that take the forms of a teacher’s questions: Do you know any related problem? Do you know an analogous problem? [Parallelograms are considered.] Here is a problem related to yours and solved before. Can you use it? Should you introduce some auxiliary element in order to make its use possible?

These ring true to this recently mustered pedantic. Questions I know I ask of my son when we contest those bouts of studying in a winner-take-all World Federation of Algebra are: Have I seen this before? Have I seen something like this before? Are parts of this exactly like other problems in the book? Is there a formula I can apply? Don’t see it? Okay, here is the formula, #$@%&!!, now apply it, so I can get back to the Packers game! Putting it all together he tells me is the hard part, especially at test time. You can’t remember which to do to which. Life happens.

Our good author would be sympathetic here. We are, often, lucky to discover “a single analogous problem,” he writes.

Polya’s patron is Pappus, the second century Greek mathematician who pursued a branch of study he called analyomenos. Rendered by Polya as ‘Treasury of Analysis” or as “Art of Solving Problems.” This brings him to a consideration of the heuristic [involving or serving as an aid to learning, discovery, or problem-solving by experimental and especially trial-and-error methods, says Mr. Webster, adding there is some kinship of the term with the Old Irish ‘fo-fúair,’ or, ‘he found.’ [See a recent consideration of heuristics in Java application server systems.]

Polya’s actual treatise is just 30 pages; the associated ‘dictionary’ definitions section is quite extended, actually, making up some 200 pages. He describes going back to first principles in problem solving. January 1, 2003 is a day perhaps to remember such back tracking is sometimes in order.

RELATED
How to Solve It on Amazon.com
Biography of Pappus
Emerging technology: Agents, algorithms, heuristics, et al. -ADTMag.com, Dec. 1, 2002

Last update: 4/12/2003; 11:47:30 AM.