The main problem with school mathematics is that there are no problems. Oh, I know what
passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here
is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad
way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question— that’s another thing.
How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a
number? How many ways can I symmetrically tile a surface? The history of mathematics is the
history of mankind’s engagement with questions like these, not the mindless regurgitation of
formulas and algorithms (together with contrived exercises designed to make use of them).
A good problem is something you don’t know how to solve. That’s what makes it a good
puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as
a springboard to other interesting questions. A triangle takes up half its box. What about a
pyramid inside its three-dimensional box? Can we handle this problem in a similar way?
I can understand the idea of training students to master certain techniques— I do that too.
But not as an end in itself. Technique in mathematics, as in any art, should be learned in context.
The great problems, their history, the creative process— that is the proper setting. Give your
students a good problem, let them struggle and get frustrated. See what they come up with.
Wait until they are dying for an idea, then give them some technique. But not too much.
Link thanks to andrewducker.