Problem solving & math as an art

Continuing on a litlte bit more with my thoughts concerning Lochart's Lament.

Lockhart starts out his lament with a comparison: WHAT IF music teaching only consisted of learning to write music, write notes on paper, and only after high school level would students be allowed to actually hear and make music? WHAT IT art instruction would consist of "paint by numbers" until high school, which is when they'd actually start applying paint...

Lockhart remarks that if he wanted to destroy a child's natural curiosity and love of pattern-making, "...I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education."

He calls school mathematics "pseudo-mathematics", where emphasis is on the accurate yet mindless manipulation of symbols.

These are, of course, very strong words. I don't fully agree... I don't feel all that's done at school would be pseudo-mathematics or mindless manipulation of symbols. Really, you CAN teach addition so that it makes sense, and we need to learn the symbols for it (5 + 6 = 11).

Lockhart uses the triangle example that I blogged about before to illustrate how mathematics is an ART:

"To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion -- not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive."

I can see where he's coming from... yet I wouldn't put down so harshly all that is done within school mathematics. There IS a place for drills, for computation practice, for studying algorithms, measuring units, and so on.

Lockhart says, "The main problem with school mathematics is that there are no problems." Here I agree. We should add GOOD PROBLEMS - or true problem solving to our lessons. I don't mean "exercises"...

"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?"

In other words, good problems do not simply practice a technique or idea that the student just saw used in an example. Notice: a good problem is such that you don't initially know how to solve it.

Now, this does not mean kids shouldn't solve "routine" problems or exercises, because ONLY by having KNOWLEDGE of techniques and concepts can new, non-routine problems be tackled.

But I'd recommend you take some time, perhaps a day every two weeks, where your math lesson consists of problem solving so that your students CAN experience this kind of problem solving process that leads them to conjecture, to investigate, to THINK hard, fail first but persevere, to justify their thoughts -- to come up with proofs.

It is of course even better if a teacher can lead the teaching with such good questions or problems even more often, such as starting the lesson with an interesting problem that leads to a new concept. But I realize not all of us can do that, and that it may take more time than direct instruction. If you can give them at least a glimpse of it sometimes, feel proud, because then you've done better than many.

... For good problems, check some resources here.