My ten-year-old has wanted to be a scientist since he was four, but he's bored by math. Paul Lockhart (PDF) helps me understand why. But what do I do now?
The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such. Everyone understands that poets, painters, and musicians create works of art, and are expressing themselves in word, image, and sound. In fact, our society is rather generous when it comes to creative expression; architects, chefs, and even television directors are considered to be working artists. So why not mathematicians?
Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science-- perhaps they help the scientists with their formulas, or feed big numbers into computers for some reason or other. There is no question that if the world had to be divided into the "poetic dreamers" and the "rational thinkers" most people would place mathematicians in the latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
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"The area of a triangle is equal to one-half its base times its height." Students are asked to memorize this formula and then "apply" it over and over in the "exercises." Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time-- there is nothing left for the student to do.[...]
Now let me be clear about what I'm objecting to. It's not about formulas, or memorizing interesting facts. That's fine in context, and has its place just as learning a vocabulary does-- it helps you to create richer, more nuanced works of art. But it's not the fact that triangles take up half their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems-- something a mere statement of fact can never give you.
By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it's actually much worse than this-- at least it's understood that there is an art of sculpture that I am being prevented from appreciating).
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the "truth" but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity-- to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs-- you deny them mathematics itself. So no, I'm not complaining about the presence of facts and formulas in our mathematics classes, I'm complaining about the lack of mathematics in our mathematics classes.
SIMPLICIO: But don't we need third graders to be able to do arithmetic?
SALVIATI: Why? You want to train them to calculate 427 plus 389? It's just not a question that very many eight-year-olds are asking. For that matter, most adults don't fully understand decimal place-value arithmetic, and you expect third graders to have a clear conception? Or do you not care if they understand it? It is simply too early for that kind of technical training. Of course it can be done, but I think it ultimately does more harm than good. Much better to wait until their own natural curiosity about numbers kicks in.
SIMPLICIO: Then what should we do with young children in math class?
SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don't worry about notation and technique, help them to become active and creative mathematical thinkers.
Teach them to read Lewis Carroll, and history, and then they can discover that the fundamental rules of the entire universe are really mathematical rules :) But at some point, they need to learn logic, and that's not easy. So playing games is good, but analyzing games is better, and designing new games based on your analysis is even better than that! Then you repeat the whole cycle!
Yesterday, I was at a talk at the local conference of the Mathematics Association of America at the University of Pittsburgh. Art Benjamin (http://www.math.hmc.edu/~benjamin/) gave a talk, and I highly recommend that you show your son some of Art's "mathemagic". After several examples, I learned a brand new way to square 2 and 3-digit numbers in my head - It's a neat trick! I learned how to make birthday magic squares, and how to show lots of things about them! Now I'm trying to figure out how to generalize the method - his final bit was to square a five digit (five different people in the audience each called out one digit) number in his head - out loud... I'm still amazed.
Bored? That can be cured :)
I do think I made some progress when I introduced my son to Scratch (a free computer programming language for kids) and helped him start creating a game of his own. I'll look into the Art Benjamin material.