### 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.

Maria,
You are so right. Lockhart's "A Mathematician's Lament," was so beautiful and poignant it almost made me cry. (OK, I'll admit it, it did make this big guy cry). Thanks for blogging about it.
By the way, I think your blog is consistently the most helpful and generally best math resource for homeschoolers on the web. Keep up the good work, colleague!
Hotcha!
Brian (a.k.a. Professor Homunculus at Math Mojo)
I think a big part of Lockhart's point is that it is rare in the extreme for our school math curricula to give students the slightest clue as to what mathematics really is from an aesthetic perspective. That is often dismissed by some progressives and traditionalists as "fluff," or as something superfluous. I've had people who favor more reform approaches telling me that while it might be nice to give students an experience of what authentic mathematics is, we don't have time. And of course in the context of schools and NCLB, they are (kind of) right, more's the pity. But Lockhart is calling for teachers to be bold and to do more than make mathematics into calculation, repetition, and tedium.

I also think that the amount of drill we "need" to have kids do is directly proportional to how badly we teach the facts we drill them on in the first place. A creative and thoughtful approach to arithmetic facts DOES get at some deep mathematical ideas. Teaching even young kids from that perspective is NOT dull and can definitely allow the exploration of real mathematical work. Take a look, for example, at the work Michael Fellows has done with early elementary students and discrete mathematics (look at the THIS IS MEGA-MATHEMATICS site and the essays via the text version). It's inspired and inspiring mathematics and teaching.
Sudoku Maniac said…
math can explain beauty ... which is what art is all about.. I guess it is all symmetry
Sam said…
Although considering math as an art is what I think as best, I also think mastering the computational and formulaic portion play an important role in early mathematics development. Without the basic math skills, students will not appreciate or understand complex manipulations. It all seems to come down to appreciating the subject math more. Just to add something, I do not advocate for students to always rely on calculators to do computations.