TITLE: Math and Computing as Art AUTHOR: Eugene Wallingford DATE: July 02, 2008 10:23 AM DESC: ----- BODY:

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.
-- Paul Lockhart

A week or so ago I mentioned reading a paper called A Mathematician's Lament by Paul Lockhart and said I'd write more on it later. Yesterday's post, which touched on the topic of teaching what is useful reminded me of Lockhart, a mathematician who stakes out a position that is at once diametrically opposed to the notion of teaching what is useful about math and yet grounded in a way that our K-12 math curriculum is not. This topic is especially salient for me right now because our state is these days devoting some effort to the reform of math and science education, and my university and college are playing a leading role in the initiative. Lockhart's lament is not that we teach mathematics poorly in our K-12 schools, but rather that we don't teach mathematics at all. We teach definitions, rules, and formal systems that have been distilled away from any interesting context, in the name of teaching students skills that will be useful later. What students do in school is not what mathematicians do, and that's a shame, because mathematicians is fun, creative, beautiful -- art. As Lockhart described his nightmare of music students not being allowed to create or even play music, having to copy and transpose sheet music, I cringed, because I recognized how much of our introductory CS courses work. As he talked about how elementary and HS students never get to "hear the music" in mathematics, I thought of Brian Greene's Put a Little Science in Your Life, which laments the same problem in science education. How have we managed to kill all that is beautiful in these wonderful ideas -- these powerful and even useful ideas -- in the name of teaching useful skills? So sad. Lockhart sets out an extreme stance. Make math optional. Don't worry about any particular content, or the order of topics, or any particular skills.
Mathematics is the music of reason. 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, damn it.
I teach computer science, and this poetic sense resonates with me. I feel these emotions about programs all the time! In the end, Lockhart admits that his position is extreme, that the pendulum has swung so far to the "useful skills" side of the continuum he feels a need to shout out for the "math is beautiful" side. Throughout the paper he tries to address objections, most of which involve our students not learning what they need to know to be citizens or scientists. (Hint: Does anyone really think that most students learn that now? How much worse off could we be to treat math as art? Maybe then at least a few more students would appreciate math and be willing to learn more.) This paper is long-ish -- 25 pages -- but it is a fun read. His screed on high school geometry is unrestrained. He calls geometry class "Instrument of the Devil" because it so thoroughly and ruthlessly kills the beauty of proof:
Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured.
His discussion of proof as a natural product of a student's curiosity and desire to explain an idea is as well written as any I've read. It extends another idea from earlier in the paper that fits quite nicely with something I have written about computer science: Mathematics is the art of explanation.
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.
I am also quite sympathetic to one of the other themes that runs deeply in this paper:
Mathematics is about problems, and problems must be made the focus of a student's mathematical life.
(Ditto for computer science.)
... you don't start with definitions, you start with problems. Nobody ever had an idea of a number being "irrational" until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction.
Problems can motivate students, especially when students create their own problems. That is one of the beautiful things about math: almost anything you see in the world can become a problem to work on. It's also true of computer science. Students who want to write a program to do something -- play a game, predict a sports score, track their workouts -- will go out of their way to learn what they need to know. I'm guessing anyone who has taught computer science for any amount of time has experienced this first hand. As I've mentioned here a few times, my colleague Owen Astrachan is working on a big project to explore the idea of problem-based learning in CS. (I'm wearing the project's official T-shirt as I type this!) This idea is also right in line with Alan Kay's proposal for an "exploratorium" of problems for students who want to learn to commmunicate via computation, which I describe in this entry. I love this passage from one of Lockhart's little dialogues:
SALVIATI:     ... people learn better when the product comes out of the process. A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own. SIMPLICIO:     Yes, but before you can write your own poems you need to learn the alphabet. The process has to begin somewhere. You have to walk before you can run. SALVIATI:     ... No, you have to have something you want to run toward.
You just have to have something you want to run toward. For teenaged boys, that something is often a girl, and suddenly the desire to write a poem becomes a powerful motivator. We should let students find goals to run toward in math and science and computer science, and then teach them how. It's interesting that I end with a running metaphor, and not just because I run. My daughter is a sprinter and now hurdler on her school track team. She sprints because she likes to run short distances and hates to run anything long (where, I think, "long" is defined as anything longer than her race distance!). The local runners' club leads a summer running program for high school students, and some people thought my daughter would benefit. One benefit of the program is camaraderie; one drawback that it involves serious workouts. Each week the group does a longer run, a day of interval training, and a day of hill work. I suggested that she might be benefit more from simply running more -- not doing workouts that kill her, just building up a base of mileage and getting stronger while enjoying some longer runs. My experience is that it's possible to get over the hump and go from disliking longs runs to enjoying them. Then you can move on to workouts that make you faster. So she and I are going to run together a couple of times a week this summer, taking it easy, enjoying the scenery, chatting and otherwise not stressing about "long runs". There is an element of beauty versus duty in learning most things. When the task is all duty, you may do it, but you may never like it. Indeed, you may come to hate it and stop altogether when the external forces that keep you on task (your teammates, your sense of belonging) disappear. When you enjoy the beauty of what you are doing, everything else changes. So it is with math, I think, and computer science, too. -----