*
So no, I'm not complaining about the presence
of facts and formulas in our mathematics classes,
I'm complaining about the lack of *

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:Mathematics isthe 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 yougaveit 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 bealive, damn it.

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.Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured.

I am also quite sympathetic to one of the other themes that runs deeply in this paper:By concentrating onwhat, and leaving outwhy, 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 isthe 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.

(Ditto for computer science.)Mathematics is about problems, and problems must be made the focus of a student's mathematical life.

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:... 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.

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. -----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 runtoward.