A Mathematician's Lament

by Paul Lockhart

  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 that 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, to 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.

  If your art teacher were to tell you that painting is all about filling in numbered regions, you would know that something was wrong. The culture informs you —there are museums and galleries, as well as the art in your own home. Painting is well understood by society as a medium of human expression. Likewise, if your science teacher tried to convince you that astronomy is about predicting a person’s future based on their date of birth, you would know she was crazy—science has seeped into the culture to such an extent that almost everyone knows about atoms and galaxies and laws of nature. But if your math teacher gives you the impression, either expressly or by default, that mathematics is about formulas and definitions and memorizing algorithms, who will set you straight?

  The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.

  And I haven’t even mentioned the lack of mathematical criticism in school. At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement; that works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter? Of course there’s no criticism going on in school—there’s no art being done to criticize!

  Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles? I think it’s simply that we as a culture don’t know what mathematics is. The impression we are given is of something very cold and highly technical, that no one could possibly understand—a self-fulfilling prophesy if there ever was one.

  It would be bad enough if the culture were merely ignorant of mathematics, but what is far worse is that people actually think they do know what math is about—and are apparently under the gross misconception that mathematics is somehow useful to society! This is already a huge difference between mathematics and the other arts. Mathematics is viewed by the culture as some sort of tool for science and technology. Everyone knows that poetry and music are for pure enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from the public school curriculum), but no, math is important.

  SIMPLICIO: Are you really trying to claim that mathematics offers no useful or practical applications to society?

  SALVIATI: Of course not. I’m merely suggesting that just because something happens to have practical consequences doesn’t mean that’s what it is about. Music can lead armies into battle, but that’s not why people write symphonies. Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind.

  SIMPLICIO: But don’t we need people to learn those useful consequences of math? Don’t we need accountants and carpenters and such?

  SALVIATI: How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers—the kind of thing a real mathematical education might provide.

  SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?

  SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful.

  SIMPLICIO: But not everyone is cut out to be an artist. What about the kids who aren’t “math people”? How would they fit into your scheme?

  SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be good at math. We are losing so many potentially gifted mathematicians—creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their time on such piffle.

  SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?

  SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.

  SIMPLICIO: So you would remove mathematics from the school curriculum?

  SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.

  SIMPLICIO: But how many mat
h teachers know enough about their subject to teach it that way?

  SALVIATI: Very few. And that’s just the tip of the iceberg . . .

  Mathematics in School

  THERE IS SURELY NO MORE RELIABLE WAY TO KILL enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it. School boards do not understand what math is; neither do educators, textbook authors, publishing companies, and, sadly, neither do most of our math teachers. The scope of the problem is so enormous I hardly know where to begin.

  Let’s start with the “math reform” debacle. For many years there has been a growing awareness that something is rotten in the state of mathematics education. Studies have been commissioned, conferences assembled, and countless committees of teachers, textbook publishers, and educators (whatever they are) have been formed to “fix the problem.” Quite apart from the self-serving interest paid to reform by the textbook industry (which profits from any minute political fluctuation by offering up “new” editions of their unreadable monstrosities), the entire reform movement has always missed the point. The mathematics curriculum doesn’t need to be reformed, it needs to be scrapped.

  All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake —it’s like rearranging the deck chairs on the Titanic! 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. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.

  The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.” You don’t need to make math interesting—it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!

  Attempts to present mathematics as relevant to daily life inevitably appear forced and contrived: “You see, kids, if you know algebra then you can figure out how old Maria is if we know that she is two years older than twice her age seven years ago!” (As if anyone would ever have access to that ridiculous kind of information, and not her age.) Algebra is not about daily life, it’s about numbers and symmetry—and this is a valid pursuit in and of itself:Suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?

  Here is a simple and elegant question, and it requires no effort to be made appealing. The ancient Babylonians enjoyed working on such problems, and so do our students. (And I hope you will enjoy thinking about it too!) We don’t need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience.

  In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide—a relief from daily life, an anodyne to the practical workaday world.

  A similar problem occurs when teachers or textbooks succumb to cutesiness. This is where, in an attempt to combat so-called “math anxiety” (one of the panoply of diseases which are actually caused by school), math is made to seem “friendly.” To help your students memorize formulas for the area and circumference of a circle, for example, you might invent a whole story about Mr. C, who drives around Mrs. A and tells her how nice his two pies are (C = 2πr) and how her pies are square (A = πr2) or some such nonsense. But what about the real story? The one about mankind’s struggle with the problem of measuring curves; about Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which is more interesting—measuring the rough dimensions of a circular piece of graph paper, using a formula that someone handed you without explanation (and made you memorize and practice over and over), or hearing the story of one of the most beautiful, fascinating problems and one of the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles for god’s sake!

  Why aren’t we giving our students a chance to even hear about these things, let alone giving them an opportunity to actually do some mathematics, and to come up with their own ideas, opinions, and reactions? What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources—beautiful works of art by some of the most creative minds in history—in favor of third-rate textbook bastardizations?

  The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.

  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?

  I can understand the idea of training students to master certain techniques—I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process—that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.

  So put away your lesson plans and your overhead projectors, your full-color textbook abominations, your CD-ROMs and the whole rest of the traveling circus freak show of contemporary education, and simply do mathematics with your students! Art teachers don’t waste their time with textbooks and rote training in specific techniques. They do what is natural to their subject—they get the kids painting. They go around from easel to easel, making suggestions and offering guidance:STUDENT: I was thinking about our triangle problem, and I noticed something. If the triangle is really slanted then it doesn’t take up half its box! See, look:

  TEACHER: Excellent observation! Our chopping argument assumes that the tip of the triangle lies directly over the base. Now we need a new idea.

  STUDENT: Should I try chopping it a different way?

  TEACHER: Absolutely. Try all sorts of ideas. Let me know what you come up with!

  So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and levels of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.

  Of course what I’m suggesting is imp
ossible for a number of reasons. Even putting aside the fact that statewide curricula and standardized tests virtually eliminate teacher autonomy, I doubt that most teachers even want to have such an intense relationship with their students. It requires too much vulnerability and too much responsibility—in short, it’s too much work!

  It is far easier to be a passive conduit of some publisher’s “materials” and to follow the shampoo-bottle instruction—lecture, test, repeat—than to think deeply and thoughtfully about the meaning of one’s subject and how best to convey that meaning directly and honestly to one’s students. We are encouraged to forego the difficult task of making decisions based on our individual wisdom and conscience, and to get with the program. It is simply the path of least resistance:TEXTBOOK PUBLISHERS : TEACHERS ::a. pharmaceutical companies : doctors

  b. record companies : disc jockeys

  c. corporations : congressmen

  d. all of the above

  The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.