A Mathematician's Lament Read online

  Table of Contents

  Title Page

  Dedication

  Epigraph

  Foreword

  PART I - Lamentation

  Mathematics and Culture

  Mathematics in School

  The Mathematics Curriculum

  High School Geometry: Instrument of the Devil

  THE STANDARD SCHOOL MATHEMATICS CURRICULUM

  PART II - Exultation

  Copyright Page

  For Stanley, who asked me to write it.

  If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.

  —ANTOINE DE SAINT EXUPÉRY

  Foreword

  IN LATE 2007, AN AUDIENCE MEMBER AT A TALK I gave handed me a 25-page typewritten document called A Mathematician’s Lament, saying he thought I might like it. Written by a mathematics teacher called Paul Lockhart, the essay had been circling somewhat erratically through the mathematics education community since its author first wrote it in 2002, but it had never been published. The audience member’s prediction turned out to be an understatement. I loved it, and felt that the words of this Paul Lockhart—whoever he was—deserved a much wider audience. And so I did something I have never done before, and probably never will again: after tracking down the essay’s author—not entirely straightforward since the essay bore no contact information—and securing his permission, I devoted an entire issue of my monthly online column “Devlin’s Angle” on the Mathematical Association of America’s web-zine MAA Online (www.maa.org) to reproducing the entire essay in its original form. It was the quickest and most effective way I knew to get it in front of the mathematics and mathematics education communities.

  When A Mathematician’s Lament appeared in my March 2008 column, I introduced it with these words:It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen.

  I was expecting a strong response. What ensued was a firestorm. Paul’s words struck a very, very loud chord that resonated around the world. In addition to many emails expressing appreciation, requests flooded in—many to me, since by agreement I did not publish Paul’s contact information—for reproduction and translation rights. (The volume you have in your hands arose in precisely this way.)

  It wasn’t that Paul was saying something that countless mathematicians and math teachers have not said before. Nor were the points he raised new to those in the sometimes divided world of mathematics education who wrote to disagree with much if not all of what he wrote. What was different was the eloquence of his words and the obvious passion he injected into them. This was not just good writing; this was great writing, coming right from the heart.

  Make no mistake about it, A Mathematician’s Lament, and this greatly expanded book version, is an opinion piece. Paul has strong views on how mathematics should be taught, and he argues forcefully for his approach, and against much of the status quo in today’s world of school mathematics education. What singles him out, besides his personal and captivating writing style, is that he brings to the thorny and much-debated issues of mathematics education a perspective that few others are able to draw upon. Paul is one of those very rare birds who began as an accomplished professional research mathematician, teaching students in universities, and then realized his true calling was in K-12 teaching, which is the career he has followed for many years now.

  In my view, this book, like the original essay it came from, should be obligatory reading for anyone going into mathematics education, for every parent of a school-aged child, and for any school or government official with responsibilities toward mathematics teaching. You may not agree with everything Paul says. You may think his approach to teaching is not one that every teacher could successfully adopt. But you should read what he says and reflect on his words. A Mathematician’s Lament is already a recognized landmark in the world of mathematics education that cannot and should not be ignored. I am not going to tell you how I think you should respond. As Paul himself would agree, that is for every individual reader to do. But I will tell you this. I would have loved to have had Paul Lockhart as my school mathematics teacher.

  KEITH DEVLIN

  Stanford University

  PART I

  Lamentation

  A MUSICIAN WAKES FROM A TERRIBLE NIGHTMARE. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made—all without the advice or participation of a single working musician or composer.

  Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

  As for the primary and secondary schools, their mission is to train students to use this language—to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”

  In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”

  In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in scales and modes, meter, harmony, and counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.” Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one. “To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable—every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician someday.”

  Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. “Of course,” he reassures himself, “no society would ever reduce such a beautiful and meaningful art form to something so mindless an
d trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!”

  Meanwhile, on the other side of town, a painter has just awakened from a similar nightmare . . .

  . . . I was surprised to find myself in a regular school classroom—no easels, no tubes of paint. “Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. “I like painting,” one of the students remarked. “They tell me what to do and I do it. It’s easy!”

  After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers,” the teacher replied. “That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations—dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters—the ones who know their colors and brushes backwards and forwards—they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.”

  “Um, these high school classes you mentioned . . .”

  “You mean Paint-by-Numbers? We’re seeing much higher enrollments lately. I think it’s mostly coming from parents wanting to make sure their kid gets into a good college. Nothing looks better than Advanced Paint-by-Numbers on a high school transcript.”

  “Why do colleges care if you can fill in numbered regions with the corresponding color?”

  “Oh, well, you know, it shows clear-headed logical thinking. And of course if a student is planning to major in one of the visual sciences, like fashion or interior decorating, then it’s really a good idea to get your painting requirements out of the way in high school.”

  “I see. And when do students get to paint freely, on a blank canvas?”

  “You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there abstract stuff. I’ve got a degree in painting myself, but I’ve never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board.”

  Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done—I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

  Everyone knows that something is wrong. The politicians say, “We need higher standards.” The schools say, “We need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “Math class is stupid and boring,” and they are right.

  Mathematics and Culture

  THE FIRST THING TO UNDERSTAND IS THAT MATHEMATICS is an art. 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.

  So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G. H. Hardy’s excellent description:A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

  So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language and culture? No, not usually. These things are all far too complicated for most mathematicians’ taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.

  For example, if I’m in the mood to think about shapes—and I often am—I might imagine a triangle inside a rectangular box:

  I wonder how much of the box the triangle takes up—two-thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is—wondering, playing, amusing yourself with your imagination. For one thing, the question of how much of the box the triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes its size from one minute to the next. That is, unless you want to talk about some sort of approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple, and consequently it is an ugly question that depends on all sorts of real-world details. Let’s leave that to the scientists. The mathematical question is about an imaginary triangle inside an imaginary box. The edges are perfect because I want them to be—that is the sort of object I prefer to think about. This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.

  On the other hand, once you have made your choices (for example I might choose to make my triangle symmetrical, or not) then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back! The triangle takes up a certain amount of its box, and I don’t have any control over what that amount is. There is a number out there, maybe it’s two-thirds, maybe it isn’t, but I don’t get to say what it is. I have to find out what it is.

  So we get to play and imagine whatever we want and make patterns and ask questions about them. But how do we answer these questions? It’s not at all like science. There’s no experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work.

  In the case of the triangle in its box, I do see something simple and pretty:

  If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must t
ake up exactly half the box!

  This is what a piece of mathematics looks and feels like. That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free!

  Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck. That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason. There is something so wonderfully transformational about this art form. The relationship between the triangle and the rectangle was a mystery, and then that one little line made it obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is all about?

  This is why it is so heartbreaking to see what is being done to mathematics in school. This rich and fascinating adventure of the imagination has been reduced to a sterile set of facts to be memorized and procedures to be followed. In place of a simple and natural question about shapes, and a creative and rewarding process of invention and discovery, students are treated to this:Triangle Area Formula: A = ½ b h

  “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.