A Mathematician's Lament

by Paul Lockhart

  SALVIATI: I’m sure you did. You probably even got to work on some nice problems occasionally. Lots of people enjoy geometry class (although lots more hate it). But this is not a point in favor of the current regime. Rather, it is powerful testimony to the allure of mathematics itself. It’s hard to completely ruin something so beautiful; even this faint shadow of mathematics can still be engaging and satisfying. Many people enjoy paint-by-numbers as well; it is a relaxing and colorful manual activity. That doesn’t make it the real thing, though.

  SIMPLICIO: But I’m telling you, I liked it.

  SALVIATI: And if you had had a more natural mathematical experience you would have liked it even more.

  SIMPLICIO: So we’re supposed to just set off on some free-form mathematical excursion, and the students will learn whatever they happen to learn?

  SALVIATI: Precisely. Problems will lead to other problems, technique will be developed as it becomes necessary, and new topics will arise naturally. And if some issue never happens to come up in thirteen years of schooling, how interesting or important could it be?

  SIMPLICIO: You’ve gone completely mad.

  SALVIATI: Perhaps I have. But even working within the conventional framework, a good teacher can guide the discussion and the flow of problems so as to allow the students to discover and invent mathematics for themselves. The real problem is that the bureaucracy does not allow an individual teacher to do that. With a set curriculum to follow, a teacher cannot lead. There should be no standards, and no curriculum. Just individuals doing what they think best for their students.

  SIMPLICIO: But then how can schools guarantee that their students will all have the same basic knowledge? How will we accurately measure their relative worth?

  SALVIATI: They can’t, and we won’t. Just like in real life. Ultimately you have to face the fact that people are all different, and that’s just fine. In any case, there’s no urgency. So a person graduates from high school not knowing the half-angle formulas. (As if they do now!) So what? At least that person would come away with some sort of an idea of what the subject is really about, and would get to see something beautiful.

  To put the finishing touches on my critique of the standard curriculum, and as a service to the community, I now present the first ever completely honest course catalog for K-12 mathematics:

  THE STANDARD SCHOOL MATHEMATICS CURRICULUM

  LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindu-Arabic symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

  MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or Math Books, are handed out, and the students learn to address the church elders as “they.” (As in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison. Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.

  ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, postmodern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

  GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. The goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

  ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

  TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.

  PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked attempt to introduce late-nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of limits and continuity are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.

  CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises that do not really apply in this setting, and that will of course not be mentioned. To be taken again in college, verbatim.

  And there you have it. A complete prescription for permanently disabling young minds—a proven cure for curiosity. What have they done to mathematics!

  There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.

  SIMPLICIO: All right, I’m thoroughly depressed. What now?

  SALVIATI: Well, I think I have an idea about a pyramid inside a cube . . .

  PART II

  Exultation

  AND SO THE SENSELESS TRAGEDY KNOWN AS “mathematics education” continues, and only grows more indefensibly asinine and corrupt with each passing year. But I don’t really want to talk about that anymore. I’m tired of complaining. And what’s the point? School has never been about thinking and creating. School is about training children to perform so that they can be sorted. It’s no shock to learn that math is ruined in school;
everything is ruined in school! Besides, you don’t need me to tell you that your math class was a boring, pointless waste of time—you went through it yourself, remember?

  So what I’d rather do is tell you more about what math really is and why I love it so much. As I said before, the most important thing to understand is that mathematics is an art. Math is something you do. And what you are doing is exploring a very special and peculiar place—a place known as “Mathematical Reality.” This is of course an imaginary place, a landscape of elegant, fanciful structures, inhabited by wonderful, imaginary creatures who engage in all sorts of fascinating and curious behaviors. I want to give you a feeling for what Mathematical Reality looks and feels like and why it is so attractive to me, but first let me just say that this place is so breathtakingly beautiful and entrancing that I actually spend a good part of my waking life there. I think about it all the time, as do most other mathematicians. We like it there, and we just can’t stay away from the place.

  In this way, being a mathematician is a lot like being a field biologist. Imagine that you have set up your camp on the outskirts of a tropical jungle, let’s say in Costa Rica. Every morning you take your machete into the jungle and explore and make observations, and every day you fall more in love with the richness and splendor of the place. Suppose you are interested in a particular type of animal, say hamsters. (Let’s not worry about whether there actually are any hamsters in Costa Rica.)

  The thing about hamsters is they have behavior. They do cool, interesting things: they dig, they mate, they run around and make nests in hollow logs. Maybe you’ve studied a particular group of Costa Rican hamsters enough that you’ve tagged them and given them names. Maybe Rosie is black and white and loves to burrow; maybe Sam is brown and enjoys lying in the sun. The point is that you are watching, noticing, and getting curious.

  Why do some hamsters behave differently from others? What features are common to all hamsters? Can hamsters be classified and grouped in meaningful and interesting ways? How do new hamsters get created from old ones, and what traits are inherited? In short, you’ve got hamster problems—natural, engaging questions about hamsters that you want answered.

  Well, I’ve got problems too. Only they are not located in Costa Rica, and they don’t concern hamsters. But the feeling is the same. There’s a jungle full of strange creatures with interesting behaviors, and I want to understand them. For example, among my favorite denizens of the mathematical jungle are these fantastical beasts: 1, 2, 3, 4, 5, . . .

  Please don’t freak out on me here. I know you’ve probably had some pretty miserable experiences connected with these particular symbols, and I can feel your chest tightening already. Just relax. Everything is going to be fine. Trust me, I’m a doctor . . . of philosophy.

  First of all, forget the symbols—they don’t matter. Names never matter. Rosie and Sam do what they do; they don’t care about your silly pet names for them. This is a hugely important idea: I’m talking about the difference between the thing itself and the representation of the thing. It is of absolutely no importance whatever what words you want to use (if any) or what symbols you wish to employ (if any). The only thing that matters in mathematics is what things are, and more important, how they act.

  So somewhere along the line people started to count (no one knows quite when). A really big step occurred when people realized that they could represent things by other things (e.g., a caribou by a painting of a caribou, or a group of people by a pile of rocks). At some point (again, we don’t know when) early humans conceived of the idea of number, of “three-ness” for instance. Not three berries, or three days, but three in the abstract. Throughout the millennia people have devised all sorts of languages for the representation of numbers—markers and tokens, coins with values on them, symbolic manipulation systems, and so on. Mathematically none of this really matters very much. From my point of view (that of the impractical daydreaming mathematician) a symbolic representation like ‘432’ is no better or worse than an imaginary pile of four hundred thirty-two rocks (and in many ways I prefer the rocks). To me the important step is not the move from rocks to symbols, it’s the transition from quantity to entity—the conception of five and seven not as amounts of something but as beings, like hamsters, which have features and behavior.

  For example, to an algebraist such as myself, the statement 5 + 7 = 12 does not so much say that five lemons and seven lemons make twelve lemons (although it certainly does say that). What it says to me is that the entities commonly known by the nicknames “five” and “seven” like to engage in a certain activity (namely “adding”) and when they do they form a new entity, the one we call “twelve.” And this is what these creatures do—no matter what they are called or by whom. In particular, twelve does not “start with a one” or “end with a two.” Twelve itself doesn’t start or end, it just is. (What does a pile of rocks “start” with?) It is only the Hindu-Arabic decimal place-value representation of twelve that starts with a ‘1’ and ends with a ‘2.’ And that’s really neither here nor there. Do you get what I’m saying?

  As mathematicians we are interested in the intrinsic properties of mathematical objects, not the mundane features of some arbitrary cultural construct. The symbol ‘69’ may look the same upside down, but the number sixty-nine doesn’t “look” any way at all. I hope you can see how this point of view is a natural outgrowth of the “simple is beautiful” aesthetic. What do I care what notation system some Arabic traders introduced into Europe in the twelfth century? I care about my hamsters, not their names.

  So let’s try to think of these numbers 1, 2, 3, et cetera, as creatures with interesting behavior. Of course their behavior is determined by what they are, namely sizes of collections. (That’s how we happened upon them in the first place!) Let’s refer to them using imaginary piles of rocks:

  This way we can observe them “in the wild,” so to speak, and we won’t be distracted or misled by some accidental artifact of notation. Now one behavior that people noticed pretty early on is that some of them (as piles of rocks) can be arranged in two equal rows:

  The numbers four, eight, and fourteen have this property, whereas three, five, and eleven do not. And it’s not because of their names—it’s because of who they are and what they do. So here is a behavioral distinction among mathematical entities: some of them do this (the so-called “even” numbers) and some do not (the “odd” ones).

  For pretty obvious reasons, I tend to think of even numbers as female and odd numbers as male. The even numbers (arranged in two equal rows) have a nice smooth profile, whereas the odd ones are always sticking something out:

  Since pushing piles of rocks together is such a natural thing to do, it’s also natural to wonder how the even/odd distinction is affected by addition. (It’s like asking whether the spotted/plain trait in hamsters is inherited.) So I play around a bit with piles of rocks and I notice a lovely pattern:Even & Even makes Even

  Even & Odd makes Odd

  Odd & Odd makes Even

  Do you see why? I especially like the way two odds fit together:

  There’s such a wonderful “two wrongs make a right” quality to this. Those annoying prongs just cancel each other out! And notice that this works for all odd numbers, not just the ones I happened to choose. In other words, this is a completely general behavior. So that’s a nice discovery. Not that there’s anything so special about using two rows. We could also investigate what happens when we arrange numbers into three rows, or four, or ten. What do our hamsters do then?

  Now I know none of this is terribly sophisticated, but I really want you to get this feeling of imaginary entities and their amusing behavior. It’s important for understanding both the attraction of the subject and its methodology (especially in the modern era). There is, however, an absolutely crucial difference between Costa Rican hamsters and mathematical entities like numbers or triangles: hamsters are real. They are part of physical reality. Mathematical objects, even if ini
tially inspired by some aspect of reality (e.g., piles of rocks, the disc of the moon), are still nothing more than figments of our imagination.

  Not only that, but they are created by us and are endowed by us with certain characteristics; that is, they are what we ask them to be. Not that we don’t build things in real life, but we are always constrained and hampered by the nature of reality itself. There are things I might want that I simply can’t have because of the way atoms and gravity work. But in Mathematical Reality, because it is an imaginary place, I actually can have pretty much whatever I want. If you tell me, for instance, that 1 + 1 = 2 and there’s nothing I can do about it, I could simply dream up a new kind of hamster, one that when you add it to itself disappears: 1 + 1 = 0. Maybe this ‘0’ and ‘1’ aren’t collection sizes anymore, and maybe this “adding” isn’t pushing collections together, but I still get a “number system” of a sort. Sure, there will be consequences (such as all even numbers being equal to zero), but so be it.

  In particular, we are free to embellish or “improve” our imaginary structures if we see fit. For example, over the centuries it gradually dawned on mathematicians that this collection, 1, 2, 3, et cetera, is in some ways quite inadequate. There is actually a rather unpleasant asymmetry to this system, in that I can always add rocks but I can’t always take them away. “You can’t take three from two” is an obvious maxim of the real world, but we mathematicians do not like being told what we can and cannot do. So we throw in some new hamsters in order to make the system prettier. Specifically, after expanding our notion of collection sizes to include zero (the size of the empty collection), we can then define new numbers like ‘-3’ to be “that which when added to three makes zero.” And similarly for the other negative numbers. Notice the philosophy here—a number is what a number does.