A Mathematician's Lament

by Paul Lockhart

  Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught by someone who doesn’t even play an instrument and has never listened to a piece of music in their lives? Would you accept as an art teacher someone who has never picked up a pencil or set foot in a museum? Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? What kind of a teacher is that? How can someone teach something that they themselves don’t do? I can’t dance, and consequently I would never presume to think that I could teach a dance class (I could try, but it wouldn’t be pretty). The difference is I know I can’t dance. I don’t have anyone telling me I’m good at dancing just because I know a bunch of dance words.

  Now I’m not saying that math teachers need to be professional mathematicians—far from it. But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it?

  If teaching is reduced to mere data transmission, if there is no sharing of excitement and wonder, if teachers themselves are passive recipients of information and not creators of new ideas, what hope is there for their students? If adding fractions is to the teacher an arbitrary set of rules, and not the outcome of a creative process and the result of aesthetic choices and desires, then of course it will feel that way to the poor students.

  Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.

  In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you can take classes in early childhood development and whatnot, and you can be trained to use a blackboard “effectively” and to prepare an organized lesson plan (which, by the way, insures that your lesson will be planned, and therefore false), but you will never be a real teacher if you are unwilling to be a real person. Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary.

  It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them. And more important, listen to them!

  SIMPLICIO: All right, I understand that there is an art to mathematics and that we are not doing a good job of exposing people to it. But isn’t this a rather esoteric, highbrow sort of thing to expect from our school system? We’re not trying to create philosophers here, we just want people to have a reasonable command of basic arithmetic so they can function in society.

  SALVIATI: But that’s not true! School mathematics concerns itself with many things that have nothing to do with the ability to get along in society—algebra and trigonometry, for instance. These studies are utterly irrelevant to daily life. I’m simply suggesting that if we are going to include such things as part of most students’ basic education, that we do it in an organic and natural way. Also, as I said before, just because a subject happens to have some mundane practical use does not mean that we have to make that use the focus of our teaching and learning. It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas. Not only would it be cruel to teach reading in such a way—to force third-graders to fill out purchase orders and tax forms—it wouldn’t work! We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math.

  SIMPLICIO: But don’t we need third-graders to be able to do arithmetic?

  SALVIATI: Why? You want to train them to calculate 427 plus 389? It’s just not a question that very many eight-year-olds are asking. For that matter, most adults don’t fully understand decimal place-value arithmetic, and you expect third-graders to have a clear conception? Or do you not care if they understand it? It is simply too early for that kind of technical training. Of course it can be done, but I think it ultimately does more harm than good. Much better to wait until their own natural curiosity about numbers kicks in.

  SIMPLICIO: Then what should we do with young children in math class?

  SALVIATI: Play games! Teach them chess and Go, Hex and backgammon, Sprouts and nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique; help them to become active and creative mathematical thinkers.

  SIMPLICIO: It seems like we’d be taking an awful risk. What if we de-emphasize arithmetic so much that our students end up not being able to add and subtract?

  SALVIATI: I think the far greater risk is that of creating schools devoid of creative expression of any kind, where the function of the students is to memorize dates, formulas, and vocabulary lists, and then regurgitate them on standardized tests—“Preparing tomorrow’s work-force today!”

  SIMPLICIO: But surely there is some body of mathematical facts of which an educated person should be cognizant.

  SALVIATI: Yes, the most important of which is that mathematics is an art form done by human beings for pleasure! All right, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them and you remember what matters to you. We have millions of adults wandering around with “negative b plus or minus the square root of b squared minus 4ac all over 2a” in their heads, and absolutely no idea whatsoever what it means. And the reason is that they were never given the chance to discover or invent such things for themselves. They never had an engaging problem to think about, to be frustrated by, and to create in them the desire for technique or method. They were never told the history of mankind’s relationship with numbers—no ancient Babylonian problem tablets, no Rhind Papyrus, no Liber Abaci, no Ars Magna. More important, no chance for them to even get curious about a question; it was answered before they could ask it.

  SIMPLICIO: But we don’t have time for every student to invent mathematics for themselves! It took centuries for people to discover the Pythagorean theorem. How can you expect the average child to do it?

  SALVIATI: I don’t. Let’s be clear about this. I’m complaining about the complete absence of art and invention, history and philosophy, context and perspective from the mathematics curriculum. That doesn’t mean that notation, technique, and the development of a knowledge base have no place. Of course they do. We should have both. If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side. But the fact is, 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. Children can write poems and stories as they learn to read and write. A piece of writing by a six-year-old is a wonderful thing, and the spelling and punctuation errors don’t make it less so. Even very young children can invent songs, and they haven’t a clue what key it is in or what type of meter they are using.

  SIMPLICIO: But isn’t math different? Isn’t math a language of its own, with all sorts of symbols that have to be learned before you can use it?

  SALVIATI: Not at all. Mat
hematics is not a language, it’s an adventure. Do musicians speak another language simply because they choose to abbreviate their ideas with little black dots? If so, it’s no obstacle to the toddler and her song. Yes, a certain amount of mathematical shorthand has evolved over the centuries, but it is in no way essential. Most mathematics is done with a friend over a cup of coffee, with a diagram scribbled on a napkin. Mathematics is and always has been about ideas, and a valuable idea transcends the symbols with which you choose to represent it. As Carl Friedrich Gauss once remarked, “What we need are notions, not notations.”

  SIMPLICIO: But isn’t one of the purposes of mathematics education to help students think in a more precise and logical way, and to develop their quantitative reasoning skills? Don’t all of these definitions and formulas sharpen the minds of our students?

  SALVIATI: No, they don’t. If anything, the current system has the effect of dulling the mind. Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.

  SIMPLICIO: Fair enough. But what about those students who are interested in pursuing a career in science or engineering? Don’t they need the training that the traditional curriculum provides? Isn’t that why we teach mathematics in school?

  SALVIATI: How many students taking literature classes will one day be writers? That is not why we teach literature, nor why students take it. We teach to enlighten everyone, not to train only the future professionals. In any case, the most valuable skill for a scientist or engineer is being able to think creatively and independently. The last thing anyone needs is to be trained.

  The Mathematics Curriculum

  THE TRULY PAINFUL THING ABOUT THE WAY MATHEMATICS is taught in school is not just what is missing—the fact that there is no actual math being done in our math classes—but what is there in its place: the confused heap of destructive disinformation known as “the mathematics curriculum.” It is time now to take a closer look at exactly what our students are up against—what they are being exposed to in the name of mathematics, and how they are being harmed in the process.

  The most striking thing about this so-called mathematics curriculum is its rigidity. This is especially true in the later grades. From school to school, city to city, and state to state, the exact same things are being said and done in the exact same way and in the exact same order. Far from being disturbed and upset by this Orwellian state of affairs, most people have simply accepted this standard model math curriculum as being synonymous with math itself.

  This is intimately connected to what I call the “ladder myth”—the idea that mathematics can be arranged as a sequence of “subjects” each being in some way more advanced, or “higher,” than the previous. The effect is to make school mathematics into a race—some students are “ahead” of others, and parents worry that their child is “falling behind.” And where exactly does this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve been cheated out of a mathematical education, and you don’t even know it.

  Real mathematics doesn’t come in a can—there is no such thing as an Algebra II idea. Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of the subject, and a teacher’s own path through the standard curriculum reinforces this myth and prevents him or her from seeing mathematics as an organic whole. As a result, we have a math curriculum with no historical perspective or thematic coherence, a fragmented collection of assorted topics and techniques, united only by the ease with which they can be reduced to step-by-step procedures.

  In place of discovery and exploration, we have rules and regulations. We never hear a student saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have teachers and textbooks presenting the “negative exponent rule” as a fait accompli with no mention of the aesthetics behind this choice, or even that it is a choice.

  In place of meaningful problems, which might lead to a synthesis of diverse ideas, to uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in mathematics, we have instead joyless and redundant exercises, specific to the technique under discussion, and so disconnected from each other and from mathematics as a whole that neither the students nor their teacher have the foggiest idea how or why such a thing might have come up in the first place.

  In place of a natural problem context in which students can make decisions about what they want their words to mean, and what notions they wish to codify, they are instead subjected to an endless sequence of unmotivated and a priori definitions. The curriculum is obsessed with jargon and nomenclature, seemingly for no other purpose than to provide teachers with something to test the students on. No mathematician in the world would bother making these senseless distinctions: 2½ is a “mixed number,” while is an “improper fraction.” They’re equal, for crying out loud. They are the exact same numbers, and have the exact same properties. Who uses such words outside of fourth grade?

  Of course it is far easier to test someone’s knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning. Even if we agree that a basic common vocabulary for mathematics is valuable, this isn’t it. How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use words like “conjecture” and “counterexample.” High school students must learn to use the secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’ a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” That this particular shorthand, a holdover from fifteenth-century nautical tables, is still with us (whereas others, such as “versine,” have died out) is mere historical accident, and is of utterly no value in an era when rapid and precise shipboard computation is no longer an issue. Thus we clutter our math classes with pointless nomenclature for its own sake.

  In practice, the curriculum is not even so much a sequence of topics, or ideas, as it is a sequence of notations. Apparently mathematics consists of a secret list of mystical symbols and rules for their manipulation. Young children are given ‘+’ and ‘÷.’ Only later can they be entrusted with ‘√,’ and then ‘x’ and ‘y’ and the alchemy of parentheses. Finally, they are indoctrinated in the use of ‘sin,’ ‘log,’ ‘f(x),’ and if they are deemed worthy, ‘d’ and ‘∫.’ All without having had a single meaningful mathematical experience.

  This program is so firmly fixed in place that teachers and textbook authors can reliably predict, years in advance, exactly what students will be doing, down to the very page of exercises. It is not at all uncommon to find second-year algebra students being asked to calculate [ f(x + h) - f(x) ] / h for various functions f, so that they will have “seen” this when they take calculus a few years later. Naturally no motivation is given (nor expected) for why such a seemingly random combination of operations would be of interest, although I’m sure there are many teachers who try to explain what such a thing might mean, and think they are doing their students a favor, when in fact to them it is just one more boring math problem to be gotten over with. “What do they want me to do? Oh, just plug it in? OK.”

  Another example is the training of students to express information in an unnecessarily complicated form, merely because at some distant future period it will have meaning. Does any middle school algebra teacher have the slightest clue why he is asking his students to rephrase “the number x lies between three and seven” as |x - 5| < 2 ? Do these hopelessly inept textbook authors really believe they are helping students by preparing them for a possible day, years hence, when they might be operating within the context of a higher-dimensional geometry or an abstract metric space? I doubt it. I expect they are simply copying each other decade after decade, maybe changing the fonts or the highlight colors, and beaming wi
th pride when a school system adopts their book and becomes their unwitting accomplice.

  Mathematics is about problems, and problems must be made the focus of a student’s mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process—having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work. Specific techniques and methods will arise naturally out of this process, as they did historically: not isolated from, but organically connected to, and an outgrowth of, their problem-background.

  English teachers know that spelling and pronunciation are best learned in a context of reading and writing. History teachers know that names and dates are uninteresting when removed from the unfolding backstory of events. Why does mathematics education remain stuck in the nineteenth century? Compare your own experience of learning algebra with Bertrand Russell’s recollection:I was made to learn by heart: “The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.” I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.