Lockhart’s Lament

A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory…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…are considered very advanced topics and generally put off till 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…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.”

Sadly, our present system of mathematics education is precisely this sort of nightmare.

So begins Paul Lockhart’s scathing critique of how mathematics is taught in this country, A Mathematician’s Lament. The book is an expansion of Lockhart’s essay of the same title. I encourage you to read the essay, buy the book, and share your thoughts in comments.

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22 Responses to “Lockhart’s Lament”


  1. 1 1 William

    Thank you so much for pointing me to this essay. I didn’t expect to read the whole thing, but it was too compelling to stop.

    I sometimes tutor students in pre-calculus, the class where the undirected pointlessness of the mathematics curriculum reaches its fullest expression. This essay does a great job of summing up my frustration.

    I just wish I had ideas for changes I could make on the margin, since I’m not in a position to scrap the mathematics curriculum as it’s currently taught.

  2. 2 2 Alan Wexelblat

    I will read the essay in a bit but I’m reminded of a conversation that I had with Marvin Minsky some years ago in which he asserted that a large part of the problem with teaching math was the paucity of vocabulary available. If you think about many other topics that get taught, they have comparably richer vocabularies (he estimated something like one order of magnitude more words) and can both express much more nuance with those rich vocabularies and can construct more story-oriented explanations. Since math lacks the vocabulary with which to make stories it has no access to a huge chunk of how the human brain learns and organizes information.

  3. 3 3 Dan Grayson

    This article is magnificent! Every point made is right on the money. Luckily, as a tenured professor in a university with good honors programs, I had the leeway to teach this way as often as I wanted to, with good results.

  4. 4 4 dWj

    I heard a story, many years ago, about young, poor students being taught calculus, and doing fairly well, despite doing poorly at “math”; they were never told that calculus was math, and their parents generally had never seen calculus.

    If this story is true, I’m curious as to how the calculus was “taught”. Oscar Wilde claimed that nothing that is worth knowing can be taught, and I think most things that are worth knowing have to be alighted upon by the student; the best a teacher can hope to do is put the student somehow in its vicinity and occasionally respond to questions.

  5. 5 5 Dan Grayson

    PS: This way of teaching is much less work for the teacher, and best of all, you don’t need a textbook. Indeed, a textbook would just get in the way.

  6. 6 6 Harold

    Wow, its a choice that negative exponents are reciprocal? Math education sure is missing a lot. I fear it is not just math. The powers that be are so worried about low standards, that they force all teaching into a narrow band in the midddle. It may prevent some very poor teaching, but it also stops the inspirational teaching at the other end.

  7. 7 7 Michael Metcalf Bishop

    Lockhart has some good points but is at once too optimistic and too pessimistic. See my critique here: http://permut.wordpress.com/2009/11/04/which-mathematics-to-teach/

  8. 8 8 Philip

    Reading Lockhart’s essay recalled to mind my love of math through high school and college statistics and then my “doneness” with it after college calculus. He has nailed why. Reflecting on this experience leads me to several observations about A Mathematician’s Lament.

    First, Lockhart mounts a powerful argument which I mostly agree with and would apply to contemporary secondary school, and much college level, teaching in general, not just in mathematics. I have a son who is very smart and usually performed poorly in school for the many of the reasons Lockhard describes.

    But my son was in private schools. Lockhart’s criticism is not confined to public schools alone. Therefore, I think his focus on largely public school “failures”, such as teaching to standardized tests, statewide curricula, teachers taking the easy way out, etc., while relevant, is also misguided.

    Today’s public school teachers carry burdens that private school teachers typically do not and public school teachers 40 years ago did not. I’m sure I don’t need to outline those differences, but they are huge hurdles to effective teaching and to learning.

    Public school teachers are also mandated to use textbooks chosen through a political process. I know this well because I’m a Texan. Because Texas has a large public school population and selects textbbooks on a statewide basis while other states typically do not, it has an outsized impact on the content of textbooks used nationwide. (Textbook publishes are driven by economies of scale.)

    Textbooks here are chosen by an elected statewide commission dominated by very conservative members who support intelligent design, oppose the teaching of evolution, support a very conservative reinterpretation of history, and other religious and ideological propaganda. (Thank god for federal courts that have frustrated many of these efforts under the 1st Amendment, a strong-government intervention many on this site would saeem to oppose. :) )

    But my more basic critique of Lockart is that teachers need to do a better job of teaching to the student. I do not believe all students are motivated by the expressive and abstract motives Lockhart ascribes to them, or when they are so motivated that mathematics is their vehicle of choice over other disciplines.

    And for those who are not motivated by the “artistic” rewards of math, the joy, satisfaction, fun that comes from mastering methematical problems is often a major motivator and can lead to a deeper appreciation of the abstract and expressive pleasures of mathematics. (My second son, who is gifted in math is an example.)

    And for other students, the practical applications of math (for me, statistics as a means of reality and hypothesis testing in policy making) can be a powerful motivator.

    Teaching only the abstract beauty of mathematics will miss these students.

    Finally, teaching math has a very practical value independent of the artistic values Lockhart emphasizes. Mathematics forges the brain’s neural pathways that contribute to logical and analytical thinking, providing a significant return to students later in life beyond the economic rewards, and a type of enjoyment all its own.

  9. 9 9 caveat bettor

    Math skills are scarce, and folks can make more money doing something else besides teaching. I can only name 1 math teacher of a 100 that I know in public schools that could teach “out of the box”.

  10. 10 10 GregS

    Lockhart reminds me of Richard Feynman. This essay pretty much describes my math education to a T. Unfortunately, our system isn’t open to reform…there’s a lot of inertia that keeps things from changing. If we could fire bad teachers and replace them with good ones, or simply open new schools with a competing curriculum, I think we’d see more Lockharts in the teaching profession. Too bad teachers are rewarded for seniority more than effectiveness (at universities, too), and the lack of a voucher system makes it unlikely that a new private school with a modern curriculum would succeed. I don’t usually give too much thought to conspiracy theories, but it strikes me that “certain people” benefit when we’re all dumb and uncreative. (That’s just a passing thought; the public education mess can be explained without an elaborate conspiracy theory.)
    I recall memorizing a long sheet of trig identities for precalculus…there were 20-something of them. I was disappointed when I later (much later) learned that they all followed trivially from Euler’s formula. Time spent memorizing nonsense could have been time spent learning a cool formula; a method for deriving all those formulas would have been infinitely more useful.

  11. 11 11 Hyman Rosen

    Most children aren’t any good at singing, playing musical instruments, or painting pictures either. Of course mathematicians think that math is beautiful and that everybody can be taught its beauty. But in fact, not everyone is going to be able or willing to learn that, just as not everyone will learn to sing, play, or draw with sufficient skill. Formal math education exists to teach people enough of the subject so they can use it when they need it. The mathematically inclined will learn to see the beauty in the subject eventually, and meanwhile everyone else’s time won’t be wasted with touchy-feely techniques that take ten times as long to get to the point.

  12. 12 12 Doctor Memory

    Wow. I immediately want to move to whatever school district it is where this man teaches, so that my daughter can attend his classes.

  13. 13 13 Al V.

    Three comments:
    – I love math, and always enjoy math and logic problems. However, my children never inherited my love of math, largely because they have been taught via rote memorization, and never had an opportunity to see the beauty – and I’m not a good enough teacher to have brought it to hem.
    – My cousin teaches high school math in the Chicago public school system. His training is in art education, but when his district ran short of math teachers, they shoved him into math. I’m sure he’s a good teacher, but I doubt he really appreciates the beauty of math, or has a native understanding of it.
    – I think many of these arguments could apply to other subjects. How many students are taught to understand the reasons behind history? Most of what I remember, and what my children experienced, is rote learning again. Facts, dates, names. The who, what, where, and when, but not the why.

  14. 14 14 John Faben

    Hyman, surely a major part of Lockhart’s point is that for the vast majority of people almost *none* of the maths they learn at school will be useful at any point in their lives, whatever and however we choose to teach them.

    Similarly, almost none of the English literature, history, economics or biology we teach them will be useful: mathematics unfortunately has the baggage that a lot of what we *could* teach in schools as maths happens to be useful to a few specialists, whereas the other arts are (almost consciously) useless.

    There may be a strong argument that 10+ years of compulsory education is just too much because it’s impossible to teach people useful stuff for 10 years. I can’t see any strong argument that if we *are* going to teach people useless stuff for 10+ years (because we think it makes them better people? because it keeps them off the streets?), we might as well teach them some useless maths, and make it interesting on the way.

  15. 15 15 Ken

    I read this essay about a month or two ago. My impression throughout was someone who pinpointed the exact problem with math education, but devolved into quite a fantasy world when presenting a ‘solution’. I think much of math is beautiful, but that is NOT why I am drawn to it. I am drawn to it because IT EXPLAINS THE WORLD. The idea that math is removed from anything practical, that it is a ‘game’ to be played, and relishing this type of navel gazing the way Lockhart does (as well as Hardy) I find distasteful and damaging. Hardy’s proud statement that he studied number theory precisely because it’s useless is the type of statement that makes me want to smack some sense it him.

    Math is important precisely because it is useful. The fact that it is often times beautiful is just gravy. But like all things that are useful, learning and doing math is a hard, arduous task. I think there should be more emphasis on actual problem solving and discovery (like Lockhart suggests students discover the rule that the area of a triangle is 1/2*bh), but if we require each generation to discover all things ALL READY known how much intellectual progress will be made? I hear people generally denigrate drill and memorization without really thinking the matter through. I know the definition of a limit. You know why? I memorized it. You know how? I had it drilled into me.

    Lockhart disparages the idea of mimicry when solving math problems saying something like students are taught certain proofs, then taught to prove other things mimicking these proofs (I don’t remember exactly what he says) as if this is a bad thing. My immediate thought was: ALMOST ALL KNOWLEDGE IS GAINED THROUGH MIMICRY. Disparaging mimicry is disparaging the human experience.

    In summary, Lockhart correctly diagnosis the idiocy of math education and makes a nice analogy through music. But then goes on to recommend an even greater idiocy by totally mischaracterizing what math is, then recommending a totally feel good, no accountability education style, which will leave us all worse off.

  16. 16 16 Neil

    Perhaps I misinterpret, but Lockhart appears to be a proponent of the “discovery method” of math education. Here in Seattle, the School District mandated discovery method math textbooks, and they were a disaster. The parents went to court to prevent a blanket requirement of textbooks with this sort because the students were not getting adequate preparation for college. The fact is, it is not efficient to motivate students by letting them rediscover Pythagoras’ theorem, or whatever. It takes too long even with help. How can we stand on the shoulders of giants if we expect every generation to rediscover results on its own? It is okay, of course, to encourage original intuitive and visual thinking in approaching math problems, as well as teaching how the “giants” did their thinking and problem solving.

  17. 17 17 lukas

    Formal math education exists to teach people enough of the subject so they can use it when they need it.

    Need it? That may be true for Arithmetic, but certainly not for the subjects deceptively taught under the names of “Algebra”, “Geometry” and “Calculus”? Most folks never need to solve a quadratic equation outside of school, let alone calculate derivatives and the like.

  18. 18 18 Hyman Rosen

    Some number of students are going to go into fields where algebra, geometry, and calculus are very necessary. If we’re not going to decide students’ futures early in their school careers, we have to give them a grounding in those subjects so that if later on they become interested in one of those careers, they have the proper background to succeed in them. So we teach the basics to everyone. As far as calculus goes, students who are studying calculus should be doing so because they need or want to, not because it’s compulsory. And run-of-the-mill calculus is neither difficult nor especially beautiful, it’s just a great way of describing how things in the world work.

  19. 19 19 John Faben

    Hyman, some number of students will go on to fields where understanding Rossian Deontology is very necessary. The question is *how much* should everyone have to learn? I agree that everyone should probably learn basic algebra (by which I mean, manipulating symbols, rather than what a mathematician would mean by algebra) but I’m not so sure about geometry or calculus.

    There are two difficult question. First, how much maths should everyone have to know? I would probably draw the line well *below* what is taught as standard in most schools. Second, how should we teach it them? I agree with Lockhart to an extent that giving them more of an idea about what the math they’re learning is *for* would help, but I’m not sure I agree with his methods for doing this (they are excellent methods for training mathematicians, but most people are not going to be mathematicians).

    If you’re going to teach elementary calculus to everyone, why not do it in a physics, or an economics course? Then you at least get to see that the “scales” that you’re practising can be put together to make some sort of tune.

    (Disclaimer, I’m still not entirely sure what I think about this issue, and I’ve been thinking about it (at least) since I first read Lockhart’s essay a good few years ago – I’m sure I’ll sort my thoughts out in the end, and discussing it here has possibly helped me to do so).

  20. 20 20 Roger Schlafly

    A Seattle judge has recently ruled that a change to math teaching was arbitrary and capricious. See http://blogs.edweek.org/edweek/curriculum/2010/02/judge_sends_seattle_back_to_dr.html.

  21. 21 21 John Thacker

    I’m a mathematician, and I highly disagree with Lockhart’s critique. I’ve had classes (and knew people that had classes) that attempted to teach mathematics according to the methods he described, and it was an utter disaster that really killed students’ love of mathematics.

    The “laboratory mathematics” style of teaching is almost never done well, and done poorly students grasp the point and the beauty of mathematics even less than they do in traditional classes.

    He’s well-meaning, but mistaken.

  22. 22 22 Walt

    It would be nice if more students had teachers with Lockhart’s motivation, subject knowledge, and ability to encourage creativity in in the classroom. If teachers could spread Lockhart’s appreciation of math far and wide, that would be wonderful. But where will these teachers come from?

    If the question is not put front and center, we enter a rhetorical world where schools are well supplied with qualified emissaries of a more coherent view of mathematics, and the only thing holding them back is devotion to outdated educational orthodoxy. Lockhart seems to exist in that world— unaware that he might not be representative of the norm, and unwilling to make the obvious inference about the practical value of his lamentations.

    My understanding of Lockhart’s essay is that it was originally intended mainly for reading by the mathematical or educational communities (however you want to define that) and that the more recent attention it has gotten grew out of various postings on the web (e.g. in Keith Devlin’s MAA column), then the book, and, by now, everywhere (e.g. its recent appearance in Steven Strogatz’s NYT blog). The shift in audience, from professional to public, makes me uncomfortable.

    I do not think the public is able to distinguish an earnest gripe by someone who has strong subject knowledge (as Lockhart does) from empty demagoguery from someone with no subject knowledge. It is ironic: Lockhart’s rhetoric of lamentation has a long history of use by politicians moonlighting as educators (or vice versa) in support of pet projects— modifications of standards, curricula, textbook lists, you name it— that made education in America exactly what Lockhart finds so troubling.

    Pick your least favorite attempt to “change mathematics education in America”. Among its proponents, do you not find someone offering the lamentation narrative? This is what’s wrong with education, this is how it should be, and isn’t it tragic that the ignorant and hidebound just don’t get it? The hallmark of this style is that no attention is paid to legitimate differences of opinion, interests and constraints that genuinely compete with one another and cannot be reconciled, or indeed anything that does not fit into the lamentation narrative.

    Reading comments on various web forums, Lockhart’s essay seems to have gone over as a sort of Rohrschach test for what’s wrong with education. Anybody can find something they agree with. Teaching means openness and honesty and loving learning, teaching means talking to students and listening to them… who could disagree with that?

    This is unfortunate. Reforming educational practice is necessarily controversial. Even assuming good faith on all sides (so rare in policy disputes), reasonable people will disagree about what should be done and compromises will have to be made. It is great when math PhDs take the time to participate in these kinds of discussions, but they have much more to offer than this.

    To me, Lockhart’s lament is shop talk among professionals that should have stayed that way. It is as if someone found a rant around the office cooler so compelling that they printed it in a book and sent it to all employees of a company. Maybe it doesn’t harm anything, but it doesn’t help, either.

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