A Mathematician’s Lament

One of the most eye-opening pieces of writing I’ve ever read is A Mathematician’s Lament” How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart. I’ve known Paul since our sons met when they were about eight years old, and I was so happy to hear that his essay (called a “gorgeous essay” by the Los Angeles Times) was printed in paperback form. This book belongs on everyone’s bookshelf.

Here’s how it begins:

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 and 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…

15 thoughts on “A Mathematician’s Lament

  1. That was very good…
    I have to admit though, that I’m getting a little more depressed every time I read a piece like that. Schooling of our children has become like throwing them into a cornfield maze, and being separated from them. We can hear them but they’re lost. A few of us see a clear passage out but how do we get our kids out of this maze?!

    If I can understand what Paul Lockhart is saying, why can’t the teachers? Why won’t they wake up!


  2. If the purpose of school was to make students hate the subjects allegedly taught there, they’d be doing a terrific job.

    I’d be curious to know what the author’s idea of a good math education is. And how would he educate the vast majority of students, who will never be mathematicians, but still need some understanding of math to use in everyday life? Also, I understand that the abstractions of math can be beautiful, but I’ve found with my daughter that she really isn’t ready for the abstract yet — it doesn’t seem to mean anything to her. She’s about to turn 12. (Or is the abstraction meaningless to her because she’s been taught badly?)

    I wonder how many of the people who are now losing their houses got taken advantage of because they didn’t understand the terms of their adjustable-rate mortgages. Did they understand how high their monthly payments would be as the interest rate went up?

    More questions than answers today …


  3. I think math concepts develop in a different way for each kid.
    …when I had a toddler a few years back..she could look at an array of 6 objects..set up like on a die, and she knew it represented the number 6. To me, that was interesting…that a configuration like that could represent a number to someone so young. She didn’t have to count it. Maybe the concept of 6 didn’t mean anything more than it being a number (not how much it was).

    Now she’s 8 and we play Yahtzee and she knows it’s six on the die and she knows that’s an amount, but the mental calculation ability isn’t there yet. It’s yet to come. For now she needs to count the dots on the dice. She can do some wicked totalling though with one hand. She can count into the 30’s and 40’s with one hand and not lose track of where she is…and she’s always right. So obviously there is some tracking system going on in her head. There’s definitely spatial art involved.

    As parents at our school, we’re discouraged from teaching our children to add and subtract the traditional way…the teachers want them to use “manipulatives” and do it all spatially before they learn the “totalling up, carry the one” etc method. I find it fascinating. But some parents worry that their kids can’t add two digit numbers in their heads by this time. If you aren’t drilling ’em, they’re not learning anything.


  4. we learned with manipulatives at my school, the unit cubes and the ten sticks and the hundred flats, and I thought that worked really well, helped me understand what was going on mathmatically instead of just using a bunch of formulas without knowing how they work. I remember even in 6th grade doing algebra at one point everyone at our table got out the risk pieces aftercare had left in the room and started using them as manipulatives to figure out one tricky problem–they’re especially good for kids who are kinetic learners/like using their hands and to make abstract concepts concrete.


  5. I learned mathematics from my uncle, who taught it to me as a game. I loved my uncle and I loved mathematics.I am a mathematician now. And I have so often thought precisely what this guy wrote about mathematics vs music education.
    I don’t do music. I seriously hate it.
    And I hate the math homework my daughter has to do for school. It’s so boring it’s insane.


  6. And I hate the math homework my daughter has to do for school. It’s so boring it’s insane.


    As FedUpMom wrote, if the goal of schools is to kill the love of learning and as this example demonstrates, assure that students grow up to hate math, then I’d say we are doing a pretty awesome job.

    Here’s the progression: child loves learning, boring busy work turns her off, she doesn’t do her homework, she is punished, teacher decides she’s (take your pick):

    1. Underachieving

    2. Not trying hard enough

    3. Difficult

    4. Exhibits poor time management (today’s education buzz phrase)

    Those are apt descriptions but dare I say they more often apply more to the teacher (or school) than the child or family. I once attended a day long program from a top educator who eschewed the traditional carrot/stick approach, proclaiming, “in ninety percent of the cases, when your students are restless and “misbehaved,” it’s because your lesson has become too boring.”


  7. Wow, I read that a couple of days ago and I am trying to find Paul’s email to congratulate him for such an excellent essay. I couldn’t possibly write it in a better way, it just put on paper the sad situation of the schools in the world. Despite its sadness, I found it also inspiring… Proving that I was not crazy. There is a problem, and common people just can’t see it. I am working on a spanish version of the essay, but I wanted to know if that has been already done or what.


  8. If anyone is interested:

    Click to access LockhartsLament.pdf

    I wholeheartedly agree with everything Lockhart has to say.

    I might want to point out that in Lockhart’s utopia of mathematics education, the homework would likely be harder than that of today’s regimented curriculum. Prove or disprove the conjecture made by one of your classmates. Come up with a more precise definition of an amorphous idea discovered in class. Solve (with justification–not meaningless two-column babble) an actually interesting problem. This is a lot more difficult, and in many cases time consuming, than doing 20 problems exactly how you did them in class. It would, however, be well worth your while, unlike the current form of mathematics homework.

    A side-note about his views on un-orthodox notation. I attended the Hampshire College Summer Studies in Mathematics (see yp17.org), where we learned about group theory calling groups “purgos” the entire time. Yes, rings were “grins”. Normal subgroups were “weird” subpurgos. I strongly recommend the program for strong high school mathematics students who love math but not necessarily high school math class. (You really do have to love math though–we did at least 8 hours of math each day.)

    Another resource for actual mathematics, as opposed to what is currently taught, is artofproblemsolving.com . Unfortunately, the resources I know of tend to be for people who are strong in mathematics and already love it and want to do more, rather than for people who have yet to be introduced to true mathematics, who are the ones that need them most…


  9. Benjamin writes:

    “I might want to point out that in Lockhart’s utopia of mathematics education, the homework would likely be harder than that of today’s regimented curriculum.”

    Benjamin, that would actually not be a problem as far as my daughter is concerned. She is highly visual spatial, as we came to learn, and especially in elementary, she had trouble with rote memorization and computation but thrived on complex material. That is why Linda Silverman’s marvelous book on this subject is entitled, “Upside Down Brilliance…The Visual Spatial Learner,” because their style is upside down. That’s us. For a more typical auditory sequential kid (auditory sequential is how traditional school is taught), maybe this wouldn’t work. All I know is that we are doing is really not working.

    My daughter also has ADD (ADD in gifted kids often presents differently. She does not have a deficit of attention, it’s an attention regulation thing) and has great difficulty with the rote tedious stuff. That takes her longer, actually. I can live with consuming when it is engaging and she is in flow. Especially in elementary, we found that when something required in-depth complexity, she got through it much faster.

    My daughter is a high school student and two years ahead in math. We find that when it’s textbook math, she doesn’t do as well. But in 10th grade, she had an awesome creative teacher. He was hard, he told me he was the hardest math teacher in the entire school. But boy, oh, boy, did we love this man. He was creative, engaging, his problems were complex and for my daughter at least, it worked out really well.

    Not all children are alike. But I can tell you for gifted visual spatial learners, the style you depict is a welcome change.

    I’m familiar with Art of Problem Solving, it’s very popular among my homeschool friends with gifted kids. Had we continued along that path into high school, it’s what we likely would have used as well.


  10. HWBlues — I am now afterschooling my daughter with Singapore Math because of the crummy math teaching she has encountered so far. I expect that will be the subject of a future rant, when we’ve had more experience with it.

    I’m thinking about another rant, entitled “Harder would be Easier.” So much of what my daughter encounters at school is difficult for the wrong reasons. It’s difficult because she doesn’t see any point to it.

    I suppose Singapore Math looks like the “drill and kill” technique that everybody hates, but I don’t see it that way. There’s some repetition, but not a whole lot, and the problems have subtle differences that bring out new ideas. And the bottom line is that if my daughter’s really going to understand fractions, she needs to work through actual equations with fractions in them. Dividing up pizzas isn’t enough, although it’s a reasonable place to start.


  11. FedUpMom, Singapore Math is very popular among a lot of homeschoolers I know, with gifted kids. I’m not on that local homeschool list anymore, I had to steel myself to get off because it was so interesting, it was stealing too much of my time. Other math programs that come to mind are Teaching Company, Math U See for younger visual spatial learners, CTY Distance Education, EPGY Stanford Distance, Thinkwell (CTY uses Thinkwell’s curriculum) and of course, Art of Problem Solving for conceptual math kids.

    FedUp, does your school use Everyday Math? I hear that’s a disaster. Our county now uses it in elementary and my friends hate it, they say their children are not learning math properly and they have to supplement at home or with private tutoring.

    My daughter for years had a mantra, “I want harder, not more, harder not more.” Of course she meant that she doesn’t want to be overwhelmed with homework just becuase she’s in a gifted program, ten math problems do the the trick nicely, no need for fifty. But she was also referring to her state of flow when occasionally (more and more rare) she was confronted with a complex in-depth assignnment. I could just see the wheels turning in her brain.

    Educators often do not get how torturous it is for a smart distractable kid to have to sit still for another long set of hours to complete boring tedious work. I’ve been thinking about that this morning. I am convinced that teachers often send home the most mind numbing stuff because they don’t want to have to deal with it in the classroom.

    So they send it home to me and I have even less experience getting a child to accomplish something meaningless and onerous. The child knows it has no educational value and you are spending hours forcing them to do it. Well, I didn’t force but that is what the school expects you to do. Get it done, do as I say, don’t question it, figure out a way. And we parents are not trained as homework coaches and teachers and now the easy boring rote tedious stuff is dumped into our laps, just when the child is tired after a long way.

    Which still begs the question I keep asking. Why can’t this stuff get done at school? Even at school, they should not be doing meaningless busy work. But what ARE they doing there for seven hours a day when so much comes home? And I don’t want to hear, teacher’s busy doing administrative paperwork. That’s not what I want you doing with my valuable tax dollars. Once again. I want the schools to teach. I can parent, I cannot teach. If I am busy teaching when I should be parenting, when do I get to parent? And when do you get to teach?

    Many people I know resort to rewards in desperation. The assumption here is that you can get a kid to do anything you want as long as you dangle that carrot. Even if that were true, which it isn’t, why should parents spend their afternoons and evenings coaxing a reluctant child for hours to do something onerous when you could be taking a walk with them, reading to them, and doing a thousand things together that are joyful and fulfilling. Why spend your days forcing it down the kids’ throats? Besides the teacher and the principal, just who benefits?


  12. HWBlues — I’m not sure what curriculum the public schools used for math in the early grades. I do know that the 5th grade “accelerated” math class used some standard textbook (Houghton Mifflin, maybe?) that was definitely classic drill-and-kill, and not well-designed, either.

    I remember a particular homework problem which had my daughter in tears. I later found out that she hadn’t yet encountered the skills she would need to solve that problem. No wonder she was frustrated. I asked the teacher about it but never got a straight answer. I now think it must have been some kind of typo.

    The Quaker school dd attends now uses Trailblazers followed by Connected Math, and even though it’s touted as “progressive”, I find myself objecting to it for all the reasons described at kitchen table math. It’s goofy, it’s vague, and a lot of it isn’t really math at all (“choose your favorite number between 10 and 100. Why is it your favorite? Write down 5 math facts about your number.”) My dd is very weak on what should be basic skills for her (but not for much longer, thanks to Singapore Math!).

    I have come to accept that the perfect school exists only in my fevered imagination, and my goal is to find a workable compromise, which I believe we have. Basically, we’ll keep dd in the Quaker schools because she goes to school with a smile on her face. The homework load is a small fraction of what she would encounter at the public school, and the quality is mostly pretty good (the infamous reading log was an outlier.) Since the homework load is small, we have time to supplement dd with subjects we feel strongly about (music, math, foreign language.)

    And that’s where we are for now! However, this is dd’s last year at this particular school, and we now have to start the whole decision-making process all over again. Plus, younger dd will be in 1st grade next year and we need to make a decision for her too. No rest for the weary!


  13. People do not realize the true price of innumeracy. Consider the task of completing your tax return (1040 in the USA) and getting the best possible refund in full compliance with tax code. This is a mathematical optimization problem, and in some cases it is complicated. Even mathematicians surrender to accountants because they lack detailed knowledge of the tax code. But even your neighborhood accountant (usually) lacks the technical knowledge to create the necessary software for the assorting jurisdictions. So a small coterie of firms gain near-monopoly power in this field. And if you venture into a realm where no widely available software solution exists (e.g. work in another country), you quickly find yourself in deep doo-doo, unless, of course, you have retained the skills to calculate things yourself.


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