09 March 2008

Comments on "Lockhart's Lament"

A Mathematician's Lament, by Paul Lockhart, on the state of mathematics education. (Via Devlin's Angle.) Lockhart has a Ph.D. in mathematics and has been a research mathematician; he currently teaches at Saint Ann's School in Brooklyn.

Lockhart begins with a terrifying metaphor -- what if classes in music were taught as exercises in rote manipulation of musical notation, or art classes were classes in Paint-by-Numbers? Yet this is the state of affairs in school mathematics education. Sometimes it astounds me that our schools manage to produce anybody who's actually any good at mathematics. (The best thing a lot of my teachers did was to just get out of my way.)

Lockhart writes (p. 3):
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.... Part of the problem is that nobody has the faintest idea what it is that mathematicians do.

I will not explain what it is that mathematicians do -- I'm preaching to the choir here, I suspect -- but I agree with this. One of the things I find myself doing, when people find out I'm a mathematician, is to try to briefly explain that mathematics is not about numbers and formulas but about abstract ideas and patterns, and that yes, it is possible to create new mathematics! (People seem to think that mathematics is complete.) I hope this missionary zeal doesn't go away -- but it very well might as I age. A lot of older mathematicians I know have said that they used to do this sort of thing but now don't bother any more.

Later (p. 6):
Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.

This reminds me of something that I overheard a friend of mine saying a few days ago. He was talking to one of the young faculty in our department, trying to get some insight about how the process of creating mathematics works. And he said that he didn't know what real mathematics was -- just graduate school mathematics! And this is a man who will quite likely be receiving a PhD in mathematics in three short years. (Let it be said that I am not saying I was not in the same situation at that point in my own graduate career.)

On sources (p. 9):
What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?

Indeed, friends of mine in the humanities are surprised to learn that we don't routinely learn from the writings of the masters. I've heard various explanations for this. First is the claim that one often doesn't want to see the original proof of some important theorem because it's a mess -- and yes, that might be true. So we write various explanations of it, and include those first in our research papers, sanding off a few rough edges here and there. Second is the fact that a lot of mathematicians just don't write that well, because the culture doesn't reward that.

Lockhart claims, in fact, that "there is no actual mathematics being done in our mathematics classes" -- I don't want to believe this, but he'd know better than I would. He's most annoyed at the fact that when school mathematics does teach the idea of "proof" -- which for traditional reasons is done in geometry classes -- the proofs that students produce are so different from real proofs as to be unrecognizable. For those of you who don't know, students in American schools are subjected to something called a "two-column proof" in which a sequence of statements is made in one column, and justifications for those statements is made in an adjacent column. I can imagine how this would be useful for very carefully checking if a proof is correct. But anybody who was actually reading such a proof in an attempt to learn something would probably attempt to translate it back to natural language first! I suspect that the real reason such "proofs" are so common in such courses is because they're much easier to grade than a proof actually written out in sentences and paragraphs. (This seems true of a lot of counterintuitive aspects of our educational system.) Lockhart advocates replacing the standard course with courses in which students actually struggle with trying to come up with their own mathematics. It sounds like it's working for him.

But Lockhart teaches at what appears to be a very good private school that has considerable latitude in choosing its students and gives considerable autonomy to its teachers. Would something like this work within the "mainstream" educational system? I don't know. I'd like to find out.

29 comments:

Blake Stacey said...

Oh, dear Lord, two-column proofs. The pain is flooding back. . . .

Something is bothering me about Lockhart's lament, and I'm having a hard time putting my finger on the exact idea. This will require additional thought.

Camilla said...

In the statement "there is no actual mathematics being done in our mathematics classes", he is using "mathematics" as a value judgment.

I find it troublesome to do so; I think boring math is still math. When people start using "mathematics" to mean "mathematics that real mathematicians agree is worthy and interesting" they will often go on to exclude fields such as combinatorics.

John Armstrong said...

People seem to think that mathematics is complete.

And mathematicians have, as a group, become used to the idea that the population at large has no idea what we do. On my way to New Orleans for the Joint Meetings last January, the plane on the last leg was pretty much full of people coming in for the conference.

The (rather noisome) guy sitting next to me was told by someone across the aisle what we were all going there for. "Math conference? What's there to talk about? Two 'n' two's still four, ain't it?" And you could feel 3/4 of the plane just tense up to hear this sentiment yet again...

And we all just let it go, when clearly the proper course would have been to rise as one and beat him to death with our various math books.

Blake Stacey said...

Indeed, friends of mine in the humanities are surprised to learn that we don't routinely learn from the writings of the masters.

A friend of mine from high school went to St. John's College, where the curriculum is focused on reading the Great Masters of whatever subject. I found that a puzzling and counterproductive way to learn science: we know more than Newton or Darwin or even Einstein, so their books shouldn't be the primary focus of a science class (and I say this as a person who thinks that Darwin and Einstein, for starters, wrote pretty darn well). It also leaves you with a perspective stuck in the past: after a month reading The Origin of Species, the friend in question didn't have a clue about abiogenesis, the application of mathematics to evolution, and other important stuff. The open questions in 1859 are not those we are asking today.

Then, too, there's an astonishing amount of baggage, of cruft which we've accumulated by passing through all that history. Just think of English units and the Fahrenheit scale and calling numbers "imaginary" — pick your favorite example of people not understanding things as well as we do now, and making choices which still clutter up our thinking.

Now, there are plenty of virtues to a historical approach: sciences were simpler when they were getting started, so you can bring students in on the "ground floor", and you can bring in the human element. But (shrugs) there's such a thing as going too far in a good direction.

Aaron said...

I found [learning from the masters] a puzzling and counterproductive way to learn science: we know more than Newton or Darwin or even Einstein, so their books shouldn't be the primary focus of a science class...

[tangent]

I have begun to feel especially this way about quantum mechanics and special relativity. Most of the "paradoxes" of special relativity, in particular, don't seem paradoxical at all if you just draw some spacetime and look at what's happening. It's only when you trade intuition for formulas that things stop making sense.

I think part of the reason that the length contraction and time dilation formulas get so much emphasis in intro SR is that they were, historically, the first mathematical results of SR, and they're used to justify the Lorenz transform because that's the way the Lorenz transform was originally derived. I can't help thinking, though, that there must be a more intuitive justification for the Lorenz transform, just like there's an intuitive justification for the Galilean transform.

Then again, maybe there isn't. We live in a Galilean world, after all...

[\tangent]

Anonymous said...

Dear "mathematicians," you have nobody but yourselves to blame for failing to explain your subject clearly to your students or making it interesting, as well as for allowing the bureaucrats and profit-driven book publishers to turn your beloved subject into homegenized artificially colored garbage. You don't even care to clearly explain what you are doing to each other, you only want to present the "proofs," no matter how ugly and obfuscating, to claim the authorship of your results. Don't blame it on "the system," you ARE the system.

Kaz Maslanka said...

If Lockhart truly thinks that mathematics is art then he obviously knows little to nothing about art.

Anonymous said...

...or Kaz Maslanka knows little to nothing about mathematics...

unapologetic said...

Anonymous 2:31: What you're missing here is that "mathematicians" and "school mathematics teachers" are almost never the same people. That's part of Lockhart's lament. I do try to make the subject interesting to my students, and I try to explain why we do what we do.

As for communications with other mathematicians, you clearly haven't been paying attention to mathematicians. We try to explain the underlying ideas all the time. Just look around all of these research-level mathematics weblogs out here.

Anonymous 4:52: seconded

Anonymous said...

i'll be reading the whole thing
later today ... meanwhile,
thanks for starting this
conversation.

actual mathematics is at least
occasionally done in *my*
mathematics classes ...
sometimes even by me
("i don't know ... let's
see if we can figure this out ...").

i like reading the masters
as well as the next guy i guess
(several papers in hawking's
enormous collection, e.g.)
but wouldn't try to organize
a lowlevel class around 'em ...

weird that this maslanka fellow
is apparently interested in
fractals & "mathematical poetry"
& _GEB_an_EGB_; wonder what point
he thinks he's making here ....

v.

Jeff said...

Ha, I had a physicist friend who said that I probably knew just about all of mathematics (I'm graduating undergraduate in May). I told him that if I spent 5 years at graduate school studying mathematics every day all day, then I might be lucky if I knew 1% of it. The misconception of how vast and growing mathematics is is huge even amongst people in subjects closely tied to it (such as physics). That's something to lament about! And if people who deal with math and mathematicians everyday don't understand what mathematics is or what mathematicians do, then how are mathematics teachers supposed to know? We need to start with the people most likely to understand and then move outward.

Anonymous said...

...John: Just look around all of these research-level mathematics weblogs out here....
And yours must be a shining example of these :-)

Anonymous said...

definitely a lot of fun reading.
but:
(1)it'd be better at a third the length.
(2) he doesn't seem to realize
that what he actually hates
is not just school *mathematics*,
but school *itself*
(long live j.t. gatto!).

unapologetic said...

anonymous 2:32: I didn't say that, and I'm not sure it's true. I do, though, try to explain the ideas. Whether I'm any good at it is up to my readers.

Daniel said...

Someone replied to the general thrust of this article as saying that there's a conflation of those who take math classes (or, for that matter, art classes, etc.) merely to fill requirements for some kind of employment track; and those who take math classes (or, for that matter, art classes, etc.) for the sake of the material being taught. Is there a reason to expect everyone to enjoy art history and method? Is there, similarly, a reason to expect everyone to enjoy abstract algebra or analysis?

I'm not sure; my opinion's a little biased. I tend to look down my nose at the engineering equivalents of math courses I am taking, have taken, or will take; but maybe the purpose-trajectory of a class should be an input in the calculation of its curriculum. My mother was a very successful commercial/industrial painter (hotels, power plants, etc.) for ~30 years, but never took a fine art class. Should she have? Similarly, data analysts in market research firms might not have completed rigorous coverage of probability theory and combinatorics. Need they have?

Is there an alternative to a stratified approach to education, which amounts to vocational training classes alongside, but not intersecting, "pure" topic classes?

Certainly, the guilty popularity of "math fear," being as it is a kind of cultural touchstone, seems an unhappy consequence of the particular means of standardization of the math curricula. Yet people seem most likely to fall toward the middle of a distribution across mathophilia, however misguided or reasonable it may be. Shouldn't they still have that choice?

Anonymous said...

What is wrong with mathematical education in America and some ideas about how to fix it: a recent post

Anonymous said...

Blake Stacey said: "Something is bothering me about Lockhart's lament, and I'm having a hard time putting my finger on the exact idea."

Maybe snobbery of the author? His stress on "purity" of mathematics? Presenting it as playing with abstractions and ignoring where these abstractions are coming from? Pretending that mathematics is not one of the experimental sciences, but something entirely different? "Defining" mathematics by a ridiculous quotation from Hardy (pages 3-4), who had done so much damage to mathematics by disparaging its roots in natural sciences and practical calculations? What else can you expect from a sterile-pure number theorist?

Or maybe this (page 8): "There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it. School boards do not understand what math is, neither do educators, textbook authors, publishing companies, and sadly, neither do most of our math teachers. The scope of the problem is so enormous, I hardly know where to begin."

I'll tell where to begin. First , realize and admit that mathematics is not about patterns, algorithms and abstractions, it is about problem solving. Second, train the teachers in problem solving and require them to teach mathematics as problem solving, also treat them as professionals, not as cheap servants. Third, adopt good textbooks and ban trashy ones, we have FDA, why not require that all the text books should meet the minimum quality standards? Fine the publishing companies that publish trashy textbooks. Fourth, stress word problems from the very beginning, from the first grade, as they do in civilized countries, it's the best way to teach students how to think.

Or maybe this (page 10): "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."

What? You can not teach teaching? Unplanned lesson is the best lesson? Teaching is a gift from heaven, and you either get it or you don't? Isn't teaching a craft that you can get better at if you work at it? Whether the schools of education actually train their students in teaching or just indoctrinate them with the latest theoretical fads, is, of course, another question.

Michael Micek said...

The problem with Lockhart's Lament (as much of it as I've read) is that the classic "mathematics" curriculum teaches skills which are useful in a fairly wide range of endeavors, including mathematics proper. It is worthwhile that schoolchildren be drilled in arithmetic, algebra, &c. with the goal that they master the algorithms (and the concepts they presuppose) that the mathematicians of ages past have provided us. It is true that these drills are not -- as scales and other fingering exercises or training in music theory are not really music -- they are not mathematics itself. And if it were possible, more children should be exposed to mathematics proper, and it is the job of mathematicians (lovers of mathematics) to do this, as Lockhart is doing; one can not expect a pedagogue to accomplish this, and it is counterproductive to attempt it: instead of a love of mathematics which inspires the mastery of skills, all that will be accomplished is confusion. I suggest, therefore, that the so-called mathematics curriculum should continue to be taught as it has been in the past decades and centuries (although I would make an effort to spare those youngsters who retain mastery of excessive review, or in general, spending any more time on it than necessary; also, perhaps it needs a new name?), while providing separate resources to encourage those who wish to participate in amateur mathematics proper.

Niel said...

Anonymous March 14: I'll tell where to begin. First , realize and admit that mathematics is not about patterns, algorithms and abstractions, it is about problem solving.

Mathematics is at least partly about problem solving, yes: but is all problem solving mathematical?

I would venture to say that all "mathematical" problem solving involves the strategic exploitation of the fact that the problem has a high degree of structure. That is, mathematics is about structure --- but not just about passively contemplating the magnificence of those structures, but rather determining the properties of them, of playing with them; this leads to solving problems about them. But that is a consequence, not the motivation, of doing math.

Second, train the teachers in problem solving and require them to teach mathematics as problem solving [...] Third, adopt good textbooks and ban trashy ones [...] Fourth, stress word problems from the very beginning, from the first grade, as they do in civilized countries, it's the best way to teach students how to think.

This problem is not going to get solved by only approving good textbooks (although that will help), or just by making students solve word problems. As far as teaching methods go, the dao that can be written is not the true dao; you have to react to what works for each student.

Personally, word problems for me were irrelevant. I could do them, but I thought they were dumb. I got off on how numbers combined with each other, the concept of divisibility, and powers of 2: word problems were just a tedious translation exercise to get at what the question was really asking. "Just make them do word problems" will help some people, but turn others off.

What is needed is to ensure we educate teachers who will want to engage students, and then give them the resources, freedom, and class-room sizes to do that.

What? You can not teach teaching? Unplanned lesson is the best lesson? Teaching is a gift from heaven, and you either get it or you don't? Isn't teaching a craft that you can get better at if you work at it?

You get better at teaching just as with mathematics. But nobody can teach you how to teach: you have to teach in order to learn how to teach, just as with math. The dao that can be taught is not the true dao.

The problem is that the system, at all levels, wants to manufacture expert systems, and to say that this won't work (or can only work badly) is not something bureaucrats are prepared to hear.

Niel said...

Anonymous March 10: definitely a lot of fun reading.
but:
(1)it'd be better at a third the length.
(2) he doesn't seem to realize
that what he actually hates
is not just school *mathematics*,
but school *itself*


(1) I'm not sure it would be. He's fighting an uphill battle when speaking to non-mathematicians; a third of the length would not allow him to make the points he needs to make.
And at 25 pages, it's not like it's J.S. Mill's On the Subjugation of Women.

(2) I think he's probably acutely aware of it, but is also acutely aware that he needs to limit the scope of his diatribe to one where he can at least claim and demonstrate his expertise.

While his solution would not be easy, his diagnosis of the problem is spot on.

TomH said...

I'm a college physics teacher who thoroughly enjoyed Lockhart's piece. Physics (and the teaching thereof) is naturally affected by the teaching of math. Over the last 25 years, I've noticed definite trends in the entering freshmen (and freshwomen). (1) They are less and less able to grasp concepts such as proportionality (e.g. the inverse square law of gravity); they seem to have to have numbers to plug into their calculators. I suspect this is because they are not allowed to play with math and therefore think that math is just numbers. (Calculators, by the way, should be banned in math classes. I better not get started on that...) (2) Students are less and less interested in learning for its own sake. They don't think it's fun to learn new things. It is harder to communicate the beauty and "coolness" of concepts in physics and math. The attitude is "I want a college degree so I can get a job." There has always been that attitude, of course, but the percentage of students who really love to learn (without worrying so much about a grade) is less than it used to be. And that kind of student is the real "reward" in teaching. I don't know what the fix is, but I'm wondering whether I should run for the local School Board. What I mean is, any real solution is going to have to happen at the local level; God forbid the state or federal government meddle any more than they already do.

LuvGod said...

Every teacher especially math teachers should read this article. There are so many truths, in fact all truths. Math in America has gone mad and Lockhart captures this in a true artistic way. NO ONE can deny the truths. We face them every day.

CaptPoco said...

The system is working as intended. Real teaching (or real "math", whichever you prefer) is guided play. This is messy, and guiding it requires talent, which is in short supply. Therefore, there is the "routine" curriculum, designed so that even talentless teachers can teach. It also allows for some flexibility, so that talented teachers can do the whole "guided play" thing.

Sometimes, however, this flexibility is taken away, and teachers are told what each day in class must "achieve". This is what Lockhart laments, and while I am sure it happens in many places, it is a rather isolated phenomenon.

You really don't need too much flexibility. Something as simple as allowing students to "test out" of homework assignments should be enough. I think that if you give Math teachers the same flexibility as, say, History teachers, then you should be fine.

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Stephen said...

Some parts I like:

"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?"

Bad teachers just pass on information. Good teachers help students to discover it themselves.

"Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you’re probably not a very good teacher."

Also, a good method can give a good, but inexperienced teacher a reasonable place to start from, and experienced teachers who have become ineffective a safe place to return to.

Some parts I don't like:

"In particular, you can’t teach teaching. Schools of education are a complete crock."

Schools of education teach education the way public schools teach math -- by taking all the creativity and art out of it, and replacing it with crude, rigid methods and terminology. Lockhart, above all others, should have been able to predict this. That doesn't mean that education cannot be taught well. It can. I do it.

"...an organized “lesson plan” (which, by the way, insures
that your lesson will be planned, and therefore false)..."

Knowing what the students will learn ahead of time, conceiving of a basic outline of how the students are going to discover it, figuring out what resources you're going to need at the ready, and anticipating problems that might arise and how to deal with them is my idea of a lesson plan, and is invaluable.

"If you don’t have enough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it?"

Sure, you should be able to talk about your subject fluently, but that doesn't refute the advantages of preparing what you're going to say. Someone preparing and delivering a speech is usually a more effective communicator than that same person speaking extemporaneously.

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Dieter Verhofstadt said...

Most problems in our society come from the sheer population. There would be no traffic regulations needed if only a few had a car. You could throw trash on the streets if there's only a few of you.

Likewise, educational organization and standardization leads to mediocrity because there is a realistic fear that if you leave it up to the teachers to organize their curriculum, according to the needs of their pupils, you'll end up with lots of uninspired teachers trying to control disinterested school kids.

Lockhart's educational critique makes for a fantastic essay, but it is at least delusionary by pretending all teachers can be ex-researchers like himself, and all pupils are of the naturally interested kind he meets at St. Ann's. Ironically, there is a reason he's allowed to do what he does there.

Far from advocating standardized curricula, I understand why societies of large populations tend towards them, as a means of risk mitigation.