Paul Lockhart’s “A Mathematician’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.  ‘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…”

So begins Paul Lockhart’s “A Mathematician’s Lament” (Lockhart 2002; expanded version, Lockhart 2009), the most thought-provoking (and, in a way, depressing) article on teaching math I’ve run across in a very long time. The opening hits me especially hard because I’m a musician myself. I’ve spent many thousands of hours working with what’s technically called “Conventional Western Music Notation”: my undergrad degree is in music composition, and — long ago — I spent years developing a music-notation score editor. What Lockhart describes is indeed a nightmare! He goes on about music for several more paragraphs; describes a painter with a similar bad dream; and then says “Sadly, our present system of mathematics education is precisely this kind of nightmare.” I regret to say that I have to agree. He goes on:

“If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

“Everyone knows that something is wrong.  The politicians say, ‘we need higher standards.’ The schools say, ‘we need more money and equipment.’ Educators say one thing, and teachers say another.  They are all wrong.  The only people who understand what is going on are the ones most often blamed and least often heard: the students.  They say, ‘math class is stupid and boring,’ and they are right.”

I’m tempted to quote more, but you really should read it yourself. Don’t miss what Lockhart calls “the first ever completely honest course catalog for K-12 mathematics”! Ouch.

Though written years earlier, Lockhart’s Lament became widely known only after Keith Devlin wrote about it in his column on the Mathematical Association of America (MAA) website (Devlin 2008). Devlin commented that the Lament is “one of the best critiques of current K-12 mathematics education I have ever seen”; me too. It’s all pretty sad. Well, Lockhart exaggerates some. While he gives math teachers credit for good intentions, I think he underestimates how well a lot of us understand what mathematics is really about. Much of what he says has been said before, and some of it has been said better — for example, by G. Polya in the very first section, “Purpose”, of his classic How to Solve It (Polya 1957). And at one point Lockhart says that people are “under the gross misconception that mathematics is somehow useful to society!”; that’s pretty silly. Still, by and large, Lockhart’s writing is brilliant, and I think his scathing criticism is justified.

“But so what?”, I can hear you saying. “In the current political climate, making radical changes in the math curriculum in the vast majority of school systems is hopeless.” True. Of course your school might be an exception. Regardless, to paraphrase what Devlin wrote in his Foreword to the book edition, the book is a “landmark in the world of mathematics education that cannot and should not be ignored… You should read what he says and reflect on his words.” I shudder to think of how I taught geometry, especially proofs, not so long ago; if I’d read and absorbed Lockhart, even under my school’s requirements, I could have done much better.

One more thing. Have you heard of this essay? It was written in 2002, and it seems to be very well-known in some circles — but, unfortunately, mostly college math teachers (e.g., MAA members), not secondary-school teachers or math-teacher educators. Well, in-service teachers are generally (and understandably) utterly focused on short-term goals like getting through the week, month, and semester; but math specialists in teacher-training programs can afford to step back quite a bit. I completed an M.S. in math education just a few months ago, and I think the program I went through was a good one. But I don’t recall Lockhart was ever mentioned, nor do two of my fellow graduates I asked! And one member of the secondary-math-education faculty I asked had never heard of it. That’s interesting, too, isn’t it?

In future posts, I’ll talk about other slants on what’s wrong with American math education and what can/should be done — for example, Vi Hart’s slant. If you haven’t seen her stuff, don’t wait for me; spend a few minutes today watching one of her wonderfully engaging “Doodling in Math Class” videos (Hart, 2012)!

References

Devlin, Keith (2008, March). Devlin’s Angle: Lockhart’s Lament. Available at http://www.maa.org/devlin/devlin_03_08.html

Hart, Vi (2012). Doodling in Math Class. Available at http://vihart.com/doodling/

Lockhart, Paul (2002). A Mathematician’s Lament. Available at http://www.maa.org/devlin/LockhartsLament.pdf

Lockhart, Paul (2009). A Mathematician’s Lament.  New York: Bellevue Literary Press. (Consists of the “Lament” proper (Lockhart 2002) plus a second
part, an “Exultation”, with a Forward by Keith Devlin.)

Polya, George (1957). How to Solve It, 2nd ed.  Princeton University Press.

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