A Mathematician's Lament

by Paul Lockhart

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Publisher: Bellevue Literary Press
Copyright: 2009
ISBN: 1-934137-33-2
Format: Kindle
Pages: 139

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A Mathematician's Lament is a rant. The author, Paul Lockhart, was a research mathematician but changed careers to be a K-12 (for non-US readers: childhood through pre-collegiate education) math teacher. The topic of the rant is standard K-12 math education, which Lockhart blames for widespread fear of and dislike for mathematics in the US, for turning popular understanding of math into a mechanical rules-following exercise that has little or nothing to do with real mathematics, and for robbing children of the aesthetic and intellectual pleasure of learning mathematics properly. In the nature of a rant, it's forceful rather than nuanced and carries its point a bit farther than might be justified, but that makes for entertaining reading.

This rant started as a 25-page paper known as Lockhart's Lament, circulated in 2002 in typewritten copies and then published by Keith Devlin (who provides the foreword to this book) in his column for MAA Online. You can still read the original to get a sample of what you'd buy in this book. This expansion both develops the argument further and provides Lockhart a chance to give the reader more examples of what he considers good mathematical education. Unlike a lot of expanded rants, it's still tight, clear, and not particularly repetitive.

Lockhart's core point is captured well by the first two paragraphs of this short book:

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 you might guess, music is an analogy for how Lockhart argues we now treat math. It has been reduced to an exercise in rote learning that bears no resemblance to the practice of mathematics as understood by a mathematician. To Lockhart, math is not a mechanical tool requiring rote learning and endless practice. It's a creative exploration of ideas and rules. We can create rules arbitrarily, but then have to faithfully follow the rules of our creation when analyzing their properties.

Some parts of this resonated strongly with me. Lockhart's love of math shines through this essay, and his examples of mathematics done properly are both fun and fascinating. His favorite parts of mathematics are a bit different from mine (I do like learning known techniques and applying them well, rather than only making up my own), but he touches the same joy of exploration and fascinating analysis that I found in George Gamow's One Two Three... Infinity at an impressionable young age. I'm now wondering how much of my ongoing delight at mathematics (not that I do much with it these days) is because Gamow's book defined mathematics for me far deeper than schoolwork ever did.

Lockhart also tackles head-on the contention that mathematics is a tool used in many other fields, and therefore needs to be taught to students the way that we teach driving or other practical skills. This is true for certain fields, but not in the way that our current mathematics education focuses on them. Most people aren't going to do complex arithmetic in their head or on paper; they're going to use a calculator, and they should! High school math isn't the source of basic geometry for carpenters, who will likely relearn the few practical bits of math they need as tools rather than rely on the jumbled and vague memory of math class. And endless memorization of times tables... well, Lockhart is a bit more strongly against the rote learning of basic arithmetic than I am, but in an age of ubiquitous cell phones, he has a point.

He would prefer math be taught like music: something that's fun in its own right, something that's part of culture and mental delight, something that doesn't need to have some utilitarian purpose. In other words, the way that practicing mathematicians treat math, which is radically different than how it is currently taught.

This got me thinking about other basic school subjects and whether we teach pre-collegiate kids any other subject in the way practitioners think about that field. I think Lockhart believes math is uniquely bad and it does seem far removed from professional practice, particularly compared to English. Students have to write fiction, reporting, persuasive essays, and analysis of books in English class, which is largely what one would do with an English degree. Science education is possibly the closest to math, since students rarely perform meaningful experiments prior to college (or even graduate school) and instead are memorizing an array of facts already discovered. But a good science curriculum does at least have students reproduce some experiments and "prove" various physical properties — somewhat artificial, but not entirely disconnected from the practice of science.

History, though, is an interesting analogous case. My own high school history education was... odd, so I may have gotten less of the practice of real history than many students, but I was taught history as a series of important events to memorize. Most of them had prepackaged lessons and morals attached. This is, of course, almost nothing like the practice of history by a historian, which involves a lot of research in original sources and attempts to reconcile contradictory or maddeningly incomplete records into a coherent story. Perhaps good high school history courses do some of this. I think they would be better for doing so, not just because it might be more interesting and engaging, but because it would call into question the pat conclusions we often draw from history. Real history is a lot messier than a textbook. Making students aware of that would, I think, make them better citizens; agreed-upon "standard" history changes, and is heavily influenced by current politics.

Lockhart makes another point that was also made in a Teaching Company course I've been listening to recently (Redefining Reality, which sadly wasn't very good): science, and math even more so, are taught almost devoid of history. Students are told what we know now, and maybe a few vague sketches of previous theories, but not how our current understanding developed. Lockhart points out that this brings math alive in a way that puts our current attempts to shame. Rather than trying to map math to artificial "everyday" problems like dividing pies, talk about the problems Archimedes or the Pythagoreans were trying to solve. It may seem less immediately practical, but people developed these techniques for reasons, and those reasons were deeply rooted in problems or theory that they were wrestling with. It's a more honest and straightforward way to add human interest.

As you can tell, I found this thought-provoking, and I think it's well worth the price and modest reading time investment. If nothing else, you'll get Lockhart's wonderfully entertaining evisceration of high-school geometric proofs, and how they make a mockery of anything a real mathematician would do in a proof. He takes his overall argument farther than I would, and I'm dubious that tool-based mathematical training is as universally useless as Lockhart portrays, but the questions he raises deserve deep examination. And it's always a pleasure to read a passionate rant written by someone with a strong sense of the absurd and some skill at skewering it.

Rating: 8 out of 10

Reviewed: 2017-12-25

Last modified and spun 2017-12-26