Review: A Mathematician's Lament, by Paul Lockhart

[Russ Allbery]

A Mathematician's Lament
by Paul Lockhart

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

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

-- 
Russ Allbery (eagle at eyrie.org)              <http://www.eyrie.org/~eagle/>