One-hundred forty pages of sheer brilliance.

I’m not sure how to properly review Paul Lockhart’s incredible little book. The jacket sleeve contains all of the superlatives from great mathematicians (presumably, because I don’t really know their names, though their titles sound impressive enough), so why should I have anything to say on the subject? In an homage to the way in which Mr. Lockhart has written his intriguing story, I’m going to come at the book through my own narrative with mathematics.

First, I had the great good fortune of having the book put into my hand by my mentor and friend, Greg Glassman, the Founder of CrossFit and CEO of CrossFit, Inc. It might be more accurate to say I lifted the book from him, but he saw me perusing it while I was standing in his study and he said, “whaddya think?” I was a few pages into it while waiting for him to come back from checking on a sleeping child.
“It’s really interesting.”
“Take it. Take it. Let me know what you think.”
“Where’d you get it?”
“Remember that conversation we had a while back about math as art?”
“Yeah…”
“This is what came up in google when I typed in the question, ‘Is Math Art?’ I think it’s the first time I’ve ever typed a question into google. So I ordered it off of Amazon.”
“Hunh. I’ll let you know.”

On Greg’s desk was also a copy of Anton, Bivens, and Davis’ “Calculus: Early Transcendentals” (10th ed.). We had been discussing taking some math classes again, several of us. I had been in engineering at Boston University until the end of my sophomore year, when – like many former ENG students – I quit to become an English major.

“Oh, you got that calc book, eh?”
“Yeah. I’m going to start in on it.”
“That seems like jumping into the deep end. Not going to start at Trig or somewhere else and work up to it?” I was nervous. I made it through differential equations, but not without struggles.
“No, no. Just jump in. It will backfill what you need or you can as you go. You’ll be better off that way. Just order a copy and dig in.”

I liked math as a kid. Loved it, in fact. I was pretty good at it, too. I was probably the best math student in any class I was in through high school. Not so in college. Not a brag, but I had a 710 SAT math score, yet somehow I lost my love and enjoyment for it in college. Completely. My boss, however, maybe the most all-around brilliant man I know, told me that he was a terrible math student. He hated it. Yet he would later tutor college calculus and other math classes. A good friend of his – and mine – who got a degree in applied mathematics and works for a very well-known tech company as an engineer, told me that he learned math from Greg.

So, how does a guy who hated “math” wind up tutoring others in it – and also wind up studying applied mathematics for much of college? And how did someone like me – a “good” math student who professed to have loved it – wind up as an English major and lawyer with a rather “math-less” adult life?

Paul Lockhart has provided an answer. The fact is that I was never doing “mathematics” in school, even though that’s what I was told I was doing. Mr. Lockhart writes an impassioned rant against math education – indeed, it could be applied to almost all of current public education on any subject – and convinces beyond cavil that mathematics has been all but removed from education in our schools, starting from the very first days when our children are introduced to “Math” in school.

At the heart of Mr. Lockhart’s criticism is the astounding (and well-proven, in my opinion) claim that mathematics is art. Period. High art, in fact, and maybe the purest (and likely oldest) art in the world.

I’ll leave that alone and rather than try to prove the merit of Mr. Lockhart’s claim in an essay, I will simply commend any reader to pick up a copy of the book. Here, I’ll even link to it on Amazon. His argument in the form of a journey will make you reconsider almost everything you thought you knew about mathematics, and then education more generally, about the joy of teaching, and then about art, and several other subjects. This book is a tour de force of ideas and if you don’t come away profoundly changed then this probably isn’t one you should read, anyway, because you’re probably part of the problem.

The central tenet to Mr. Lockhart’s book helped me understand why I thought I was good at “mathematics” and why Greg thought he hated “mathematics.” We were using the same word to mean two very different things – and neither of us was correct. One quotation hit me in the chest like a horse-kick:

Many a graduate student [of mathematics] 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. [Uh-oh.] Math is not about following directions, it’s about making new directions. [double-gulp]

Paul Lockhart has hit on exactly why Greg told me he “hated math” and why I thought I was “good at math.” Greg intuitively recognized, as Lockhart points out, that the “main problem with school mathematics is that there are no problems.” The hell you say? “Oh, I know what passes for problems in math classes,” Lockhart goes on, “[…] these insipid ‘exercises.’ ‘Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.’ What a sad way to learn mathematics: to be a trained chimpanzee.”

The most compelling case for math as art comes in Lockhart’s wonderful analogy between mathematics and other arts, such as painting or music.

Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught music by someone who doesn’t even play an instrument and has never listened to a piece of music in their lives? Would you accept as an art teacher someone who had never picked up a pencil or set foot in a museum? Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students?

Some might find this a bit presumptuous or off-putting, but Lockhart makes a compelling case that the state of math education has devolved into nothing more than the regurgitation of rote memorization of certain techniques, dryly applied across a series of exercises that are completely divorced from the context from which these techniques were derived. I’m embarrassed to admit I was the rote memorizer – and really good at it. It was only after reading Lockhart’s book that I re-discovered, remembered little slivers of joy I had in early middle school when Mr. Wnuk would attempt to explain to our sixth grade class the original historical problems from which the “math” techniques that we were studying would be derived.

There are more gems in Paul Lockhart’s book than I can possibly do justice to. For example, there is this gem that simultaneously crushes the teaching profession (and will make certification  and standards people lose their minds), while revealing a truth that I’ve always believed and known from my own life experiences, but could never quite articulate:

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.

(P.47, Bold mine.)

As a brief diversion, I think about this point now in so many contexts. Long before our children go to school, we – their parents – are their first teachers. Who teaches a child to walk? You do, of course. But where is your degree in exercise physiology? Biomechanics? Kinesiology? Where is your teaching certificate? Who said you were qualified to teach an infant to walk?
Who teaches a child to talk? You do. But where is your degree in speech pathology? Linguistics? Do you even have any relevant experience in teaching words and vocabulary? Grammar? Language syntax and structure?
In the earliest development of a child, years before they go to school, We – their parents – are the principle instructors of ethics, philosophy, metaphysics, language, manners, morals, phys ed, mathematics, spelling, literature, art, love, and just about everything else under the sun. Few of us are “qualified” to do so, yet we think nothing of it.

Somehow, magically, when they turn five or six, however, we decide that now we have to have all manner of degrees, certifications, certificates, and licenses from the government that allegedly confer expertise to convey information to our children – as if prior to that moment never happened. It’s silly when you think about it.

I teach jiu jitsu, or I used to, anyway, pretty regularly, both to adults and kids. I have no teaching credentials. I have only a brown belt in Brasilian Jiu Jitsu under Prof. Pedro Sauer, yet I’m perfectly confident that I know what I’m doing and that I have a sense of both what must be taught, how to teach it, and how to do so safely and effectively. Would some pedagogical methods help me be a better teacher? Sure, probably. But the inverse isn’t true. You could have all the teaching degrees in the world, but if you have never been on the mat, you’re useless and can convey nothing of import to anyone, child or adult.

This is just one small sampling of what Lockhart’s book offers anyone with the willingness to open their mind to what he’s saying. The mathematics discussion is simply icing on an otherwise delicious cake. The book is worth buying and reading for the last 20 pages alone, when Lockhart provides examples of the kinds of “problems” that mathematics raises and he asks us to think about possible solutions and then he shows us his “proof,” or, as he calls each of them, “a narrative of some kind that helps us to understand why this pattern is occurring.”

Lockhart saves the best for last, with moments when he reveals what real mathematics is.

[…] this kind of mathematical experience goes to the heart of what it means to be human. And I’ll go even further and say that mathematics, this art of abstract pattern-making – even more than story-telling, painting, or music – is our most quintessentially human art form. This is what our brain do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are.

(P. 117).

Bravo, Mister Lockhart. I’m about to return to mathematics, but this time I won’t be a mere regurgitator of what I know the teacher wants. Instead, I will strive to be a musician, to struggle with the technique, but only in the context of a creative endeavor to find solutions to old problems and maybe even to find some new ones.

In short, I intend to be an artist this time through. Thank you. And for whatever it’s worth, I’ve ordered 10 copies of the book for my own staff, all of whom profess to hate mathematics. After all, they’re lawyers and everyone knows lawyers hate math… I think I now know why.