## Book Review: A Mathematicians Lament, by Paul Lockhart

This book is fantastic. I recommend it to all those people who, upon hearing from me that I do math, have replied, “Oh, I suck at math” or “Oh, I always hated math in school.” For years I’ve encountered recurring frustration at the fact that, whenever I tell people that I’m studying mathematics, I tend to discover that they have an almost completely wrong impression of what it is that I do (or at least try to do). It is not always easy to correct this impression. At my best, I try to tell them: you hate math because you don’t know what it is. You hate it because the stuff they taught you in school was not math, but almost something else entirely. Real math is fun and interesting and makes you think both logically and imaginatively, it is both beautiful and surprising. But I never quite feel like I get the whole truth across.

This book gets the whole truth across, or at least a good portion of it. It says everything I’ve ever wanted to say (as well as many things I hadn’t thought of) in regards to the question of what mathematicians actually do, but does a much better job than I ever could. He accurately portrays exactly how it is that people commonly misunderstand what mathematicians do, the common misperceptions that they tend to have. Then he goes on to critique what is pretty obviously at the root of all this misunderstanding: the state of math education itself. Through a few clever analogies and a bit of simple explanation, the author demonstrates how the average student’s assessment of the “mathematics” that we are taught in school is completely accurate: it is arbitrary, stupid, and boring. The analogy he opens the book with is worth quoting:

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 for the primary and secondary schools, their mission is to train students to use this language–to jiggle symbols around according to a fixed set of rules…

In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in scales and modes, meter, harmony, and counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school…” (p. 15-17)

Lockhart goes on to clearly and succinctly articulates the tragedy of the public math education system, and how it poisons the general public’s understanding of what mathematics is, and never even comes close to giving the general public a real sense of what the subject is even about. He gleefully and accurately (and also quite humorously) tears to shreds the current K-12 curriculum, exposing it’s idiocy.

But the book doesn’t stop there; after showing us what is wrong with the current state of mathematics education, he goes on to give a brief and wonderful little demonstration, through several very accessible examples, of what mathematics really is, the types of things that mathematicians actually think about, and why they are interesting. I found this to be wonderful–the best part of the book.

Here is one of his examples. Suppose I just decide to start adding up the odd numbers:

1+ 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

Notice a pattern? Yes, each of these sums is a perfect square! Does this pattern continue? It seems to. But why should it? What should odd numbers and squares have to do with each other? Can I show that the pattern goes on forever? Lockhart explains how this is the sort of thing that fascinates mathematicians: when there seems to be a pattern somewhere, or a relation between things that we never imagined would be related. We then try to discover the heart of why that pattern or relation exists–this is how mathematical proofs are born. Lockhart’s proof of this particular fact is a gem; you’ll have to read the book to see it. I also encourage the reader (as Lockhart does), to try to prove it himself.

The whole book is superbly written; his choice of words had me laughing out loud at times.

Finally, reading this book actually made me feel somewhat ashamed of my sometimes aloof personal attitude towards my area of study; my tendency to think of my subject as something intractably esoteric and advanced, inaccessible to the general public (because, of course, only the intellectual elite such as myself are capable of comprehending it). At one point he makes note of the fact that many people make it most of the way through graduate school believing (because they’ve always been told) that they are good at math, only to discover, when they attempt to do some real mathematics, that all they were really good at is following directions. Ouch. I hope that isn’t me…

So again, I recommend this book to all my friends and family who wonder what it is that I’m trying to do. I may even be buying it for some of you, come Christmas time.

Perhaps I will read this, since as you know, I am one of those people who hated math in school.

Yes, this book is definitely for you.