A Mathematician's Lament by: Paul Lockhart

A brilliant research mathematician who has devoted his career to teaching kids reveals math to be creative and beautiful and rejects standard anxiety-producing teaching methods. Witty and accessible, Paul Lockhart’s controversial approach will provoke spirited debate among educators and parents alike and it will alter the way we think about math forever.


So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy’s excellent description:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. So mathematicians sit around making patterns of ideas.

What sort of patterns? What sort of ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language and culture? No, not usually. These things are all far too complicated for most mathematicians’ taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary. For example, if I’m in the mood to think about shapes— and I often am— I might imagine a triangle inside a rectangular box:

I wonder how much of the box the triangle takes up? Two-thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing, amusing yourself with your imagination. For one thing, the question of how much of the box the triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes its size from one minute to the next. That is, unless you want to talk about some sort of approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple, and consequently it is an ugly question which depends on all sorts of real-world details. Let’s leave that to the scientists. The mathematical question is about an imaginary triangle inside an imaginary box. The edges are perfect because I want them to be— that is the sort of object I prefer to think about. This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.

Peer Review: (Mr. Sorochak April, 2017)

A mathematician's Lament takes to task the standard Math curriculum in today's public schools. Lockhart does go on a rant, but what he says rings true to me. I enjoyed math in school, was pretty good at it, but after finishing I was left feeling like: "I know how to do things, I just don't understand what those things really are." Lockheart takes on this feeling, explaining that the reason for it is that students are taught to do operations, but are not taught fundamentally what math it. To him, Math is an art which requires creativity and imagination, not processes and formulas.

Although his rips on math teachers are, I believe, unfair, his explanations for his desire to see the math curriculum change are clear and compelling representing good "food for thought." I highly recommend this book not only for those in math and science, but English and art as well.

Web Reviews:

When I began to read Lockhart's Lament, I was skeptical -- particularly with his view of mathematics as more of an art than a science. I am an applied mathematician, and I most enjoy teaching applied mathematics, but after serious and humble reflection, I came to fundamentally agree with Lockhart. Mathematics was developed as an expression of human creativity, and teaching it as such is really the only viable option for most students to be able to appreciate it and therefore fully apply it (if they ever need or want to).

As a relatively new mathematics teacher, I appreciate Lockhart's observations of the mathematics curriculum. I taught (college) trigonometry just before reading his Lament for the first time, and I was blown-away (and a little devastated) by the accuracy of his scathing description of that course:

"Two weeks of content are stretched to semester length by masturbatory definitional runarounds... students must learn to use the secant function, 'sec x,' as an abbreviation for the reciprocal of the cosine function, '1 / cos x' (a definition with as much intellectual weight as the decision to use '&' in place of 'and.') That this particular shorthand, a holdover from fifteenth century nautical tables, is still with us... is mere historical accident... Thus we clutter our math classes with pointless nomenclature for its own sake."

This book is an absolute necessity for anyone who wants to make sure their students actually enjoy mathematics. But be warned, if you view teaching mathematics as just a job, this book probably isn't for you.

More of what the web is saying:

Once in a while we read books that we just know are especially important, and that we know we will be thinking and talking about long after reading them. This book is one of them for me.

I am a returning adult student, and I am about to finish my training to become a math teacher. Having gone through my education program, my enthusiasm was just about completely drained, and I've been having trouble remembering why I ever wanted to become a math teacher in the first place. Why would anyone?

Paul Lockhart knows, and his book has reawakened my desire to help students discover the joy of mathematics. His argument is concise, and he makes it forcefully. His book is a joy to read, mainly because his understanding of the subject and his passion for it are clear in every page. He reinforces ideas I already had about how school sucks the life out of math (and all subjects), but he also challenges some of my opinions. I think this will happen with most people who read it.

Once he finishes making his argument about math education in about the first two-thirds of this short book, he devotes the remaining section to describing what he finds wonderful about mathematics itself. This section should make just about anyone want to become either a mathematician or a math teacher.

I want people to read the book for the specifics of his arguments, but I want to discuss one important point that he makes. Many people in math education claim that in order to make math more understandable and interesting to students, we need to show how practical it is and how it is used in everyday life. I've always felt like this idea was wrong, or at least limited in its usefulness in that regard. Well, Lockhart demolishes the idea, essentially claiming that practical uses are simply by-products of math, and that the real excitement and beauty of mathematics is in the abstract, imaginary, and creative world of mathematical ideas that have no specific connection to the everyday. By-products and applications can make math seem boring and secondary to the uses it serves. I agree with him--and much more now after having read his argument.

I honestly think just about everyone should read this book. Of course math teachers should, as should anybody involved in math education in any way. But I think people outside of math education should read it too. The specific mathematical ideas discussed in the book do not require a strong mathematical background, and I can't think of a better book that so concisely conveys the nature of the subject and the way it is viewed and misunderstood in society. I'm still not sure I agree with Lockhart's every point, but I love this book. (And I might agree with his every point after more thought and experience in the classroom.)

One who is a skeptic as well:

I teach math at a public high school. I agree with the sentiment presented in much of the book. However, I have at most 5-6 students a year that have the drive and curiosity to explore the ideas behind the mathematics. For many of them, it is a forced march through something they have to do. Much of this has to do with the way the curriculum is structured, but much of it has to do with the fact that many, many people do not find mathematics fascinating, just like many people are not interested in painting pictures. My teaching them the art and beauty behind the math, which I do when I can, for the most part causes moaning and eye rolling and confirms for the students that I am mostly crazy.

I also find it interesting that the back cover says that the author has now dedicated himself to teaching K-12 students... at an exclusive private school in New York. The reality in a regular public school in America will be much different than what goes on in that environment.

Leave you with a sample of what Lockhart is talking about:

From the book: So the mathematical landscape is filled with these interesting and delightful structures that we have built (or accidentally discovered) for our own amusement. We observe them, notice interesting patterns and try to craft elegant and compelling narratives to explain their behavior.

Read on to see an example:

One puzzle that needs a solution:

From the Book: What happens when you add up the first few odd numbers?

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81

Noticing a pattern? They always equal a square.

Will this go on forever? Why is this happening? Can you come up with a simple and elegant explanation which would excite and "move" the wonder of a child?

If the Preceding Challenge intrigued you, Perhaps you should pick this book up in the WMRHS Library (510.4 loc 2009)


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