For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.
In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.
Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.
Paul Lockhart teaches mathematics at Saint Ann’s School in Brooklyn, New York. He is the author of Arithmetic, Measurement, and the essay A Mathematician’s Lament.
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Reality and Imagination
There are many realities out there. There is of course the “physical reality” we find ourselves in. Then there are those imaginary universes that resemble physical reality very closely, such as the one where everything is exactly the same except I didn’t pee in my pants in fifth grade, or the one where that beautiful dark-haired girl on the bus turned to me and we started talking and ended up falling in love. There are plenty of those kinds of imaginary realities, believe me. But that’s neither here nor there.
I want to talk about a different sort of place. I’m going to call it “mathematical reality.” In my mind’s eye there is a universe, where beautiful shapes and patterns float by and do curious and surprising things that keep me amused and entertained. It’s an amazing place, and I really love it.
The thing is, physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in this universe is necessarily a rough approx-imation. It’s not bad, it’s just the nature of the place. The smallest speck is not a point and the thinnest wire is not a line.
Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I will never hold a circle in my hand, but I can hold one in my mind. And I can measure it. Mathematical reality is a beautiful wonderland of my own creation and I can explore it and think about it and talk about it with my friends.
Now there are lots of reasons people get interested in physical reality. After all, here we are! Astronomers, Biologists, Chemists, and all the rest are trying to figure out how it works. To describe it.
I want to describe mathematical reality. To make patterns. To figure out how they work. That’s what mathematicians like me try to do.
