Measurement

by Paul Lockhart

Overview

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.

Product Details

ISBN-13:	9780674284388
Publisher:	Harvard University Press
Publication date:	05/12/2014
Edition description:	Reprint
Pages:	416
Sales rank:	248,609
Product dimensions:	5.50(w) x 8.25(h) x 1.12(d)

About the Author

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.

Read an Excerpt

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.

Table of Contents

Reality and Imagination 1

On Problems 5

Part 1 Size and Shape 21

In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves.

Part 2 Time and Space 199

Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.

Acknowledgments 399

Index 401

Editorial Reviews

This book forced me to use mental muscles I haven’t exercised in a long time, but it felt fantastic! Paul Lockhart is a mathematics evangelist; his passion for his subject is evident on every page, in every line. Looking at the subject of Measurement, he takes the reader on a journey that covers geometry, algebra, trigonometry, and on through differential calculus. He has a conversational tone and self-deprecating humor that sets the reader at ease. He understands that many people have been turned off of mathematics. His attitude is playful and joyous… Math is usually taught in such a compartmentalized way that it loses any meaning or coherence, and certainly any sense of wonder or beauty, but Measurement restores the connection. Paul Lockhart feels that math is the most beautiful, abstract and pure art form, and that it is actually fun! By the end of the book, you come to agree with him.

Sacramento Book Review

There are many books available these days on what mathematicians do, and this is one of the best… Lockhart’s approach is fresh and effective.

Choice

A conversational book about mathematics as an art that invites the reader to join in the fun. Sounding every bit the teacher whose love for his subject is infectious, he guides us through exercises in geometry and calculus—giving information and hints along the way while always encouraging us to ask, and answer, ‘Why?’ Lockhart does not try to make math seem easy; instead he wants his readers to understand that the difficulty brings rewards.

Scientific American

This invitation to tackle mathematical questions is infused with the joys of the rarefied reality of maths. Paul Lockhart largely avoids complex formulae and the wilder shores of jargon, opting instead for simple geometric drawings, lucid instructions and honest warnings about the hurdles. Covering size, shape, space and time, Lockhart, a maths teacher, gets through scores of problems, from showing that a cone in a hemisphere occupies half the volume to determining the size of the largest circle that can sit at the bottom of a parabola. Elegant, amusing and challenging.

Nature

In place of the usual boxed and high-lighted formulas and tricks, Measurement offers questions to be pondered. Lockhart invites readers to trade tutorial fake problems about actual objects, which lead students to abhor school mathematics, for real problems about fantastical objects, which lead mathematicians to love math.

Science

This invitation to tackle mathematical questions is infused with the joys of the rarefied reality of maths. Paul Lockhart largely avoids complex formulae and the wilder shores of jargon, opting instead for simple geometric drawings, lucid instructions and honest warnings about the hurdles. Covering size, shape, space and time, Lockhart, a maths teacher, gets through scores of problems, from showing that a cone in a hemisphere occupies half the volume to determining the size of the largest circle that can sit at the bottom of a parabola. Elegant, amusing and challenging.

Nature Lib

In place of the usual boxed and high-lighted formulas and tricks, Measurement offers questions to be pondered. Lockhart invites readers to trade tutorial fake problems about actual objects, which lead students to abhor school mathematics, for real problems about fantastical objects, which lead mathematicians to love math.

Science

There are many books available these days on what mathematicians do, and this is one of the best...Lockhart's approach is fresh and effective.

Choice - C. A. Gorini

This book forced me to use mental muscles I haven't exercised in a long time, but it felt fantastic! Paul Lockhart is a mathematics evangelist; his passion for his subject is evident on every page, in every line. Looking at the subject of Measurement, he takes the reader on a journey that covers geometry, algebra, trigonometry, and on through differential calculus. He has a conversational tone and self-deprecating humor that sets the reader at ease. He understands that many people have been turned off of mathematics. His attitude is playful and joyous...Math is usually taught in such a compartmentalized way that it loses any meaning or coherence, and certainly any sense of wonder or beauty, but Measurement restores the connection. Paul Lockhart feels that math is the most beautiful, abstract and pure art form, and that it is actually fun! By the end of the book, you come to agree with him.

Sacramento Book Review - Gretchen Wagner

Prospective readers should rest assured that while aimed at the nonexpert, Lockhart's writing is sophisticated and mathematically modern...In place of the usual boxed and high-lighted formulas and tricks, Measurement offers questions to be pondered. Lockhart invites readers to trade tutorial fake problems about actual objects, which lead students to abhor school mathematics, for real problems about fantastical objects, which lead mathematicians to love math.

Science - Brie Finegold

Lockhart is famous in the math world for a 2002 essay about the state of mathematics teaching. He described it as akin to teaching music by forcing children to transcribe notation without ever touching an instrument or singing. Measurement is his attempt to change the equation: a conversational book about mathematics as an art that invites the reader to join in the fun. Sounding every bit the teacher whose love for his subject is infectious, he guides us through exercises in geometry and calculus—giving information and hints along the way while always encouraging us to ask, and answer, "Why?" Lockhart does not try to make math seem easy; instead he wants his readers to understand that the difficulty brings rewards.

Scientific American - Evelyn Lamb

No matter what mathematical education you had, or didn't have, you will be delighted by this enticing book if you take up Paul Lockhart's invitation to engage in the mathematical sensibility that radiates from its pages, and try your own hand—not only at answering, but even more fruitfully, at formulating questions as you explore the world of mathematics.

Barry Mazur

The book is a love song and a philosophical manifesto about the pleasures and frustrations, but mainly the pleasures, of doing math.

Steven Strogatz

In A Mathematician’s Lament, Lockhart bemoaned the state of mathematics education in America. Taking up where he left off, Lockhart presents math as an art and argues that just as there is no systematic way to create beautiful and meaningful art, there is also no method for producing beautiful and meaningful mathematical arguments. Doing mathematics, according to Lockhart, is to make a discovery (by, say, physical objects like string or rubber bands) and then to explain it in the simplest and most elegant way possible. Using illustrations of various shapes and mathematical formulas, he leads readers through several problems step by step, encouraging them to collaborate with others in working through the problem. Measuring, for example, is relative because it involves comparing the object being measured to another object. Measurement is only one of the many rivers in the “vast, ever-expanding jungle” of mathematics, which for Lockhart satisfies our need to find patterns as well as our curiosity. “In my mind’s eye, there is a universe where beautiful shapes and patterns float by and do curious and surprising things,” he writes. His playful and ingenious approach not only takes the fear out of math but also elegantly illustrates that universe and the joy he finds in it. 16 line illus. Agent: req (Sept.)

Publishers Weekly

A love song and a philosophical manifesto about the pleasures and frustrations, but mainly the pleasures, of doing math.

In place of the usual boxed and high-lighted formulas and tricks, Measurement offers questions to be pondered. Lockhart invites readers to trade tutorial fake problems about actual objects, which lead students to abhor school mathematics, for real problems about fantastical objects, which lead mathematicians to love math.
-- Science

A conversational book about mathematics as an art that invites the reader to join in the fun. Sounding every bit the teacher whose love for his subject is infectious, he guides us through exercises in geometry and calculus—giving information and hints along the way while always encouraging us to ask, and answer, ‘Why?’ Lockhart does not try to make math seem easy; instead he wants his readers to understand that the difficulty brings rewards.
-- Scientific American

This invitation to tackle mathematical questions is infused with the joys of the rarefied reality of maths. Paul Lockhart largely avoids complex formulae and the wilder shores of jargon, opting instead for simple geometric drawings, lucid instructions and honest warnings about the hurdles. Covering size, shape, space and time, Lockhart, a maths teacher, gets through scores of problems, from showing that a cone in a hemisphere occupies half the volume to determining the size of the largest circle that can sit at the bottom of a parabola. Elegant, amusing and challenging.
-- Nature

This book forced me to use mental muscles I haven’t exercised in a long time, but it felt fantastic! Paul Lockhart is a mathematics evangelist; his passion for his subject is evident on every page, in every line. Looking at the subject of Measurement, he takes the reader on a journey that covers geometry, algebra, trigonometry, and on through differential calculus. He has a conversational tone and self-deprecating humor that sets the reader at ease. He understands that many people have been turned off of mathematics. His attitude is playful and joyous… Math is usually taught in such a compartmentalized way that it loses any meaning or coherence, and certainly any sense of wonder or beauty, but Measurement restores the connection. Paul Lockhart feels that math is the most beautiful, abstract and pure art form, and that it is actually fun! By the end of the book, you come to agree with him.
-- Sacramento Book Review

There are many books available these days on what mathematicians do, and this is one of the best… Lockhart’s approach is fresh and effective.
-- Choice

Lockhart presents math as an art and argues that just as there is no systematic way to create beautiful and meaningful art, there is also no method for producing beautiful and meaningful mathematical arguments. Doing mathematics, according to Lockhart, is to make a discovery (by, say, physical objects like string or rubber bands) and then to explain it in the simplest and most elegant way possible. Using illustrations of various shapes and mathematical formulas, he leads readers through several problems step by step, encouraging them to collaborate with others in working through the problem. Measuring, for example, is relative because it involves comparing the object being measured to another object. Measurement is only one of the many rivers in the ‘vast, ever-expanding jungle’ of mathematics, which for Lockhart satisfies our need to find patterns as well as our curiosity… His playful and ingenious approach not only takes the fear out of math but also elegantly illustrates that universe and the joy he finds in it.
-- Publishers Weekly

No matter what mathematical education you had, or didn’t have, you will be delighted by this enticing book if you take up Paul Lockhart’s invitation to engage in the mathematical sensibility that radiates from its pages, and try your own hand—not only at answering, but even more fruitfully, at formulating questions as you explore the world of mathematics.

From the Publisher