(A few days later)
YOU: I think I’ve discovered something! I was looking for prime triplets, and what I noticed is that whenever you have three odd numbers in a row, one of them is always a multiple of three. Like with 13, 15, 17, the middle number is 5 × 3.
ME: That’s fantastic! And it certainly explains why 3, 5, 7 is the last of the prime triplets—the only prime which is a multiple of three is three itself. So now you just have to figure out why three odds in a row must always contain a multiple of three.
YOU: Does this process ever stop? Does math ever come to an end?
ME: No, because solving problems always leads to new problems. For instance, now you’ve got me wondering whether five odd numbers in a row must always contain a multiple of five . . .
This is how math problems arise—just from sincere and serendipitous exploration. And isn’t that how every great thing in life works? Children understand this. They know that learning and playing are the same thing. How sad that the grownups have forgotten. They think of learning as a chore, so they make it into one. Their problem is intentionality.
So let me leave you with the only practical advice I have to offer: just play! You don’t need a license to do math. You don’t need to take a class or read a book. Mathematical Reality is yours to enjoy for the rest of your life. It exists in your imagination and you can do whatever you want with it. Including nothing, of course.
If you happen to be a student in school (and you have my condolences), then try to ignore the pointless absurdity of your math class. If you want, you can escape from the tedium by actually doing mathematics . It’s nice to have interesting things to think about while you’re staring out the window and waiting for the bell to ring.
And if you are a math teacher, then you especially need to be playing around in Mathematical Reality. Your teaching should flow naturally from your own experience in the jungle, not from some fake tourist version with a car on tracks and the windows rolled up. So throw the stupid curriculum and textbooks out the window! Then you and your students can start doing some math together. And seriously, if you have no interest in exploring your own personal imaginary universe, in making discoveries and trying to understand them, then what are you doing calling yourself a math teacher? If you don’t have a personal relationship to your subject, and if it doesn’t move you and send chills down your spine, then you need to find something else to do. If you love working with children and you really want to be a teacher, that’s wonderful—but teach something that actually means something to you, about which you have something to say. It’s important that we be honest about that. Otherwise I think we teachers can do a lot of unintentional harm.
And if you are neither student nor teacher, but simply a person living in this world and searching as we all are for love and meaning, I hope I have managed to give you a glimpse of something beautiful and pure, a harmless and joyful activity that has brought untold delight to many people for thousands of years.
First published in the United States in 2009 by Bellevue Literary Press, New York
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Copyright © 2009 by Paul Lockhart
All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system now known or to be invented without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in a magazine, newspaper, or broadcast.
This book was published with the generous support of Bellevue Literary Press’s founding donor the Arnold Simon Family Trust, the Bernard & Irene Schwartz Foundation and the Lucius N. Littauer Foundation.
Library of Congress Cataloging-in-Publication Data
Book design and type formatting by Bernard Schleifer
eISBN : 978-1-934-13733-8
A Mathematician's Lament Page 8