Claudia Zaslavsky - Number Sense and Nonsense: Games, Puzzles, and Problems for Building Creative Math Confidence

Copyright 2019 by Claudia Zaslavsky

All rights reserved

Second Edition

Published by Chicago Review Press Incorporated

814 North Franklin Street

Chicago, Illinois 60610

ISBN 978-1-64160-245-7

Library of Congress Cataloging-in-Publication Data

Is available from the Library of Congress.

Cover design: Lindsey Cleworth Schauer

Interior design: Jonathan Hahn

Interior Illustrations: Rattray Design

Printed in the United States of America

5 4 3 2 1

To all our children. I hope they will enjoy figuring out the difference between number sense and nonsense.

I want to thank the many educators and authors whose ideas I have adapted for this book. In particular, I am indebted to the Family Math book for the activity Number Rectangles on

This book is about numbers, all kinds of numbers. Numbers have different personalities, just like people. Numbers have relationships, just as people do. A person might be a daughter or son, a sister or brother, a cousin, or a friend. Numbers are also related to other numbers in various ways.

Some ways of connecting numbers make good number sense. Others may make no sense at all; they are just nonsense. In this book you will meet young people who are trying to tell the difference between number sense and nonsense. Often they need some common sense to help them decide. I think you will enjoy their discussions and debates. You may want to have such discussions with your classmates, friends, and even grown-ups. These discussions will help you to build up your own number sense. You will develop a good head for numbers, the ability to see patterns in groups of numbers, and an appreciation of how useful numbers are and how much fun you can have with numbers.

Most of the topics in each chapter include ideas for activities that you can carry out either alone or with other people. There are also many questions for you to think about, questions that sometimes have no definite answers. Many of the activities are easy for you to check; you will decide whether they are correct. (See for the answers to some of the activities, although you probably wont need to refer to the answer pages very often. Its much more satisfying to work out the solutions yourself, even if it takes time.)

Some activities are harder than others and may be too difficult for you to complete at present. Try them anyhow. Discuss them with your friends or a grown-up. Your number sense will grow as you work on them, even if you dont get the final answer correct. And you will feel so proud when you do work out a hard problem. But if you cant finish some of the activities now, come back to them next year, or the year after next.

is all about odds and evens.

introduces other relationships among numbers. You will have a chance to work on an idea, called a conjecture, about even numbers that no one has been able to prove either true or false, although mathematicians have worked on it for over 200 years.

In you will learn why zero is a very special number. For example, we cannot divide a nonzero number by zero. When I try to divide six by zero on my calculator, the display shows the word error. Zero is special in other ways too.

deals with money, decimal points, and measures. You will learn how the United States lost hundreds of millions of dollars because of a mix-up in units of measurement.

Riddles and puzzles are the subject of , although you will also find some riddles and puzzles in other chapters. You will learn clever tricks with numbers that will make people think you are a genius!

How did numbers begin? How have people used numbers in the past? Thats what is all about. You will read about finger-counting, number words in several languages, signs for numbers, and ways of calculating that have been used in different parts of the world over thousands of years.

You may wonder, what about the calculator? Why bother learning about numbers when the calculator can do all the work? Not true! The calculator has no number sense. It takes a human being to tell the calculator what to do. The same is true of a computer. But the calculator can be very useful, as you will find out in . You will learn about negative numbers, repeating and terminating decimal fractions, and a lot more.

is about big numbers. Would you like your allowance to take the form of one cent on the first day of the month, two cents on the second day, four cents on the third day, eight cents on the fourth day, and double the amount every day until the end of the month? Guess how much money you would receive.

Happy reading, thinking, and talking about math!

This book introduces groups of children having a wonderful time discussing, reflecting on, and arguing about mathematical ideas. These children discover that some ideas make good number sense, while other ideas do not; they are just plain nonsense.

Math is for everyone! Research has shown that every child can learn math. When experienced through challenging activities, math can be a source of great joy and satisfaction to children and to adults. No longer can parents excuse their children by saying, I was poor in math, so I am not surprised that Jennifer has trouble learning the subject. No longer can teachers justify their students failure with the excuse, Those kids cant learn.

We now know a great deal about how children learn. The British neuropsychologist Brian Butterworth believes that the brain has specialized circuits that he calls the number module. In his fascinating and comprehensive book What Counts: How Every Brain Is Hardwired for Math (New York: Free Press, 1999) he writes, There is nothing intrinsically dull or hateful about mathematics. It will be, and can be, fun, as long as children understand what they are doing and feel pride of ownership in mathematical ideas. He advises teachers to use group discussions, encourage different solution strategies, acknowledge pupils own intuitions and knowledge to stimulate inventiveness. Of course, this advice is equally relevant for parents.

According to the Principles and Standards for School Mathematics, published in 2000 by the National Council of Teachers of Mathematics (NCTM), Students learn more and better when they can take control of their learning. When challenged with appropriately chosen tasks, students become confident in their ability to tackle difficult problems, eager to figure things out on their own, flexible in exploring mathematical ideas and trying alternative solution paths, and willing to persevere. In other words, children must learn mathematics with understanding.

Unfortunately, much school instruction in mathematics has been based on rote memorization of facts and procedures, rather than on understanding. Memorized procedures are easily forgotten or confused, but the ability to reconstruct methods and arrive at solutions on the basis of a mastery of the underlying concepts remains with a person over the years.

As I write these words, educators and the public are caught in a dilemma. On one side are parents who feel that their children are being shortchanged because many of the current math programs do not call for extensive drills on multiplication tables and addition factsthe so-called basics. On the other side are those who object to the kind of rote learning advanced by the older math programs, still used in many schools, and by teachers who teach as they have been taught, using drill and kill methods. With the increasing emphasis on standardized test scores, teachers are pressured to spend a great deal of time drilling for tests, in spite of the fact that research shows that when children learn the basics in the course of doing challenging math work, they are far better prepared to cope with new ideas than if they had learned by rote memorization methods.