Arithmetic for Parents: A Book for Grown-Ups About Children's Mathematics: Revised Edition

211 Pages·2015·2.21 MB·English

by  Ron Aharoni

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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Aharoni, Ron.
Arithmetic for parents : a book for grown-ups about children's mathematics / Ron Aharoni, Technion, Haifa,
Israel. -- [2015 edition].
pages cm
Translated from Hebrew.
Previously published: El Cerrito, Calif. : Sumizdat, 2006.
ISBN 978-9814602891 (hardcover : alk. paper) -- ISBN 978-9814602907 (pbk. : alk. paper)
1. Mathematics--Study and teaching (Elementary) 2. Arithmetic--Foundations. I. Title.
QA135.6.A37 2015
372.7'2--dc23
2014047935
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
the publisher.
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
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Contents
Preface vii
Introduction 1
Part One: Elements 5
What is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . 7
The Three Mathematical Ways of Economy . . . . . . . . . . . . 10
The Secret of Mathematical Beauty . . . . . . . . . . . . . . . . . 14
Layer upon Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Meaning and Calculation . . . . . . . . . . . . . . . . . . . . . . . 30
The Decimal System . . . . . . . . . . . . . . . . . . . . . . . . . 35
What is Learned? . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Part Two: The Road to Abstraction — Principles
of Teaching 47
Conveying Abstractions . . . . . . . . . . . . . . . . . . . . . . . . 49
Diversity and Fixation . . . . . . . . . . . . . . . . . . . . . . . . 51
Why Teaching is Difficult . . . . . . . . . . . . . . . . . . . . . . . 53
Mediation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Magic Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
The Calculator and Other Aids . . . . . . . . . . . . . . . . . . . 63
The Courage for Simplicity . . . . . . . . . . . . . . . . . . . . . . 67
Part Three: Arithmetic from First to Sixth Grades 69
A. Meaning 71
The Meaning of Addition . . . . . . . . . . . . . . . . . . . . . . . 72
The Meaning of Subtraction . . . . . . . . . . . . . . . . . . . . . 76
The Essence of Multiplication . . . . . . . . . . . . . . . . . . . . 79
The Two Meanings of Division . . . . . . . . . . . . . . . . . . . . 88
Meaning and Word Problems . . . . . . . . . . . . . . . . . . . . . 97
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vi Arithmetic for Parents
B. Calculation 99
The Calculation of Addition . . . . . . . . . . . . . . . . . . . . . 100
Subtraction: Loan or Reorganize? . . . . . . . . . . . . . . . . . . 106
The Calculation of Multiplication . . . . . . . . . . . . . . . . . . 110
Vertical Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 113
Memorize or Calculate Anew? . . . . . . . . . . . . . . . . . . . . 117
Calculation of Division Begins on the Left . . . . . . . . . . . . . 119
C. Fractions 127
Division and Fractions . . . . . . . . . . . . . . . . . . . . . . . . 128
Fractions Turn Division into Multiplication . . . . . . . . . . . . . 136
A Conversation on Division by Fractions . . . . . . . . . . . . . . 144
Solved Exercises on Fractions . . . . . . . . . . . . . . . . . . . . 147
The Common Denominator . . . . . . . . . . . . . . . . . . . . . . 149
The Least Common Denominator . . . . . . . . . . . . . . . . . . 154
Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
D. Decimals 161
Decimal Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Calculation in Decimal Fractions . . . . . . . . . . . . . . . . . . . 168
Percentages — a Universal Language for Fractions . . . . . . . . . 176
E. Ratios 181
Proportionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
More Ratio Problems . . . . . . . . . . . . . . . . . . . . . . . . . 188
Afterword 193
Appendix: Turning Points in the History of Modern
Mathematical Education 195
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Preface
Most adults have long buried their memories of studying mathematics.
All they really want is to forget the trauma. They accept their past
incomprehension as a tolerable, albeit painful, fact. “You don’t really need
toknowmathematics,” theyconsole themselves.Untilonedaytheneeddoes
arise and the old anxieties resurface. This happens when their child begins
dealing with the same experiences.
Many people would like to help their children with their arithmetic
studies,butaretooafraid toreturntoasubjectthatwas sopainfulforthem
as children. However, they forget that they have acquired many new tools
since their schooldays. The adult has more forbearance, abstraction skills,
the capability to deal with complex sentences, and the patience to wait until
the whole picture emerges. All these skills can be used to deal more easily
and quickly with the principles of elementary school mathematics.
The purpose of this book is to provide the guidance required to do so.
It offers assistance to the parent who wishes to be an active participant in
hisorherchild’sarithmeticstudies.Asamatteroffact,thatishowthebook
was born. The parents at my son’s school asked for written instructions so
thattheycouldhelptheirchildrenwithmathematics.Whatbeganasmodest
notes slowly evolved and eventually reached the structure and proportions
of a book.
Books, like ideas, have a life of their own. Sometimes they lead their
writer no less than he leads them. And so it was that this book gradually
took on a different form. One of the insights I came by while teaching in
elementary schoolisthatelementary mathematics isn’tsimpleatall.Besides
beauty, it has depth. This message slowly found its way into the book and
gave it an additional direction: A description of the beauty of elementary
mathematics and, consequently, mathematics in general. Thus my original
target audience expanded to include the reader who wishes to return to his
childhood mathematics, but from a different angle. For this group of readers
the book provides a second chance. Thosewho have learned how to multiply
fractions or how to perform long division, but never understood why it was
done exactly so, are invited to take a look from a new, mature viewpoint.
The book is also aimed at a third audience, one no less important for
me: teachers andeducators. Tothem thebooksendsaclear message: Proper
teaching of mathematics depends more on an understanding of the math-
ematical principles than on educational tricks. It requires familiarity with
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viii Arithmetic for Parents
the way the fine mathematical layers lay one upon the other. Furthermore,
it is best done by direct, concrete experience with the concepts, with no
intermediaries.
The first part of the book, Elements, describes the fundamentals of
elementary mathematics from the point of view of a teacher. That is, it
deals with the questions of what mathematics is, what material should be
taught in elementary school, and what makes mathematics beautiful and
similar to the arts.
The parent who wishes to help his or her child must, beyond
mathematics, be familiar with basic teaching principles, and to these I have
dedicated the second part of this book. It includes the principal rules of
progressing from the concrete to the abstract. In addition, parents must
also be familiar with the educational trends that determine the educational
character of their child’s school. For this reason, the book includes an
appendix describing, in a nutshell, the major developments in mathematical
education over the last fifty years.
The third part of the book is devoted to the nitty-gritty of elementary
mathematics, the details of the material taught in school, stage by stage.
Although geometry constitutes between 10% and 20% of the usual curricu-
lum,Idecidedtorenounceitandstick toarithmetic,thatis,thestudyofthe
propertiesofnumbers.Besidesthecentrality ofarithmeticinthecurriculum,
the reason for this choice is that arithmetic is a uniform, polished body of
knowledge, not unlike a fine diamond, and worthy of a book dedicated to
it alone.
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First Lesson
You never get a second chance to make a good first impression.
An American saying
In education as in life, first impressions are important. The way a subject is
introduced for the first time will determine to a great extent the student’s
future attitude towards it. Will it remain a pleasant memory, or a painful
one? Will it arouse a feeling of “I understand,” or of “this is difficult”?
For this reason, chapters titled First Lesson are interspersed throughout
the book. They provide suggestions of possible ways to present subjects for
thefirsttime.Thereisnever asinglemethod,butitisalways helpfultohave
a few options. The ideas are presented as pointers for teachers, but parents
may also find them useful.
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