Two and Two Make Zero
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Two and Two Make Zero Two and Two Make Zero ❖ The Counting Numbers, Their Conceptualization, Symbolization, and Acquisition H.S. Yaseen Copyright © 2011 by H.S. Yaseen. Library of Congress Control Number: 2010914196 ISBN: Hardcover 978-1-4535-8445-3 Softcover 978-1-4535-8444-6 Ebook 978-1-4535-8446-0 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner. This book was printed in the United States of America. To order additional copies of this book, contact: Xlibris Corporation 1-888-795-4274 www.Xlibris.com [email protected] 87515 CONTENTS PREFACE .............................................................................................................9 I WHAT’S IN A NUMBER? ............................................................................11 I-1. The Idea ........................................................................................................11 I-2. Describing ‘Three’ ........................................................................................11 I-3. Why Numbers? Three Approaches to Size Assessment ...............................12 II NUMBER APPLICATION ..........................................................................16 II-1. Numerical Evaluation of Concrete Magnitudes ..........................................16 II-2. Calculation ..................................................................................................19 III NUMBER PERCEPTION ..........................................................................22 III-1. The Perceptible Number ............................................................................22 III-2. Number Illusions .......................................................................................24 III-3. Subitation ...................................................................................................27 III-4. Estimation ..................................................................................................32 IV Number Cognition and Symbolization ......................................................35 IV-1. The Dichotomy of Number Concepts ........................................................35 IV-2. General Principle of Conceptual/Symbolic Number—Sequence ..............37 IV-3. ‘Sums’ versus ‘Units’ .................................................................................40 V A HISTORY OF NUMERICAL NOTATIONS ..........................................43 V-1. The Gap .......................................................................................................43 V-2. The Import of Notational Symbols .............................................................49 V-3. The Three Methods of Visual Representation .............................................53 V-4. The Hindu Numerals Breakthrough ............................................................58 V-5. The New Arithmetic ....................................................................................61 V-6. New Arithmetic versus Old Arithmetic .......................................................63 V-7. Instrumental versus Conceptual Use of the Hindu Numerals .....................65 VI THE ORIGIN OF NUMBER .....................................................................69 VI-1. The Search for Number Sense ...................................................................69 VI-2. Number as a Process ..................................................................................87 VI-3. From Adjective to Noun ............................................................................89 VII THEORIES OF CHILDREN’S ACQUISITION OF NUMBER CONCEPTS .................................................................92 VII-1. Alfred Binet’s Pioneering Studies ............................................................92 VII-2. Piaget and the Origin of Number in Children ..........................................95 VII-3. Piaget’s Theory Reviewed......................................................................100 VII-4. Gelman and Gallistel: The Child’s Understanding of Number ..............107 VII-5. Gelman and Gallistel’s Theory Reviewed..............................................113 VIII CHILDREN’S NUMBER-CONCEPT ACQUISITION REVISITED ........................................................................................120 VIII-1. Symbols First ........................................................................................120 VIII-2. Parental Inputs into Children’s-Number Development ........................123 VIII-3. The Role of Counting in the Acquisition of Primal Numerical Concepts .................................................................129 VIII-4. The Limitations of Counting as a Tool of Arithmetic Education ..........135 VIII-5. The Nature of Number-Concept Development .....................................137 VIII-6. Steps in the Acquisition of Number Concepts ......................................141 LIST OF FIGURES Figure II-1: Comparative measurements...........................................................18 Figure II-2: Comparative measurements...........................................................19 Figure V-1: Comparison of Verbal, Hindu, Egyptian, Roman, Greek Alphabetical, Standard Chinese, and Abacus numerals ................47 Figure V-2: Egyptian numeral hieroglyphs .......................................................48 Figure V-3 Babylonian Numeral system..........................................................48 Figure V-4: Mayan numeral system ..................................................................49 Figure V-5: Roman numeral system .................................................................57 Figure V-6: Classical Greek alphabet numerals ................................................58 Figure V-7: Chinese numeral system ................................................................58 Figure V-8: Abacus ...........................................................................................63 Figure VI-1: The training-phase stimuli .............................................................84 Figure VI-2: Meck and Church’s experiment results .........................................85 Figure VI-3: The three modes of operations ......................................................85 Figure VI-4: The model of the counting-and-timing information processor .....86 References .........................................................................................................147 Index ..................................................................................................................155 PREFACE The nature of the things is perfectly indifferent, of all things it is true that two and two make four. -Alfred North Whitehead “Five plus three is really zero ,” explained mischievous Sarai to her second-grade teacher, “ . because,” she argued, “ . numbers are nothing!” Somewhat bewildered and concerned, her teacher related this incident to me in a parent-teacher conference. I smiled to myself realizing that my little Sarai had discovered that numbers are abstract ideas, not physical things. The charming way she articulated this most fundamental and necessary understanding inspired the title of this work. 9 I WHAT’S IN A NUMBER? I-1. THE IDEA ‘O ne,’ ‘two,’ ‘three,’ ‘ten,’ ‘hundred,’ and ‘thousand’ are number words with which we count, measure, and calculate. Each of these words communicates a discrete idea of size, as do any of the number words between or beyond them. This size idea is envisioned as, and defined by a fixed sum of units or, if you wish, ‘ones.’ It is because ‘five’ comprises more ‘ones’ than ‘three’ and fewer ‘ones’ than ‘six’ that ‘five’ is larger than ‘three’ and smaller than ‘six.’ What makes ‘five’ ‘five’ is the exact sum of its constituent units. Add one unit and it is no longer ‘five’ but ‘six’—take one away and it becomes ‘four.’ There are, of course, an infinite number of possible discrete sizes of this kind, many more than there are things to be counted or measured. After all, one may always add units to, multiply, or raise to another power any assembly of units one may wish to consider. In the words of Edward Kasner and James R. Newman , “Mathematics is man’s own handiwork, subject only to the limitation imposed by the laws of thought.”1 I-2. DESCRIBING ‘THREE’ Like all other numbers, “three” describes a numerical attribute: three. Insofar as the numerical attribute of ‘three’ itself is ‘three,’ ‘three’ is a self-descriptive or self-referential concept. Because numbers are self-referential all attempts to define or describe them result in a tautology. Take for example the mathematician philosopher Bertrand Russell ’s definition of a number as “the class of all classes that are similar to the given class,”2 which can be roughly translated into: “three is three.” His explanation that “every collection of similar classes has some common 1 Kasner and Newman , 1989, p.359 2 Russell , 1952, p. 208; 1996, p. 115 11 12 H.S. YASEEN predicate applicable to no entities except the class in question,”3 does not help to undo the circularity. But it is because of this self-descriptive, self-referential property that each number constitutes a wholly coherent and complete meaning or mental presentation on its own. A sentence such as: “five plus three equals eight,” is entirely intelligible and meaningful,