NCTM members, single copy only; $12 to general This ... · Deborah Mullins, and Judy Chapman, for the editorial work, page layout, and production; the designers and artists, for overall

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Payne, Joseph N., Ed.Mathematics Learning in Early Childhood.National Council of Teachers of Mathematics, Inc.,Washington, D.C.75300p.; NCTM 37th YearbookNational Council of Teachers of Mathematics, Inc.,1906 Association Drive, Reston, VA 22091 ($11 forNCTM members, single copy only; $12 to generalpublic; 2-9 copies, 10 percent off the list price; 10or more copies, 20 percent off the list price)

EDRS PRICE MF-$0.76 HC Not Available from EDES. PLUS POSTAGEDESCRIPTORS Activity Learning; *Curriculum; Elementary Education;

*Elementary School Mathematics; Geometry;Instruction; *Learning; Literature Reviews;*Mathematics Education; Measurement; Number Concepts;Problem Solving; *Yearbooks

IDENTIFIERS *National Council of Teachers of Mathematics; NCTM

ABSTRACTThis yearbook presents many aspects of mathematics

learning by children between the ages of three and eight. Addressedto teachers of primary school children, the book begins with chaptersdiscussing learning and cognition, the primary curriculum, andresearch on mathematics learning at this age level. Eight subsequentchapters deal with the teaching of specific mathematics content:problem solving; mathematical experiences; number and numeration;operations on whole numbers; fractional numbers; geometry;measurement; and relations, number sentences, and other topics. Thefinal chapter discusses curricular change. A major theme throughoutthe bookis the importance of experience to learning, and thebuilding of new knowledge on the foundation these experiencesprovide. The book is designed to provide easy reference to bothgeneral information, such as answers to research questions, andsuggested classroom activities related to specific topics. Manyillustrations, the use of two-color printing, and a detailed indexaid the user in this. regard, (SD)

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NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

I

r

mathema=tics learningin early childhood

thirty-seventh yearbook

national council ,1 teachers of mathematics

1975

editor joseph n. payne, university of michigan

editorial panel mary folsom, university of miami (florida)

e. glenadine gibb, university of texas

1. doyal nelson, university of alberta

edward c. rathmell, university of northern iowa

caul r. trafton, national college of education

design and artwork irene b. tejuda, university of michigan

joanna c. payne, stephens college

nctm staff james d. gates

charles r. hucka

charles c. e. clement::

Copyright CI 1975 byTHE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. INC.

1906 Association Drive. Reston. Virginia 22091All rights reserved

Library of Congress Cataloging in Publication Data:

Main entry under title:

Mathematics learning in early childhood.

(37th YearbookNational C_ ouncil of Teachers ofMathematics)

Includes bibliographies and index.1. MathematicsStudy and teaching (Preschool)

2. MathematicsStudy and teaching (Primary)I. Payne. Joseph Neal. II. Title. III. Series: Na-tional Council of Teachers of Mathematics. Yearbook;37th.QA1.N3 37th (QA135.5J 510'.7s (372.7J 75-6631

Printed in the United States of America

contents

ABOUT THE BOOK v

1 learning and' cognitionharold w. stevenson, university of michigan

1

2 the curriculum 15pawl r. trafton, national college of education

3 research on mathematics learning 43marilyn n. suydam, ohio state universityj. fred weaver, university of wisconsin

4 problem solvingI. doyal nelson, university of albertajoan kirkpatrick, university of alberta

69

5 experiences for young children 95e. glenadine gibb, university of texasalberta m. castaneda, university of texas

6 number and numerationJoseph n. payne, university of michiganedward c. rathmell, university of northern Iowa

7 operations on whole numbersmary folsom, university of miami (florid#

8 fractional numbersarthur 1. coxford, university of michiganlawrence w. ellerbruch, eastern michigan university

iii

125

161

191

iv CONTENTS

9 geometryg. edith robinson, university of georgia

10 measurementg. edith robinson, university of georgiamichael I. mahaffey, university of georgiaI. doyal nelson,,university of alberta

11 relations, number sentences, andother topics

henry van engen, emeritus, university of wisconsindouglas grouws, university of rnissouri

12 directions of curricular changealan r. osborne, ohio state universitywilliam h. nibbelink, university of iowa

INDEX

os.

205

227

251

273

295

about the book

Children, ages three to eight: their mathematics, their learning, their teacher, theirachievement, assessing their knowledge and giving guidance to their thoughts, planningtheir experiences, and cultivating their ability

... to solve problems

... to learn mathematics.. to be successful with mathematica

... to see uses of mathematics

... to have fun and satisfaction with mathematics

that's what this yearbook is about.It is a book for teachersfor teachers of young childrendesigned to help them in

making important decisions about teaching and about curriculum. It was planned to be asource of knowledge and a source of activities.

Knowledge about children's general cognitive development, about the way they learnmathematics, and about curriculum design is reflected in chapters 1, 2, and 3. In these earlychapters''s established a general framewOrk for learning,and for curriculum. Knowledgeabout learning and teach.ng specific mathematical content is found in chapters 4 through 11Knowledge special curriculum projects and direction for change are outlined in chapter12.

Activities for children permeate the "content" chapters, chapters 4 through 11. Theseactivities are designed to engage children and to elicit thoughtful responses to produceeffective learning. Most will stimulate teachers to go further in designing their own activities.

Three themes are stressed: problem solving, relating mathematics to the real world of thechild, and a developmental point of view toward learning.

"Problem Solvingwas deliberately placed first among the content chapters. This primeplacement suggests both the importance of problem solving as a major goal and the use ofproblem solving in all the content areas.

Relating mathematics to the real world of the child requires at least two things: (1) relatingthe mathematics to be learned to that which a child already knows; and (2) relating themathematics to be learned to the child's own perceptions and views of concretemanipulative objects and situations in the real world. All content chapters stress the need tobegin with concrete objects. Questions are then asked about the features of the objects,leading to the abstraction of the mathematical idea. Suggestions are given for helpingchildren relate symbols and verbal language to the mathematical ideas in a meaningful way.

The developmental nature of learning should be evident throughout the book. Eachauthor took seriously the charge to try to show how given content relates to, and grows from,that which a child already knows. A careful analysis of each child's learning is emphasized.The teacher is viewed as a planner of experiencesexperiences executed with a balancebetween teacher direction and child initiation. Incidental experiences are to be used, butthey are not to be the only learning experience.

The informality of the learning atmosphere is reflected in the activities and in thesuggestions to teachers. Furthermore, it was a major factor considered in the overall designof the book. The design, the large pages, the use of a second color all should combine to

V

vi ABOUT THE BOOK

reflect an attractive and inviting atmosphere, which should be sought with children in theclassroom.

The variety of the content chapters suggests a wide range of objectives for children agedthree to eight. These chapters contain a careful analysis of familiar ideas as well as freshviews of some content areas.

Special concern was noted for the mathematics learning of children at an early age. Togive emphasis to this concern, chapter 5 was devoted to mathematics experiences for youngchildren. Most of the other content chapters also have specific suggestions that are usefulwith young children. What should be striking to the reader of chapter 5 and the otherchapters is that the mathematics learning of very young children is much more that, justcounting (too often the only goal of early childhood education).

The book is designed to be used by teachers on an independent basis, by teachersengaged in continuing education, by preservice teachers in courses on teachingmathematics to children, and by graduate students in seminars on mathematics learning.The book should be helpful to school systems active in curriculum development at theprimary school level and to preschool programs such as Head Start. It is a hope that thepractical suggestions will make the book of continuing use to a teacher in the classroom.Further, it is hoped that the research and the theoretical work will be of continuing use toresearchers and scholars in mathematics education.

The time to edit and produce this book has meant that the two chapters stressingresearch and the chapter on newer curricula reflect the knowledge up to two years ago.Since research and curricular results do change over time, the reader is alerted to seeksubsequent publications in these areas.

Many people deserve special thanks for their contribution to the book. As editor, I saw agreat array of talent and energy given freely by many people. Among this large group, I wishto thank these especially:

the Editorial Panel, for overall planning, selection of authors, and critical reading ofmanuscripts;

the authors, both for the initial writing of the manuscript and for rewriting at (eastonce in view of editorial comments;

the NCTM editorial staff, Charles Hucka, Charles Clements, Jean Carpenter,Deborah Mullins, and Judy Chapman, for the editorial work, page layout, andproduction;

the designers and artists, for overall design of the book and page layout, for designof the cover and the chapter openers, and for the artwork;

these people for specific jobs: James Watson and Lawrence Ellerbruch, for criticalreviews and editorial work; Leslie Steffe, Sheila Rosenbaum, Marilyn Hall,Graham Jones, Calvin Irons, and Timothy Parker for critical reading; RowanMurphy, for special work on the chapter openers; the University of AlbertaFaculty of Education Audio Visual Media Center and Westbrook ElementarySchool in Edmonton, for photographs used in chapter 4, on the cover, and inother parts of the book; and James Macmillan, for photographs-in chapter 6.

It has been fun to work on the book. I hope that some parts of it elicit for the readerthe same feelings of excitement and satisfaction that have gone into its production.

Joseph N. Payne, Editor

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major conclusions of theresearch

''The following generalizations are rather wellsubstantiated by experimental data. Some areimplicit in the teaching techniques already inpractice in most schools; others may be new toeducators. Still others may contradictor mayappear to contradictassumptions underlyingcurrent teaching practices.

individual differences

Wide individual differences exist in the abili-ties of children to learn and to solve problems,and these differences are complex a:id difficultto determinean obvious statement, perhaps,and one that most teachers would confirm. Yetits infallibility is basic to any consideration ofchildren's learning. Whatever the group, what-ever the task and its presentation, children tendto learn at different speeds. The rate of acquir-ing a conditioned response differs among new-borns just as the rate of learning differentialequations differs among twenty-year-olds. Thepervasiveness of individual differences pre-cludes the possibility of producing equallyrapid progress by all children through any setof materials, The more we discover about indi-vidual differences. the more complex the prob-lem seems to become.

A commonly held hypothesis about theof differential rates of learning is that learningability is determined by the child's level of in-telligence. Everyday experience indicates thatbrighter children tend to learn more rapidly. es-pecially in the early grades. The difficulty inreaching conclusions about the relation be-tween learning and intelligence from what goeson in classrooms is that children do not startlearning with equal amounts of information andexperience The child who has a good vocabu-lary is more likely to be able to use language ef-fectively in school, the child who has broad ex-perience and can identify a large number ofcommon objects will be able to relate this expe-rience to his classroom studies. Because there

LEARNING AND COGNITION 3

is so much transfer from the child's everydayexperience to what occurs in school, the class-room is probably not the best place to try to de-termine the relationship between learning andintelligence.

Perhaps a more revealing approach wouldbe to use tasks in which there is less transferfrom everyday life. This can be done with manyof the materials used in laboratory studies. Wecan test children's ability to learn to associatethe names of two animals, to remember the lo-cation of cards of different colors, to learn anew code, or to learn a new principle. Althoughdifferences in past experience are not elimi-nated, these tasks are less dependent on whatchildren have already learned. The results of anumber of studies using such tasks have beenreported, and the correlation between 10 andrate of learning in these laboratory tasks oflearning and problem solving is rarely morethan .50 (i.e., r = .50). This means that in-telligence is related to children s ability to learn,but that rarely is more than 25 percent (r2 = .50x.50, or 25 percent) of the variability in per-formance on the learning task attributable tovariability in intelligence. It is not appropriate,therefore, to discuss learning and intelligenceas identical functions. Many factors besides in-telligence play an important role in determiningthe rate of learning. For example, the widelydiffering levels of anxiety with which childrenenter learning situations have been found to besignificantly related to their rate of learning.The style with which they approach problemsalso differssome respond rapidly and impul-sively, others are more cautious and reflec-tiveand some learning situations require oneor another style of approach more or less ex-clusively. Furthermore, the nature of children'smotivation to achieve and their level of aspira-tion have been found to play an important rolein determining how well they will perform inlearning tasks.

It would, therefore, seem that the teachermust be a psychodagnostician to comprehendal the factors that may contribute to differ-ences in performance among children in aclassroom. This suggestion is impracticable, ofcourse, but it does emphasize the importanceof the teacher's close attention to the charac-teristics of each child as well as to the material

4 CHAPTER ONEC

presented. On the basis of what is known aboutindividual differences in learning, it seems thatthe teacher's efforts should be directed in-creasingly toward presenting to children mate-rials that will capitalize on their individualstrengths.

attention to irrelevances

Children may make errors because they at-tend to the irrelevant attributes of a situation.Some of the most frequently quoted examplesof inadequate cognitive development are foundin the responses of young children to the con-servation problems developed by Piaget. Thechild is shown two identical beakers filled withequivalent amounts of liquid. Aft .r he hasjudged the amounts to be the same, the liquidin one beaker is poured into a third, narrowerbeaker. The child is asked whether the twobeakers now contain the same amounts of liq-uid, and young children typically say that theydo not. Their explanation is that the water leveldiffers in the two beakers. Their judgment isbased, therefore, on the attribute of heightrather than volume. In another example thechild is asked to count out ten beans and placethem in two parallel rows, with each bean inone row directly opposite its counterpart in theother row. Is the number of beans in each rowthe same? Yes, the child concludes. Then thebeans in one row are placed close together sothat they occupy a shorter length than thebeans in the other row. Now are there equalnumbers of beans in each row? The youngchild is likely to answer no. Why? Because onerow is longer than the other. He appears unableto comprehend that differences in length do notalways produce differences in quantity.

Do results like these mean that the child be-lieves that variations in one attribute implychanges in a second attribute? Or are the re-sults more readily explained by the manner inwhich young children deploy their attention?Can young children be taught to ignorechanges in an irrelevant attribute and basetheir judgment on what happens to the relevantattribute? Gelman (1969) has presented eyi-.dence that they can.

From the adult's po!nt of view only theamount of liquid is relevant in problems dealingwith the conservation of liquid quantity. But onthe test trials, children notice that the beakersmay differ in size, shape, height, and width.Gelman attempted to force children to concen-trate only on the quantitative relations and toignore other attributes. Standard tests of con-servation of length, number, liquid, and masswere given to five-year-olds, and only thosewho failed to demonstrate conservation wereretained for the study. For training, these chil-dren were given experience with oddity prob-lems in which they had to select the odd stimu-lus in a set of three. Number wcs the relevantattribute, and spatial configuration was irrele-vant; that is, the child was to learn to choose thestimulus that differed in number from the othertwo members of the set. The stimuli were care-fully constructed so that on one trial there mayhave been two different patterns of five dotswith a third, still different pattern of only threedots (). On the next trial there may have beenone set of five dots arranged in a short line buttwo sets of four dots spread out in different ar-rangements.

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The children soon learnec to seiect the stim-ulus that was "odd' in its nut Cher of elements,regardless of its spatial configuration. Theywere then retested for conservation of lengthand number, and nearly all showed perfectconservation. In addition, when they were re-tested for conservation of liquid and mass.

which they also had failed earlier, over half thechildren demonstrated conservation. There-fore, both specific and generalized transfer oc-curred as a result of the oddity training. Whenthe children were later presented with manydifferent examples of quantitative equalitiesand differences and were required to respondon the basis of these examples, they performedeffectively in tests of conservation. Their in-adequacies appeared, not in their thinking, butin their attending.

We know that attention is a prerequisite forlearningif the child is not attending to the ma-terial being presented, it is impossible forlearning to occur. We should be equally awarethat his attention must be directed specificallyto those aspects of the situation that are criticalfor the solution of the problem. For the teacher,what is relevant and what is irrelevant is obvi-ous. We cannot assume, however, that what isrelevant for the teacher will attract the child'sattention. What appears to be poor conceptual-ization by the child may simply be a result of theteacher's failure to present the problem so thatits critical features are highlighted.

distraction by irrelevantinformation

Young children are easily distracted by thepresence of irrelevant information. It is very dif-ficult for us to see the world as it is perceived byyoung children. Vast amounts of experiencehave enabled us to respond to critical featuresof our environment and to ignore those aspectsthat have no immediate significance. Youngchildren do this only with difficulty; for them, in-cidental features of the environment may be assalient as those that have some importance.The ability to observe selectivelyto categorizethe environment into what is critical and what isnotdevelops rather late; evidence indicatesthat not until the child is ten or twelve years oldis he able to do this spontaneously. Theyounger child, hoWever, can do it with help orspecial training, but in designing materials forhim, we run into a paradox. In an effort to makethe material interesting, unessential details areoften includedscenes contain more than the


central figures, workbook fofmats vary frompage to page. The net effect is that the addition-al details introduced to heighten interest oftenact at the same time as distracters and lead thechild to fail to note the central information thatis being imparted.

This point is illustrated in a study by Lubkerand Small (1969), who presented third andfourth graders with an oddity task of the typedescribed earlier. The problem looked simple.All the child had to do was choose from a set offour stimuli the one that differed from the oth-ers in color. But in some of the tasks irrelevantinformation was presentfor example, theforms differed in brightness, size, or thickness.Adults would rapidly learn to ionore the irrele-vant information, for they could quickly deter-mine that it was of no help in attaining the cor-rect response. For children, however, thepresence of the irrelevant information acted asa distraction, and their performance suffered.When tested with stimuli that contained one ortwo irrelevant dimensions, they performed onlyslightly above chance at the end of the training.When no irrelevant dimensions were present,over 90 percent of their responses were cor-rect.

We can make learning easier if we eliminateas much irrelevant information as possible, foryoung children have a hard time doing this bythemselves. At the same time, we can be help-ful if we heighten the differences among stimuliby having them differ consistently in more thanone respect. That is, irrelevant information maybe deleterious to learning, but redundant infor-mation may be helpful. Learning to discrimi-nate a large black square from a small whitecircle would be easy, but learning to discrimi-nate a square that was sometimes black andsometimes large from a circle that was alsoblack at times and large at times would be verydifficult.

transfer of information

Transfer is facilitated for the child if each rulehas a wide variety of examples covering a widerange of extremes. The goal of most teaching is

6 CHAPTER ONE

to provide information that can be used appro-priately in new settings. Therefore, we are inter-ested in improving the child's ability to transferinformation from one context to another. Wecommonly teach the child a rule and then askhim to use the rule with new materials, a taskthat is often difficult for young children. Al-though they are able to learn the ru'e for solv-ing the original problem, they approach thechanged situatio.. as if it posed anew problemand fail to apply the rule.

An example of this difficulty can be seen inthe behavior of young children in problems oftransposition. where the child is asked to learna relational rule and transfer it to new sets ofstimuli. The child is taught to choose from a setof stimuli the one that differs in a property suchas size. Original learning can occur in a trial-and-error fashion. Each time the child chooses,say, the middle-sized stimulus in a set of three.he is rewarded; choices of the smallest orlargest stimuli do not lead to reward. When the

T:77:737-77-",71:1

middle-sized stimulus is chosen consistently,the experimenter introduces a new set of stim-uli that differ in absolute size from the trainingset but bear the same within-set relation.Young children have difficulty with this type ofproblem and fail to apply the rule they have justlearned. Could it be that they fail to demon-strate transfer, not because they did not learnthe rule well, but because they failed to under-stand that a response learned in one situationis applicable to other situations? If this is true, itshould be possible to demonstrate to the childduring the original training that the response isnot restricted to stimuli with certain absoluteproperties. Beatty and Weir (1966) have donethis with three- and four-year-olds. childrenwho typically have difficulty in transferring con-cepts such as largest, smallest, and middle-sized.

The stimuli were sixteen squares with arearatios of 1.3 to 1. Training was conducted withstimuli 4-5-6 and 14-15-16 (the squares are

tdr77-77-71

1

numbered from 1 to 16 in order of increasingsize for convenience of description). The sets ofstimuli were presented in random order. Thistype of training should increase the child's un-derstanding that the choice of intermediate sizeis appropriate over a broad range of stimuli. Af-ter they learned the correct response to thetraining stimuli, a new set, 9-10-11, waspresented without comment. Would the chil-dren choose at random, or would their firstchoice be that of the stimulus of intermediatesize? Over three-fourths of the children dem-onstrated transferthey chose the stimulus ofintermediate size.

This study offers an interesting insight intohow we can improve the child's ability to trans-fer information. If we teach the child from thebeginning that the same rule is applicable towidely divergent instances within the same


class of problems, we should increase the like-lihood that the child will be able to use the rulewith still different instances. We usually try toselect similar examples for use during thechild's first exposure to a problem in an effortto aid him in his original learning, but this mayhave the unexpected effect of restricting hisability to use the information in other situations.

language and abstract thought

Language may be helpful, but it is unneces-sary for abstract thought. An enormous amountof effort has been expended in attempts to un-derstand the relationship between them. Somepsychologists view abstract thought as a prod-uct of language; others believe that language

8 CHAPTER ONE

development and cognitive development areparallel but not interdependent processes.There is no question that language may be ofhelp in conceptualization, but can concepts belearned and used without the intervention oflanguage? For older children and adults it isnearly impossible to separate the two proc-esses. Language is such a highly practiced skillthat one can translate nearly all experience intowords. Young children, however, are still in theprocess of learning language. Are there ways inwhich we can demonstrate that they are ca-pable of abstract thinking, as exemplified intheir correct application of certain concepts,but are unable to tell us how they solved theproblem or to describe the concepts they em-ployed?

We can use a study by Caron (1968) as a ref-erence. Three-year-olds do not know words todescribe the concepts of roundness or angu-larity. Furthermore, without prior training it isextremely difficult for them to use these con-cepts. Caron sought to develop pretraining ex-periences that might lead these young childrento employ the concepts correctly. Many sets offigures were constructed in which the differ-entiating attribute was the roundedness orpointedness of a portion of the figure. The fig-ures were paired in a discrimination problemwhere the correct choice was dependent on theconsistent selection of a figure that containedone of these characteristics ( ). Some childrenconsistently had to pick the stimulus with arJunded portion, and others had to pick thestimulus with a pointed portion. For some of thechildren the figures were initially presentedonly in part. Rather than use the fully repre-sented figure, only the portion of each figurethat contained the distinctive attribute was vis-ible. Very gradually, over a tong series of trials,the full figure was faded in. By seeing at firstonly the critical feature, three-year-olds wereable to learn the discrimination. They gaveclear evidence of having used the concept: butthere was no indication that the concepts hadbeen represented in words. The children couldnot tell the experimenter at the end of the studyhow they had solved the problem, nor couldthey pick out the ':round" and "pointed" figureswhen they were directed to do so.

r\n"

The same results were obtained with a differ-ent pretraining procedure. Other groups ofchildren were asked to fit the stimulus figuresinto a hollow V shape. The figures with an an-gular portion fitted into the V, but the others didnot. The child was to go through the stimuli,plac..1g the figures that fitted the shape into onepile and those that did not into another. Whenthe children later were required to learn thediscrimination task, they were highly success-ful. Again, they could not give a verbal ex-planation of how they solved the problem. norwere they able to identify the figures that pos-sessed the attribute described by the adult.From studies such as these we can concludethat children are capable of using conceptsthey cannot verbalize.

We may be expecting too much of children,especially during the early years when lan-guage is still undergoing rapid development, ifwe require that they should always be able totell us how they solved a problem. Words arethe natural means for transmitting knowledgeamong adults, but they are not always the mosteffective medium for instructing children. Yetclassrooms are highly verbal environments, afact that may indicate why scores on the verbalportion of IQ tests are better predictors ofschool success than scores on the nonverbalportions of the tests. Perhaps our classroomsin the early elementar, years are too full ofwords. Learning may ti aided if children aregiven greater opportunities to learn in otherways.

learning by observation

Children learn well through observation. Inmost formal learning situations the child is ex-pected to act rather than observe. We tend tothink that we are not performing effectively asteachers unless something is being activelytaught. unless the child is making some formof verbal or motor response throughout thelesson Each child is given a workbook and ex-pected to learn by solving each successiveproblem. Our children come to expect thatwhile they are solving problems at the boardthey are supposed to be learning, but otherwisethey are supposed to be waiting their turn pas-sively This is in direct contrast with what hap-pens in everyday life, where a significantamount of learning occurs through the obser-vation of other persons in the home, at play, oron television Children learn styles of dress byobserving what others are wearing: they learncertain types of speech by hearing what otherpeople say Young children learn complexgames by watching older children performthem. Active participation by the child is ofgreat importance in producing many types ofbehavioral change. but we know that the childcan also learn effectively by observing the be-havior of other persons. In fact, in some situ-ations learning may be more effective throughobservation than through direct participation.

A study by Rosenbaum (1967) illustratesmany of the features of observational learning.An observer and a performer (children fromgrades 1 through 6) participated in each exper-imental session. The performers solved twentyposition discriminations in which one of fourpositions was correct. They responded by in-serting a stylus into holes of an 80-hole matrix(20 rows, 4 columns). They were required to lo-cate the correct hole before proceeding to thenext row. Both performers and observers werethen given a printed duplicate of the matrix andasked to mark the position that was correct ineach row The observers demonstrated a sig-nificant degree of learning, even though theirexperience had been limited to viewing anotherperson's efforts and the consequence of his re-sponse The scores of the observers not only


were above chance at all grade levels but ex-ceeded those obtained by the performersthemselves. Spared the chores faced by theperformers of following directions, inserting thestylus at the correct times, and rememberingwhich holes had and had not been tried, the ob-servers apparently were able to stand back andview the performer's efforts in a casual, but ef-fective, manner.

Although great concern is shown over howyoung children learn the wrong things andwrong values by passively watching television,not enough regard is shown for the value of ob-servational learning in the classroom. (Even-tually, of course, we must ask the child to per-form, for otherwise we have no indication ofhow much he has learned.) During the acquisi-tion process, especially in topics that may bedifficult for some children, we could use obser-vational learning in creative and constructiveways. For example, in the early phases ofteaching addition could we not arrange condi-tions so that the child could learn through ob-servation rather than solely through perform-ance? Using games and educational materialsin the classroom offers good opportunities forchildren to learn through the observation oftheir own efforts and the efforts of others. For-mal instruction that "a cube is a figure with sixsides" or that "two one-fourths equal one-half"may cause young children much more difficultythan having an opportunity to observe the rela-tions and to hear incidental verbalizations.

inappropriate hypotheses

Children use hypothese5, but their hypothe-ses may be inappropriate for the .problembeing presented. At one time we were led tobelieve that children functioned pretty much ina stimulus-response manner. get the responseto occur in the presence of the relevant stimu-lus, reinforce the response, and learning willtake place. According to this view, children arepassive respondents to their environment, con-trolled by the contingencies between responseand ieinforcement. The: e is no doubt that wecan exert a strong influence over what childrenlearn and think when we have full control of the

10 CHAPTER ONE

resources that are available to them, as in insti-tutions, or when we are able to offer them theironly access to highly desired rewards. In mosteveryday environments, however, we have fewopportunities to exert this kind of influence.

Many child psychologists currently view 'rechild as an active participant in the construc-tion of his environmentan individual who re-sponds with hypotheses and expectations,preferences and biases. Children behave in ahighly systematic manner, even at very earlyages; they appear to act on their environment,not merely to respond to it. At times, however,children do not seem to operate in such a so-phisticated fashiontheir behavior appears tobe uncomprf rending and inappropriate. It iseasy to attribute this.behavior to dullness or alack of interest. Careful scrutiny of what chil-dren are doing, however, may lead us to differ-ent conclusions. They may be responding inthe best way they know, developing hypothesesand strategies that to them seem to be reason-able avenues to successful performance. Theproblem may be that the hypotheses and strat-egies they devise are too simple or too complexfor the task at hand.

For example, Weir and Stevenson (1959)studied children of ages three, five, seven, andnine, who were required to learn the correctmember of each of five pairs of pictures ofcommon animals. Each pair of pictures ap-peared once in each block of 5 trials and a totalof 140 trials was given. This is not a difficulttask, and we would expect increasingly rapidlearning with increasing age. All the child hadto do was discover and remember which animal in each of five pairs the experimenter hadarbitrarily assigned as correct. For three-year-olds the problem should have been difficult,and it was; only about half of their total numberof responses were correct. Performance im-proved for five-year-oldsthey were able toname the animals and use the names as cuesfor remembering which animal in each pair wascorrect. But at age seven performancedropped, and by age nine the children weremaking approximately the same number ofcorrect responses as the three-year-olds. Theproblem had not changed and the childrerwere older, but performance was unbelievablypoor.

We could conclude that the older childrenwere not paying attention, were uninterested inthe task, or were an unrepresentative sample ofnine-year-olds. None of these explanationsseemed to hold up. The children did appear tobe interestea in the problem and evidenceddisappointment when their choices were incor-rect. Why, then, should they have performed sopoorly? A due to the cause of their difficultywas found in what they had to say at the end ofthe task when they were asked how they hadknown which animal was correct. The olderchildren made statements like "I tho...gh.t it wasgoing to be a pattern" or "I thought you weregoing to change them all around." They hadformed comple:: hypotheses, and these hy-potheses had hindered their progress in reach-ing the simple, correct solution.

We can make erroneous interpretationsabout children's abilities if we do not take thetime to investigate the basis of their mistakes. Ifthey do not behave the way we think theyshould, we may consider them far less capablethan they really are.

recognition of components

Difficult problems can be solved more read-ily if they are broken down into successivelysimpler components. One of the great contri-butions of programmed instruction and of be-havior modification has been the demonstra-tion of how complex problems can bemastered, often without error, if the problem isbroken dov.in into its components. Elementarycomponents are presented first, and after eachof these is mastered, successively more com-plex components are introduced.

Just as children make assumptions aboutwhat the teacher expects for an answer, teach-ers make assumptions about what children al-ready knowand in both cases the assump-tions may be inappropriate. For example, wegive children a problem in which they are tomake judgments of "same" or "different" aboutgeometrical figures. The children confusesquares and rectangles, circles and ellipses arefrequently judged to be the same. We repeatthe instructions, but errors continue. Finally, we

realize that our request has been misinter-preted The children had been defining "same"to mean having a common characteristic, butwe had meant identical. Had we attempted toanalyze the task, we would have seen that ourfirst step should have been to demonstratewhat we meant by "same" and "different."Once this step is understood, the problem be-comes much clearer. This may be a trivial ex-ample, but it illustrates the importance of ex-amining our assumptions concerning whatchildren know before proceeding to more com-plex problems.

p)

erhaps a better example is found in a studyby Bijou, reported in Bijou and Baer (1963),which required children of ages three to six tomake matching-to-sample responses thatwould be difficult even for adults. The child wasasked to match a complicated geometric figureappearing at the top of a screen with one of fivealternative figures that differed in structure andin angle of rotation. (See II .)

O

cil3 LP cg

How were children this young brought tosuch a remarkable level of proficiency? Bijoubroke the problem down into its simplest com-ponents and required that each step be under-stood before the next step was introduced. Atfirst only a simple figure, such as a circle, ap-peared as the sample. Directly below it was amatching circle along with a square and a tri-angle. The identity of both the sample and the


correct alternative was emphasized by present-ing simple geometric figures in close proximity.Gradually, in successive trials, the matchingfigure was moved from directly below thesample, the figure became more complex, andthe number of choices was increased to five.The children proceeded through the training attheir own pace, but each problem had to besolved correctly before the next, slightly moredifficult problem appeared.

Many other examples could be used, but thepoint is the same: we can often produce supe-rior performance if we identify the componentsof a task, present the components in succes-sively more complex combinations, and assureourselves that the child has mastered each ofthe steps. We should make no assumptionsabout what the child is capable of doing untilafter he has demonstrated his level of com-petence. However obvious this may seem, ittakes a study such as Bijou and Baer's to im-press us with how much can be accomplishedwith this .procedure.

remembering the components ofthe problem

Children may fail to solve problems becausethey cannot remember the components of theprobleg:. When a child is presented with asimple word problem and fails to come up withthe correct answer, we usually conclude that heis incapable of making the appropriate infer-ences from the information contained in theproblem. In other words, he cannot solve theproblem even though he understands its com-ponents. This conclusion is common amongteachers and psychologists. For example, in histheory of cognitive development Piaget hasmade much of the fact that children under theages of seven or eight are incapable of makingtransitive inferences. In his studies, childrenwere given problems of the following, familiartype: If Mary is older than Jane, and Jane isolder than Sue, who is older, Mary or Sue? Tosolve this problem, it is necessary to combinetwo separate pieces of information, the rela-tions between the ages of Mary and Jane and

t 43

12 CHAPTER ONE

the ages of Jane and Sue From these two rela-tions the child must infer a third relation that isnot directly specified. If it is true, a Pugetholds, that young children have great difficultyin making such inferences before close to theend of the first grade, we may wonder how theycan comprehend elementary principles ofmeasurement where comparisons require thiskind of transitive thinking.

A recent study by Bryant and Trabasso(1971) has cast doubt on the validity of claimsthat transitive inference does not occur at earlydevelopmental periods. They asked whetherchildren's poor performance on inferenceproblems was due to their cognitive immaturityor to their failure to remember the first and sec-ond relations. If the difficulty was due to prob-lems of memory rather than of thought, itshould have been possible to find transitive in-ference at much younger ages, provided thatappropriate care was taken to insure that thechild remembered the components of theproblem.

Bryant and Trabasso used five wooden rodsof different lengths (A through E in order of de-creasing length). and the subjects were four-,five-, and six-year-olds. The rods protrudedone inch from the top of a box and were colorcoded The child was asked the color of the rodthat was longer (or shorter) within a pair. Afterhe responded, the rods were removed and thechild could observe whether his response wascorrect. (See .) Training was given on fourcomparisons. A B, B C, C D, and D > E,presented repeatedly in this fixed order or itsreverse Trading continued in this manner untilthe child was correct at least eight of ten times,after this, the four pairs were presented in ran-dom order until the child responded correctlysix times in succession This extensive proce-dure was adopted to insure that the child hadthoroughly learned the initial relations. Immedi-ately following this training, each child wastested four times on all ten possible pairs ofcolors Memory was assessed from the re-sponses to the four original comparisons, abil-ity to make inferences was assessed from theresponses to the six other comparisons. Tran-sitive inference was found at all ages. Four-year-olds, for example, gave correct responses

on at least 90 percent of the trials (except in thecompanson B D, where they were correct on78 percent of the trials).

a

This whole procedure, with one modification,was repeated with a different sample of chil-dren. Rather than allowing them to see which ofthe rods was longer after each response duringthe training period, the adult simply told themthe correct answer. The results were practicallyidentical to those of the first study.

These studies lead to an important point. Be-fore concluding that children have failed toreach a level of cognitive development wherecertain types of mental operations can takeplace, one must be sure that basic conditionsfor successful response have been met. One ofthese conditions is that the child remember thecontent of the problem.

The same line of argument has beenpresented by Kagan and Kogan (1970). It hasbeen asserted that children under the ages ofseven or eight are incapable of performingadequately on class-inclusion problems. Typi-cally, a problem of the following type ispresented orally. "See these beads? These areall wooden beads. Some of them are brownand some of them are white. Tell me. are there

more wooden beads or more brown beads?"Young children typically say that there aremore brown beads. Such results have been in-terpreted as evidence that the child is unable tothink simultaneotr,ly of both the whole and theparts, of classes and subclasses, and that thisis an index of cognitive immaturity. Kagan andKogan report that IThen six-year-olds wereasked a problem of t type orally, only 10 per-cent were able to give the correct answer.When the problem was presented in writtenform and all elements of the problem wereavailable to the child as long as he wished, 70percent gave the correct response. This is an-other example where poor performance is a re-sult, not of inadequacies in thought, but of aninability to retain all the required information.

teacher/child relationship

The relationship between teacher and childis an important determinant of the child's per-formance. Every teacher faces a basic di-lemma: With so little time and so much materialto cover, how can the teacher worry about re-lating to each individual child? Is the teacher'sprimary responsibility to impart informationand to teach skills, with the building of positiveinterpersonal 'relationships merely an in-cidental goal'? The answer must always be r o.The kind of relationship that exists between ateacher and a pupil will determine, in part, howmuch and how well the child will learn. What-ever effort that is spent in developing soundteaching procedures can lead to only partialsuccess unless the child perceives the teacheras a potentially interested and supportive per-son This is especially true in teaching youngchildren, for they have not reached the pointwhere formal school material itself begins tohave sufficient inherent interest to insure theirmaking persistent efforts to learn.

Many studies have demonstrated that theadult's effectiveness in influencing children'sbehavior depends on the quality of their inter-action. We know that children will try harderand that learning will be aided if the adult re-wards the child's efforts with praise or other


forms of supportive response. Even praise,however, may have differing degrees of influ-ence, depending on the role the adult estab-lishes with the child. McCoy and Zigler (1965)have demonstrated this experimentally. Anadult experimenter attempted to establish dif-ferent roles with six- and seven-year-old boys.She took the boys in groups of six to a class-room where they were given attractive art ma-terials with which to work. With one group sheattempted to be as neutral as possible, keepingbusy at her desk and trying not to elicit bids forsocial interaction. In a more positive setting, asecond group of boys was again allowed towork with the art materials, but this time theadult was diligent in her efforts to interact witheach boy, and she tried to be complimentary,helpful, and resporiSive. Three sessions, heldone week apart, were conducted in this man-ner. One week after the third session theseboys, along with a third group who had had noprior interaction with the experimenter, playeda game with her in which she made supportivecomments about their performance twice aminute. The boys were allowed to play thegame as long as they wished. The boys forwhom the adult was a stranger terminated thegame after an a "erage of 2.5 minutes. The boyswith whom th ,:xperimenter had behaved in aneutral mannz.-1 in the earlier sessions re-mained at the game for an average of 9.6 min-utes, and those with whom she had interactedpositively remained for an average of 13.4 min-utes.

Other studies show that adults differ greatlyin their ability to influence the behavior ofyoung children and that these differences in-crease as children grow older. Still other stud-ies offer examples of how adults can be trainedto adopt roles that reduce these differences. Ingeneral, adults can heighten their effectivenesswith children if they are enthusiastic, involved,and responsive. The critical, punitive, aloofadult may be effective in some situations butrarely in interacting with young children. Muchof the motivation of young children to learnsubjects such as mathematics depends on thehuman factorchildren work in part to pleasethe teacher. Ideally, we hope for situationswhere learning is directed by the interests ofthe child, but in the early years these interests

14 I CHAPTER ONE

are usually not developed sufficiently to moti-vate the child without a teacher's enthusiasmand encouragement.

concluding remarks

Some of the ideas that can be derived fromrecent research on children's learning andthought have been presented here. Undoubt-edly, a great many more could be developed.

references

Beatty. William E. and Morton W. Weir. "Children's Per-formance on the Intermediate Size Problem as a Func-tion of Two Training Procedures." Journal of Experimen-tal Child Psychology 4 (1966):332-40.

Bijou. Sidney W.. and Donald M. Baer. Some Method-ological Contributions from a Functional Analysis ofChild Development.' In Advances in Child Developmentand Behavior, edited by Lewis P. Lipsitt and Charles C.Spiker, vol. 1, pp. 197-231. New York. Academic Press.1963.

Bryant. P E.. and T. Trabasso. "Transitive Inferences andMemory in Young Children." Nature 232 (1971).456-58.

Caron. Albert J "Conceptual Transfer in Preverbal Chil-dren as a Consequence of Dimensional Training." Jaw-nal of Experimental Child Psychology 6 (1968).522 -42

Flavell, John H. Concept Development.' In Manual ofChild Psychology. edited by Paul H. Mussen. pp. 1061-162. New York: John Wiley & Sons, 1970.

Gelman. Roche!. "Conservation Acquisition: A Problem ofLearning to Attend to Relevant Attributes." Journal of Ex-perimental Child Psychology 7 (1969):167-87.

Kagan. Jerome. and Nathan Kogan. "Individuality and Cog-nitive Performance' In Manual of Child Psychology, ed-

Because of limited space, however, it has beennecessary to present these ideas in skeletaloutline. Further information about research onchildren's learning can be found in Stevenson(1972); more on children's cognitive devel-opment can be found in Flavell (1970) and Roh-wer (1970). We hope that in the future, psycho-logical research with children will be linkedmore closely to education. From such mutualefforts it should be possible to develop a soundscientific basis for teaching practices.

it;.icl by Paul H. Mussen. pp. 1273-365. New York: JohnWiley & Sons, 197G.

Lubker, Bonnie J.. and Melinda Y. Small. "Children's Per-formance on Dimension-abstracted Oddity Problems."Developmental Psychology (1969):35-39.

McCoy. Norma. and Edward Zigier. 'Social Reinforcer Ef-fectiveness as a Function of the Relationship betweenChild and Adult.- Journal of Personality and Social Psy-chology 1 (1965):604-12.

Rohwer William D., Jr. "Implications of Cognitive Devel-opment for Education." In Manual of Child Psychology.edited by Paul H. Mussen. pp. 1379-454. New York. JohnWiley & Sons. 1970.

Rosenbaum, Milton E. The Effect of Verbalization of Cor-rect Responses by Performers and Observers on Reten-tion." Child Development 38 (1967):615-22.

Stevenson. Harold W. Children's Learning. New York: Ap-pleton-Century-Crofts. 1972.

Weir, Morton W., and Harold W. Stevenson. The Effect ofVerbalization in Children's Learning as a Function ofChronological Age." Child Development 30 (1959),143-49.

thecurriculum

)Iv

*

paul trafton

16

What are the goals or purposes of instruc-tion in mathematics?

What mathematical concepts and skillsshould children learn?

What factors contribute to the successfullearning of mathematics?

How can reflective, analytical thinking bedeveloped in mathematics?

How can children be helped to use theirknowledge in recognizing, managing,and interpreting the mathematical as-pects of their world?

The continuing process of developing well-balanced, effective programs of instruction foryoung learners involves many variables, in-

-cluding mathematics, children, and ways of re-lating mathematics to children. Raising broad,basic questions, such as those above, is an in-tegral part of the process. Our perceptions onsuch questions greatly affect the nature of anymathematics program for children. The posi-tions taken affect such matters as the range ofcontent included, the allocation of priorities,curriculum emphases, ways of sequencingcontent and learning experiences, types of in-structional procedures employed, and ways oforganizing instruction for adjusting learning toindividuals.

The list of questions fundamental to thelearning of mathematics and the developmentof appropriate programs can be greatly ex-panded: What kinds of experiences, and itwhat amounts, are necessary to help childrerdeal with the abstractions inherent in the learning of mathematics? What is the relative importance of mathematical concepts and specificskills? How can these two aspects of learninj

be related so that each complements and sup-ports the other? How much emphasis shouldbe placed on the learning of computationalskills and how is computational proficiency de-veloped? What are appropriate problem-solv-ing experiences and how can they be made anintegral part of the curriculum?

The rapid development of programs for thepreschool child in recent years has given rise toa new set of questions from those personsworking at this level: Should explicit emphasisbe placed on certain mathematical notions? Issuch emphasis consistent with broader child-development goals at this level? What mathe-matical ideas should be explored? What typesof foundational experiences in mathematicspromote desired thinking patterns and under-standings in the young child? How can themathematical aspects of more general experi-ences be put to maximum use?

Many important curriculum questions arecompex, and simple, conclusive answers tothem are difficult to provide. Prior to makingcurriculum decisions, however, those inter-ested in the learning of young children shoulddiscuss such questions extensively. More spe-cific questions are posed by those responsiblefor the learning of children on a daily basis:What is the significance of developmental ex-periences with basic ideas and relationshipssuch as classification and matching? Shouldsuch work receive precedence in preprimarywork over direct emphasis on counting andrecognizing numerals? Is counting still an im-portant skill and how does it fit into work withsets stres:ng cardinality? How does under-standing the meaning of addition and its prop-erties relate to developing proficiency in findingsums? Is it important to develop the relation-ship between addition and subtraction? Whatare effective techniques for helping childrenlearn basic facts? Why is it important to employa developmental approach to computationalprocedures such as the subtraction algorithm?Why should informal experiences with non-standard content such as geometry be in-cluded and how can such work be fitted into acrowded curriculum? Such specific questionsrequire pragmatic, albeit tentative, answers ifthe helpful guidance sought by teachers is tobe provided.

THE CURRICULUM 17

Understanding the implications of curricularquestions and curricular emphases rests heav-ily on an understanding of the nature and pur-poses of a well-balanced mathematics pro-gram. Yet communicating the nature of suchprograms has been difficult for several rea-sons. Many educators see only the arithmeticaspect of the program, not comprehending thepurpose or value of other mathematical com-ponents. Even the arithmetic aspects areviewed diversely. Some view arithmetic interms of computational skills, with proficiencyin them an end in itself, they see proficiency asthe result of an emphasis on procedural as-pects of computing alone. Others focus on theuse of arithmetic understandings and skills infamiliar practical situations, they argue thatnumber aspects of applied situations are suf-ficient to develop these understandings andskills. At times the curriculum is organized pri-marily around one of these viewpoints despitethe lack of evidence supporting either position.

The relative importance of sequential. sys-tematic instruction and exploratory or prob-lem-solving experiences is difficult to deter-mine. Frequently the meaningful learning ofnew content is dependent on prior understand-ings and skills, a fact that implies a role for se-quential instruction focusing on specific ideas.But this aspect of instruction is often ignored byteaching pieces of mathematics in isolation andnot teaching specific content with care, and it isoften abused by imposing too rigid a structureon the curriculum. Specific content can oftenbe put in a broader curriculum setting. The roleof intuitive, exploratory work, as well as oppor-tunities to apply and extend learning in a van-ety of ways, could be increased. Carefully se-quenced learning is often extended to areasthat need not be highly structured, such as ge-ometry.

It is often difficult to see the long-range impli-cations of early experiences with mathematicalideas and relationships. Preprimary- and pri-mary-level programs include experiences thatmay be unfarr,liar to many educators. Pupilshave early experiences with the language oflogical thinking, with mathematical relations,with aspects of informal geometry. and withstructural properties of operations. In additionto their present value for children, these experi-

18 CHAPTER TWO

ences also have strong implications for futurework in mathematics. For exarnp.e, implicit inclassification experiences u: the notion of anequivalence relation. Informal work with thisidea at an early age builds a 'oundation for fu-ture applications and work.

Broadening the mathematical component ofthe curriculum through an infusion of new con-tent and placing the familiar arithmetic pro-gram in a broader mathematical context haveclouded the goals of the curriculum. Con-sequently. teachers have felt an increasingsense of inadequacy or insecurity toward math-ematics, a feeling that has led many of them toavoid new content. Sometimes, however, thecontent, language, and symbolism of a newprogram are clear, but their underlying pur-poses, as well as the relation of familiar ideas tonew approaches, remain obscure. Further-more. a limited mathematics background mayrestrict the teacher's ability to modify or extendhis pupils' learning experiences. Thus, al-though an understanding of the purposes of amathematics curriculum is essential for re-sponding adequately to significant curriculumquestions, several factors may cause educa-tors to misunderstand or misinterpret the cur-riculum.

Today the mathematics curriculum is underintense scrutiny. with particular attention beinggiven to broad questions related to the teach-ing and learning of mathematics. This is due toseveral factors. There is abundant evidencethat the learning of mathematics is still not asuccessful experience for many children. It hasbeen difficult to implement promising class-room practices on a widespread basis. Weak-nesses of the reform movement of the 1960sare now more apparent. The continuing de-velopment of experiences appropriate forchildren and representative of many areas ofmathematics causes stress over the allocationof instructional priorities and emphases. Find-ings about children's intellectual developmentraise interesting but difficult questions abouthow those findings may affect instruction and,in turn, the content and structure of the curricu-lum. (The fact that the implications of presentfindings are not immediately obvious, togetherwith the continuous input of new evidence frominvestigations of children's thinking and learn-

ing. make it hazardous to base firm decisionson the present state of knowledge.) Finally,questions about the curriculum and the natureof meaningful learning in matnematics are anatural consequence of the current broaderdiscussion of goals and purposes in preschooland elementary education.

I his chapter will discuss many of the ques-tions and issues dealing with curriculum plan-ning and classroom instruction. The purpose isto provide a broad basis for discussions of cur-rent issues, to identify important elements ofeffective instruction and curriculum devel-opment, and to provide a framework that canguide daily practice for those who work withchildren.

The chapter is developed in two major sec-tions. In the first section. it is suggested thatcurrent issues are not unique to the present erabut are continuing ones. Issues basic to twomajor reform periods in mathematics educa-tion are identified and related to the present sit-uation, for a knowledge of past attempts to dealwith important concerns can be beneficial topresent discussions. In the second section, theimportance of balance is established in thebuilding of effective programs for young learn-ers, with balanced approaches discussed forseveral areas. It is often difficult to keep asense of perspective regarding the com-plexities of learning and instruction at a timewhen a bewildering array of approaches andalternatives are being advocated.

The content presented is pertinent to bothpreprimary and primary levels, although thechapter zit times may speak more directly to is-sues at tt,e primary level. Chapter 5 contains acomprehetlswe discussion of preprimary pro-grams.

continuing issues inmathematics curriculumdevelopment

The direction of the elementary school math-ematics program has undergone manychanges in this century. In the past forty yearsalone there have been two major periods of re-

form. The first of these periods began in theearly 1930s and extended into the 1950s and isfrequently referred to as the "meaningfularithmetic" era. The second reform period,known as the "modern mathematics" move-ment, had its beginnings in the decade of thefifties.

In each of these periods, many of the issuesthat are of concern today were an integral partof the discussions then and influenced the di-rection of the mathematics curriculum. Ananalysis of past discussions can set current de-bate in a broader perspective and providehelpful input for understanding pervasive is-sues more clearly.

issues in the development ofmeaningful arithmetic programs

The setting for reform. For an extended pe-riod beginning in the early 1930s, the arithmeticprogram of the elementary school was the sub-ject of intense scrutiny, the result of which wasa great transformation in the emphasis of in-struction. Initially the curriculum centered onthe learning of skills with a strong reliance ondrill as the principal mode of instruction, but bythe erd o' the period, strong support had de-veloped for presenting arithmetic in a "mean-ingful" way. The issues and discussions of thisperiod can be traced by the interested readerthrough the three yearbooks devoted to theteaching of arithmetic at that time (NCTM 1935;NCTM 1941; NSSE 1951) as well as through theanalysis of this era in the Thirty-second Year-book of the National Council of Teachers ofMathematics (1970).

When the period opened, the prevailingviewpoint was that arithmetic was a tool subjectconsisting of a series of computational skills.The rote learning of skills was paramount, withrate and accuracy the criteria for measuringlearning This approach became labeled as the"drill theory" of arithmetic and was describedby William Brownell as follows (1935, p. 2):

Arithmetic consists of a vast host of unrelatedfacts and relatively independent skills. The pu-pil acquires the facts by repeating them overand over again until he is able to recall them im-mediately and correctly He develops the skills

THE CURRICULUM 19

by going through the processes in question un-til he can perform the required operations auto-matically and accurately. The teacher need givelittle time to instructing the pupil in the meaningof what he is learning....

This approach, as well as the psychology onwhich it was often based, was attackedthroughout the period. Numerous exampleswere cited of its weaknesses: pupils performedpoorly. neither understanding nor enjoying thesubject; they were unable to apply what theyhad learned to new situations; forgetting wasrapid; learning occurred in a vacuum, since thecurriculum related only rarely to the real world;and little attention was paid to the. needs, inter-ests, and development of the learner. At thesame time, a movement to abandon systematic"ntruction occurred as a reaction against thesterile, formal programs of the day. Whereasthe economic conditions of the depressionsupported the "tool subject" conception of thedrill approach, the movement away from pro-grams of instruction in arithmetic had its rootsin the child-development movement of thetimes. The thesis was that systematic instruc-tion was unnecessary, since pupils would learnarithmetic "incidentally" through informal con-tact with number in daily life and through thequantitative aspects of activity-oriented or ex-perience-oriented units. It was felt that as pu-pils worked on broad units dealing with suchtopics as transportation or communication, sit-uations requiring arithmetic would arise. Insuch settings pupils would learn skills readily,with the applied situation providing the motiva-tion for learning. The notion of "meaning" wasof central importance to the incidentalists,compared to the slight attention it receivedfrom the advocates of drill approaches. The in-cidentalists saw meaning being supplied by thesocially significant settings in which numberwork originated. This point of view was a popu-lar one, fitting well into the prevailing educa-tional thought of the day. In many schools sys-tematic instruction was abandoned, at leastofficially, and in other schools "formal" ap-proaches (i e , drill approaches) were deferredfor the first few years of schooling.

The nature of "meaningful" arithmetic. It wasin this setting of widely divergent conceptual-izations of the nature of an arithmetic curricu-

20 CHAPTER TWO

lum that the "meaning" theory of arithmetic in-struction was proposed. The crucial ingredientJrider meaningful instruction was that r,,upilszhould understand or see sense in what theylearned. Also, a new view of arithmetic itselfcame to be associated with meaningful instruc-tion. In the Tenth Yearbook of the NCTMBrownell wrote, "The 'meaning' theory con-ceives of arithmetic as a closely knit system ofunderstandable ideas, principles, and proc-esses" (1935, p. 19). He further explained thatnow the true test of learning was having "an in-telligent grasp upon number relations and theability to deal with arithmetical situations withproper comprehension of their mathematicalas well as their practical significance" (p. 19).

The meaningful approach, representing areasonable alternative to two extremes, thusproposed a view of meaning that was related toideas inherent in the subject matter. The ap-proach received support and nurture from thegrowth of a new psychology of learning thatemphasized the role of insight in learning.

Throughout the era, however, the notion ofmeaningful arithmetic retained an elusivenessand a level of generality that obscured dis-cussion and increased the difficulty of both im-plementing and assessing the effects of mean-ingful programs. Eventually, attempts weremade to describe more explicitly what mean-ingful arithmetic involved. In 1947 Brownell, themost articulate spokesman for the approach,provided a more detailed explanation (p. 257).

"Meaningful" arithmetic, in contrast to"meaningless" arithmetic. refers to instructionwhich is deliberately planned to teach arithmeti-cal meanings and to make arithmetic sensibleto children through its mathematical relation-ships. . . . The meanings of arithmetic can beroughly grouped under a number of categories.I am suggesting four.

1. One group consists of a large list of basicconcepts. Here, for example, are the meaningsof whole numbers, of common fractions, ofdecimal fractions, or per cent, and most per-sons would say, of ratio and proportion. . . .

2. A second group of arithmetical meaningsincludes understanding of the fundamental op-erations. Children must know when to add,when to subtract, when to multiply and when todivide. They must possess this knowledge and

they must also know what happens to the num-bers used when a given operation is employed.

3. A third group of meanings is composed ofthe more important principles, relationhips,and generalizations of arithmetic, of which thefollowing are typical: when 0 is added to a num-ber, the value of that number is unchanged. Theproduct of two abstract factors remains thesame regardless of which factor is used asmultiplier. The numerator and denominator of afraction may be divided by the same numberwithout changing the value of the fraction.

z A fourth group of meanings relates to theunderstanding of our decimal numbe; systemand its use in rationalizing our computationalprocedtires and our algorisms.

Dealing with instructional issves. In thetwenty years following the publication of theNCTM's Tenth Yearbook in 1935, the notion ofmeaningful programs in mathematics w: keptbefore educators and slowly gained accept-ance as a valid approach to the teaching andlearning of arithmetic. Spokesmen for a mean-ingful approach continued to point out the sim-ilarities to the other approaches, although theystill noted the differences. The extreme empha-sis on the mechanical learning of skills, preva-lent under a drill theory, was consistently re-jected under meaningful programs. Alsorejected was the conception of arithmetic assimply a tool or skills subject, since this notionseemed to promote an emphasis on teachingan isolated set of skills apart from the under-standings inherent in the subject and withoutrecopnizing the factors involved in making skilllearning meaningful.

Nonetheless, the learning of skills and theuse of drill was accorded a place. Computa-tional skills were accepted as a major com-ponent of the program because of their useful-ness in the real world. The complexity of skilllearning was recognized in the careful attentiongiven to the development of children's thinkingand to the sequencing of instruction in thisarea. This was due to the realization that mean-ingful development of computational skillswould result in more effective initial learningand retention. Thus skills were carefully relatedto concepts of the basic operations and ra-tionalized through reliance on physical models

and properties of the decimal numeration sys-tem. A limited use of drill techniques was ac-cepted as valid for helping pupils fix learningafter understanding had first been established.

During this period the drill approach becamediscredited (although it remained alive in manyclassrooms). and the.discussion shifted to howthe curriculum should be organized to supportmeaningful instruction in arithmetic. The keyquestion was. "Shall arithmetic be taught as asystematic subject. or should the pupils ac-quire arithmetical abilities incidentally, i.e., inconnection with other subjects. or only as theybecome a part of purposeful life activities?"(McConnell 1941, p. 282). Although proponentsof meaningful approaches were generally sup-portive of many of the aims of incidental pro-grams, they rejected the abandonment of sys-tematic instruction inherent in such programs.The central contention was that incidental ap-proaches did not provide the careful, sustaineddevelopment required for pupils to understandmathematical concepts. To support this claimseveral factors wera cited: that number aspectsof a project would be treated superficially in theimmediacy of the situation, that opportunitiesfor quantitative thinking would not be recog-nized owing to a lack of content knowledge,that heavy demands were placed on teachersto recognize and develop the quantitative as-pects of a situation, that most units containedinsufficient number experiences, and thatarithmetic was again treated as a tool subjectconsisting of a series of isolated skills.

Underlying the support for planned pro-grams was the viewpoint expressed by theNCTM Committee on Arithmetic. The com-mittee, echoing Brownell, stated that "arithme-tic is conceived as a closely knit system of un-derstandable ideas, principles, and processes

." (Morton 1938, p. 269). This led to the argu-ment that systematic, sequenced instructionwas necessary in order to achieve the requiredunderstanding, reasoning ability, and skills.The report of the Commission on Post-WarPlans of the NCTM supported this positionwhen it stated, "Mathematics, includingarithmetic, has an inherent organization. Thisorganization must be respected in learning.Teaching, to be effective, must be orderly andsystematic; hence, arithmetic cannot be taught

THE CURRICULUM 21

informally or incidentally" (1945, p. 202). Bus-well noted that "competence in quantitativethinking warrants purposeful teaching of themost effective type that can be devised" (Bus-well 1951, p. 3). However, systematic instruc-tion was distinguished from formal instruction,and it was carefully shown how systematic in-struction made provisions for the needs, inter-ests, and developmental level of the child. Thiswas to be accomplished through buildinglearning on experiences familiar to the child,providing many informal experiences at earlystages of learning, involving the child actively,and providing many opportunities for usingarithmetic in socially significant situations.

The debate regarding the organization of thecurriculum involved the more fundamentalquestion of the source of meaning in learning.Is meaning found within mathematics or doesmeaning emerge from the mathematical as-pects of an applied situation? In the eyes ofmathematics educators, there was little doubton this issue. According to Brownell, "Meaningis to be sought in the structure, the organiza-tion, the inner relationships of the subject itself"(1945, p. 481).

As implied earlier, proponents of plannedprograms recognized that the curriculum hadto relate to the real world. The report of theCommission on Post-War Plans also noted, "Ifarithmetic does not contribute to more effectiveliving, it has no place in the elementary curricu-lum" (1945, p. 200). Thus recognition was givento two aims of instruction, the mathematicaland the social. The question of the source ofmeaning was broken down into the dual ideasof meaning and significance. Meaning dealtwith the mathematical aims of instruction,whereas significance came from relating learn-ing to the real world. Buckingham, in a key ar-ticle, put it this way (1938, p. 26):

The teacher who emphasizes the social as-pects of arithmetic may say that she is givingmeaning to numbers. I prefer to say that she isgiving them significance. In my view, the onlyway to give numbers meaning is to treat themmathematically. I am suggesting, therefore, thatwe distinguish theSe two terms by allowing,broadly speaking, significance to be social andmeaning to be mathematical. I hasten to say,however, that each idea supports the other.

22 CHAPTER TWO

This viewpoint became accepted and pro-vided a workable resolution of a difficult issue.It resulted, nonetheless, in a static view of whatcontent should comprise the curriculum. Thestatement of the NCTM Committee on Arithme-tic in 1938 reflected the prevailing point of view(Morton 1938, p. 269):

Arithmetic is an important means of inter-preting children's and adults' quantitative expe-rience and of solving their quantitative prob-lems. Consequently, the content should bedetermined largely on the basis of its socialusefulness and should consist of those con-cepts and number relationships which may beeffectively used.

This position limited both the range of con-tent and the range of applications included. Italso had the effect of limiting the mathematicalorientation of the curriculum. Operations withwhole numbers and fractional numbers com-prised a large portion of the curriculum. Al-though emphasizing the mathematical aspectsof such work was encouraged, in practice thiswas frequently not done, and the definition of"mathematical aspects" would today be con-sidered quite limited. However, the position didgive the work wits computation a broadermathematical and social perspective, whichprovided a stronger motivation for such learn-ing.

Assessing the impact. This first period of re-form was a rich one in which today's educatorcan find much literature that is still significant.Two of the key issues were obviously the inter-related ones of the origin of meaning and theorganization of the curriculum. Recognizingand establishing roles for mathematical andapplied aspects was an important contribution.Although the resolution was not ideal, it mustbe noted that educators still seek an appropri-ate balance between these two aspects of in-struction. The recognition of the importance ofdirect, planned instruction in achieving certaininstructional objectives was another contribu-tion. Such a position was central to the reformmovement of the 1960s, where it was often ar-ticulated primarily for mathematical reasons.Yet at this earlier time the support was basedon the recognition that such instruction wasnecessary to develop the level of quantitativethinking and computational proficiency essen-

tial for a functional use of mathematics in thedaily lives of people. Those stating the positionwere often general educators and seldom hadstrong connections to the field of mathematics.

During the period little change occurred inthe content of the curriculum. Computation re-mained of central importance, with a greaterrationalization of procedures and a greater so-cial orientation sometimes accompanying it.However, the question of how to make learningmole effective was extensively and significantlytreated, with major contributions being maderegarding children's learning of mathematicalideas. The influence of this work is still felttoday. Strong emphasis was placed on the psy-chology of the learner and, as a result, on thedevelopmental aspects of learninga fact thathad several effects on curriculum and instruc-tion. The progressive nature of learning wasrecognized and implemented through the spi-raling of content. The use of physical materialsin learning received great support, as did theactive involvement of pupils in formulatingideas and discovering relationships. Concernfor children's learning drew attention to theproblem of providing for individual differencesamong learners. As guiders of learning, teach-ers were encouraged to be aware of each pu-pil's development and to provide necessary in-dividual guidance. It should also be noted thatthe focus on broader purposes of instructionhad an impact on standardized testing. At thistime problem-solving sections on tests oc-curred, and later, items on concepts appeared.

The literature of this period should hold par-ticular interest for those concerned with theearly aspects of children's learning of arith-metical ideas. Many writers centered their dis-cussions of meaningful learning on children'searly schoolwork with number and operations.The many insightfui observations regardingchildren's learning and the perceptive sugges-tions for working with children give the litera-ture a richness and practicality that make ituseful even today. The roots of an active ap-proach to readiness are also found at this time,along with recommendations that pupils haveexperiences with number ideas from theirearliest exposure to schooling so that they canuse and build on the knowledge they alreadypossess.

In summary, the meaningful arithmetic peri-od is a significant one. The issues consideredwere important and held major and continuingimplications for children's learning. They dealtwith fundamental questions that are of concerntoday. Discernible progress was made in for-mulations about curriculum emphases and thenature of learning in arithmetic. A better-bal-anced curriculum was the result. It is obviousthat the issues were not totally resolved andthat recommended practice did not result in thewidespread change in classroom practice thatwas necessary. However, a foundation was es-tablished on which future reformers could buildnew approaches to continuing issues.

issues in the development ofmodern programs

The setting for reform. The period beginningaround 1955 was one of extensive reform inmathematics education, with the greatestchanges occurring at the elementary schoollevel in the early-to-mid-1960s. In many re-spects, this era continues up to the present, butwithout the continuing major content reorgani-zation and the spotlight of national attentionthat was part of the initial years. An intense pe-riod of curriculum reform seldom occurs with-out an accompanying complex of forces thatcreate and support the climate of change. Inthe previous period, the impetus was the reac-tion against a narrow conception of arithmetic,together with the rejection of a psychology oflearning. After 1955, the contributing factorswere an awareness of the need for greatermathematical substance in the curriculum tomeet the nation's scientific and technologicalneeds, a dissatisfaction with the quality of in-struction in mathematics, and a general dis-satisfaction with the quality of the mathematicsitself. Many viewed the movement as a first at-tempt to bring meaning to the learning of math-ematics. This was not so much a reflection onwhat had been advocated in past years; rather,it was a reaction against prevailing instructionalpractices, pointing up the continuing problemof translating change into widespread practice.

The nature of reform. The heart of the move-ment was a new conception of meaningful pro-

THE CURRICULUM 23

grams and meaningful learning in mathemat-ics. Now the idea of meaning was closely linkedto the mathematical validity of the content ofthe curriculum. There was no dichotomy be-tween meaning and mathematics, a mathemat-ically valid program would be a meaningfulone. In the previous period, the focus on mean-ing had been primarily methodological; that is,the content was given a new setting rather thanchanged. Now, however, both the content ofthe curriculum and ways of presenting it effec-tjvely were being considered.

Content considerations centered on the ideaof structure, with "structure" being interpretedin two ways. First, structure was applied to theidea of building the curriculum around thebroad, unifying themes of mathematics. In-stead of being a collection of individual topicsonly slightly related (as in ), the curriculumwould be organized around a series of mathe-matical themes woven or spiraled throughoutthe curriculum (0 ).

24 CHAPTER TWO

As a result, mathematical ideas could be ex-panded and extended as pupils had continuingexperiences with them. It was argued that thiswould result in a more natural flow in the devel-opment of mathematical content. Specifics ofmathematics could be related to basic ideas,and mathematical skills in particular could beput into a more conceptual framework. One ob-vious result of this conception was an expan-sion of the content of the curriculum. At anearly age pupils were exposed to elements ofsets, open sentences, properties, geometry,and graphing.

There were pedagogical as well as mathe-matical purposes for this approach to contentorganization. It was felt that learning would bemore natural and continuous, since new under-standing could easily be built on previouslearning. New approaches sometimes came tohave significant pedagogical value apart frommathematical considerations. For example, theearly use of primitive set ideas was often sup-ported because of their significance as a foun-dational, unifying role in mathematics and themore logical basis they created for the devel-opment of number and operation ideas. In ad-dition, set ideas have provided a way to lookwith more care at the idea of cardinal number,with the result that greater precision can bebrought to the development of this notion. Setand subset ideas underlie general classificationactivities, which are often developed withyoung learners prior to classifying sets on thebasis of "same number." Similarly, the early in-troduction of geometry has come to be seen asa basis for helping a child interpret an inher-ently geometric environment.

The second application of the idea of struc-ture was the more mathematically recognizeduse of the term, namely, the explicit use ofstructural properties of the real-number sys-tem. It was assumed that an early developmentof such ideas as commutativity, associativity,and the identity elements would contribute to adeeper comprehension of the nature of opera-tions on sets of numbers and to an awarenessof the distinction between operations and tech-niques for computing answers. Not only wereproperties identified early, but they formed thebasis for justifying techniques and rationalizingalgorithms. One example occurs in the use of

grouping by ten to solve addition facts withsums greater than ten. As shown in , theprocess is justified in step-by-step fashion byusing the associative property. In , a formalrationalization of the addition algorithm at theprimary-grade level is presented. This methodrelies heavily on structural properties as well asplace-value ideas.

fitle4i:ii /46 4, A8 5 = g (2,+3)

= +2) t 3= to +3

13

20 ++410 +7

60 -t-15

O

= 60 +- (10 +5)

= (60 I- 1 0) t5

= 75

Although this use of structure had wide sup-port initially, doubts were eventually raisedabout the appropriateness of such deductiveapproaches. Some felt that this approach wastoo formal and involved too much sym-bolization. They wondered whether such tech-niques might even contribute to problems withlearning skills rather than resolve them.

A strong pedagogical thrust to the reformmovement called much attention to ways of de-veloping ideas meaningfully. The phraseologywas often similar to that of the previous era. Pu-pils were to be actively involved in the learningprocess through their use of materials, through

the discovery of relationships, and through dis-cussion with their peers and teachers. One no-table aspect was the emphasis on exploratory,discovery-oriented experiences in which thethinking process used by the pupils was con-sidered of equal or greater importance than theparticular content under consideration.

For example, pupils might be asked to find apattern in adding pairs of odd numbers. Afterdiscovering that each sum was an even number( 0), some pupils generalize that this would al-ways happen. Then they engage in a discussionand demonstration of why this must be so, as in

Odd Numbers

1, 3, 5,7, .9, 3+5=13,15,17,49, 7+11=1821, 23 . . . 13 +17 =30

0

d

The mastery of the generalization "The sumof two odd numbers is an even number" wouldnot be stressed in this experience, althoughnaturally it was hoped that some would remem-ber and be able to use the general:zation. Thelesson was justified on the basis of the experi-ence it provided in discovering relationships.From such experiences, pupils would developinsight into their use of numbers.'" The-pedegogical thrust had other aspects.Initial developmental work was again high-lightedlessons were to begin at an informallevel, making use of materials when appropri-

THE CURRICULUM 25

ate and connecting the new elements in alearning situation to ideas with which the pupilswere familiar. Such an approach to learningwas considered essential if the use of symbolswas to have meaning. Mathematics educatorsalso examined how content might be more ef-fectively sequenced to promote learning.

Curriculum and instructional issues. It hasbeen established that this movement dealt in amajor way with issues of content and method-ology. Other aspects of curriculum and instruc-tion also received consideration, as well asthose elements of the new movement causingconcern. The learning of computational skillswas one of these (as it had been before). Thesignificant position that computational skillscontinued to occupy is attested to by the greatemphasis in the early 1960s on explaining andimplementing this aspect of reform.

Again there was the concern that computa-tional skills occupied too central a position inthe curriculum. Although the goal of reason-able computational proficiency was still sup-ported, reformers placed skill work in abroader mathematical framework and built acase for exploratory work in the early stages oflearning facts. Essentially the approach to skillsconsisted of two basic tenets: first, that allowingpupils to discover basic facts and the relation-ships between them would result in quick, effi-cient fact learning; and second, that extensivemathematical rationalization of algorithmswould result in their being effectively learned.Some thought that it was not even necessaryfor children to learn basic facts.

As a result of these tenets and of the asser-tion that pupils could learn more mathematicsearlier, computational skills came to be in-troduced too early in many programs. Manyfirst-grade programs included all the additionand subtraction facts, and renaming in two-andthree-digit addition and subtraction examplesbecame standard topics in second-grade textmaterials. Although the intent may have beento provide earlier exploratory work, thus allow-ing greater time prior to teaching for mastery,in practice this increase in content resulted insignificantly less time being devoted to devel-opmental work. Unfortunately, the Lssumptionswere overstated, and the learning of compute-

26 CHAPTER TWO

tional skills remained a problem. The result.was that computation skill continued to occupytoo great a share of the curriculum. This was acurriculum area about which teachers weresensitive, and many, failing to receive answersto their questions, stopped listening and re-verted to old ways.

Another issue considered again was the roleof applications and real-world problem-solvingsituations. In the earlier reform movement, thelearning of mathematics was considered pur-poseful only in its social applications. This gavemathematics its significance and itsreason forexistence. During the modern mathematicsera, a broader conception of "problem solving"was articulated. Problem solving could occurtotally within the setting of mathematics, as indiscovering a pattern or learning a new ideathrough reliance on previous learning. Further,even though it was important for mathematicsto relate to the physical world, such appli-cations were not considered necessary to lendpurpose or significance to the study of mathe-matics This Was contained within the mathe-matics itself. It was also argued that an under-standing of mathematical relationships wouldresult in a high level of transfer to applied situ-ations Concern was expressed about this posi-tion, and it continued to undergo modification.Today, two aspects of a necessary and properrole of applications are considered. (1) the rela-tion of number ideas and skills to real-worldsettings, and (2) the role of real-world experi-ence in the initial learning of concepts. The firstis particularly appliCable at the primary level,the second is of concern at both the preprimaryand the primary levels.

Questions were eventually raised regardingthe appropriateness of certain topics and ap-proaches for young children. Thought wasgiven to how ideas might be treated in a man-ner more congruent with the thinking of chil-dren. Examples of such concern were the for-mal application of properties and a formalpoint-line-plane approach to geometry. Somefelt that the approach to structure, togetherwith the earlier introduction of topics, mighthave intensified existing problems. Also, find-ings from developmental psychology did notappear to support the level of formalism some-times found in the reform programs.

24

Educators began to make a distinction be-tween content that was necessary to know andcontent that could be considered "nice" toknow. From a curriculum that had once beenlabeled intellectually impoverished, the prob-lem now was determining which experiencesshould be included and to what extent particu-lar content should be emphasized. This was ofparticular concern to classroom teachers, whoultimately had to make the decisions. Unfortu-nately, teachers seldom received the specific,practical guidance they sought.

Assessing the impact. The modern mathe-matics era had its greatest impact during the1960s. It was accompanied by a highly optimis-tic feeling that the changes advocated wouldresolve many of the problems of teaching andlearning mathematics. Yet the message of re-form was frequently distorted, with manyeducators focusing on the "new" content andits accompanying language and symbolism.Often the spirit of the content did not comethrough. The heavy emphasis on the mathe-matics when training teachers was seldom ac-companied with imaginative ways of devel-oping it with children. Some approaches to thecontent seemed to be more appropriate formiddle or high achievers than for the totalspectrum of learners. When it became appar-ent that problems still existed, many becamediscouraged, unaware of the scope of the taskinvolved. Some labeled the movement a fail-urea premature and shortsighted con-clusionand suggested that educators muststart anew the task of creating effective pro-grams.

Yet some significant, positive changes didoccur, and considerable progress was made indealing with recurring issues. The curriculumbecame revitalized in its content and ped-agogy. A better conception of the role of con-tent in programs for children was created. Newcontent was introduced, and it appears to haveestablished a firm place for itself in the curricu-lum. In particular, geometry has finally been ac-cepted, after many statements supporting itspotential over a 160-year period and after sev-eral earlier, unsuccessful attempts to imple-ment programs in it. The momentum in this di-rection encouraged mathematics educators toinvestigate and introduce other appealing new

content as the movement progressed. Exam-ples of this activity at the early childhood levelinclude social graphs and probability. Also atthis time informal, planned programs of in-struction gained acceptance at the kindergar-ten level The blend of content and method-ology resulted in a great number of teachersand pupils finding mathematics more excitingto teach and to learn. Further, the climate of in-novation promoted creative efforts on the partof teachers and encouraged a host of individ-uals representing many kinds of input to be-come interested in the curriculum and to makemany contributions to curriculum devel-opment. Investigations into several aspects ofchildren's thinking and learning in mathematicsgrew at a rapid rate. Finally, unprecedented at-tention by educators nationwide was focusedon fundamental questions of curriculum andlearning. By the late 1960s. however, the atten-tion of the schools had turned to other majorproblems, and this climate no longer existstoday.

That problems remain is not so much a markof failui:e in the reform attempt as it is a re-minder of the difficulty of creating a changethat takes into account all the variables that de-termine effective learning, the difficulty of evenidentifying those factors, and the difficulty oftranslating promise into practice.

issues of current concern

Diverse questions from a variety of sourcesare being raised today about school mathemat-ics. Many of the issues identified in the preced-ing sections should seem famikar. Concernabout curriculum content and about ways ofdeveloping ideas in a manner appropriate to achild's level of development is ever present.Three particular sources of concern are appar-ent at the present time.

One such concern is the learning of compu-tational skills Most individuals would still ac-cept reasonable proficiency with computationaltechniques as one important aspect of elemen-tary school programs. Yet the changing em-phases of the past era probably caused moreconfusion in this aspect of the program than in

THE CURRICULUM 27

any other area. In an attempt to define and at-tain goals in the area of computation today,educators show a disturbing tendency to ig-nore concepts and instead focus on the endproducts of learning without recognizing theprocess by which children develop proficiency.Some would again define the curriculum as aseries of isolated skills and employ drill as theprimary technique for teaching these skills.

A second concern or issue involves the roleof applications and problem-solving experi-ences in mathematics. Some urge that system-atic instruction be abandoned and replacedwith general problem-solving experiencesdrawn from the real world. At times this posi-tion closely resembles that of the incidentalistsearlier. Some advocate emphasizing experi-ences selected from mathematical areas thatclosely relate to the physical world and lendthemselves to manipulative experiences, suchas measurement and geometry. Althoughproponents of this viewpoint recognize the im-portance of the strands of numbers and opera-tions on numbers, they are often unclear onhow an understanding of them and a com-petency with them are to be developed.

A third current issue concerns ways of or-ganizing instruction to provide for individualdifferences. At one extreme, individualized in-struction is equated with self-paced instruction.The same goals are set for all students, with theimportant variable being the rate at which pu-pils master tasks. At the other extreme, an ap-,proach is advocated in which pupils selectlearning experiences on the basis of interest.Much of the discussion of this third issue ap-pears to take place without consideration ofwhat content is worth knowing and the vari-ables that affect meaningful learning. In partic-ular, little consideration is given to the role ofguided instruction in making learning enjoy-able and successful and in developing in-sightful patterns of thought.

Each of these issues is a valid one and de-serves extended and comprehensive dis-cussion. Each deA.3 with a real problem. Eachhas a bearing on the learning of children. Yetnone of the issues is new. It is interesting to re-late current discussions to past attempts atdealing with the same issues. Whereas it is true

ft

28 CHAPT-R TWO

that issues must be interpreted in the context ofthe periods in which they arise, it is nonethelessunfortunate that a greater awareness of the his-torical counterparts of present issues does notexist Such awareness could not only sharpenthe focus of the debate but also reveal weak-nesses inherent in extreme formulations of theissues.

In this section, several issues and their influ-ence on curriculum development have beenidentified In the following section, several as-pects of these and other issues are developedmore fully. The discusjon will center on the im-portance of balanced approaches to key con-cerns. As one becomes aware of the many fac-tors that must be considered in creatingeffective programs, the complexity of tha taskbecomes obvious. Looking at a single variablein isolation is not enough. Desired goals can berealized only by deliberate, sensitive consid-eration of many variables. The analysis of pre-vious attempts provides at least partial supportfor this notion.

considerations in buildinga balanced cureculum

Each of the two reform efforts cited pre-viously can be characterized as a search forbalancebalance in the content, emphases,and methodology of the curriculum, and bal-ance among the needs of the discipline, theneeds of society, and the needs of the individ-ual. Current curricular efforts partially reflectthe need to establish better balance in pro-grams.

Although the reform of the 1930s and 1940sresulted in a better-balanced curriculum at thattime, the strong emphasis on social appli-cations did not result in balance with respect tothe role of mathematical content. In the 1960sthe reform movement attempted to achieve abetter balance by highlighting content andformulating a broader conception of problemsolving. In so doing, the reformers may havecreated an imbalance in the opposite direction.

Frequently, new efforts seem to be con-structed solely on the rationale of being differ-

ent from existing "bad" practices rather thanbeing the result of broadly based consid-erations of the many components involved inmaking learning effective. Today, a new at-tempt is being made to find a workable balancein the curriculum. Evidence indicates that fo-cusing on a few variables or on narrow inter-pretations of them will not result in the desiredprogress. Rather, the attempt must take intoaccount the many factors involved, their com-plexity and interdependence, the many kinds oflearning in mathematics, and the full implica-tions of what it means to learn mathematics.Even though ultimate solutions to continuing is-sues may never be realized, some factors thatcontribute to effective programs can be identi-fied, and this section will discuss some of theconsiderations that can guide curriculum plan -.ners and classroom teachers.

a broadened view of curriculumcontent

A balanced program with respect to content involvesbroadening the scope of the content beyond whole-number ideas and reexamining the treatment of wholenumbers.

Number ideas have long received the majorattention at the kindergarten and primary lev-els. It is true that young children become awareof number ideas at an early age and have acontinuing, expanding need to be able to usenumbers in a variety of ways. Perceptive teach-ers are well aware of the gradual, devel-opmental process of building understandingand competency with whole numbers. Thus it isnot unusual for work with whole numbers tocomprise the entire mathematics curriculum.

In recent years, many new content areashave been introduced into the curriculum in or-der to expose children to the many facets ofmathematics and the many connections be-tween mathematics and the real world. Much ofthis new content has a strong potential for pro-viding productive and satisfying learninc ..xpe-riences for children. For example, many geo-metric ideas have been found to be quit?appropriate at the early childhood level. Idt-asof symmetry and congruence Lien be explored.

1

(

11MMr

Solid shapes can be discussed and classified ina variety of ways: some solids roll, and othersslide: some have corners, some have straightedges, and others have curved edges. Geome-try provides a vehicle for showing the closeconnection between mathematics and the realworld.

In turn, work with geometry provides a betterway to conceptualize measurement. Many de-velopmental experiences with initial measure-ment ideas are possible while building a foun-dation for understanding measurement.Another source of experience is found in col-lecting and organizing data. Graphing orderedpairs of numbers is productive for upper-pri-mary levels and relates to science activities inwhich quantitative relationships are graphed.Preliminary work with probability shows prom-ise of being another appropriate area for infor-mal exploration.

Broadening the curriculum content canmake several contributions to children's !earn-ing. First, early informal experience with impor-tant mathematical ideas can contribute togreater success in future treatments of themfor example. an understanding of certain basicfractional-number ideas at the early childhoodlevel would provide a firmer basis for laterwork Second, mathematical ideas have an in-herent interest for children and are the sourceof imaginative activities appropriate to childrenat this level Third. ail exposure to a wide rangeof mathematical ideas makes children aware ofthe pervasiveness and usefulness of mathe-matics in daily life Finally, these topics contrib-ute to general instructional goals, such as de-veloping a problem-solving mind set, showingthe relationship between mathematics and thereal world, creating an orientation toward find-ing patterns and relationships, and makingchildren active participants in learning experi-ences Goals in this latter area are particularlyappropriate ones around which preprimaryprograms can be constructed. Tnus the newcontent emphases have relevance for the fullrange of early childhood years.

Balancing program content involves reex-amining most aspects of the important workwith number, particularly at the primary level.That teachers consider this aspect of the cur-riculum overcrowded is, in part, the result of

THE CURRICULUM 29

the recent downward shift in the grade place-ment of computational topics. Reexamining thework with numbers could alleviate severalproblems and increase the time available forbroadening the range of the content. Factors tobe considered include (1) reassessing the ap-propriateness of the current grade placementsof topics, (2) finding more effective ways of de-veloping number ideas, (3) identifying those as-pects of number work that are basic and thosethat can be reserved for advanced students ateach level, and (4) helping teachers determinewhich experiences should be considered ex-ploratory and which should result in mastery.

a balanced view of learning andinstruction

'Nu; curriculum should include a variety of types oflearning and thinking strategies. Each has implica-tions for the organization of learning experiences andinstructional strategies.

The study of mathematics involves manykinds of learning. learning content, learningspecific techniques, learning relationships andgeneralizations, learning problem-solvingstrategies. learning to relate new ideas to pre-vious knowledge, and learning to apply and ex-tend ideas. The process of learning mathemat-ics cannot be separated from the thinkingemployed in the various types of learning. It isimportant to recognize that our view of the na-ture of learning in mathematics can influencethe mental set that children develop towardlearning mathematics.

The broad variety of types of learning impliesthe use of different instructional emphases toattain goals. For purposes of discussion theymay be partitioned into two categories. Onecategory centers on a range of guided learningtechniques designed to help learners identifyand focus on specific elements. The other cate-gory includes several techniques in which lessguidance is provided. Obviously it is importantto match the requirements of learning situ-ations with the strategies employed.

The range of types of learning has implica-tions for the organization of the curriculum. At

I, .4

r')

30 CHAPTER TWO

times carefully sequenced instruction is need-ed, involving related series of lessons de-signed to build understanding and help thelearner "fit the pieces together." At other timesless sequencing is required, and more unstruc-tured approaches may be appropriate. Oftenpupils need a broad background of experienceon which to construct more precise notions. Inboth situations a clear sense of purpose andcareful planning are necessary to maximize thepotential for learning.

The distinction between the sequential, hier-archical aspects of learning mathematics andthe less structured dimensions of the subject isdifficult for many to perceive. The diagram hereprovides a focus for further discussion. Thenarrow cylinder represents the sequential as-pect; the wide cylinder represents broader as-pects of learning, or the curriculum setting.Placing the narrow cylinder inside the othershows how the two relate to each other; that is.learning can be put into a broader context evenwhile building understanding systematically.However, establishing a role for each aspectand keeping them in balance is one of the moredifficult tasks in curriculum planning and in-struction.

Exploredwy Experiences

Problem Solving

Patten! Finclbv

Applications and Extension

Several illustrations of learning situations arenow provided in order to focus on the types oflearning and the roles of instruction. A majorgoal in early instruction on a new operation is tohelp the learner relate mathematics to particu-lar models from which the mathematics can beabstracted and to enable him to move readilyfrom one realm to the other. This is illustratedwith multiplicationextended work at the"translation" level builds a firm foundation on

which other mathematical ideas and computa-tional techniques, such as division, can bemeaningfully constructed.

In building a connection between the modeland the mathematics, the teacher must keep inmind that extended discussion and carefulguidance is necessary. Three steps in concep-tualizing multiplication are shown next. The firststep is to help children describe the array, us-ing the informal language "3 fives." Next, theconnection between this step and the fact that aset of 5 dots is shown three times is made. Thisleads to the description "3 times 5," and then "3. 5." Finally, these experiences need to be re-lated to the complete number sentence and theideas of factor and product. Without such guid-ance, symbols such as 3 A 5 may have littlemeaning. The importance of such guidance isindicated by the student who remarked. Thethree 5s I get, but what does 'times' mean?' Af-ter a redevelopment of the first two stages, fo-cusing on a set of 5 shown one time, two times,

0l ^ )

1

:

3 fives

three times, he exclaimed. "Oh, is that whatmultiplication is all about!"

A set of 5shown 3times

7lwie are '3 tines 5"or 3 x 5 dots.

15 dotsin all

3 X 5 = IS'ador factor product

Many early primary pupils need guidance insymbolizing subtraction. A child may under-stand the idea of subtraction ,but be confusedabout how to write the number sentence. Theteacher may need to help him recognize thatthe numeral for the sum is written in front of theminus sign.

How many 112 (6)

Where do yoci show the 6In the number sentence?

Another type of learning involves the use ofstrategies for solving numerical situations.Children frequently develop intriguing methodsof their own. The following dialogues show thetypes of thinking that children employ:

Six-year-old: I know what 100 and 100 is.... 200.Adult: How do you know?Six-year-old: Because 1 and 1 is 2.

A Six-year-old is determining the total shown on apair of dice.

Child: 5 , .. 6. 7. 8.

Child: 8.

Child: 11.

Adult: How did you know it was 11?Child: Because 6 and 6 is 12.

THE CURRICULUM 31

Eight-year-old: How many days is 4 weeks?Adult: Can you figure it out?Eight-year-old: . . 28.Adult: Why?Eight-year-old: Well, 7 and 7 is 14, and if you

double 14. you get 28.

Seven-year, old: Wn at's 16 and 16?Adult: Why do you want to know?Seven-year-old: I'm doubling numbers. You

know-1 and 1 is 2. 2 and 2 is 4. 4 and 4 is 8. 8and 8 is 16.

Adult. Can you figure out 16 and 16 by yourself?Seven-year-old: ... its 32.Adult: How do you know?Seven-year-old: Well. 16 and 16 is two lOs and two

6s Two 10s is 20: 20 and 6 is 26 27. 28. 29.30.31, 32.

A parent and a six-year-old boy are discussing aswimming class that will meet twice a week for 5weeks.

Parent: How many times would you go in all?"Child. pausing and looking at each finger on one

hand as he counts to himself, answers: 10 times.Parent: How did you figure it out?Child: Well. I counted 1 on my fingers and 2 in my

head. The child demonstrates, pointing to a fin-ger as he says each number: 2. 4.6. 8. 10.

One of the contributions teachers can maketo children's learning is to encourage such pat-terns of thinking and provide instruction de-signed to develop a variety of thinking strate-gies. Such instruction will appeal to the naturalinterest children have in number ideas. Workwith basic facts provides an excellent opportu-nity to do this, but in striving for mastery, theteacher frequently overlooks this aspect of thework In fact, such activity needs to be viewedas one element in building proficiency withfacts. Four illustrations of thinking strategiesthat can be developec. Are shown next. Empha-sizing various approaches to thinking aboutfacts helps children organize their learning andbuilds an awareness that this work is a rational,meaningful activity.

4+4=8so +5= ?

32 CHAPTER TWO-0 s 2 3 4 5 k, / 8 9 lo

You know that 6 + 4 = 10.Which are greater than 10?6+3 6+5 6+1 6+76÷4=10,so6+5= ?

1 ten = 10so 2fives = ?

1 ten= 2 fives

2 tens = 202 tens = 4 fives, so

4 fives = _?

Many instances of exploratory and problem-solving experiences are given In chapter 4 andchapter 5, but one additional situation is pro-vided in . Symmetry is very much a part ofthe child's world and is not a difficult notion.This idea can be explored in a great number ofinteresting activities, many of which requirelittle guidance and can be extended in a varietyof ways.

The intent of this section has been to demon-strate that learning in mathematics is not all ofthe same kind. The variety of types of learning,ranging fro;n the specific to the very broad, in-dicates a corresponding variety of approaches

AA

so 3 fives is 10and 5 more.3 fives = ?

2 fives = 10,

in teaching mathematics. The nature of manymathematical ioeas requires some guidance ifthey are to have clear, continuous devel-opment. Therefore, a well-balanced curt ;alummakes provisions for many kinds of learningand matches the type of guidance given to thenature of the learning.

a balanced emphasis oncol1cepts, skills, and applications

Establishing a batance among these three elementshas important implications for the quality of mathe-

40

Fold to Ind linesof symmetry.

Complete so the redOne is a line

of symew.rhy.

matics programs. A balanced approach to computa-tion is particularly important.

The teaching of mathematics involves work-ing with children in three distinct, yet frequentlyrelated, areas. The first area involves the learn-ing of mathematical contentthe ideas, con-cepts, and relationships that are the heart ofthe subject. Teachers are responsible for help-ing children form concepts and for guiding thedevelopment of concepts from hazy, tentativenotions into more mature, productive under-standings. The second area involves learningspecific techniques and skills that are neces-sary for being able to move facilely in the worldof mathematical ideas and in the world of ev-eryday experience. Finally, from mathematicalideas and skills come applications that relatemathematics to a variety of other situations.Such situations can provide a motivation for thelearning of mathematics as well as stimulate re-flective thought.

Developing a well-balanced view of thesethree dimensions has proved difficult. Many in-dividuals have seen only the skills dimension ofthe subject. The mathematics program thenbecomes a curriculum of computational andmeasurement skills studied in isolation from itsother aspects. One result is a premature focuson skill work At other times, as a result oftrying to counterbalance the undue focus onskills, the content or applications aspects be-come the lens through which the entire subjectis viewed.

Teachers should therefore be aware that thestudy of mathematics involves ,earning in allthree areas and that distorting the curriculumby emphasizing only one phase of it hinders

LI +2,=6

6 -2, =al

THE CURRICULUM 33

children's learning. Since a curriculum neces-sarily deals with number concepts and opera-tions on numbers, it is particularly importanthere to maintain an overall balance amongcontent, skills, and applications. The inter-dependence of these elements and the signifi-cance of a balanced approach for effectivelearning should be recognized.

Two facets deserve further comment. First,instruction may emphasize particular aspectsat a given time. For example, preprimary pro-grams develop a cluster of concepts related tothe idea of cardinal number, and little is gainedby emphasizing such skills as forming numer-als or reciting addition facts. And in geometry,too, few specific skills or techniques are to bemastered. Thus instruction can maximize theconceptual and application aspects.

Second, it is important to recognize the rela-tionships that exist between these elements,particularly in the area of number. One way ofrelating them is pictured here, showing the im-portance of basic understanding for futurework.

Concepts --Skills ---> Applications

An example of this relationship is provided inthe next figure. Early work should center on es-tablishing an understanding of the operationsof addition and subtraction. Pupils learn toconstruct number sentences to describe situ-ations. Next. children learn to compute and toapply their knowledge to real-world situations.

However, the interrelationship of these ele-ments needs to be interpreted more broadly

b +3 =9

8 -5 =3

9-3

24

+7 How n-u4ch more money

does Se4e need to bc,f_y

the coloring b1'ok?

34 CHAPTER TWO

their interactions can operate in more than onedirection:

Concepts

.SkSkills

e%Applications

For example, physical situations frequentlygive rise to number concepts. The early "count-ing on" experiences found in some games or inusing dice provide an informal background forthe study of addition. Pairing shoes. or pairingchildren for a classroom game suggests theideas of "coming out even" or "one left over."Such activities can be the basis for developingeven and odd numbers. The hypothetical prob-lem of packing Ping-Pong balls in boxes leadsto finding the greatest number of fives and theremainder. This in turn would lead to inter-preting division as finding a quotient and a re-mainder.

A rich conceptual background does facilitatethe learning of arithmetic skills, but it is alsotrue that skills can contribute to the learning ofideas, a point that is sometimes overlooked. Anexample is provided in the pattern developedbelow. Skill in adding may be an importantfactor in successfully discovering and usingsuch patterns; possessing necessary skills canfree children to concentrate on the conceptualaspects of a situation.

No formula can direct the proper mixing ofthese three aspects. Additional evidence isneeded before more explicit guidance can beprovided. Nonetheless, an awareness of theimportance of each of the three aspects andtheir mutually supporting roles can bringgreater sensitivity and flexibility to the instruc-tional process.

0 0 0 0 0

5+5=10

0 0 0. 0

6+4=10

PAUL'S PING-PONG BALLPACKING COMPANY

Put 5 in

each box

13 Ping-Pong

What is the greatest number

of boxes Paul can fill?

How many balls leftover?

= X 5) + ?

5 r1:3-

A closer look at computation. Much of thediscussion in these last two sections has directrelevance for the teaching of computation.Many persons feel that computation should bemore than the mere manipulation of numbers.They argue that computation should be taughtso that-

1. computational techniques are related tobasic concepts and properties (computa-tion is one aspect of the larger topic ofoperations on numbers);

8+8=16 20+20=40so _9 + 7 = _? 21 + 19 = ?

10 + 10 = 20 300 + 300 = 600so 11 +9= ? 305+295= ?

You know Mat 40 + 40 =

Which equals- 80 42.+ _28 42_+ 36 42+ 3P

2. computation is used as a vehicle for de-veloping mathematical thinking and spe-cific thinking strategies;

3 computational techniques are related tocomponent concepts;

4. a variety of experiences in applying com-putation are provided.

r7 and 3 of Ihe 5 makes 10.

I still must add 2:

10 and 2 is 12

THE CURRICULUM 35

Several aspects of computation have beentreated already. Two additional elements willnow be discussed. The first deals with the roleof formal rationalization in developing informalthinking strategies. Two thinking patterns andformal rationalizations of them are presented inthe next figures.

Thinking Pattern

7 + 5 = ?

Formal Rationalization

7+5=7+ (3+ 2)= (7 + 3)+ 2=10+2=12

:1' E at 11M ?: Dan1 121.1

c m m ::

Renaming

Associative Property

Thinking Pattern___.-

I know 3 fours is 12,

so 3 x 4 tens is 12 tens.

12 tens is 120,

so 3 X 40 = 120.

3 X 40 = n

Formal Rationalization

3 X 40 = 3 X (4 X 10) Renaming= (3 X 4) X 10 Associative Properly= 12X 10= 120

36 CHAPTER TWO

3 31-i

-FLI +itAdding Tens

SI+2-I2,

Adding Tens&- Ones

In recent years, formal rationalizations basedon a deductive use of structural propertieshave been used to develop approaches tocomputation The assumption has been thatthese approaches build a thinking pattern. It isargued, however, that it is important to distin-guish between establishing an informal, verbalpattern and demonstrating mathematically at alater time why the pattern works. Children arecapable of grasping relationships intuitivelyand informally. Frequently they are abie to ap-ply a pattern when they cannot appreciate thevalue of a formal rationalization A systematic,step-by-step application of properties mayeven prevent children from seeing the patternitself. Although at some point it may be helpfulfor them to see how certain approaches can bejustified logically on the basis of a few unifyingideas, establishing and applying patterns ofthinking must be the primary aim. Formal ra-tionalizations should occupy a subordinate rolewith a carefully defin,:d use, and the distinctionbetween the two approaches needs clearerrecognition than it has received in recent years.

A second aspect of computation is the role ofsequencing instruction in developing computa-tional algorithms. Since the learning of al-gorithms has many complexities, it is importantto provide systematic instruction designed bothto relate this work to previous learning and toclarify the ideas involved. Sensitive, perceptiveteaching that builds on basic conceptual un-derstanding is required, and instruction should

0

.38 633+i-t8 2-1--Ti

Reniaming Onesas Tens

Renaming Tensas Hundreds

proceed carefully and deliberately, with a rec-ognition of the child's level of maturity.

In the first section of the chapter, the impor-tance of setting systematic instruction in abroader framework was discussed. This can bedone when teaching computational algorithms.Such work can continually be related to prob-lem situations, opportunities can be providedto explore various number patterns, and pupilscan be encouraged to use informal computa-tional techniques built around a variety of esti-mation and mental arithmetic approaches.Such a setting gives a more holistic frameworkto the focus on algorithmic details.

There are tvt,o ways in which sequencing isinvolved in work with computation. First, the hi-erarchical ordering ofIskills should be recog-nized. For example, learning to solve 38 - 48follows lower levels of operations, this skill inturn precedes renaming with three-digit nu-merals (111 ). Recognizing and building instruc-tion around the various levels of difficulty incomputing can help learning to proceed moresmoothly. Understanding at one level is impor-tant for learning at the next.

Second, sequencing is also an importantfactor in developing a particular algorithm. Ini-tial instruction on the renaming algorithm insubtraction might proceed as follows:

1. Development of the concept that sub-traction is not always possible in the set ofwhole numbers(, )

6 marbles on a fable. Can you

pick tip 5 marbles? 6 marbles?

7 marbles? 8 marbles? What is

the greatest number you can pick

up? Is there a whole number

that makes 6 7 = n true?

2 Development of renaming around theprinciple that a number may be renamedas "1 less ten and 10 more ones"

111111C

71111

1111111111111

M4 fens 2 ones

3 tens 12 ones

3. Use of a developmental algorithm thatjh lights these two ideas as well as more

general grouping and place-value notions

11111111111

1111111 Illitallillfilimooi IT

3 12JhLens Zones

2 tens 8 ones

Does 2 b' namea whole number?

How do you rename?

4. A transition to the final, compact form

3 12 3 121-fens 42- ones AZ2 tens X ones 28

5 A focus on when it is appropriate to re-name

In which do yeti need to rename?

65 74 87 81 60 3248 35 36 42 20

THE CURRICULUM 37

An unhurried, careful development closelylinks this procedure to physical and mathemat-ical ideas and provides an opportunity for theirintegration in the final algorithm.

Developing proficiency with computationalskills is not an easy task; however, ignoring thecomplex problems of instruction in this area oradopting simplistic positions will not producemore effective learning. In the development ofmore effective programs, all the many ele-ments involved must be considered.

a balanced view of the use ofphysical materials

The use of physical materials can make many contri-butions to early learning. Yet physical materials arenot a panacea. Understanding the instructional pur-poses they may serve can help them to be used pro-ductively.

The judicious use of physical materials at theearly childhood level plays an important role in'he learning of mathematics, particularly in theinitial forination of some mathematical content.Today the use of a wide variety of materials isreceiving strong supportsupport that some-times has led to such statements as "Learningcannot occur without the physical manipulationof materials." Although this probably over-states the case, it is unfortunately true thatmathematics is often taught with little or no useof materials or other perceptual supports.

Even though there is general support for theuse of materials, a careful description of howthey influence learning is difficult at this time formany reasons. The amount of experience nec-essary to effect learning is hard to determine.More must be known about bridging the gapbetween the materials and the ideas they de-velop. Knowing which materials are the mostappropriate for particular concepts is difficult.The relative roles of physical experiences andpictorial representations is not clear. Some seethe manipulation of materials as an end in itself;others concentrate on their use as a means toan end. Additional knowledge in this areawould be invaluable in providing a auide todaily practice.

In the meantime, it is important to stress thatpupils' involvement in physically oriented activ-

A

38 CHAPTER TWO

ities can contribute both to learning and to theestablishment of a positive learning climate.One guide to the productive use of such experi-ences is an awareness of the purposes forwhich the manipulation of materials can be em-ployed:

1. Materials can be used in exploratoryexperiences where the purpose is to providereadiness for later work. For example, kin-dergarten children can explore the idea ofshape by making familiar figures on thegeoboard as well as examples of variouspolygons ( ).

2. Materials can be used in the devel-opment of a specific mathematical idea. Todevelop the pattern of renaming used in thesubtraction algorithm, teachers should havepupils regroup physical materials and record

n:=1 i--1-=-rfc-1

ffr-nazu3 tens, ones

You can rename 6y thinking

1 less ten

and 10 more ones

the results. At this level specific guidance isneeded if pupils are to see the connectionwith the idea being developed. The activityshould lead to the generalization that anumber can be renamed as 1 less ten and10 more ones ( ).

3. Materials can be used to provide addedexperience with an idea that has alreadybeen developed. Many preprimary experi-ences with early number work are of thistype. For example, in dealing with groupingand place-value ideas, the teacher mightplace around the room envelopes containingsets of buttons, beads, paper clips, tooth-picks, and so on. Pupils move about theroom and empty the contents of each enve-lope, group them by tens, record the numberof tens and the number of ones left over, andthen write the standard numeral (0).

tt

,

2 tens, ? ones

353 tens 5 ones

2 tens 15 onesA

7.3

0

4. Materials can be used as the source ofexperiences in which pupils extend theirknowledge or apply it in a variety of ways.The experiences with symmetry cited earlierin the chapter illustrate this use.

An awareness of the many uses to which ac-tivities can be put can enable a teacher moreeffectively (a) to determine the purpose for agiven activity. (b) to provide help in selectingactivities appropriate for a given instructionalpurpose, and (c) to give guidance in how theactivity should be structured. Activities that arecarefully and purposefully integrated into in-structional programs can be of great value, butthey are not cure-alls. An intelligent and bal-anced use of activities, as well as an awarenessof when they contribute to learning mathemat-ics, is required.

a balanced approach to theorganization of instruction

Effective instruction for all pupils involves the consid-eration of many approaches to organizing instruction.Such consideration must take into account many vari-ables related to learning and instruction. A variety ofapproaches may need to be employed.

Adjusting instruction to the needs and abili-ties of individuals is receiving renewed and ap-propriate interest today. A concern for the de-velopment of each child, for the quality of hislearning, and for the level of success he attainsmust be at the heart of the educational process.Basically, individualizing instruction implies a

THE CURRICULUM 39

point of view or attitude toward students that isindependent of an organizational pattern. A ba-sic element of this point of view is the accept-ance of each child as an individual worthy ofadult respect. Other important ingredients in-clude an acceptance of the child's ideas, a pro-vision of opportunities for pupil input in devel-oping and selecting learning experiences, aconcern for the quality of the child's intellectualdevelopment, and a willingness to take time toknow the child as an individual.

Concern for pupils also involves concern forthe quality and extent of their learning. Closelyrelated to this concern is the attempt to find ap-propriate content and to develop it appropri-ately. Factors that promote learning must berecognized and considered when fitting in-struction to children. In order to help childrenlearn effectively and enjoyably, teachers musthave a knowledge of the learning process inmathematics and use it in making decisionsabout organization for instruction.

Individualizing instructionthat is, person-alizing instructionand adjusting it in order tomaximize the probability of successful learningnecessarily involves organization for instruc-tion. Yet plans for classroom managementare a means to an end, they are techniquesfor promoting more effective learning in apleasant, accepting atmosphere.

Individualized instruction is frequently equa-ted with self-paced instruction. It involveslearners moving independently through pre-determined lessons, receiving guidance as re-quested, and progressing at their own rates. Attimes the curriculum emphasis is on the skill di-mensions of learning, with little developmentwork provided and with the measures of learn-ing and understanding limited to correct re-sponses to written exercises. Although inde-pendent progress approaches may be used atappropriate times, identifying individualized in-struction with a single organizational patternmay cause significant considerations about thelearning process to be ignored.

Balance in this area can be achieved by us-ing techniques appropriate to particular situ-ations. When wisely implemented, group devel-opmental lessons take into account individualcapabilities and provide means for individualinput. Interest in learning in general can be in-

40 CHAPTER TWO

creased by stimulating interest in the learningof new content and presenting it informally.Children can be helped to organize their ideasand see the important elements in new learn-ing. Opportunity can be provided for dis-cussion. with child-initiated questions provid-ing additional input for a lesson or altering itsdirection. Interacting with peers and concernedadults is satisfying to children and is an impor-tant part of learning.

In the process of developing a topic, pupilsmight work with materials either in smallgroups or individually for certain periods oftime. At times, a variety of groupings may beemployed. Pupil performance with specificcontent is often the basis for such groupings.Sometimes pups lo need to work independentlyon particular weaknesses or on more advancedwork with a topic. A flexible program would of-ten provide the opportunity for pupils to selecttheir own learning experiences.

Classroom organization for instruction is oneof the elements that must be considered whenattempting to provide for individual differencesin learning. Yet having organizational patternsserve more basic goals requires considerationand sensitive Judgment about the nature of par-ticular content, purposes of particular in-stoction, the learner, and the learning proces_.Instruction organized in a flexible and balanceamanner is likely to produce effective progress.

references

Brownell, William A. The Place of Meaning in the Teachingof Arithmetic. Elementary School Journal 47 (January1947): 256-65.

. "Psychological Considerations in the Learning andTeaching of Arithmetic " In The Teaching of Arithmetic.pp. 1 -31 Tenth Yearbook of the National Council ofTeachers of Mathematics. New York: Bureau of Publica-tions. Teachers College, Columbia University, 1935.

. "When Is Arithmetic Meaningful?" Journal of Edu-cational Research 38 (March 1945) 481 98.

Buckingham. B. R. Significance, Meaning. InsightTheseThree. Mathematics Teacher 31 (January 1938). 24 30.

summary

This chapter has identified some of the ideasand issues related to effective instructional pro-grams for young children. It has been arguedthroughout that curriculum deliberationsshould focus on basic questions that deal withthe learning of mathematics in a comprehen-sive and realistic fashion. Several issues of cur-riculum and instruction have been discussed inthe context of their influence on curriculumover an extended period of time. They havealso been discussed in terms of the notion thatbalanced emphases in several areas are im-portant for 'earning. Each of the five aspects ofa balanced curriculum should receive carefulattention, each is an important aspect of pro-viding better perspective on the mathematicscurriculum.

The task of building and implementing effec-tive programs is not an easy one. It is nevercompleted. Yet through thoughtful consid-eration of significant questions, careful devel-opment of new approaches, wise implementa-tion of ideas ...th demonstrated potential. and acommitment to excellence can come progressthat will result in meaningful learning by thechildren we serve.

Buswell. Guy T. Introductton, In The Teaching of Arithme-tic, pp. 1 5. Fiftieth Yearbook, pt. 2. of the National So-ciety for the Study of Education. Chicago: University ofChicago Press. 1951.

Commission on Post-War Plans of the NCTM. "Second Re-port of the Commission on Post-War Plans." Mathemat-ics Teacher 38 (May 1945): 195-221.

McConnell. T. R. 'Recent Trends in Learning Theory. TheirApplication to the Psychology of Arithmetic. In Arithme-tic In General Education, pp. 268-89. Sixteenth Yearbookof the National Council of Teachers of Mathematics. New

C8

York Bureau of Publications. Teachers College. Colum-bia University. 1941.

Morton. Robert L. "The National Council Committee onArithmetic." Mathematics Teacher 31 (October 1938).267 72.

National Council of Teachers of Mathematics. Arithmetic inGenera! Education Sixteenth Yearbook. New York. Bu-reau of Publications, Teachers College. Columbia Uni-versity. 1941.

THE CURRICULUM 41

. A History of Mathematics Education in the UnitedStates and Canada. Thirty-second Yearbook. Washing-ton. D.C.: The Council. 1970.

. The Teaching of Arithmetic. Tenth Yearbook. NewYork: Bureau of Publications. Teachers College. Colum-bia University. 1935.

National Society for the Study of Education. The Teachingof Arithmetic. Fiftieth Yearbook. pt. 2. Chicago. Univer-sity of Chicago Press. 1951.

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factors that may be learnable but not directlyteachable, yet those who work with childrenmust be continually aware of their vital impor-tance. The limited number of studies in the af-fective area are reviewed first, followed bystudies (Ai ,:le developmental aspects of math-ematics learning and mathematics learningfrom systematic instruction.

affective factors ofmathematics learning

The affective domain pertains to attitudes.interest, self-concepts, and factors such asanxiety and frustration, all of which influence orinteract with motivation. These various con-comitants that children carry with them to eachmathematics lesson affect the way they feelabout mathematics. Research on mathematicseducation, however, has looked carefully atvery few of these concomitants.

what is the role of attitudes inmathematics?

Teachers of young children realize, possiblyeven more than other teachers. the importanceof attitudes in leaning. Young children tend toindicate overtly their feelings of enjoymeit, in-terest, and enthusiasm; they also readily in-dicate their dislike or apathy. Yet surprisinglylittle research has been done concerning theirattitudes toward mathematics. Probably thisfact is due in large part to (1) the difficulty ofmeasuring attitudes when dealing with youngchildren (e.g.. their responses on group teststend to be of questionable validity and (2) theinstability of young children's attitudes fromday to day.

A concern with attitudes and other affectivefactors has developed because it seems rea-sonable to assume that the way children feel

RESEARCH ON MATHEMATICS 'LEARNING 45

about, and react to, mathematical content andideas is related to the quantity and quality oftheir learning. This logical relationship has ledto continual concern with "motivation," thoughexactly what motivation is has been the subjectof some debate. However, it may be assumedthat motivation includes whatever the teacherdoes to increase children's interest in learning.One hopes that increased interest will lead toincreased achievement, though research hasnot provided evidence that this relationship isas strong as might be expected (see, for ex-ample. Deighan (1971 )). Numerous reportshave been made about games, materials, andtechniques that teachers have used success-fully to increase interest. It seems reasonablethat the manner in which the teacher presentsan activity or idea is closely related to the reac-tion of children to that activity or idea. Whatteachers sayand how they say ithas beenfound to be particularly important. Not surpris-ingly, praise has been found to be a highly ef-fective way to motivate. For instance, Masek(1970) studied the behavior of children whenteachers provided such positive reinforcementas verbal praise, physical contact, and facialexpressions indicating approval. Significant in-creases in mathematical performance and intask orientation occurred when such reinforce-ment was frequent; lower performance wasnoted when reinforcement was stopped, andperformance increased once more when rein-forcement was again used.

Teachers and other mathematics educatorsgenerally believe that children learn more ef-fectively when they are interested in what theylearn and that they will achieve better in mathe-matics if they like mathematics. Therefore, con-tinual attention should be directed towardcreating, developing, maintaining, and rein-forcing positive attitudes.

what are the attitudes of youngchildren toward mathematics?

Researchers who are interested in measur-ing attitudes toward mathematics have tended

46 CHAPTER THREE

to begin their study in grade 3. A large per-centage of both boys and girls in grades 3tdrough 6 have indicated that they liked mathe-matics (Greenblatt 1962; Stright 1960). The evi-dence on whether boys like it better than girls,or vice versa. is conflicting.

Very tittle is known from research about chil-dren's attitudes toward mathematics prior tograde 3. Generally, teachers find that the atti-tudes of children in the early grades are posi-t.ve toward both school and mathematics. Tomost observers, it is apparent that the ideas ofnumber and geometry hold a fascination forthese children. It is commonly believed thatfrom grade 3 to grade 6. however, children s at-titudes toward mathematics become less posi-tive, anc research provides some support forthis belief Or example. see Deighan [1971 D.Yet, other investigations have shown that chil-dren's ai.titudes toward mathematics becomemore positive from grade 4 to grade 6 (for ex-ample. see Crosswhite (19721).

It seems likely that a relationship would existbetween the teacher's attitude toward mathe-matics and his pupils' attitudes toward mathe-matics. Yet only a weak relationship betweensuch attitudes has been observed in grades 3through 6 (Caezza 1970. Deighan 1971; Wess1970). This finding, which conflicts with com-mon sense. may be accounted for by the way inwhich the attitudes were measured in thesestudies. Other rationales could also be devel-oped. of course.

developmental aspects ofmathematics learning

Children begin to acquire mathematicalideas almost from the time they become awareof the world around them. Researchers inter-ested in the development of children have at-tempted to ascertain both what ideas are devel-oped and how they are developed.

what are the findings of researchrelated to children's developmentand their implications formathematics education?

IIg and Ames (1965). with others at the GesellInstitute, have long been engaged in longitudi-nal studies of how children develop and the im-plications of these studies for the sGhool. Theypropose that a thorough analysis of the child'sdevelopmental level would aid teachers in pro-viding an individualized program for each child.Their tests are designed to ascertain devel-opmental factors (including the development o:mathematical ideas) rather than knowledge ac-quired from school instruction. They point outthat educators must keep in mind the meshingof three factors: the child at a certain age orlevel of growth, the child as a unique individual.and the child in a certain environment.

A vast amount of research has been directedtoward testing various aspects of the theory ofPiaget In his theory. Piaget depicts intelligenceas an active. organizing process whereby thechild attempts to structure his world and givemeaning to it. Piaget has identified four impor-tant factors involved in the development of in-tellectual capacity (1) maturation. (2) encoun-ters with the physical environment, (3) socialexperience. including both interaction with. andinstruction by. adults and peers; and (4) a proc-ess he calls equilibration, by which the child in-corporates new knowledge. Not only has Piagetidentified these four factors, but he has also-

1. identified certain stages of developmentand established approximate chronologi-cal age boundaries for each stage;

2. ascertained that a child's development isinvariant in terms of sequencingthat is.he passes from one stage to another inmuch the same order as all other children;

3. developed certain procedures. oi proto-cols, for ascertaining the child s stage ofdevelopment, involving a specific set ofquestions to be asked about a specific setof materials.

An overwhelming amount of Piagetian-in-spired research has been concerned with threequestions:

1. Are the stages proposed by Piaget valid?Most research has confirmed the existence ofthese stages: that is, they do occur, and theyoccur by and large in the order Piaget suggest-ed Piaget maintains that a major shift in thenature of children's thinking occurs at aboutage seven. At this point, the child becomesless prone to distractions than he was earlier,can manipulate ideas more readily, is awareof contradictions, and can correct himself whenhe makes mistakes. Thus the period of earlychkihood marks the transition from thoughtthat is perceptual and subjectively oriented tothought that is conceptual, objective, and sys-tematic The child develops increasing trust inreasoning rather than reacting almost entirelyon the basis of what he perceives.

Research findings on when the stages occurhave indicated more variability than Piagetsuggested. For instance. Sawada and Nelson(1967) reported that the threshold age for con-servation of length appeared to be betweenages five and six, not ages seven and eight asPiaget found.

2. What is the effect of training on advancingthe age at which the stages are reached? Pia-get's interest was in determining how childrendevelop spontaneously and in interaction withtheir environment, but without plannedchanges in that environment. He contends thatthe child develops mathematical concepts notonly from instruction but also independentlyand spontaneously from his own experiences(Piaget 1953) Other research indicates that bythe use of certain procedures, children can betrained to evidence a stage earlier than wouldbe expected f they were untrained. However.there is no evidence that training changes'theoverall developmental level of the child. Thepositive effect of training is rarel!, retained un-less the child is already in transition from onelevel to another (see. for example. Coxford[19641).

3. Are Piaget's protocols accurate me3s-ures9 Research generally indicates. with littlevariation, that unless the questiors and materi-als of Piaget's protocols are used, results may

RESEARCH ON MATHEMATICS LEARNING 47

differ. (For example, Sawada and Nelson[1967 j used a nonverbal method of assessmentrather than a verbal one, a fact that may havecontributed to their findings, contradictingthose of Piaget.)

The key question that we as teachers ofmathematics find of greatest concernWhatare the implications for the teaching of mathe-matics?has been the subject of much dis-cussion. Sinclair succinctly states the con-clusion of many (1971, p. 2):

Educational applications of Piaget's experi-mental procedures and theoretical principleswill have to be very indirectand he himselfhas given hardly any indication of how onecould go about it. His experiments cannot bemodified into specific teaching methods forspecific problems, and his principles shouldnot be used simply to set the general tone of aninstructional program.

Comparatively little research has exploredthe classroom implications of Piaget's theory.The topic on which the most research has beendone is conservationthe realization of theprinciple that a particular aspect of an object ora set of objects may remain invariant underchanges in other, irrelevant aspects of the situ-ation. Conservation of number involves aware-ness that the number of objects in a set re-mains unchanged in spite of changes in thearrangement of the objects. Piaget believesthat the child must grasp the principle of con-servation before he can develop a reliable con-cept of cardinal number. He found that the de-velopment of this concept takes place at aboutseven to eight years of age. He believes thatnumber concepts involving cardination and or-dination, which are aspects of classification(the grouping of objects according to their sim-ilarities) and serration (the ordering of objectsaccording to their differences), cannot betaught to younger children, since the child isconceptually unable to understand.

Conservation of number has been shown byseveral researchers to be related to work w;thcounting and with additior, and subtraction.Almy, Chittenden. and Mills. (1966) noted thatchildren who conserve at an early age do betterin beginning arithmetic than those who are latein acquiring conservation. Robinson (19681

48 CHAPTER THREE

also found significant relationships between achild's ability to conserve, serrate, and classifyand his level of achievement in grade 1. Heconcluded that conservation may be necessarybut not sufficient for mathematics achievement.Others have found that first graders who areconservers tend to do better at certain mathe-matical problem-solving tasks. Steffe (1967) re-ported that the tests of conservation of numer-ousness used in his study provided anexcellent prediction of success in solving addi-tion problems and learning addition facts forchildren entering first grade. Using the sametest in a companion study on subtraction. Le-Blanc (1968) reported a similar conclusion.Dodwell (1961) and Wheatley (1968) have alsoindicated that the tests of number conservationthat they used may be meaningful measures ofarithmetic readiness.

Ambiguous findings were reported by Almyand her associates (1970). Second-grade chil-dren who had no prescribed mathematics les-sons in either kindergarten or first grade per-formed about as well on certain Piaget-derivedtasks as second-grade children who did haveprescribed mathematics lessons beginning inkindergarten But the latter groups performedbetter than the second-grade children forwhom prescribed lessons did not begin untilfirst grade The following conclusions werenoted (Almy et al. 1970. p. 170):

The notion that the instruction should matchor pace the children's cognitive developmentappears logical, but much is st.11 to be learnedabout how to assess such development as wellas about pacing instruction from it.

The problem is not only that developmentappears to be uneven, and that progress as theteacher observes it may be a matter of advanceand retreat rather than the forward-moving se-quence so often inferred from Piaget s theory,it is also that individual children differ so muchnot only in cognitive level, but also in their atti-tudes. their interests, and their concerns.

It may also be true that the type of teachingthe child has had may affect the pace of his de-velopment. and the type of materials he and histeacher use may be of importance. The learn-ing style of the child may also be a factor toconsider For instance, Simpson (1971) foundthat the child's ability to conserve substance,

Au

weight. and volume was affected by whether hehad a reflective or an impulsive learning style.

Although an overwhelming quantity of thePiagetian-oriented research has concentratedon conservation, attention is gradually shiftingto other facets of Piaget's theory, such asclassification and seriation. Piaget stated thatthe child must first construct simple collectionsof materials before he is able to manipulate ob-jects and classes within hierarchical systems.Nowak (1969) found that third graders bene-fited from instruction on two simple hierar-chical operations. Johnson (1970) reported thatthe kindergarteners he studied were able tocategorize by using almost exclusively a per-ceptual focusing strategy that involved a seem-ingly random choice of a focus picture and aone-to-one search for identical pictures. Thetask was extremely simplified when the focus-ing picture was supplied. Bonney (1970) as-sessed the serration skills of first graders andfound that they did not tend to generalizeacross types of content (concrete or verbal.quantitative or interpersonal).

Thus it appears that a child s mathematicalachievement may be aided by instruction that isbased on an assessment of his developmentallevel and that includes content that helps him tofocus on relevant aspects of such factors asconservation. serration, and classification. Pia-get's protocols can aid in assessment, but theyare not intended to be used for instruction. Al-though they could provide training guides,there seems to be little long-range payoff fromtraining. Instead, the emphasis should be onproviding experiences that help the child gainin understanding, most of the chapters in thisyearbook focus on these various experiences.

what has been ascertained aboutthe mathematical knowledgechildren have when they enterschool?

Through the years, many surveys have as-sessed the mathematical knowledge acquiredby children during their first five or six years(see, for example, Buckingham and MacLatchy[1930 j, Brownell [1941 j, Ilg and Ames [1951 ];

Bjonerud [19601; Rea and Reys (1970)). Ananalysis of these surveys provides informationon both the extent and the variability of attain-ment Unfortunately, the generalizations thatcan be made from such studies are limited forseveral reasons:

1. Usually the groups were relatively small,and readily available groups, rather thangroups representative of the total popu-lation, were used.

2. Few surveys have been replicated, mostwere conducted at only one point in time,as well as in only one locality. As withmost surveys, the question ariseswhether the findings would be accuratefor other children in other places.

3. The methods of collecting informationvaried. Some used individual interviews;others used group tests. Some used onequestion to affirm a particular aspect;others asked many. Also the depth of thequestions varied from those requiringsingle-word responses to those requiringmanipulation of objects or, rarely, rea-sons for answers.

But most important of all in analyzing theusefulness of these surveys is the fact that whatteachers really need to know is not what otherchildren know. but what their children know. AsBrownell stated, -Research findings tell theteacher ... little about the class as a whole. butthey tell very much less about the number abili-ties of particular children" (1941, p. 61). Eachteacher must plan experiences and base in-struct r1" n an accurate knowledge of the

dens of cwcf

.entur. of I, CJ tits S tit tAlcfPerhaps one of the greatest contributions of

surveys assessing the knowledge of young chil-dren has been the development of proceduresand instruments for assessment. Tests such asthose developed by Rea and Reys (1970) orSchwartz (1969) can serve the teacher as valu-able tools in assessing children's mathematicalknowledge.

Tests designed to ascertain the child's statusin relation to different aspects of Piaget's the-ory are also being developed. If it is true, for in-stance. that the child must conserve number

RESEARCH ON MATHEMATICS LEARNING- 49

before he can deal meaningfully with counting,addition, subtraction, and other mathematicalideas, then it becomes important that teachersascertain his level of conservation as they takeinventory of his mathematical ideas.

Some conclusions can be drawn from the re-search. It is clear that children have acquiredmany mathematical ideas before they beginformal instruction in school (Rickard 1967,Heimgartner 1968). Moreover. in the range andextent of their knowledge, there is relativelylittle difference between children today andchildren decades ago. Also, as the age, in-telligence, or socioeconomic level of his par-ents increases, the child'S knowledge of mathe-matical ideas appears to increase.

The variability of the methods of assessingthe knowledge of children entering kindergar-ten makes it difficult to affirm precisely whatchildren in general know. A summary of thedata on five-year-olds without prior schoolingindicates that-

1. many children can count and find thenumber of objects to ten. and some areable to count to at least twenty;

2. some can say the number names for tensin order (that is, ten, twenty. 'thirty ...),but far fewer can say the names whencounting by twos and fives;

3. most know the meaning of "first," andmany can identify ordinal positionsthrough "fifth";

4 many can recognize the numerals from 1to 10. and some can write them;

5 most can give correct answers to simpleaddition and subtraction combinationspresented verbally either with or withoutmanipulative materials;

6. most have some knowledge about coins.time, and other measures, about simplefractional concepts. and about geometricshapes.

It is not only when a child enters kindergar-ten (or first grade) that teachers need informa-tion on his mathematical knowledgethe be-ginning of each year is a particularly importantpoint of assessment, for it provides the teacherwith a point of beginning and a base line-guide

50 CHAPTER THREE

for planning the year's work. It indicates to theteacher what prerequisite skills and under-standings the child has and thus indicates whathe is probably ready to learn. But assessingreadiness is an almost continuous process. Theteacher needs to know when the child is readyto proceed with, and profit from, experiencesor instruction at each step in the mathematicalprogram. Decisions must be reached aboutwhen to introduce a new process or a new as-pect of a process. A teacher cannot expect tobe effective unless readiness is continually as-sessed.

mathematics learning fromsystematic instruction

Not unexpectedly, the bulk of the researchon mathematics education has been con-cerned with the effects of systematic instruc-tion. This fact can be considered under twoheadings. (1) the organization of the programand other general methodological consid-erations and (2) the content and how it may betaug ht.

organization and method

what are the effects of various types ofkindergarten experiences?

Research has indicated that children whohave kindergarten experience score higher inlater grades than those who do not have kin-dergarten experience (Haines 1961). Most. butnot all, studies indicate that children who wereolder when they entered kindergarten or grade1 achieved more than their younger class-mates.

The specific type of program for kindergar-ten has had, and will continue to have, much re-search focused on it. One type of research isconcerned with creative ways of developingmathematical ideas. For instance. Fortson(1970) taught beginning mathematics to five-year-cids through multiple-stimuli techniques

using concrete materials, bodily movement,and sound. The children saw, heard, and feltpatterned numerical relationships. They scoredhigher after this program than pupils not hav-ing it.

Another type of study is concerned withmore formal processes. Carr (1971) studied theeffects of a program modeled after the Ber-eiter-Engelmann preschool program (seechapter 12). Bereiter-Enaelmann materialswere used in kindergarten with children whohad had zero, one. or two years of the specialtraining. No significant differences were foundon four tasks of the Piaget type involving abilityto conserve number, to discriminate, to seriate,and to enumerate. A significant correlation wasfound, however, between performance onthese tests and on achievement tests. It wasconcluded (1) that instruction of this type ismost effective when the child has reached acertain level in his cognitive development thatmight be described as number readiness and(2) that tests of the Piaget type can be used toassess a child's readiness for systematic in-struction in arithmetic.

Kindergarten programs vary greatly. It is fre-quently averred that a planned program doesnot need to be a formal program_The teachercan provide experiences and materials that en-able the child both to explore mathematicalideas in his environment and to resolve hismathematical questions as they arise. Text-books and workbooks do not need to be part ofsuch a program.

what factors need to be considered inplanning for instruction?

It may seem to many educators that two fac-tors of particular importance in planning for in-struction are how the school is organized andhow the classroom is organized. Certainly anextensive amount of research (particularly atintermediate-grade levels) has been devoted toattempts to ascertain the "best" patterns ofschool and classroom organization. But theclearest generalization that can be drawn fromthis research is that apparently no one patternper se will increase pupil achievement in math-ematics. Perhaps the most important implica-tion of the various studies is that good teachers

7

are effective regardless of the nature of the pat-tern of school or classroom organization.

The methods of instruction that the teacheruses are of particular importance, thoughprobably no one method or procedure is effec-tive for all teachers with all content. In general,however, the importance of teaching withmeaning has been confirmed. Dawson andRuddell (1955) summarized studies of variousaspects of meaning. They concluded thatmeaningful teaching generally leads to greaterretention, greater transfer, and increased abil-ity to solve problems independently. They sug-gested that teachers should (1) use more mate-rials. (2) spend more class time ondevelopment and discussion, and (3) provideshort, specific practice periods. More recentstudies have supported these findings. (For amore extensive discussion of the role of mean-ing in mathematics. see Weaver and Suydam(19721.)

Many teachers have noted that children failto retain knowledge well over the summer va-cation. The amount of loss vanes with thechild's ability and age, but the length of timebetween the presentationCof the material andthe start of the vacation is of particular impor-tance Practice during the summer and reviewconcentrated on materials presented in thespring have been shown to be especially im-portant.

Transfer. the ability to apply somethinglearned from one experience to another. ap-pears to be facilitated by instruction in gener-alizing. that is by teaching children to set, pat-terns and to apply procedures to newsituations In most studies there is the implica-tion that transfer is facilitated when teachersplan and teach for it. And children need toknow that transfer is one of the goals of instruc-tion.

Many studies have provided evidence thatthe type of test is important in measuring out-comes This point was emphasized during the1960s when there was much concern about theeffect of **modern" programs compared with"traditional" programs. Generally, it was foundthat pupils who had had a "modern** programscored higher on a "modern" test and thatthose who had had a "traditional" programscored higher on a "traditional" test. In addition


to published, standardized tests, Ashlock andWelch (1966), Flournoy (1967), and Thompson(1969), among others, have reported paper-and-pencil tests designed to measure achieve-ment with contemporary programs, with theirstress on concepts and not merely on skills.

what factors associated with the learnerinfluence achievement in mathematics?

Research has shown that the ability to learnspecific mathematical ideas is highly related toage and intelligence. Socioeconomic level alsois related to achievement, but not as strongly.Achievement does increase as the socioeco-nomic level of *the parent increases (see, for ex-ample, Unkel 11966 ]).

The majority of studies with the environmen-tally disadvantaged are status studies, provid-ing descriptive information on how studentswere achieving at the time of the study. Somestudies, however, have compared the achieve-ment of pupils from two or more levels. Thus,Montague (1964) reported that kindergartenchildren from a high socioeconomic areascored significantly higher on an inventory ofmathematical, knowledge than pupils from alow socioeconomic area. Dunkley (1965) sim-ilarly reported that the achievement of pupilsfrom disadvantaged areas was generally belowthat of children from middle-class areas. Differ-ences were greater in first grade than in kinder-garten, which suggests that being dis-advantaged may have a cumulative effect.

Johnson (1970) noted that kindergarten chl-dren from low socioeconomic backgroundsdemonstrated less ability to categorize consist-ently on attribute resemblance. They showedsimilar stages of development regardless ofsocioeconomic level, however, they apparentlyproceed through the stages at a slower pace.

In other studies, it has been suggested thatpersonality factors or emotional difficulties maybe more relevant to a lack of success in mathe-matics than intelligence. In one of the few exist-ing studies that deals specifically with the rela-tionship of cognitive style to mathematicalachievement. Cathcart and Liedtke (1969)tested fifty-eight children in second and thirdgrades. They reported that children in secondgrade tended to be impulsive (to jump quickly

52 CHAPTER THREE

to conclusions), whereas those in third gradewere comparatively reflective (more inclined toreserve judgment) The relationship betweenintelligence and cognitive style was not statisti-cally significant, although there was a trend forthose with high intelligence to be more reflec-tive than those with lower intelligence. No sig-nificant interaction was noted between cogni-tive style and sex, grade, age, intelligence, orconservation However, reflective pupils at-tained higher mathematics achievement scoresthan impulsive

Although some researchers have reportedthat boys and girls score differently both on

'tests of reasoning and on tests of fundamentals(for example, see Wozencraft (1963)), mosthave concluded that what little difference existsis not sufficient to influence curriculum deci-sions.

what is the role of diagnosis inmathematics instruction?

The purpose of diagnosis is to identifystrengths as well as weaknesses and in thecase of weakness. to identify the cause andprovide appropriate remediation. As part ofthis process. there have been many studiesthat ascertained the errors that pupils make.For instance, Roberts (1968) analyzed compu-tation items from a third-grade standardizedachievement test and classified errors into fourmajor categories wrong operation. computa-tion error. "defective" algorithm. and undis-cernible errors The inaccurate use of al-gorithms accounted for the largest number oferrors: errors due to carelessness or lack of fa-miliarity with addition and multiplication factswere fairly constant for all levels. Roberts sug-gested that teachers must carefully analyze thechild's method and give specific remedial help.

Ilg and Ames (1951) concluded that less em-phasis should be put on whether answers areright or wrong and more on the kinds of errorsthe child makes The kind of error provides agood clue to the intellectual process the child isusing, since errors often are not idiosyncratic,the same types of errors occur over and over.

The individual-interview technique has beenused with success in many studies. The proce-dure, as described by Brownell (1944). is to

face a child with a problem, let him find a solu-tion, and then, in order to elicit his highest levelof understanding, challenge him. The intent isto ascertain not only what the child knows buthow he thinks about the mathematical proce-dure he is using.

Computer-assisted instruction is presentlybeing used in some elementary school mathe-matics classes. Suppes has reported exten-sively on the use of both tutorial and drill-and-practice programs (see, for example, Suppesand Morningstar [1969 j). The computer readilycollects data on how children are responding,thus facilitating a diagnosis of their difficultiesas well as increasing one's knowledge of howthey learn. The dnll- and - practice materials,with some provision for individual needs, resultin at least equivalent achievement in less timethan it would take the classroom teacher usingconventional methods.

how necessary is the use of concretematerials?

Materials have generally been categorizedinto three types. concrete, or manipulative;semiconcrete, or pictorial, and abstract, orsymbolic. Much research has focused on as-certaining the role of each of these, the threestudies cited here illustrate three approachesto the question.

Ekman (1967) compared the effectiveness ofthree ways of presenting addition and sub-traction ideas to children in third grade. (1)presenting the ideas immediately in algorithmform, (2) developing the ideas with pictures be-fore presenting the algorithms, or (3) devel-oping the ideas with cardboard disks, manipu-lated by the pupils, before presenting thealgorithms. On the "understanding and "trans-fer" scales, there was some evidence that thethird group performed better. On the "skill"scale. both the second and the third groups,each-using some form of materials, performedbetter immediately after learning, although nosignificant differences were noted in the threegroups by the end of the retention period.

In a study with more precision. Fennema(1972) investigated the relative effectiveness ofa meaningful concrete and a meaningful sym-bolic model in learning a mathematical prin-ciple at the second-grade level. Learning was

measured (1) by tests of recall, in which prob-lems were stated in the symbols used duringinstruction; (2) by two symbolic transfer tests,which included problems that were untaughtsymbolic instances of the principle and onwhich pupils could use either the model theyhad learned or familiar concrete aids, and (3)by a concrete transfer test, which measured theability to demonstrate the principle on an unfa-miliar concrete device. She found no significantdifferences on any of the tests, the childrenwere able to learn a mathematical principle byusing either a concrete or a symbolic modelwhen that model was related to knowledge thechildren had achieved. Making the teachingmeaningful appeared to be as important as thematerials used.

Knaupp (1971) studied two modes of instruc-tion and two manipulative- models for present-ing addition and subtraction algorithms and theideas of base and place value to second-gradeclasses He found that both teacher-demon-stration and student-manipulation modes witheither blocks or sticks resulted in significantgains in achievement.

what types of manipulative materials havebeen found to be effective?

Earhart (1964) used an abacus to teach first-,second-, and third-grade children whoseteachers received in-service help, other groupsreceived instruction without the use of an aba-cus. On tests of reasoning no significant differ-ences were noted. but on tests of fundamen-tals. the group using the abacus performedsignificantly better. It is difficult, however, to tellwhether the abacus or the in-service help wasthe basis for this difference.

In a study by Hershman, Wells, and Payne(1962), first-grade children were taught for oneyear with programs of varying content, using ei-ther (1) a collection of inexpensive commercialmaterials, (2) a set of expensive commercialmaterials, or (3) materials provided by theteacher Teachers in the first two groups re-ceived in-service training in the use of the ma-terials. When significant differences in achieve-ment were observed, they were always in favorof the third program It was concluded that highexpenditure for manipulative materials does


not seem justified and that different materialsshould perhaps be used with different IQgroups.

Weber (1970) studied the effect of the rein-forcement of mathematics concepts with firstgraders. She used (1) paper-and-pencil follow-up activities or (2) manipulative and concretematerials for follow-up activities, No significantdifferences were noted on a standardized test,however, a definite trend favored the groupsusing manipulative materials, and these groupsscored significantly higher on an oral test of un-derstanding.

In another investigation in grade 1, Lucas(1967) studied the effects of using attnbuteblocks, which are varied in shape, color, andsize. He found that the children using theblocks showed a greater ability _both to con-serve number and to conceptualize addition-subtraction relations than those children notusing them.

Much research has been focused on the ef-fectiveness of the Cuisenaire materials. Inthese studies, the materials were used as theentire program instead of being used in con-junction with some other program. They servedas a manipulative material, and it is not alwaysclear whether students in the comparisongroups used any manipulative materials.

In studying a group of first graders using theCuisenaire program, Crowder (1966) reportedthat (1) the children learned more e,onveretionalsubject matter and more. -cepts and sk,i`, t',an I, este t ,

tional program, Qi ;1,i rage t i Int),pupils profited most from the Cuisenaire pro-gram, and (3) sex was not a significant factor inrelation to achievement, but socioeconomicstatus was. Working with first and second grad-ers, Hollis (1965) also compared the use of aCuisenaire program with a conventional ap-proach. He concluded (1) that children learnedconventional subject matter as well with theCuisenaire program as they did with the con-ventional program and (2) that pupils taught bythe Cuisenaire program acquired additionaiconcepts and skills beyond the ones taught inthe conventional program.

Brownell (1968) used tests and extensive in-terviews to analyze the effect of three mathe-matical programs on the underlying thought

54 CHAPTER THREE

processes of British children who had studiedthose programs for three years. He concludedthat (1) in Scotland, the Cuisenaire programwas in general much more effective than theconventional program in developing mean-ingful mathematical abstractions and that (2) inEngland, the conventional program had thehighest overall ranking for effectiveness in pro-moting conceptual maturity, with the Dienesand Cuisenaire programs ranked about equalto each other. Brownell inferred that the qualityof teaching was decisive in determining the rel-ative effectiveness of the programs.

Other studies have been concerned with theeffect of using Cuisenaire materials on a partic-ular topic for shorter periods of lime. Lucow(1964) and Haynes (1964) studied the use ofthese materials in teaching multiplication anddivision concepts for six weeks in third grade.Lucow attempted to take into account the effectof prior work in grades 1 and 2. He concludedthat the Cuisenaire materials were as effectiveas regular instruction. Haynes used pupas whowere unfamiliar with the materials, he found nosignificant differences in achievement betweenpupils who used the Cuisenaire materials andthose who did not.

Prior background, length of time. and thespecific topic may account for differences inthe success of the Cuisenaire materials. It hasbeen suggested that the program might bemore effective in grades 1 and 2. with its effec-tiveness dissipating during third grade. At anyrate. the findings of these studies tend to rein-force the findings of other studies that the useof manipulative materials aids primary-gradechildren in developing mathematical ideas.

are mathematics programs that aredesigned for specific groups successful?

Much attention has been directed in recentyears to the use of special programs for the en-.vironmentally disadvantaged. It is not at all sur-prising to find studies reporting that programsdesigned to provide special treatments andemphases for such pupils result in higherachievement when compared with "regular"programs that include no special provisions forsuch pupils.

A mathematics program designed for Mexi-can-American children in first grade capitalizedon the intrinsic motivation of success by takinginto accoun, their need and desire to progressfrom their low levels of understanding andknowledge (Castaneda 1968). The group usingthis program showed greater gains than agroup using a program with no such specialprovisions. In another study with Spanish- andEnglish-speaking children. Trevino (1968)found that among first-grade children tested inarithmetic fundamentals and among third-grade children tested in arithmetic reasoning.those taught bilingually did significantly betterthan those taught exclusively in English.

In pilot project undertaken to evaluate theuse of School Mathematics Study. Group(SMSG) materials by disadvantaged pupils inkindergarten and grade 1. Liederman. Chinn.and Dunk ley (1966) reported wide variability inachievement. The variability was great withinany given class and between classes. Similarvariability is well documented for all groups ofchildren, a persistent reminder of the fact of in-dividual differences.

Heitzman (1970) used extrinsic motivation, inthe form of plastic tokens that were exchange-able for toys and candy, to reward skills-learn-ing responses of a group of primary-grade pu-pils from migrant families. These childrenachieved significantly higher scores on a skillstest than a group not receiving the tokens.

Watching the tele,. .ion program SesameStreet had a measurable effect on theachievement of kindergarten children in recog-nizing and using numerical symbols and ontheir knowledge of geometric form (Carrico1971).

Although most Head Start programs werenot by intention academically oriented, somestudies attempted to measure the effects ofsuch programs on later achievement. For in-stance, Mackey (1969) reported that childrenwho participated in a Head Start program gen-erally scored significantly higher on arithmetictests at the end of first grade than qualified pu-pils who did not participate in Head Start. Ema-nuel (1971) found no differences for first-gradechildren, but those with Head Start experiencehad higher achievement scores in grade 2 andhigher marks in both grades 2 and 3.

r

what is the role of vocabulary andlanguage?

The arithmetic vocabulary used by kinder-garten children was analyzed by Kolson (1963).He identified 229 words, which represented 6percent of the total number of different wordsused by the children. Quantitative vocabularyaccounted for 70 percent of the total arithmeticvocabulary. Johnston (1964) analyzed the vo-cabulary of four- and five-year-olds and foundthat number words comprised. approximately50 percent of all the mathematical vocabulary;measurement words, 10 percent; positionwords, 30 percent, and form words, less than10 percent. No relationship was found betweenaptitude and mathematical vocabulary use.

Smith and Heddens (1964), in a report on thereadability of mathematical materials used inthe elementary school, reported that averagereading levels tended to be considerablyhigher than the assigned grade level of the ma-terials. particularly at grade 1.

In another type of study comparing the vo-cabularies of reading and mathematics text-books. Stevenson (1971) identified 396 mathe-matical words that were used in third-grademathematics textbooks. Of these, 161 werecommon to all the mathematics books, but only51 were used In both the reziding and the math-ematics textbooks studied. Willmon (197found that the 473 words she listed fromtwenty-four textbooks used in grades 1 through3 were generally used less than twenty-fivetimes. It seems evident that teachers mustspend some time teaching the vocabulary usedin textbooks.

Beilin and Gillman (1967) studied the rela-tionship between a child's number-languagestatus and his ability to deal with problem-solv-ing tasks in grade 1. Number language wasfound to be related to the cardinal-ordinalnumber task Knight (1971) reported that usingsubculturally appropriate language to teach aunit on nonmetric geometry enabled pupils toperform more successfully than those taughtand assessed by means of standard language.Teachers should consider the needs of chil-dren even .to the point of thinking carefully


about the words they use when talking aboutmathematical ideas.

content

how should operations with wholenumbers be conceptualized?

Research pertaining to this question has fo-cused principally on the operations of sub-traction. multiplication, and division. Althoughinstruction on addition is vital, its conceptual-ization has not concerned researchers to theextent that the conceptualization of the otheroperations has. Addition appears to be theeasiest operation for most children. especiallysince it is so readily conceptualized as a"counting on" process and since children en-counter so many situations in their daily lives inwhich they combine groups of objects.

Subtraction. Gibb (1956) explored ways inwhich pupils think as they attempt to solve sub-traction problems. In interviews with thirty-sixchildren in second grade, she found that pupilsdid best on "take away" problems and pooreston "comparative" problems. For instance,when the question was 'How many are left?'the problem was easier than when a was Howmany more does Tom have than Jeff?' Addi-tive" problems. in which the question might beHow many more dues he need?" were of me-

dium difficulty took more time. She re-ported that the k-hiikiren solved the problems interms of each situation rather than conceivingthat one basic idea appeared in all appli-cations.

This same relative order of difficulty amongsubtraction situations was observed by Schelland Burns (1962) for twenty-three pupils in sec-ond grade. However, performance levels on thethree types of subtraction problems were notsignificantly different (in a statistical sense),even though the pupils themselves considered"take away" problems to be the easiest.

Coxford (1966) and Osborne (1967) foundthat an approach using set partitioning, withemphasis on the relationship between additionand subtraction, resulted in greater under-standing than the "take away" approach. It is

56 CHAPTER THREE

important that those who want to develop set-subset concepts as a strand in the curriculumconsider this finding.

Multiplication. Traditbrally the multiplicationof whole numbers has been conceptualized forchildren in terms of the addition of equal ad-dends. For instance, "4 X 7" has been inter-preted to mean "7 + 7 + 7 + 7." But logical dif-ficulties are inherent in this interpretation whenthe first factor in a multiplication example is 0or 1.

Some recent research has investigated thefeasibility of using other conceptualizations ofmultiplication. Hervey (1966) reported that sec-ond-grade pupils had significantly greater suc-cess in conceptualizing, visually representing,and solving equal-addend problems than Car-tesian-product problems. Cartesian-productproblems were conceptualized and solvedmore often by high achievers than by lowachievers, more often by boys than by girls,and more often by pupils with above-averageintelligence than by pupils with below-averageintelligence. Hervey was not able to determinethe extent to which her findings might have

-been-influenced by the nature of prior instruc--tion or by differences inherent in the mathe-matical nature of the two conceptualizations.

Another conceptualization of multiplicationmay be associated with rectangular arraysei-ther independent of. or in conjunction with,Cartesian products. At the third-grade levelSchell (1964) investigated the achievement ofpupils who used only array representations fortheir introductory work with multiplication ascompared with pupils who used a variety ofrepresentations. He found n. conclusive evi-dence of a difference in achievement levels.

Division Two kinds of problem situations fordivision need to be distinguished:

1. Measurement problemsFind the num-ber of equivalent subsets.

Example: If each boy is to receive 3 ap-ples, how many boys can share 12 ap-ples?

2. Partition problemsFind the number ofelements in each equivalent subset.

Example: If there are 4 boys to share 12apples equally, how many will each boyreceive?

Each boy begins by taking an apple. Thisis repeated until all the apples aretaken.

Zweng (1964) found that partition problemswere significantly more difficult for secondgraders than measurement problems. She fur-ther reported that problems-in which two setsof tangible objects were specified (as in thepreceding problems) were easier than thoseproblems in which only one set of tangible ob-jects was specified.

Generally speaking, the conceptualization ofan operation is closely associated with real-world problem situations to which that opera-tion may be applied. Since the same operationmay be applied to problem situations that differin significant respects, a single interpretationmay not be adequate for children until they areable to conceptualize in less particular ways.

is learning facilitated by introducing"inversely" related operations oralgorithms at the same time?

Considering how frequently this questionarises (especially in relation to the Piagetian

concept of reversibility), it is somewhat surpris-Mg to find that little recent research has beendone on it.

Spencer (1968) reported that some intertaskinterference may occur when the addition andsubtraction operations are introduced more orless simultaneously but that emphasis on therelationship between the operations facilitatesunderstanding.

Wiles. Romberg, and Moser (1972) investi-gated both a sequential and an integrated ap-proach to the introduction of two currently usedalgorithms for addition and subtraction exam-ples that involve renaming:

37 = 30 + 7+16-10+ 6

40 + 13 =50 + 3 = 53

53 - 50 +3 = 40. 13-16 - - (10 + 6) - (10 4. 6)

30 + 7 = 37

There was no evidence to support any 4,/an-tage of an integrated approach (introducing thetwo algorithms more or less simultaneously)over a sequential approach (introducing firstthe addition algorithm, then the subtraction al-gorith m).

Research pertaining to the (more or less) si-multaneous introduction of multiplication anddivision is lacking.

what procedures are effective indeveloping basic number facts?

Basic number facts (e.g., 6 + 3 -= 9, 11 4 =7, 5 Y 4 - 20, 42 -: 6 7) are important fortwo quite different reasons: (1) they providesimple. "small number" contexts for the devel-opment of mathematical ideas pertaining to anoperation, and (2) pupils' memorization and re-call of such facts are essential to efficient com-putational skill with larger numbers.

Teachers know that children pass through aseries of stages from counting to automatic re-call in their work with basic number facts. Pu-pils use various ways to obtain answers to com-binationsguessing, counting, and solving


from known combinations, as well as immedi-ate recall. Brownell has stated that 'children at-tain 'mastery' only after a period during whichthey deal with combinations by procedures lessadvanced (but to them more meaningful) thanautomatic responses" (1941, p. 96).

We know from the research of some yearsago that drill per se is not effective in devel-oping mathematical concepts. Programsstressing relationships and generalizationsamong the combinations have been found tobe preferable for developing understandingand the ability to transfer (see, for example,Thiele [19381; Anderson [19491; Swenson[19491).

Brownell and Chazal (1935) summarizedtheir research work with third graders by con-cluding that drill must be preceded by mean-ingful instruction. The type of thinking that isdeveloped and the child's facility with the proc-ess of thinking are of greater importance thanmere recall. Drill in itself makes little contribu-tion to growth in quantitative thinking, since itfails to supply more mature ways of dealingWith numbers. Pincus (1956) also found thatwhether drill did or did not incorporate an em-phasis on relationships was not significant,when the drill followed meaningful mstruction.

Another procedure was checked by Full-erton (1955). He compared two methods ofteaching "easy" multiplication facts to third'graders. (1) an inductive method by which pu-pils developed multiplication facts from wordproblems, using a variety of procedures: and(2) a "conventional" method that presentedmultiplication facts to pupils without involvingthem in the development of such facts. In thisinstance a significant difference in favor of theinductive method was found on a measure ofimmediate recall of taught facts as well as onmeasures of transfer and retention.

Teachers know that the number of specificbasic facts to be memorized is reduced sub-stantially if pupils are able to apply the proper-ties of an operation. This is illustrated by thefollowing examples'

1. 3 ), 5 15; so 5 ), 3 15 (commutativeproperty of multiplication)

2. 8 Al 8; so 1 A 8- 8 (identity propertyof multiplication)

58 CHAPTER THREE

3. 7 t 0 0, so 0 7 0 (zero property ofmultiplication)

4. 8 >, 6 8 (5 t. 1) (8 5) t (8 y 1)(distributive property of multiplicationover addition)

5. 8 3 (2 4) 3 2 >. (4 3) (asso-ciative property of multiplication)

Little research attention has been directedexplicitly toward the use of properties. Hall's(1967) research on teaching selected multi-plication facts to third-grade pupils appears tosupport an emphasis on the use of the commu-tative property.

In an investigation with third-grade pupilsand their beginning work with multiplication,Gray (1965) found that an emphasis on distrib-utivity led to "superior" results when comparedwith an approach that did not include work withthis property. The superiority was statisticallysignificant on three of four measures: a posttestof transfer ability, a retention test of multi-plication achievement, and a retention test oftransfer. On the remaining measure. a posttestof multiplication achievement, children whohad worked with distributivity scored higherthan those who had not, but the difference w.c-not statis: cal'/ significant

Gra, s findings t i iarti,y,body of evidence on aspected from instruction that empl, ,

emahcal meaning and understanding. Theseadvantages may not always be particularly evi-dent in terms of skills-achievement immedi-ately following instruction: rather, the payoff ismuch more clearly evident in relation to factorssuch as comprehension, transfer, and reten-tion.

ex-

what do we know about the relativedifficulty of basic facts?

At one time. especially when stimulus-re-sponse theories of learning were prevalent,there was great interest in ascertair..ng the rel-ative difficulty of the basic number facts orcombinations. Textbook writers as well asclassroom teachers used the results of such re-search to determine the order in which fats

would be presented for mastery. The assump-tion was that if the combinations were se-quenced appropriately, the time needed tomemorize them could be reduced.

Although there was variability in these stud-ies on relative difficulty, some generalizationscan be drawn:

1. Subtraction combinations are more difficultfor children to learn than addition com-binations.

2. An addition combination and the com-bination with its addends reversed tend tobe of comparable difficulty.

3. The size of the addends the principal in-dicator of difficulty, rather than the size ofthe sum.

4. Combinations with a common addend ap-pear to be similarbut not equaldiffi-culty.

5. The "doubles" and those combinations inwhich 1 is added to a number appear to beleast difficult in addition, whereas thosecombinations with differences of 1 or 2 areeasiest in subtraction.

6 Multplicatioti k ofTlbimitiotis involving 0present tinlk_;o0. (1!tfli_iptle

1 he -.tie 4. product is positively corre-lated with

Recently. Suppes (1967) used the data-gath-ering potential of the computer to explore therelative difficulty of mathematical examples, in-cluding the basic facts. A drill-and-practiceprogram served as the vehicle to determine asuggested order of presentation and a sug-gested amount of practice.

In one aspect of his study on the strategiesand processes used by pupils as they learnmultiplication, Jerman (1970) reported data onthe difficulty level of multiplication com-binations. He found support for the findings ofBrownell and Carper (1943) that children douse different strategies for different com-binations and that the strategy used may be afunction of the combination itself. Certain com-binations, such as n x 0. n x 5, n n, n (n1), and n (n I 1), appear to be easier to re-tain.

.111MIMMI AN= mt Mi 1=1,1MMIll NMI =111111111111PROMM

A number of years ago, Swenson (1944)questioned whether results on the relative diffi-culty of factsresults that are obtained underrepetitive dr,II-oriented methods of learningare valid when applied to learning situationsthat are not so definitely drill oriented. Whensecond-lrade children were taught by drill, bygeneralization, and by a combined method, itwas found that the order of difficulty s.emed tobeat least in parta function of the eaci)ingmethod. Thus research that aims at estab-lishing the difficulty of arithmetic skills andprocesses should probably do so by means ofa clearly defined teaching and learningmethod.

what is the role of mathematicalsentences?

Within the past few years. much attentionhas been given to open sentences, such as 0

7 11, 9 -0 3,6 0 24,0 2 14.Some research on the relative difficulty of suchopen sentences is just now beginning to ap-pear. Within the context of basic facts havingsums between 10 and 18, sentences of the form

b c or c 0 - b are significantly moredifficult than sentences of the form 0 + b cor c 0 b (Weaver 1971). We may expectmore definitive research along these lines.Weaver also found that the position of theplaceholder in the sentence affected the diffi-culty of the sentences. That is. sentences of theform a s 0 c were less difficult than sen-tences like 0 x b c.

Grouws (1972) studied sentences of the formNtb c,a+N c, N - b c, and a - N c.He found that sentences using basic facts withsums between 10 and 18 were significantly eas-ier than similar open sentences using addendsand sums between 20 and 100. Thus the diffi-culty appears to be a function not only of theform of the sentence but also of the magnitudeof the numbers.

Engle and Lerch (1971) surveyed a group offirst graders who had had no instruction onclosed number sentences (that is, sentences ofthe form 3 4 7, to which the children re-spond "true" or "false"). They found that thepupils could respond at about the same level ofcorrectness to closed number sentences as


they could to exercises of the computationaltype. It seems logical to conclude that a varietyof forms of mathematical sentences and inher-ent interrelationships should be emphasized.

what algorithms are most effective?

In order to compute efficiently with largernumbers, algorithms are essential. When ap-plied to whole numbers, research on thesecomputational forms and procedures has fo-cused almost exclusively on subtraction and di-vision.

Over the years, researchers have been veryconcerned with procedures for teaching sub-traction that involves renaming (once com-monly called "borrowing"). The question ofmost concern has been whether to teach sub-traction by equal additions or by decomposi-tion. Consider the example

91

-24

67

With the decomposition algorithm, the sub-traction is done this way:

11 - 4 = 7 (ones); 8 - 2 6 (tens)

With the equal additions algorithm, it is done asfollows:

11 - 4 7 (ones); 9 - 3 = 6 (tens)

In a classic study, Brownell and Moser (1949)investigated the comparative merits of thesetwo algorithmsdecomposition and equal ad-ditionsin combination with two methods of in-structionmeaningful (or rational) and me-chanical:

meaningful mechanical

decomposition a b

equal additions c d

They found that at the time of initial instruc-tion-

1. meaningful decomposition [a j was betterthan mechanical decomposition [1:1 I onmeasures of understanding and accu-racy;

60 CHAPTER THREE

2 meaningful equal additions jc I was signif-icantly better than mechanical equal ad-ditions [d I on measures of understand-ing;

3. mechanical decomposition [b J was not aseffective as either equal addition_, proce-dure [c or d I;

4. meaningful decomposition [a I was supe-rior to each equal additions procedure [cand d) on measures of understandingand accuracy.

It was concluded that whether to teach theequal additions or the decomposition algorithmdepends on the desired outcome.

In recent years, the decomposition proce-dure has been used almost exclusively in theUnited States, since it was considered easier ioexplain in a meaningful way. However, somequestion has recently been raised about thismethod. With the increased emphasis in manyprograms on properties and on compensationin particular, the equal additions method canalso be presented with meaning. For instance,pupils are learning that, in essence,

a - b 0 means that 0 + b a,

or b I+ 0 a.

They are also learning that, in essence,

a b (a , k) - (b ; k).

The development of such ideas should facili-tate the teaching of the equal additions proce-dure. Whether there will again be a shift towarda wider use of this procedure remains to beseen. Evidence from other studies indicatesthat its use leads to greater accuracy andspeed.

Ekman (1967) reported that when third grad-ers manipulated materials before an additionalgorithm was presented, both understandingand ability to transfer increased. Using materi-als was better than either using pictures alonebefore introducing the algorithm or developingthe algorithm with neither aid.

Trafton (1971) found that third-grade chil-dren who had experienced a "general" ap-proach based on the main concepts of sub-traction and using the number line as an aid tosolution before working with the decompositionalgorithm had no greater understanding of, or

performance with, the decomposition agorithmthan children who had had no such "general"approach but instead had had a prolonged de-velopment of the algorithm.

Variants of two algorithms for division havebeen used in the schools:

"Distributive" algorithm "Subtractive" algorithm

3 1

29

87 3 [87-6 30 10

27 5727 30 10

2721 7

66 2

29

In one investigation comparing the use of theconventional (or distributive) and the sub-tractive forms, Van Engen and Gibb (1956) re-ported that there were some advantages foreach. They evaluated pupil achievement interms of understanding the process of division,transfer of learning, retention, and problem-solving achievement. Among their conclusionswere these:

1. Children taught the subtractive methodhad a better understanding of the proc-ess or idea of division than childrentaught with the conventional method. Theuse of this algorithm was especially effec-tive for children with low abilitythosewith high ability used the two methodswith equivalent effectiveness.

2. Children taught the conventional (distrib-utive) method achieved higher problem-solving scores (for the type of problemused in the study).

3. The subtractive method was more effec-tive in enabling children to transfer to un-familiar liut similar situations.

4. The two procedures appeared to beequally effective on measures of the re-tention of skill and understanding. The re-tention factor seems to be related more toteaching procedures than to the methodof division.

5 Children who used the distributive al-gorithm had greater success with parti-tion situations; those who used the sub-tractive algorithm had greater successwith measurement situations.

Scott (1963) used the subtractive algorithmfor measurement situations and the distributivealgorithm for partition situations. He suggestedthat (1) the use of the two algorithms was nottoo difficult for third-grade children; (2) two al-gorithms demanded no more teaching timethan only one algorithm; and (3) children taughtboth algorithms had a greater understanding ofdivision.

should children check their computations?

Children are encouraged to check their workbecause teachers believe that checking con-tributes to greater accuracy. Although there issome research evidence to support this belief,Grossnickle (1938) reported data that are con-trary to this idea. He analyzed the work of 174third-grade children who used addition tocheck subtraction answers. He found that pu-pils frequently "forced the check," that is. madethe sums agree without actually adding. Inmany cases. the checking was perfunctoryGenerally. only a chance difference existed be-tween the mean accuracy of the group of pupilswhen they checked and their mean accuracywhen they did not check.

What does this indicate to teachers? Obvi-ously, children must understand the purpose ofchecking and must know what to do if the solu-tion in the check does not agree wilt, 'he origi-nal solution.

what can young children learn aboutfractions?

Surveys of what children know about mathe-matics on entering school indicate that many ofthem can recognize halves, fourths, and thirdsand have acquired some facility in using thesefractions. Gunderson and Gunderson (1957) in-terviewed twenty-two second-grade childrenfollowing their initial oxperience with a lessoncn fractional parts of L. -.les. The irn, estigatorsconcluded that fractions could be introduced at


this grade level if manipulative materials wereused and if the oral work made no use of sym-bols.

A planned, systematic program for devel-oping fractional ideas seems essential as read-iness for work with symbols. The use of manip-ulative materials is vital in this preparation.

Sension (1971) reported that area, set-sub-set, and combination representations of in-troducing rational-number concepts appearedto be equally effective on tests using two typesof pictorial models with second-grade children.

what can children be taught aboutgeometry and measurement?

Little. research on geometry and measure-ment has been done with young children. Froma set of tests administered after two weeks ofteaching, Shah (1969) reported that childrenaged seven to eleven learned concepts asso-ciated with plane figures, symmetry, reflection,rotation, translation, bending and stretching,and networks.

Williford (1971) found that second- andthird-grade children can be taught lo performparticular transformational geometry skills(such as producing slide images, turn images,etc.) to a greater degree than they can betaught to apply such skills toward the solutionof more general problems.

Four- and five-year-olds exhibit wide differ-ences in their familiarity with ideas of time, lin-ear and liquid measures, and money, with littlemastery evident. In a survey with first graders,Mascho (1961) reported that as their age,socioeconomic level, or mental ability in-creased, the children's familiarity with meas-urement also increased. Familiarity was greaterwhen the terms were used in context. It wassuggested that (1) some ideas now consideredappropriate for first grade should be consid-ered part of the child's knowledge when he en-ters school and (2) teachers need to study thecomposition of their classes in terms of age,socioeconomic level, and mental ability whenplanning curricular activities with measure-ment.

62 CHAPTER THREE

miscellaneous content

Certain content areas (for example, proba-bility) are new to mathematics programs in theelementary school, and at present there is verylittle agreement on how much of such areascan, or should oe included. Most of the verylimited relevant research has been in the formof status or feasibility studies, which have var-ied widely in the particule concepts or sub-topics investigated within a given domain ofcontent.

Probability. Gipson (1972) developed a set oflessons for teaching children in grades 3 and 6the concepts of a finite sample space and theprobability of a simple event. She found thatpupils at each of these two grade levels could"learn" the two concepts that were taught, al-though some subordinate ideas were easierthan others for the children to grasp. McLeod(1971) observed that in grades 2 and 4, chil-dren who studied a unit on probabilityusingeither a "laboratory participation" approach ora "teacher demonstration" approachscoredno be) tter on immediate and delayed post-experimental measures than pupils who hadnot studied the instructional unit. This observa-tion was equally true for both boys and girls.

Logic. If a child is to learn to think critically, itis important that he make logically correct in-ferences, recognize fallacies, and identify in-consistencies among statements. Hill (1961)concluded that children aged six through eightare able to recognize valid conclusions derivedfrom sets of.given premises. There seems to bea "gradual, steady growth which is neat uni-form for all types of formal logic" (p. 3359). Dif-ferences in difficulty were associated with thetype of inference, but these difficulties werespecific to age. Difficulties associated with sexwere not significant. Children can learn to rec-ognize identical logical form in different con-tent, although the addition of negation very sig-nificantly increased the difficulty in recognizingvalidity.

O'Brien and Shapiro (1968) confirmed Hill'sfindings. except that little growth was observed

between ages seven and eight. Using a modifi-cation of Hill's test, they found that children ex-perienced great difficulty in testing the logicalnecessity of a conclusion and showed slowgrowth in this ability. The investigators sug-gested that Hill's research should be inter-preted and applied with caution. Hypothetical-deductive ability cannot be taken for granted inchildren of this age. This seems consistent withPiaget's theory on the attainment of formallogic.

In a later study, Shapiro and O'Brien (1970)reported that at all age levels children's abilityto recognize logical necessity was significantlyeasier than their ability to test for it.

McAloon (1969) used units in logic with third-and sixth-grade classes. These units includedsuch topics as sets, the relation of sets, state-ments; truth-falsity. class logic, such as themeaning of all, some, and none; disjunctionand conjunction, negations and implications;and basic ideas of validity and invalidity. Thegroup that was taught logiceither interwovenwith mathematics or-separate from mathemat-icsscored significantly higher on the mathe-matics achievement test and on reasoning teststhan the group that was not taught logic. Takingtime from mathematics did not decrease math-ematical achievement but in fact increased it.Symbolism and the principles of transitivity inboth class and conditional reasoning were easyfor both grades, although in general class rea-soning was easier than conditional reasoning.Fallacy principles and the principles of con-traposition ana converse were more difficult toteach in grade 3 than in grade 6. Items with acorrect answer of "maybe" were answeredleast well.

The identification of conjunctive concepts in-volves noting attributes common to positive in-stances. the identification of disjunctive con-cepts requires noting attributes nevercontained in negative instances. McCreary(1965), working with third graders, and Snowand Rabinovitch (1969). studying children atages five through thirteen, found that dis-junctive tasks were more difficult than con-junctive tasks. Weeks (1971) reported that theuse of attribute blocks had a strong positive ef-fect at grades 2 and 3 on both logical and per-ceptual reasoning ability.

what procedures are effective in teachingverbal problem solving?

Although problem solving may be viewedbroadly, its consideration here is restricted tothose situations commonly referred to as "wordproblems" or "story problems."

Most of the relevant research on verbalproblem solving has been done with childrenabove the third-grade level. This is under-standable in light of the highly related readingfactor, although this factor con be controlledwith younger children by giving them wordproblems orally:

Steffe (1970) investigated the relation be-tween the ability of first-grade pupils to makequantitative comparisons and their ability tosolve addition problems. He found that childrenwho failed on a test of quantitative comparisonsperformed significantly lower on a test of addi-tion problems. LeBlanc (1968) reported a sim-ilar finding with quantitative comparisons andsubtraction problems.

Somewhat conflicting findings were notedregarding whether or not children were able tosolve more easily those problems in which adescribed action was embedded in the prob-lem statement (Steffe 1970; Steffe and Johnson1971). Such studies have agreed, however, thatthe presence of manipulative objects facilitatedproblem solving. Other studies also agree withthis finding (see, for example, Bolduc [19701).

There are some verbal problem-solvingtechniques for which there is little explicit re-search evidence. even with older children.Common sense, however, says that these tech-niques may be applicable at many points in theproblem-solving prograrri. Among the tech-niques that researchers suggest are the follow-ing, whereby the teacher might-

1. provide a differentiated program, withproblems at appropriate levels of diffi-culty;

2. have pupils write the number- questionor mathematical sentence for a problem;

3. have pupils dramatize problem situ-ations and their solutions;

4. have pupils make drawings and c,a-grams and use them to solve problemsor to verify solutions to problems;


5. have pupils formulate problems forgiven conditions;

6. use problems without numbers;7. have pupils designate the process to be

used;

8. have pupils test the reasonableness oftheir answers;

9. have pupils work together to solve prob-lems;

10. encourage pupils to find alternate waysin which to solve problems.

concluding observations

The research cited in this chapter is not asdefinitive as one would like. Many questions re-main to be answered. However, the researchdoes give the teacher some broad guidelinesfor instructional practices and some broadboundary conditions within which to exploreand experiment informally. Teachers are en-couraged to use the research to focus explora-tion and experimentation in directions thathave potential for improving classroom instruc-tion and to avoid pathWays that have little or nopromise.

references

Almy. Millie. and Associates. Logical Thinking in SecondGrade. New York: Teachers College Press. 1970.

Almy. Millie. E. Chittenden. and P. Miller. Young Children'sThinking. New York. Teachers College Press. 1966.

Anderson. G. Lester. Quantitative Thinking As Developedunder Connectionist and Field Theories of Learning." InLearning Theory in School Situations. University of Min-nesota Studies in Education. no. 2. pp. 40 /3. Minneapo-lis: University of Minnesota Press. 1949.

Ashlock. Robert B.. and Ronald C. Welch."A Test of Under-standings of Selected Properties of a Number System."Indiana University School of Education Bulletin 42(March 1966): 1-74.

Bailin. Harry. and Irene S. Gillman."Number Language andNumber Reversal Learning." Journal of ExperimentalChild Psychology 5 (June 1967): 263-77.

64 CHAPTER THREE

Bjonerud, Corwin E "Arithmetic Concepts Possessed bythe Preschool Child Arithmetic Teacher 7 (November1960): 347-50.

Bolduc. Elroy J.. Jr. A Factorial Study of the Effects ofThree Variables on the Ability of First-Grade Children toSolve Arithmetic Addition Problems. (University of Ten-nessee. 1969.) Dissertation Abstracts International 30A(February 1970): 3358.

Bonney. Lewis A. Relationships between Content Experi-ence and the Development of Serration Skills in FirstGrade Children. (University of Arizona. 1970.) Dis-sertation Abstracts International 31A (November 1970).2167.

Brownell. William A. Arithmetic in Grades I and A CriticalSummary of New and Previously Reported Research.Duke University Research Studies in Education. no. 6.Durham. N.C.: Duke University Press. 1941.

. "Conceptual Maturity in Arithmetic under DifferingSystems of Instruction " Elementary School Journal 69(December 1968): 151-63.

. The Development of Children s Number Ideas in thePrimary Grades. Supplementary Education Mono-graphs, no 35 Chicago University of Chicago Press.1928.

. "Rate. Accuracy. and Process in Learning :' Journalof Educational Psychology 35 (September 1944). 321-37

Brownell, William A.. and Doris V. Carper. Learning theMultiplication Combinations. Duke University ResearchStudies in Education, no. 7. Durham. N.C.: Duke Univer-sity Press. 1943.

Brownell. William A . and Charlotte B. Chazal. The Effectsof Premature Drat in Third-Grade Arithmetic :' Journal ofEducational Research 29 (September 1935): 17-28:Supplement (November 1935): 1-4.

Brownell. William A . and Harold E Moser Meaningful vs.Mechanical Learning- A Study in Grade Ill SubtractionDuke University Research Studies in Education, no. 8.Durham, N.C.: Duke University Press. 1949.

Buckingham B R and Josephine MacLatchy. The Num-ber Abilities of Children When They Enter Grade One." InReport of the Society's Committee on Arithmetic.Twenty-ninth Yearbook of the National Society k theStudy of Education. pt 2. pp 473-524. Bloomingion..11..Public School Publishing Co.. 1930.

Caezza. John F "A Study of Teacher Experience. Knowl-edge of and Attitude toward Mathematics and the Rela-tionship of These Variables to Elementary School Pupils'Attitudes toward and Achievement in Mathematics."(Syracuse University. 1969.) Dissertation Abstracts Inter-national 31A (September 1970): 921-22.

Carr. Donna H The Development of Number Ccincept AsDefined by Piaget in Advantaged Children Exposed tothe Bereiter-Englemann Pre-school Materials and Train-ing." (University of Utah. 1970.) Dissertation Abstracts In-ternational 31A (February 1971): 3947-48.

Carrico. Mark A. An Assessment of the Children s Televi-sion Program 'Sesame Street' in Relation to the Attain-ment of the Program's Goals by Kindergarten Children inthe Sioux Falls. South Dakota Public Schools." (Univer-sity of South Dakota, 1971.) Dissertation Abstracts Intpr-national 32A (November 1971): 2297

Castaneda, Alberta M. M. The Differential Effectiveness ofTwo First Grade Mathematics Programs for Dis-advantaged Mexican-American Children." (The Univer-sity of Texas, 1967 ) Dissertation Abstracts 28A (April1968): 3878-79.

Cathcart, W George, and Werner Liedtke. Reflectiveness/Impulsiveness and Mathematics Achievement.' Arithme-tic Teacher 16 (November 1969): 563-67.

Coxford, Arthur F , Jr The Effects of Instruction on theStage Placement of Children in Piagers Serration Experi-ments "Arithmetic Teacher 11 (January 1964) :4 -9.

'The Effects of Two Instructional Approaches onthe Learning of Addition and Subtraction Concepts inGrade One (University of Michigan, 1965.) DissertationAbstracts 26 (May 1966): 6543-44.

Crosswhite, F Joe Correlates of Attitudes toward Mathe-matics. NLSMA Reports, no. 20, edited by James W. Wil-son and Edward G. Begle. Stanford, Calif.: School Math-ematics Study Group, 1972.

Crowder, Alex B., Jr. "A Comparative Study of Two Meth-ods of Teaching Arithmetic in the First Grade." (NorthTexas State University. 1965.) Dissertation Abstracts 26(January 1966): 3778.

Dawson. Dan T., and Arden K. Ruddell. The Case for theMeaning Theory in Teaching Arithmetic." ElementarySchool Journal 55 (March 1955): 393-99.

Deighan. William P. An Examination of the Relationshipbetween Teachers' Attitudes toward Arithmetic and theAttitudes of Their Students toward Arithmetic.' (CaseWestern Reserve University. 1970.) Dissertation Ab-stracts International 31A (January 1971): 3333.

Dodwell, P. C. "Children's Understanding of Number andRelated Concepts." Canadian Journal of Psychology 14(1960): 191-203.

Dunkley, M. E. "Sorre Number Concepts of DisadvantagedChildren: Arithmetic Teacher 12 (May 1965): 359-61.

Earhart. Eileen. Evaluating Certain Aspects of a New Ap-proach to Mathematics in the Primary Grades.' SchoolScience and Mathematics 64 (November 1964): 715-20.

Ekman, Lincoln G. "A Comparison of the Effectiveness ofDifferent Approaches to the Teaching of Addition andSubtraction Algorithms in the Third Grade. 2 vols. (Uni-versity of Minnesota. 1966.) Dissertation Abstracts 27A(February 1967): 2275-76.

Emanuel. Jane M. "The Intelligence. Achievement, andProgress Scores of Children Who Attended SummerHead Start Programs in 1967.1968. and 1969: (Univer-sity of Alabama. 1970.) Dissertation Abstracts Inter-national 31A (April 1971): 5031-32.

Engle. Carol D.. and Harold H. Lerch. "A Comparison ofFirst-Grade Children's Abilities on Two Types of Arith-metical Practice Exercises. School Science and Mathe-matics 71 (April 1971): 327-34.

Fennema, Elizabeth H. The Relative Effectiveness of aSymbolic and a Concrete Model in Learning a SelectedMathematical Principle. Journal for Research in Mathe-matics Education 3 (November 1972). 233-38.

Flournoy, Frances. "A Study of Pupils' Understanding ofArithmetic in the Primary Grades. Arithmetic Teacher 14(October 1967): 48185.

Fortson. Laura R. A Creative-Aesthetic Approach to Read-iness and Beginning Reading and Mathematics in theKindergarten. (University of Georgia. 1969.) DissertationAbstracts International 30A (June 1970): 5339.

Fullerton. Craig K. 'A Comparison of the Effectiveness ofTwo Prescribed Methods of Teaching Multiplication ofWhole Numbers (State University of Iowa. 1955) Dis-sertation Abstracts 15 (November 1955): 2126-27.

Gibb, E Glenadine "Children's Thinking in the Process ofSubtraction " Journal of Experimental Education 25(September 1956): 71-80.

Gipson. Joella H. Teaching Probability in the ElementarySchool' An Exploratory Study." (University of Illinois atUrbana-Champaign, 1971 ) Disserta:'m Abstracts In-ternational 32A (February 1972): 4325-26.

Gray. Roland F. An Experiment in the Teaching of In-troductory Multiplication Arithmetic Teacher 12 (March1965): 199-203.

Greenblatt. E L An Analysis of School Subject Prefer-ences of Elementary School Children of the MiddleGrades" Journal of Educational Research 55 (August1962): 554-60.

Grossnickle. Foster E The Effectiveness of Checking Sub-traction by Addition Elementary School Journal 38(February 1938): 436-41.

Grouws. Douglas A. "Open Sentences. Some InstructionalConsiderations from Research." Arithmetic Teacher 19(November 1972): 595-99.

Gunderson. Ethel. and Agnes C. Gunderson. 'FractionConcepts Held by Young Children."Arithmetic Teacher 4(October 1957): 168-73.

Haines, Leeman E 'The Effect of Kindergarten Experienceupon Academic Achievement in the Elementary SchoolGrades." (University of Connecticut. 1960.) DissertationAbstracts 21 (January 1961): 1816-17.

Hall. Kenneth D. An Experimental Study of Two Methodsof Instruction for Mastering Multiplication Facts at theThird-Grade Level." (Duke University. 1967.) DissertationAbstracts 28A (August 1967): 390-91.

Harshman. Hardwick W.. David W. Wells. and Joseph N,Payne. "Manipulative Materials and Arithmetic Achieve-ment in Grade 1. Arithmetic Teacher 9 (April 1962). 188-92.

Haynes. Jerry 0. "Cuisenaire Rods and the Teaching ofMultiplication to Third-Grade Children.' (Florida StateUniversity. 1963.) Dissertation Abstracts 24 (May 1964):4545.

Heimgartner. Norman L "Selected Mathematical Abilitiesof Beginning Kindergarten Children. (Colorad,.., StateCollege. 1968.) Dissertation Abstracts 29A (August1968): 406-7.

Heitzman, Andrew J. "Effects of a Token ReinforcementSystem on the Reading and Arithmetic Skills Learningsof Migrant Primary School Pupils." Journal of Education-al Research 63 (July/August 1970): 455-58.

Hervey. Margaret A. "Children s Responses to Two Typesof Multiplication Problems." Arithmetic Teacher 13 (April1966): 288-92.

Hill, Shirley A "A Study of the Logical Abilities of Children.'(Stanford University. 1961.) Dissertation Abstracts 21(May 1961): 3359.

Hollis Loye Y "A Study to Compare the Effect of TeachingFirst and Second Grade Mathematics by the Cuisenaire-Gattegno Method with a Traditional Method." School Sci-ence and Mathematics 65 (November 1965): 683-87.

11g. Frances and Louise B. Ames. "Developmental Trendsin Arithmetic Journal of Genetic Psychology 79 (Sep-tember 1951): 3-28.

RESEARCH ON MATHEMATICS LEARNING

. School Readiness. New York. Harper & Row. 1965.Jerman, Max. "Some Strategies for Solving Simple Multi-

plication Combinations." Journal for Research in Mathe-matics Education 1 (March 1970): 95-128.

Johnson. Roger T.. Jr. "A Comparison of Categorizing Abil-ity in High and Low Socioeconomic Kindergarteners."(University of California. Berkeley, 1969.) DissertationAbstracts International 31A (July 1970). 225.

Johnston, Betty B. K. An Analysis of the Mathematical Vo-cabulary of Four- and Five-Year-Old Children." (Univer-sity of California, Los Angeles. 1964.) Dissertation Ab-stracts 25 (December 1964): 3433-34.

Knaupp. Jonathan E. "A Study of Achievement and Attitudeof Second Grade Students Using Two Modes of Instruc-tion and Two Manipulative Models for the NumerationSystem," (University of Illinois at Urbana-Champaign,1970) Dissertation Abstracts International 31A (June1971): 6471.

Knight. Genevieve M. The Effect of a Sub-Culturally Ap-propriate Language upon Achievement in MathematicalContent." (University of Maryland, 1970.) DissertationAbstracts International 31B (June 1971). 7433-34.

Kolson, Clifford J. The Oral Arithmetic Vocabulary of Kin-dergarten Children." Arithmetic Teacher 10 (February1963): 81-83.

LeBlanc. John F. The Performances of First Grade Chil-dren in Four Levels of Conservation of Numerousnessand Three I.Q. Groups When Solving Arithmetic Sub-trai-tion Problems." (University of Wisconsin. 1968.) Dis-sertation Abstracts 29A (July 1968): 67.

Leiderman, Gloria F., William G. Chinn. and Mervyn E.Dunkley The Special Curriculum Project. Pilot Programon Mathematics Learning of Culturally DisadvantagedPrimary School Children. SMSG Reports, no. 2. Stan-ford, Calif.: School Mathematics Study Group. 1966.

Lucas, James S. The Effect of Attribute-Block Training onChildren's Development of Arithmetic Concepts. (Uni-versity of California. Berkeley. 1966.) Dissertation Ab-stracts 27A (February 1967): 2400-2401.

Lucow, William H. "An Experiment with the CuisenaireMethod in Grade Three." American Educational Re-search Journal 1 (May 1964): 159-67.

Mackey. Beryl F. "The Influence of a Summer Head StartProgram on the Achievement of First Grade Children.(East Texas State University. 1968.) Dissertation Ab-stracts 29A (April 1969): 3500-3501.

Mascho. George. "Familiarity with Measurement." Arithme-tic Teacher 8 (April 1961): 164-67.

Masek, Richard M. The Effects of Teacher Applied SocialReinforcement on Arithmetic Performance and Task-Ori-entation." (Utah State University, 1970.) Dissertation Ab-stracts International 30A (June 1970). 5345-46.

McAloon, Sister Mary deLourdes. 'An Exploratory Studyon Teaching Units in Logic to Grades Three and Six."(University of Michigan. 1969.) Dissertation Abstracts In-ternational 30 (November 1969). 1918-19.

McCreary, Joyce Y. B. "Factors Affecting the Attainment ofConjunctive and Disjunctive Concepts in Third GradeChildren. (University of Pittsburgh. 1963.) DissertationAbstracts 26 (September 1965). 1794,95.

McLeod, Gordon K. "An Experiment in the Teaching of Se-lected Concepts of Probability to Elementary SchoolChildren. (Stanford University. 1971.) Dissertation Ab-stracts International 32A (September 1971): 1359.

e

65

66 CHAPTER THREE

Montague David 0 "Arithmetic Concepts of KindergartenChildren in Contrasting Socio-Economic Areas." Ele-mentary School Journal 64 (April 1964): 393-97.

Nowak Stanley M "The Development and Analysis of theEffects of an Instructional Program Based on PiagetsTheory of Classification (State University of New York ofBuffalo. 1969 ) Dissertation Abstracts International 30A(November 1969). 1875.

O'Brien. Thomas C.. and Bernard J. Shapiro. "The Devel-opment of Logical Thinking in Children.' American Edu-cational Research Journal 5 (November 1968). 531 42.

Osborne. Alan R. The Effects of Two Instructional Ap-proaches on the Understanding of Subtraction by GradeTwo Pupils (University of Michigan. 1966.) DissertationAbstracts 28A (July 1967): 158.

Puget, Jean Now Children Form Mathematical Con-cepts "Scientific American 189 (1953). 74-79.

Pincus. Morris. An Investigation into the Effectiveness ofTwo Methods of Instruction in Addition and SubtractionFacts (New York University. 1956.) Dissertation Ab-stracts 16 (August 1956): 1415.

Rea. Robert E.. and Robert E. Reys. Mathematical Com-petencies of Entering Kindergarteners." ArithmeticTeacher 17 (January 1970): 65-74.

Rickard. Esther E. S. An Inventory of the Number Knowl-edge of Beginning First Grade School Children. Basedon the Performance of Selected Number Tasks." (In-diana University. 1964 ) Dissertation Abstracts 28A (Au-gust 1967) 406.

Roberts. Gerhard H. The Failure Strategies of Third GradeArithmetic Pupils." Arithmetic Teacher 15 (May 1968):442-46.

Robinson. Inez C. The Acquisition of Quantitative Con-cepts in Children. (University of Southern California.1967.) Dissertation Abstracts 28A (February 1968). 3038.

Sawa da. Daiyo. and L. Doyal Nelson. "Conservation ofLength and the Teaching of Linear Measurement. AMethodological Critique. Arithmetic Teacher 14 May1967): 345-48.

Schell. Leo M: "Two Aspects of Introductory Multiplication:The Array and the Distributive Property. (State Univer-sity of Iowa. 1964.) Dissertation Abstracts 25 (September1964): 1561-62.

Schell. Leo M., and Paul C. Burns. Pupil Performance withThree Types of Subtraction Situations. School Scienceand Mathematics 62 (March 1962). 208-14.

Scott, Lloyd. A Study of Teaching Division through the Useof Two Algorithms." School Science and Mathematics 63(December 1963). 739-52.

Schwartz. Anthony N. Assessment of Mathematical Con-cepts of Five-Year-Old Children "Journal of Experimen-tal Education 37 (Spring 1969): 67-74.

Sension. Donald B. A Comparison of Two ConceptualFrameworks for Teaching the Basic Concepts of RationalNumbers. (University of Minnesota. 1971.) DissertationAbstracts International 32A (November 1971): 2408.

Shah. Sair A "Selected Geometric Concepts Taught toChildren Ages Seven to Eleven Arithmetic Teacher 16(February 1969): 119-28.

Shapiro, Bernard J.. and Thomas C. O'Brien. "LogicalThinking in Children Ages Six through Thirteen ChildDevelopment 41 (September 1970): 82329.

Simpson. Robert E The Effects of Cognitive Styles andChronological Age in Achieving Conservation Concepts

of Substance, Weight, and Volume." (University of Ala-bama. 1970.) Dissertation Abstracts International 31A(January 1971): 3353.

Sinclair. Hermine. 'Plaget s Theory of Development. TheMain Stages." In Piagetian Cognitive-Development Re-search and Mathematics Education, edited by Myron F,Rosskopf. Leslie P. Steffe. and Stanley Taback, pp. 1-11.Washington. D.C.: National Council of Teachers of Math-ematics, 1971.

Smith. Kenneth J., and James W. Heddens. The Read-ability of Experimental Mathematics Materials. Arithme-tic Teacher 11 (October 1964): 391-94.

Snow. Catherine E.. and M. Sam Rabinovitch. Conjunctiveand Disjunctive Thinking in Children. Journal of Experi-mental Child Psychology 7 (February 1969): 1-9.

Spencer, James E. Antertask Interference in PrimaryArithmetic.' (University of California. Berkeley.) Dis-sertation Abstracts 28A (January 1968). 2570 71.

Steffe. Leslie P. The Performance of First Grade Childrenin Four Levels of Conservation of Numerousness andThree I.Q. Groups When Solving Arithmetic AdditionProblems. (University of Wisconsin. 1966.) DissertationAbstracts 28A (August 1967): 885-86.

"Differential Performance of First-Grade ChildrenWhen Solving Arithmetic Addition Problems. Journal forResearch in Mathematics Education 1 (May 1970). 144-61.

Steffe. Leslie P.. and David C. Johnson. Problem-solvingPerformances of First-Grade Children." Journal for Re-search in Mathematics Education 2 (January 1971). 50-64.

Stevenson. Erwin F. "An Analysis of the Technical andSemi-Technical Vocabulary Contained in Third GradeMathematics Textbooks and First and Second GraceReaders" (Indiana University. 1971.) Dissertation Ab-stracts International 32A (December 1971): 3012.

Stright. Virginia M "A Study of the Attitudes towardArithmetic of Students and Teachers in the Third. Fourth.and Sixth Grades." Arithmetic Teacher 7 (October 1960):280-86.

Suppes. Patrick "Some Theoretical Models for Mathemat-ics Learning Journal of Research and Development inEducation 1 (Fall 1967): 5-22.

Suppes. Patrick. and Mona Morningstar. Evaluation ofThree Computer-Assisted Instruction Programs. Techni-cal Report no. 142, Psychology Series. Stanford. Calif..Institute for Mathematical Studies in the Social Sciences.Stanford University. 1969.

Swenson. Esther J. "Difficulty Ratings of Addition Facts AsRelated to Learning Method." Journal of Educational Re-search 38 (October 1944): 81-85.

"Organization and Generalization as Factors inLearning. Transfer. and Retroactive Inhibition. In Learn-ing Theory in School Situations. University of MinnesotaStudies in Education. no. 2. pp. 9-39. Minneapolis. Uni-versity of Minnesota Press. 1949.

Thiele. C. Louis. Contribution of Generalization to theLearning of the Addition Facts. Contributions to Educa-tion, no. 673. New York: Bureau of Publications. Teach-ers College. Columbia University. 1938.

Thompson. John W.. Jr. 'A Measurement of Achievementin Modern Mathematics in the Primary Grades.' (NorthTexas State University. 1968.) Dissertation Abstracts 29B(April 1969): 3839.

Trafton, Paul R "The Effects of Two Initial Instructional Se-quences on the Learring of the Subtraction Algorithm inGrade Three." (University of Michigan, 1970.) .Dis-sertation Abstracts International 31A (February 1971).4049-50.

Trevino, Bertha A G "An Analysis of the Effectiveness of aBilingual Program in the Teaching of Mathematics in thePrimary Grades." (University of Texas, Austin. 1968.) Dis-sertation Abstracts 29A (August 1968): 521-22.

Unkel Esther R "A Study of the Interaction of Socioeco-nomic Groups and Sex Factors with the Disciepancy be-tween Anticipated Achievement and Actual Achievementin Elementary School Mathematics." (Syracuse Univer-sity. 1965 ) Dissertation Abstracts 27A (July 1966): 59.

Van Engen, Henry, and E Glenadine Gibb General MentalFunctions Associated with Division Educational ServicesStudies, no 2 Cedar Falls, Iowa Iowa State TeachersCollege. 1956.

Weaver J Fred Some Factors Associated with PupilsPerformance Levels on Simple Open Addition and Sub-traction Sentences Arithmetic Teacher 18 (November1971): 513-19.

Weaver J Fred. and Marilyn N. Suydam. Meaningful In-struction in Mathematics Education. Columbus, Ohio:ERIC Center for Science. Mathematics. and Environmen-tal Education, 1972.

Weber. Audra W. "Introducing Mathematics to First GradeChildren' Manipulative vs. Paper and Pencil." (Universityof California. Berkeley 1969) Dissertation Abstracts In-ternational 30A (February 1970): 3372-73.

Weeks Gerald M "The Effect of Attribute Block Training onSecond and Third Graders' Logical and Perceptual Rea-


soning Abilities. (University of Georgia, 1970.) Dis-sertation Abstracts International 31A (May 1971). 5681-82.

Wess, Roger G. "An Analysis of the Relationship of Teach-ers' Attitudes as Compared to Pupils' Attitudes andAchievement in Mathematics." (University of South Da-kota. 1969.) Dissertation Abstracts International 30A(March 1970): 3844-45.

Wheatley, Grayson H., Jr. "Conservation. Counting, andCardination as Factors in Mathematics Achievementamong First-Grade Students.' (University of Delaware,1967.) Dissertation Abstracts 29A (November 1968).1481-82.

Wiles, Clyde A.. Thomas A. Romberg, and James M. Mo-ser The Relative Effectiveness of Two Different Instruc-tional Sequences Designed to Teach the Addition andSubtraction Algorithms. Technical Report no. 222. Madi-son, Wis.: Wisconsin Research and Development Centerfor Cognitive Learning, University of Wisconsin, June1972.

Williford, Harold J. A Study of Transformational GeometryInstruction in the Primary Grades." (University of Geor-gia. 1970.) Dissertation Abstracts International 31A (June1971): 6462.

Willmon, Betty. "Reading in the Content Area. A New MathTerminology List for the Primary Grades." ElementaryEnglish 48 (May 1971): 453-71.

Wozencraft, Marian. 'Are Boys Better wan Girls inArithmetic? Arithmetic Teacher 10 (Decerriber 1963).486-90.

Zweng, Marilyn J. Division Problems and the Concept ofRate. Arithmetic Teacher 11 (December 1964). 547-56.

6

problemsolving

doyal nelson

joan kirkpatrick

70'0

THE central purpose of any instruction in math-ematics at the early childhood level is to helpthe child see order and meaning in the situ-ations and events that occur in his day-to-dayactivities. Once the child learns how to relate asituation to its mathematical representation, hecan see new meaning in it and begins to exer-cise some control over it. In so doing, he learnsto change or modify it so that his needs andpurposes are better served.

Mathematics instruction should emphasizestrategies for relating events and situation.: totheir mathematical models. For example, wehelp a child to estimate numbers, distances.and capacities and to apply these estimatingabilities to other situations in his real world. Weteach him to develop criteria comparing twolengths and for ordering a series of objects ac-cording to length. We introduce him to certainaspects of the coordinate plane, assist him inmoving about on it, and then deliberately relatesuch moves to the trips he makes through hisown neighborhood; the regularities of thecoordinate system when applied to his realworld make the child's movements in that worldmore controlled and easier to modify.

In order to help the child see the relationshipbetween day-to-day events and their mathe-matical structure, we must somehow developfor him problems that are derived from real-world situations and events that make sense tohim. In such problems, a real situation occurs(or is created or simulated), and the child is re-quired to make some modifications or imposesome order on the situation, which can sooneror later be summarized in mathematical form.

Consider, for example, the following situ-ation An empty egg carton that holds twelveeggs is on a table. A child wants to fill the car-ton from a basket of eggs on the floor. He can,of course, fill the carton by putting eggs in itone at a time, thereby taking twelve trips to ac-complish the task. He could increase his effi-ciency by taking two eggs at a time. Or he couldfurther increase his efficiency by making whatmay be thought of as a nonmathematical modi-fication on the situationhe could move thecarton down beside the basket. In any event,the problem provides him with the opportunityof seeing that there is a definite relation be-tween the number of eggs the carton will hold,the number of eggs taken in each trip, and thenumber of trips required. Eventually, of course,the mathematical model for the situation will befully developed, and the child can use if to pre-dict precisely how many trips are associatedwith any particular number of eggs per trip for acarton of any reasonable size. However, oncehe has gained this measure of control over thesituation, it is no longer a problem for him.

It is not the purpose here to discuss at anygreat length the steps the child goes through insolving a problem. It is more important to ob-serve the behavior of the child in problem situ-ations and to make inferences from his actionsabout how he is thinking. Our task is to createand select significant problems for the child. Todo this, we must know the characteristics of agood problem.

some characteristicsof good problems

Perhaps some popular misconceptionsabout the nature of problem solving should firstbe dispelled It is quite common even today tohear someone claim that a child faced withpages of computation exercises is solvingproblems. This activity is not problem solving,for practice exercises do not establish any kindof relation.between a real situation and itsmathematical model.

Another common activity that goes under the

PROBLEM SOLVING 71

name of problem solving is the verbal descrip-tion of a situation for which the child is ex-pected to write a mathematical model. Con-sider the following example:

Jim had some shells. His sister gave him 4more. Now .Iim has 11 shells. How manyshells did Jim have at first?

The child's response to this verbal descriptionmay be something like this:

A

fin, had 7 shells at first:

In the early stages of thinking about the ver-bal descriptions and their mathematical struc-tures, the child may indeed be involved in aform of problem-solving activity, particularly ifthe described event can be simulated in a real-istic way. In the later stages, when he is re-quired merely to write down mathematical sen-tences that fit contrived verbal descriptions ofimagined events, his involvement in a problem-solving sense is likely to be minimal. This is notto say that associating the verbal description ofsuch an event, however contrived. with a math-ematical description of the event is undesir-ableit is an essential form of mathematicalactivitybut it is better defined as mathemati-cal modeling than as problem solving.

If the purpose of problem solving is to helpthe child learn to see a relationship between anevent and its mathematical model, what thenare the characteristics of a good problem de-signed for young children?

1 The problem should be of significancemathematically. The potential of a situ-ation as a vehicle for the development ofmathematical ideas determines whetherwe choose one particular problem situ-ation over another.

2. The situation in which the problem occursshould involve real objects or obvioussimulations of real objects. The problemmust be comprehensible to the child andeasily related to his world of reality.

72 CHAPTER FOUR

3. The problem situation should capture theinterest of the child. This is done by thenature of the materials, by the situation it-self, by the transformations the child canimpose on the materials, or by some com-bination of these factors.

4. The problem should require the childhimself to move, transform, or modifythe materials. It would be difficult to over-emphasize the role of action in earlychildhood learning. Most of the mathe-matical models at this level are what mightbe called "action models."

5. The problem should offer opportunitiesfor different levels of solution. If the prob-lem allows the child to move immediatelyfrom the problem situation to an ex-pression of its mathematical structure, it isnot a problem.

6. The problem situation should have manyphysical embodiments. Whatever situ-ation is chosen as the particular vehiclefor the problem, it shoule be possible tocreate other situations having the samemathematical structure. It may not bepossible for a child to generalize a solu-tion to a certain kind of problem until theproblem has come up in a variety of situ-ations.

7. The child should be convinced that he cansolve the problem, and he should knowwhen he has a solution for it. If the child issomehow required to react with, or trans-form, materials used in the problem situ-ation, it is usually easy to determinewhether the problem meets the criterionor not.

Two examples of good problems for youngchildren follow. They illustrate the character-istics listed above.

sample problem 1The process of division is usually a source of

difficulty for the young child. The difficultycould possibly be eased if at an early age thechild could be provided with problems in-volving actions that could be associated withthe division process. For example, the following

problem is designed to help the child see therelation between certain actions he must carryout :rt real-world situations and the process wecall division. It is apparent that the first charac-teristic of a good problemthat it have somemathematical significanceis present in divi-sion problems.

Let us examine the basic structure of a situ-ation related to division. Suppose we have fif-teen objects and we wish to know how manygroups of three objects each can be formedfrom them. We would regroup them as shownin .

* 61 *to 0* 0

*0 . 0*

, 4*

The action involved is usually a successivesubtraction of groups of three until all the ob-jects in the original group have been used up.The mathematical statement of what has hap-pened is usually written as follows:

15 ÷ 3 = 5

The 5 refers to the number of groups of three.Here, division is used in what is called a meas-urement situation.

But another type of situation also uses divi-sion. Consider, for example, the same fifteenobjects, but suppose this time the objects areto be placed in three boxes so that each boxcontains the same number of objects, as shownins.

0 V00

cV'

shows the situation, but in order to com-plete the action and answer the question, theobjects would have to be partitioned among the

boxes. This partitioning, or sharing, would con-tinue until all the objects were in the boxes. Thesharing action would ensure that each box con-tained the same number of objects. Again wewould write

15 3 - 5.The 3 in the mathematical statement refers tothe number of objects used up in each round ofthe sharing. The 5 refers to the number of ob-jects in each box. Here, division is used in apartitive situation.

The mathematical structure of such situ-ations is clear, and it is also evident that chil-dren often get into play situations that would in-volve these division actions. However, if wedesign problems for the child around such situ-ations, they must be more interesting thanthose that have appeared in our discussion sofar. Thus it remains for us to devise a situationthat would appeal to the child.

Suppose we collect fifteen toy cars and sim-ulate on a table model a river on which a ferrymoves back and forth. (See .) The child mightbe asked a problem like the following:

River

ferry must be used to get these carsacross the river. If it can carry only threecars at a time, how many trips must theferry make to get the cars across?

Even at the age of three, a child shows someunderstanding of the problem and is interestedin performing the actions that will get the carsacross the river. It is common, however, for achild this young to forget the requirement that

PROBLEM SOLVING 73

4

the ferry must carry three and only three cars ata time. He may load one car and ferry it across,and for the next trip he may pile on five cars, asthe three-year-old in the photograph is doing.Yet some measure of control over the situationis indicated by this kind of solution: the child al-most always uses the ferry and somehow getsthe cars across. But it is obvious that the three-year-old still has many facets of the problem tolearn.

In general, the four-year-old solves the prob-lem at a much more advanced level. He seldomforgets, for example, that the ferry must carryexactly three cars. However, he often becomesso involved in making the action completely au-thentic that he forgets the question he set out toanswer. For example, here the child has care-fully turned the ferry around and has driveneach car on over the rear of the boat. Likewise,the unloading is just as authentic and as closeto reality as he can make it.

J

74 CHAPTER FOUR

A five-year-old tends to give less emphasisto the action and will generally answer thequestion about the number of trips. This childhad to perform all the actions, but he could re-member in the end how many trips were re-quired to get the cars across.

By the time the child is six he may have lostinterest in going through all the motions. He*ends to give his full attention to what is re-quired in the problem. One six-year-old asked,"May I get the cars ready?" She proceeded togroup them in threes, and then said, "I wouldhave to make five trips." Her only action wasgrouping the cars; not one car was physicallyput on the ferry. This child had stripped the

-it, t

problem of all actions except those absolutelyessential to the solution. When a child hasreached this stage, the next step is to help himlearn how to write the mathematical model forthe situation. The timing of this last step is leftto the wisdom of the teacher.

With only slight modification, this problemcan be changed into a partitive rather than ameasurement situation. Partitive situations areoften more difficult for the child to understand.Although he may have had extensive experi-ence in sharing, he has not usually generalizedthe action.

In , the diagram and its problem suggest apartitive situation for division.

These five ferries can all be used at onceto move the cars across the river. Eachferry must carry the same number of cars.How many cars would be on each ferry?

The three- or four-year-old child has difficultyseeing how sharing would work in this problem.He can use only trial and error to equalize thenumber of cars. Even if the number of cars andthe number of ferries were reduced, the three-or four-year-old child does not have a strategyfor solving this type of problem.

The older child with some instruction beginsto see that sharing is not an action restricted toa narrow range of experiences. He usually par-titions the set of cars among the ferries if he isgiven a start. Once he uses sharing in this situ-ation, he seems to be able to take it out ofsimple sharing situations and use it readily tosolve any partitive problem. The girl here is us-ing partition to get the loads ready for the fer-ries to take across.

Cr

PROBLEM SOLVING 75

It can be seen that the problem presentedhere meets most of the requirements of a goodproblem:

It is significant mathematically.It involves real objects.The child is interested in the prqblem.The child must make modificat:ons in the

situation.Several levels of solution are available.The readiness with which the child at-

tempts a solution indicates that he isconvinced he can solve the problem.

One important requirement, however, hasnot yet been satisfied. In order for the child tobe able to generalize division solutions, hemust be given varying problem situations withthe same mathematical structure. The followingexamples illustrate the ease with which teach-ers can vary the problem situation:

A *1

The builder wants to move all the bricksto the top of the building. The tray on thecrane can hold only 5 bricks. How manytrips will it take to get all the bricks up?

76 CHAPTER FOUR

There are 5 trucks to move the cratesfrom the terminal to the warehouse. Eachtruck carries the same number of crates.How many crates will be on each truck9

1:1

I I

t

A*

fe

Each girl can take 3 cookies from thetray. How many girls can get 3 cookies?

The 30 students in this class are going toa picnic in 6 cars. Each car is taking thesame number of students. How many stu-dents will go in each car?

01,0%sj-1111%4;

-r

It is important that the child be faced with asmany variations as he needs to ge &ralize theactions associated with division. The total num-ber of objects and the number in subgroupscan be varied as easily as the type of objects.

Situations in which there is a remainder havenot been suggested. Problems that help thechild think about how to handle remainderscould form another kind of variation to be usedlater.

sample problem 2

It is important that the child become flexiblein his ability to classify. Classifying objects onthe basis of some physical attribute is a corn-mon activity not only for young children but foradults as well. The ability to form classes on thebasis of number, however, requires the child toignore all physical attributes of the objects inthe classes. consider the following photographand problems.

Ftl

15 4 j"."4-41":

I7

41,

r

You must not move anything that is .on aplate to another plate. Try to sort the platesso that there are two or more croups ofplates. Explain why you groupeu the platestogether as you did.

tn.

If-the child.has a rudimentary idea of one-to-one correspondence and is flexible in his abilityto classify, he will sort out the plates somewhatin the manner shown, which would enable theteacher to determine the child's ability to rec-ognize sets that have the same cardinal num-ber. This is not to say, however, that the exer-cise is a good problem.

A better problem in classifying could be pro-vided by supplying the child with a fairly largecollection of objects that can be sorted in a va-riety of ways. For example, attribute blocks,which are commercially available, can besorted by color, shape, size, thickness, texture,and so on. Appropriate problems in classifyingdo not necessarily depend on such sets; theycan be built around common materials likebeads, buttons, spools, blocks, containers, anda host of others.

For example, a child is provided with a pile ofobjects made up of a set of farm animals, a setof plastic cars, and a set of prehistoric animals.

Sort the things into two piles. Explainhow you did it. Next make three piles, thenfour piles. Now make as many piles as youcan There must be at least two things ineach pile when you have finished. Can youexplain how you sorted them out as youdid?

PROBLEM SOLVING 77

One solution to the problem by a five-year-old child is shown here.

It is not difficult to see that a great variety ofproblems like this can be designed from readilyavailable materials Such problems shouldgreatly increase the child's flexibility in recog-nizing and forming classes.

selected problems forearly childhood

The characteristics of a good problem havebeen outlined, and examples have been usedto show how problems can be generated withthese characteristics as a guide. The next taskis to select a sample of good problems for earlychildhood and describe a setting for them. Forconvenience, the selection will appear in twoparts--those problems appropriate for the be-ginning stages of early childhood, covering ap-proximately the ages of three to five years, andthose problems for the older child. approxi-mately five to eight years of age.

i/ 3

78 CHAPTER FOUR

problems for thebeginning stages

comparing and ordering weights

Materials:

11,Variations:

A beam balanceSix or eight clay balls that look

identicalA steel ball to conceal in one of

the balls of clay

Problem: Use the balance -to find the ballthat is heavier than the others.

Collect a variety ofobjects that differmark6dly in appearance and size but onlyslightly in weight. and have the child use thebeam balance to find the heaviest object.

Have the child use the beam balance to or-der a collection of objects. from heaviest tolightest.

Have the child use the beam balance as anaid in making one ball of clay just as heavy asanother one.

Give the child a ball of clay or other objeccand a handful of identical steel washers andhave him use the beam balance to find 'nowmany washers would weigh about the sane asthe ball of clay. (NOTE: Spring scales Shouldnot be used in these problerm fc.: two rea-st.,... first the ^4".' ;....,zreedd the number onthe scale; second, the spring scale is not adirect indication of weight, whereas the bal-ance is.)

making inferences

Materials: A metal container with a coverthat can be sealed

Six beads (four black, two white)A length of masking tape for seal-

ing the can

Problem: What is in the Can?For the basic problem, the six beads are put

in the container, and the cover is affixed but not

sealed. The chill should be permitted to ma-nipulate the container-inany way he pleases.but he should not remove the.cover. Then lethim open the container and draw out only onebead. He must not see the others, and he mustreplace the bead after he has looked at it. Allowseveral draws and get the child to make con-jectures about the contents after each draw. Inmaking his responses, the child tends to jumpto conclusions. For example. if he draws ablack bead on the first draw, he is likely to con-clude that all the beads are black. The object isto get him to modify his conclusions as he gains -

more information. In this problem he shouldcome to the conclusion after -repeated draw-ings that there are some black beads and someite beadS and probably more black beads

than white ones.Variations:

Place a single object in the container andhave the child make inferences about its natureafter he has manipulated the container.

Place an object in the container and seal itwith masking tape. Leave it in the classroom fora day. Each child should be encouraaed tohandle the container and to make-conjectures',bout its contents.

rudimentary graphs

Materials: A collection of pictures repre-senting television programs

One 1-inch-cube for each child inthe classroom

A-table to which the pictures maybe attached

Problem: What are the favorite televisionprograms of students in this classroom?

Rij

To solve the problem. each child places hisblock .. love the picture representing his favor-ite program.

Variations:Use a piece of graph paper with 1-inch

squares and have children take turns coloring asquare over a picture of their favorite pet. food.game. color. and so on.

Have the child keep records of weatherwith the same kinds of charts: for example, hecould determine whether the month of Marchhas more days than sunny days.

Have the children prepare graphs thatshow the number of people in their classoomwith blue eyes and the number with browneyes.

Draw graphs or charts to record informa-tion about children in- the classroom: theirbirthdays. hair color. color of toothbrush, colorof socks, size-of families. ways of coming toschool, distance from school. time taken to getto school. height, reach. and so on.

patterns in number

Materials: A train of cardobard rectanglesfastened together with paperclips. string, or some other de-vice

A collection of small counterssuch as cubes, beans. buttons

Prhblem: Muke a pattern of counters on thetrarystittai all the counters you have are usedup.

Suppose there are seven_cars and you gavethe child eleven counters. One solution mightlook like this:

122tcr-74E7-11:7-1:1:,1':i

Variations:Place counters on the cars as shown:

is

PROBLEM SOLVING 79

Give the child additional counters and ask himto put the correct number of them on the emptycars. This problem could be arranged so thatmany alternate solutions are possible.

Provide colored shapes and set up a pat-tern that the child must continue. A simple ex-ample might look like this:

Rule off a strip of cardboard into a seriesof squares and put counters or shapes or col-ored squares or some other objects on all thespaces in some pattern. Then cover somespaces and have the child determine what is onthose spaces.

spatial relations

AA' Nki`,Aka.IA* 14.ii

Materials: Several, sets of -seven 1-inchcubes

Problem: How many-different buildings canyou make, using Seven blocks in each build-ing?

are some sample responses:

Variations:Vary the number of blocks in the sets. or

vary the rules: for example. all the buildingsmust be L-shaped.

Once the buildings are completed, havethe child group them in classes.

Take a set of blocks and construct a build-ing: ask the child to build one just like it.

;1j

80 CHAPTER FOUR

patterns in the coordinate plane

Materials: A- geoboard or spoolboard atleast 4 X 4

At least sixteen colored beadswith large holes or coloredspools:

e

9 O 9

Problem: Can you complete the pattern?Variations:

Give one:child. red. be-ads. and.--the otherblue one. Have them move in -turn: =The firstchild to get lock: in 8 -row is the winner. Thesketch shows a Win for red. (It is important inthis problem to develop a strategy for blockingan opponent.)

Use the geoboard as a parking lot and letthe child code the plaCes on the board as if theywere places to park. He 0/4 the code (for ex-ample,-fourth car in red rovOCranother childwho then tries to "park- in the right place.

Have the child Use'beada athree colors toconstruct as many patterns-as he can on theboard. Require-that all places 'on the board areto be used up.:

manyto-one.correspondonce

Materials: A-SuPOY:of---papee -plates or flatboices.

A-supply of sugar cubes. beans.or colored shapes

The child is given five paper plates and fif-teen sugar cubes:

Problem: Are:there enough sugar cubes sothat yoti cab-Put two on a plate? Can you putfour- on -each plate? On how'_inany plates canyou put five sugar cubes?

Variations:Give the chilittfive plates and seven each of

circles..trienglesi;and squares:-Firstask him tofind ongi ofeadh shape for each plate_and thento get_as many more plates as ;he needs to useup all thethapes.

-l_et=eath plate contain o nly..Cife, triangularshaPe, one square shape;fandlone-cirOle. Ask ifthereare-,the sath u rribef..af. ih a peS. oh eachplate. The Ska*stiauld vary in %gar and size.

a proigern with shapes _

Materials: A set ofshapes as follows:

A Amiga.A A si 0

AThree spinners like. these:

A set of cards ruled to makeplaying board:

PROBLEM SOLVING 81

a Have the child try to arrange pieces on theboard so that all'the spaces are filled and all theobjects in the rows and columns are alike insome way.

Have the child start with a particular pieceon the center space of the board, for example.the small red triangle. The problem is tocomplete the rows and diagonals so that thepieces in each are alike in some way. Here isone possible solution:

Have the child flip the spinners in turn.choose the piece indicated, and put it on theboard wherever there is an empiy space. (Thedials on the diagram shown indicate a smallred triangle.) Then have him spin again andplace that object in an empty space. This goeson until he has objects that are in some wayalike in a row. column. or diagonal. He can thenrecord a win.

Problem: Place the pieces on the board sothat you can win in the least number of flips onthe spinners.

The diagram shows a win on both diagonals.(The pieces on the top right-to-left. diagonal areall large. The pieces on the top left-to-rightdiagonal-are ail red.) The board does not haveto be full to declare a win.

A

Variations:Have the child use only one spinner. for

example, the one indicating color. He canchoose an object of any shape or size to put on -the board, so long as it is the color the spinnerindicates. Again, it is up to the child to choosewhere he will put the piece. but once he haschosen a piece and a position for it. it cannot bechanged.

ma ...Amor

All Triangles

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Bv

vAI

III

All Red

_ _ All Small

NAll Red

rudimentary measurement

Materials: Sheets of paper on which linesegments of various lengthshave been drawn

A supply of beans, candies, sugarcubes, or blocks to be used asUnits of measure

itinifip&red end io endalong bile segments

Problem: On which line could you put themost beans?

Variations:Ask the child to try to draw a line segment

that will hold just five candies.

,

fl

82 CHAPTER FOUR

Ask the child if there is a way of puttingmore beans on the line. (For example, he capchange the unit by laying beans across the lineinstead of end to end, or he may stand candiesuprigift if he uses candles of the right shape.)

a sorting device

Materials: Various diagrams of branchingroads

Shapes used in the previousproblem

A red sign means only red things can pass. a0 means only circles can pass. and so on.

Problem: Can you take each shape down theroad and show where it comes out?

Variations:Draw a path with chalk on tLe floor. As they

file through. five-year-olds-take path A; all oth-ers take path B.

Give these directions. If your name beginswith A, B, C...., k, take path A. If it begins withany other letter, take path B.

Give directions such as this: If you are aboy and blond, take path B, otherwite, takepath A.

Let the child make more complicated sort-ers: he can make his own paths and the rulesfor following them.

The problems above and their variationsrepresent only a small-- sample of :.:11 those thatmight have been selected, and few of them arecompletely original. It is hoped. however, thatthese situations wili prove helpful to thosewanting to provide good problems for youngchildren.

problems for the laterstages

In the ages from five to eight, the child showsincreasing. ability to .make estimates, to keep arecord of his observations, to make statementsof a mathematical nature that fit events or situ-ations, and even to make some decisions on.the basis of what may happen. This does notmean that the problems should be more ab-stract or that their solution should depend lesson the actual manipulation of things in the realworld. It does mean, however, that more atten-tion should be given to having the child keepwritten records of the observations and maneu-vers that are involved in the solution of prob-lems. A child will learn through appropriateproblems how to express as written statementsmathematical Models related to many aspectsof reality. For example, when he sees a groupof five birds loin a group of three birds on afence, he may think 3 5 8. Or he may be

10 1

able to calculate the number of 10-centimetersquare tiles heeded- to Cover a 60-by-70-cen-tirneter area by thinking or writing 6 x 7 = 42.1tis with thosp developments in the older child inmind that the following selection of sampleproblems is presented.

loading and unloading

Materials: A toy truckTwelve or fifteen blocks or ob-

jects that will all fit in the truckat once

A table top that will accommodatefive or six (or more) stations forthe truck.

WrD

iron

.

Problem: At how many stations can thetruck pick up three blocks if ,the truck holdstwelve blocks?

Variations:Set up the problem so that the driver starts

with eight blocks and drops off three blocks atevery second Jtation. At the first station and ev-ery additional one after that he picks up twoblocks. The luestion is how many stationscould he visit before the truck is -empty. Thenumber of blocks to begin with, the numberdropped, and the number picked up can all bevaried.

Have twelve blocks (for example) in thetruck at the beginning. The driver drops offoneblock at the first station and at each successive

LlaragildiakraommlimomimmeirR9

-\-tfa

PROBLEM SOLVING 83

station two more blocks than at the station be-fore. Ask the child to determine at which stationthere will not be enough blocks to make a drop.The number of blocks on the truck can be var-ied, and the child can keep a record of thenumber and of which loads have just enoughblocks for the last drop.

Have the child pick up two blocks, for ex-ample. at each station and predict how manyblocks will be on the truck after four stops. Hecould write down the number of blocks in histruck after two stops, four stops, six stops. andso on.

Permit the child to devise his own sched-ules for picking up and dropping off blocks andto keep a record of his findings.

t- Using different materials, have the childgather things off plates around a table or servethings in some kind of progression or pattern.

making stacks

Materials: Twenty to thirty cubes arranged

_1

in uneven stacks

Problem: Make all the stacks the sameheight in as few moves as possible.

Here is an example with its solution:

84 CHAPTER FOUR

Variations:Present a different arrangement of

stacked blockS, for example. 7, 3, 3, 3, whichmust be equalized in the least number ofmoves.

Give the child 18 blocks ,:rif; arrange themwith 6, 4, 2, and 6 blocks in four stacks.Then ask him to make all the staoks the sameheight in the least number ol moves. (Ofcourse, there is no way to solve this problem.The child soon learns that making all stacks thesame height requires that the total number ofblocks be some multiple of the number ofstacks.)

Vary the number of stacks with.which thechild works. With proper record keeping andflexibility in stacking, the child can discover thefactors of many composite numbers, thoughthe word composite need not `be introduced.He can discover, too, that if the total number is7, 11, 13, 17 (prime numbers), there is no wayto make equal stacks. .

Start with four stacks and continue to re-quire the child to make equal stacks with theleast number of moves, but allow him tochange the number of stacks as he thinks nec-essary. For example, if the stacks number 3. 4.3. 2, he would keep four stacks because onemove would be sufficient to make them allequal. If they number 3, 1, 3, 3, -he couldchange to either two or five stacks because the10 cubes cannot be made into four equalstacks. It would take the same number ofmoves to make two stacks of the same heightas it would to make five stacks of the sameheight. Or, when they number 2, 4. 3, 2. theproblem cannot be solved if there is to be morethan one stack and at least 2 cubes in a stack.

tessellationsMaterials: A sheet of colored paper, 8 1/2

inches by 11 'nchesA variety of regular and irregular

shapes (triangles, quadrilater =-als, pentagons, hexagons, rectangles, squares), with. enough estimatingof each to cover the sheet of k,

paper. The area of each shapeshould be between one and twosquare inches.

Problem: Pick out all the shapes of one kind.Can you put the-se shaPis to-gether so that youcan cover the colored sheet of paper withoutleaving any gaps or spaces?

The beginning of a solution using equilateraltriangles is shown here.

L

Variations:Instead of using- a large number of units.

prepare a template of the desired shape; thechild can then trace around this shape to fill thecolored sheet.

Permit the child to make his own shape forihe tessellating.

Use the process to compare the area oftwo dissimilar shapes: each !shape-is coveredwith units, and the numbers of units requiredto cover each one are compared.

Materials: Several sheets of paper of vary-ing dimensions squared off in' equal units

--Problem: Estimate the number of squares ineach sheet. Do not count all the squarestryto find a shortcut that will get you close to thecorrect number without having to count everyone.

Variations:Ask the child to estimate the number of

beans in a jar. He should be provided with ma-terials such as measuring cups, scales, rules.other quantities of beans, and so forth. The ob-ject is for him to make progressive improve-ments in the estimate.

Have the child make estimates of thelength of the classroom, the playground. thegymnasium, the number of windows in theschool. the number of tiles in the floor or ceil-ing. his own heignt and that of other children,weights. what can be done in one minute, howlong it takes to tie his shoes. and so on.

Have the child keep a record of situationsin which an exact count or measure is not avail-able or even possible to get. Require only anestimate (attendance at games. population fig-ures, cost estimates, time and movement esti-mates, crowds, etc.).

Require estimates in situations like the fol-lowing:

a) Is two hours enough time to drivefrom New York V: Washington?

b) Plastic cars can be bought for 35cents, 20 cents, and 25 cents. Achild is given a dollar and told hecan buy four cars. Which can hebuy? Or the child can buy as manycars as he wants, but he can spendonly 50 cents. Which can he buy?

0) Is this shelf long enough to hold thebooks piled on the table?

c!) How long will the batteries last inthis flashlight?

PROBLEM SOLVING 85

area-perimeter relations

Materials: A geoboard (at least 6 x C) foreach child

A supply of rubber bandsPaper for recording

Problem: How is the distance around a rec-tangle related to the number of squares in-side?

A sample procedure might work like this:

I. I

i Ii1 I

. .i I}s 1

i f II I

v1

I

I. ,... ___.

The dotted line around A represents a rubt)erband. The child counts the units around (pe-rimeter) and the number of squares inside therectangle (area) and keeps a record of thecount. In this case, the perimeter is,10 and thearea is 6. The child then tries to change theshape of the rectangle, keeping the same num-ber of units in the perimeter. For example, hemight change to a one by tour (B), which wouldhave the same perimeter (10) but a differentarea (4). A variety of situations would have tobe investigated before the relation would beapparent.

The child will find, of course, that the great-est area will occur when the shape is nearest tobeing square. In the example chosen, a squareis not possible, but the area is greater when thedimensions are 2 x 3 than when they are 1 x 4.The investigation will undoubtedly lead thechild to situations in which the.square is a pos-sibility (i.e., the number of units in the perime-ter is exactly divisible by 4).

The child may decide to keep the area thesame but to change the shape of the rectangleand then see what happens to the perimeter. It

86 CHAPTER FOUR

will be found that the perimeter is least whenthe shape is a square (or nearest to being asquare). He may even come to attach somespecial significance to 4, 9, 16, 25 as areas,since they_can be made into squares.

Variations:Have the child start with squares (2 x 2, 3

x 3, 4 X 4) and record the perimeter and thearea. Then have him change the shape to a rec-tangle, keeping the perimeter the same andnoting any change in area.

Once the child becomes involved in the in-vestigation, he will think of keeping the areaconstant and varying the perimeter. The childmight require some assistance in keeping asystematic record. Verbal summaries shouldnot be forced.

Let the child make up a problem for therest of the class, using the data he has gath-ered.

Conduct an investigation to find the rela-tionship between people's height, weight,reach, length of shadow, size of neck, waist,foot, ankle, wrist, and so forth. Other relation-ships could be investigated, such as the weightof materials in air and their weight in water.

investigations in probability

Materials: A pair of dicePaper for recordingFifteen 1-inch squares made of

cardboard painted black onone side. The first square has azero written on both sides, thesecond has a one written onboth sides, and So on, until allthe squares are numbered onboth sides up to fourteen.

The child is taught a game procedure. Thesquares are laid out in numerical order, somewith white sides up and some with black. Be-fore he rolls the dice, the child chooses a color,either black or white. If the dice total a numberof the color he chose, then he winS. Suppose hehas chosen white and the dice show a 3 and a

2. The total is 5; so the child records a win be-cause 5 is white. Suppose on the next throw hechooses black and the total on the dice is 8. Hethen records a kiss, since 8 is white.

Problem: Try to learn the best way to playthis game.

Variations:Change the squares so that a different

proportion of black and white squares is used.Permit the child to arrange the squares as

he likes. but there must not be more than eightof one color turned up.

Allow the child to arrange the squares inany way he pleases to help him win.

(Nom The child should discover. that num-bers like 7, 6, and 5 are much more likely to oc-cur than 2, 3; or 12. He should discover thatsome numbers-0, 1, 13, 14can' never comeup. It is not expected, however, that he make acomplete analysis of the probabilities.)

traffic counts

Materials: A timerA sheet of paper for recording

Problem: How many cars pass through thetraffic lights at the corner?

It is clear that the problem will have to be re-stricted in some way. For example, observa-tions must be limited to a certain period of time,and "cars" must be explicitly defined. it is bestto have the child impose those restrictions hethinks necessary. As the child gains experiencein activities of this kind, he will improve hismethods of tallying,and recording.

Variations:Activities like this one do not have to be

conducted entirely during school hours. For ex-ample, have the child compare the number ofpeople entering a place of business during theweek with the number entering on Saturday.

Show the child how to measure pulse rate;he can compare rates after different forms ofexercise and report the results.

If circumstances permit, allow the child towatch as cars come to a gasoline pump; hecould record the kinds of cars and the amountsof gasoline bought. The results could form agood basis for a report.

predictions of heads and tails

Materials: Three penniesA tally board, which can be drawn

on a sheet of paper and dupli-cated

tails heads

siati;

Problem: Learn the best way to play thisgame.

Explain to the child that he will toss the threecoins all at once and observe whether theycome up heads or tails. Whenever a headcomes up, he will move one place to the righton the board; when a tail comes up, he willmove one place to the left. He will record hisactual position on the board by writing thenumber of the toss (1 for the fir-SI toss, 2 for thesecond, etc.) in the proper space. Give thechild some practice in tossing the pennies andobserving the possible outcomes. Then askhim to mark with an X the place on the tallyboard where he thinks he will be after he hastossed the three coins ten times.

Here is how the first toss would be recordedif two tails and a head come up. (The mostlikely place for him to be after ten tosses, or anynumber of tosses, is back at START.)

I I

PROBLEM SOLVING 87

heads

illVariations:

This game can be played competitively be-tween two or more players. Give each player 10points at the beginning of the.game. After eachten tosses, the child finds the number of spacesbetween where he is on the board and wherehe thought he would be and subtracts thisnumber from 10. When a player has no pointsleft, he is out of the game. The last player in thegame is the winner.

Vary the number of,coins used in tossing.Scoring need not be based on the estimate

but on the number of spaces from START. Donot require the child to make an estimate ofwhere he will be after a given number of plays.The score is the actual number of spaces fromSTART.

classroom organization forproblem solving

The problem examples introduced here havebeen described as if only one child were in-volved. It is highly unlikely that all problem solv-ing could be arranged on a one teacher-onechild basis in all the years of early childhood.Indeed, the development of the child's ability todeal with problems probably depends in somemeasure on his interaction with other children.In any event, it is important that teachers havesome ideas on how the classroom can be orga-nized for activities that might be called prob-lem-solving sessions. It is equally importantthat teachers choose those approaches thatare best adapted to their own interests andabilities as well as to those of the children theyteach. There are many ways of organizing theclassroom for such activities, but only three arepresented here. Each involves a child workingwith other children either in pairs or in smallgroups. This is not to suggest that the child willalways work with someone else, for there aremany opportunities for individual work withproblemsindeed, it is often needed.

88 CHAPTER FOUR

working in groups on the same problem

The photographs in the following seriesshow a first-grade classroom organized intoeight groups of about four children in eachgroup. The groups are all working in the gen-eral area of classification. Some are workingwith 'attribute blocks and some with collectionsof toys or objects. However, all are working onthe same problems; which were posed orally bythe teacher as follows:,

1. Sort out the objects on your desk sothat you have two piles and so that eachpile has things that go with one another.

2. Now sort the objects into three piles so

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that each pile still has things that go to-gether.Now try sorting them into four piles.Everyone take an object by turns untilthe objects are all used up. Now, play a"one difference" game: the first personto take a turn puts out one of his ob-jects. The next person must put out anobject that is different from the first inany one way. The next person puts outan object that is different in any oneway from the last object put out. Play inturns until One person has used all hisobjects.

90 CHAPTER FOUR

working in pairs on thesame problem

The desks in the second-gradeclassroom pictured here are usual-ly arranged in rows; so it was aneasy matter to have adjacent rowsmoved together to allow children towork in pairs. This is often a con-venient arrangement if materialsare to be shared or if the childrenneed partners for an activity. In thesituation pictured here, the chil-dren are working on tne "investiga-tions in probability" problem,outlined earlier. The teacher hasgiven the directions to the wholegroup and is thus able to spendtime with pairs of children, watch-ing, listening, and discussing.

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working in groups, each group ona different problem

The children in the third-grade class shownhere are working at stations on problems thatdiffer greatly from one another. The problemsare in the following general areas. tessellations.estimations. stacking. comparing and orderingweights. measuring with different units. trafficcounts. and probability.

The problems for each group are on cardsand are somewhat open ended. As a generalrule. problem cards provide minimal direc-tions. and children must do something. thenrecord their results for class discussion later.The problem cards being used by this class arepictured below.

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411

PROBLEM SOLVING 91

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Each group works at a station for a specifiedlength of time. then moves to another station insome orderly way. In this class, each groupworked on a problem for one class period. Thiscould be done on consecutive days or on oneday a week until each group worked each prob-lem card. The conclusion of the activity is in theform of a presentation and discussion by eachgroup about its findings at one station. Somegroups or individuals within groups will presentfairly sophisticated solutions: other groups willpresent solutions that are quite rudimentary.The teacher may serve as a discussion leaderor may let each group handle the discussion inits own way. The teacher generally helps thechildren sum up their conclusions and arrive atdesired generalizations.

The actual problems this class is working onand the materials needed for each one arelisted below.

Problem1. Arrange the differ-

ently shaped tiles onthe colored paper soas to leave no emptyspaces.

2. How many grains ofpuffed rice are in the

Materialscolored paper, about

8 1/2 inches by 11inchessets of geometricshapessquares, cir-cles, triangles. pen-tagons. hexagons.parallelogramsjar containing puffedrice. sealed tightly

92 CHAPTER FOUR

jar? (The actual num-ber of grains ofpuffed rice is writtenin a secret place inthe room. When allthe groups havemade estimates. youwill be given someclues to help you findthe secret place.)

3. Make four stacks ofblocks with-

6 blocks in onestack:3 blocks in onestack:2 blocks in onestack:5 blocks in onestack.

Can you make all thestacks th-es-i-meheight in two moves?(You may move morethan one tlock at atime if you wish.)

Now make stacks of7. 3. 3. 3 blocks. Canyou make all thestacks the sameheight in threemoves?

Now make stacks of6. 6. 1.3 blocks. Whatis the smallest num-berof moves neededto make all the stacksthe same height?Now make stacks of6. 4. 2. 6. How manymoves will it take tomake all the stacksthe same height?Make up some stack-ing problems of your,own.

4. Look at each of thethings in the box, andmatch each with aname card. Guesswhat order the ob-jects would be infrom lightest toheavi-est and put the cardsin that order.

extra puffed riceextra jarruler. length of-string.

small box, and smallcup

18 wooden 1-inchblocks

balance scalebox containing lead

weight. crayon box.large piece of Styro-foam. pair of scissors.plastic toy. woodenblock

name card for eachitem in the box

Now use the balancescale to help you putthe objects them-selves in order. light-est to heaviest. Howmany did you guesscorrectly?

5. Use the things on thetable and see howmany ways you canfind the followingmeasurements.Make a chart thatshows your measure-ments.

How tall are you?How high can youreach?How big are yourhands?How big are yourfeet?

6. List all the things withwheels that you thinkmight pass theschool on FortiethAvenue. Then watchthe road for ten min-utes and keep arecord of the thingsyou see passing.

7. You are going to tossthree coins and keepa record of how manyheads and tails youget. For every head.you must move oneplace to the right: forevery tail, one placeto ttie left. To recordyour position at theend of each toss.write the number ofthe toss in the properspace on your tallyboard. To work theproblem. mark withan X the space on thetally board where youthink you will be afteryou have tossed thethree coins ten timesand see how closeyou are after your tentosses.

ruler. tape measure.ball of string. jar ofpennies. book. shoe-box. large-squaredpaperpaper for a large.chartpens

-paper for a large chart--pens

three coinspaper cuptally sheets

concluding remarks

In early childhood, the main purpose ofmathematics teaching is to help the child learnhow to relate a situation or event that occurs inhis daily life to a mathematical representationof that situation or event. Since the most effec-tive way to accomplish this purpose is to pro-vide carefully selected, significant problems, itis more important to consider the basic charac-teristics of good problems at the early child-hood level than to trace the steps in the prob-lem- solving process itself. Thesecharacteristics are summarized as follows:

1. The problem should have demon-strable significance mathematically.

2. The situation in which the problem oc-curs should involve real objects or sim-ulations of real objects.

3. The problem should be interesting tothe child.

4 The problem should require the child tomake some modifications or transfor-mations in the materials used.

PROBLEM SOLVING 93

5. The problem should allow for differentlevels of solution.

6. Many physical embodiments should bepossible for the same problem.

7. The child should believe he can solvethe problem and should know when hehas a solution for it.

The problems included as examples were cho-sen because they met these criteria and be-cause they have been effective in classroomand individual use.

When selecting or designing appropriateproblems, it is very important that one have aparticular child with a particular background inmind. After a good problem has been designedin this way, it can easily be modified and ad-justed for the whole classit is much more dif-ficult to create a good problem than it is tomodify a good one for classroom use. What-ever the method of design may be, however,problem solving is of such basic importance inearly childhood that the selection and design ofgood problems is worth any amount of creativeenergy that teachers can devote to them.

experiencesforyoungchildren

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glenadine gibbt)

alberta castaneda

96

THE search for effective education programsfor very young children has been heightened bytwo developments: (1) the rapidly growingnumber of preschools, kindergartens, and day-care centers and (2) a growing awareness ofthe importance of early experiences in the totaldevelopment of the human learner. Tradition-ally, preschool programs have concentrated onthe development of a healthy, well-adjustedpersonality. Spurred by a concern for childrenfrom low-income homes and from minority eth-nic groups, attention has been directed in-creasingly to the intellectual growth of children.As a result of this emphasis on intellectualgrowth, attempts have been made to use pre-school experience as a guarantee of futureschool success, which in turn has producedsome programs directed so exclusively towardacademic achievement that other aspects ofthe young child's life have been ignored.

When the interrelatednessone might evensay the inseparabilityof physical, social, andintellectual development is accepted and actedon, it is possible to devise a balanced programwith a broadly intellectual orientation. JeromeBruner emphasized this in a statement aboutmathematics for the young child (1965, p.1009):

The more "elementary" a course and theyounger its students, the more serious must beits pedagogical aim of forming the intellectualpower of those whom it serves. It is as importantto justify a good mathematics course by the in-tellectual discipline it provides or the honesty itpromotes as by the mathematics it transmits. In-deed, neither can be accomplished without theother.

The work of Piaget has contributed much to-ward producing programs that emphasize thedevelopment of broad intellectual power ratherthan the attainment of narrow academic skills.Piaget's research has significance for thosewho guide the mathematical learning experi-ences of young children. It indicates that onecannot teach children ideas such as number,length, time, and shape in isolation from natu-ral experiences in their environment. This, ofcourse, does not preclude the planned experi-ence of direct instruction.

planning instruction

In planning any instruction, one must choosecontent and instructional strategies that arebased on the characteristics of the learnersandof the subject matter.

The young child as a learner relies on sen-sory information of a concrete or pictorial na-ture. Since mathematical concepts are ab-stract, however, it is essential that he relate thesensory experiences and the mathematical ab-stractions For example, a child cannot experi-ence adding numbers in the same sense thathe can experience melting things. But he canmanipulate a set of objects and gradually,through much experience, form in his mind amathematical idea or model that is abstract.

PRENUMBER KTIVITIES

rClassyying

[Comparing

Ordering

rI

Measuring

LOrganizing Data

1Shape and Space

EXPERIENCES FOR YOUNG CHILDREN 97

This "abstractness," this quality of being onestep removed from the real world, is what isdistinctive of even beginning mathematics con-cepts. In order to communicate these con-cepts, some language is needed, and this lan-guage must be based on sensory data.

It is with these stepsthe accumulation ofsensory data, its organization into concepts,the acquisition of the appropriate language,and the development of an intuitive back-ground for further symbolizationthat a begin-ning mathematics program must be con--cerned. In fact, a beginning mathematicsprogram can best be thought of as one that em-phasizes the development of conceptS and theacquisition of oral language. Great cautionmust be exercised in moving to new languageor to any written symbolization.

The activities suggested in this chapter havebeen chosen because they take into accountthe young child's need for perceptual informa-tion and yet do not distort the mathematicalideas being generated. It is a basic assumptionof the authors that all experiences, includingplanned activities, must be directed towardbuilding the young child's trust and confidencein his own perception and cognition, his trust inhis own ability to learn, and his image of himselfas a learner and a valued human being.

The activities, organized around a series oftopics and abilities, are listed in the chart be-low. All the activities suggested for any one

ACTIVITIES WITH NUMBERS

Numbier .) Classifyingf"---

Comparing

Ordering

Measuring

Seeing Patterns

Organizing Data

Shape and Space

Seeing Patterns

98 CHAPTER FIVE

topic need not be exhausted before anothertopic is begun, a teacher might well considermoving back and forth between topics. How-ever, one obvious break in the sequence mustbe notedthe point where number is concep-tualized and named. The chart shows this pointby the arrow running from the first three topicsto "Number." Activities in classifying, compar-ing, and ordering must precede the naming ofnumber. Other activities on the left -hand sideof the chart may or may not precede the intro-duction of number but must precede the cor-responding activities on the right-hand sidethat do use number.

prenumber activities

Classifying, comparing, and ordering are.hree processes that underlie the concept ofr umber. Once the child is able to classify ob-jects and see the collections thus formed as en-tities, then he can classify sets of objects asequivalent or nonequivalent and order them onthe basis of numerousness. Yet in order to as-sociate number with these processes, the idenrtities and attributes of the objects must be dis-regarded. Hence experience in classifying.comparing, and ordering provides the neces-sary background for the higher degree of ab-straction required for number.

classifying

Very young children classify. Applying iden-tity labels such as ball and sock is a form ofclassification. Whenever a child accuratelynames an unfamiliar object belonging to a fa-miliar identity class, he is classifying. For ex-ample, he may never have had a banana oat-meal cookie, but if given one, he may readilyidentify it as a cookie. He has a concept and alabel for "cookieness," and he has classifiedthis object as being a cookie.

Classifying is a part of everyday life and ofmany activities of the preschool program. It isoften presented to -children as a sorting task.These sorting tasks may be informal. "Let's put

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the blocks in this basket" or "The rocks go inthis drawer and the shirts in that drawer." Orthey may be carefully planned and sequencedPlanned sorting tasks may be either indepen-dently chosen and carried out by the child, orteacher directed, they may be performed eitherwith a small collection of objects varying in fewand obvious attributes, or with a large collection varying in several attributes, they may beplanned either on the basis of a criterion of thechild's own choice, with or without later de-scription of the criterion, or on the basis of astated, predetermined criterion.

early activitiesChildren can classifythat is, put together

things that are alike or that belong togethereven before they have the precise languagewith which to label the objects, describe thelikeness, or justify the "belongingness." Butthey cannot do this before they have an under-standing of "put together," "alike," and "belongtogether" when these words are used in in-structions. Any classification is dependent onhaving concepts and labels for identities, attri-butes, purposes, locations, positions, and soon. The acquisition of such concepts and labelsshould be an integral part of early mathematicslearning.

Discussing the differences and likenessesfound in the following activities will usually leadto the discovery of other similarities and willprevent children from becoming tied to the tra-ditional classifications of size, shape, and color.

Have boxes of assorted "junk" freely avail-able for the children to handle, use, or group asthey wish. On occasion ask a child about thegroupings he has made. Do not try to imposeany correct or acceptable criterion; just talkwith him to see how he has done the grouping.

Assemble a small "junk pile" and chal-lenge the children to find two objects that are"the same." Ask them to tell how they are thesame.

Set one stamp aside from a pile of stampsand ask the children to find another one that isin some way "like" the first stamp.

Have a child select two children who are in

, EXPERIENCES FOR YOUNG CHILDREN 99

some way "the same." Have other children tryto guess the similarity, that is, how the two chil-dren are "the same,"

later activitiesCriteria for sorting may include such attri-

butes as simple identity (blocks), larger identi-ties (toys), properties (blue), utility (things towrite with), composition (things made of wood),and combinations of attributes (things that arered and roll). The teacher should be careful tochoose materials that the child can talk aboutfreely.

Assemble a collection of cars, pencils, andblockssome red, some blue, and some yel-low, for instance, and some larger than others.Have the children put them in piles so that thethings in each pile are alike in at least one way.

Display a collection of cutouts with dis-tinctive shapes and ask the children to'arrangethe cutouts in two piles so that the cutouts ineach pile are alike in some way.

Set up a collection of doll furniture andeating utensils. Tell the children to put all thechairs together in one pile, all the beds in an-other pile, all the forks in another,'and so on.Later, they can sort the furniture into one pileand the eating utensils into another pile.

Put together a collection of shells, driedlegumes, rocks, dry pasta, buttons, beads,samples of fabric, or any other materials amen-able to sorting. Have the children put togetherthe things that are alike or the same. Then askthem how they decided to put them togetherthe way they did.

Take one of the collections that has beenpartially sorted and ask the children to decidewhich group the next object will be part of andwhy, or ask them to describe how the thingsthat are together are alike.

Classifying can be used to develop con-cepts and language related to sets of objects.Although set language is rarely a part of every-day conversation, children have had experi-ences with sets of objects. They may not have

identified building blocks, their fingers, a pairof socks, their crayons, or their mother's dishesas sets, but they have considered them assingle entities; when children think of thesethings as single entities, they are using the con-cept of set.

early activities

Initial, planned experiences and the accom-panying introduction of some set languagemight include opportunities for the child to con-clude (1) that a set can be named by describingor listing its members, (2) that the name of a setdetermines its membership, (3) that thephrases "a set of" and "the set of all the" canname different sets, and (4) that an object canbe a member of more than one set.

In introducing the language "this is a set of,"teachers should not use sets with one mem-berit is awkward and perhaps confusing tospeak of "a set of pencils" when only one pencilis in view. Also, the number in the sets shouldbe varied to prevent the child from concludingthat all sets have some given number of mem-bers.

Show the children several objects, such aschalkboard erasers, and say, "This is a set oferasers." Show other sets of objects and namethem, leaving them in view: Show other familiarobjects, such as shells, and ask, "What is this?"The children may reply "a set of shells" or just"shells." In either event, agree that they areright and say, "They are shells. This is a set ofshells." After many sets have been named andare still visible, ask again about each set andsay, "What is the name of this set?" Then askthem to show you the set of pencils, cups, andso on.

Building on earlier classification skills, thechildren readily use the words "this is a set of"to describe sets whose members have com-mon qualities with which they are familiar. If themembers of those sets are distinguishable, it ispossible to name the individual members of theset. Once this is possible, children also canname the members of a set where no common

100 CHAPTER FIVE

quality exists. These experiences can leadthem to conclude that members of a set neednot be related.

Sometimes a child is inclined to think that aset of objects is inherently what he sees andnames it to be. If so, the following activitiesmight be used to show that a set of objects canbe named in more than one way and moreoverthat the name will determine what other objectsmight be members of the set.

From a collection of objects, select somethat have two easily identified common quali-ties, for example, red airplanes, animals withwings, or toys with wheels. Display the ob-jectsfor example, a robin, a chicken, and aduckand name them by using one of thecommon qualities: "This is a set of animals."Have the children find other members for theset of animals. Again show the robin, thechicken; and the duck, and name the set in adifferent way: "This is a set of things withwings." Now have the children find other mem-bers of the set of things with wings. Ask themwhy they choose a dog, cat, and rabbit to gowith the robin, chicken, and duck the first timeand an airplane and ladybug the second time.

Whisper to one child the name for a set ofobjects. He then decides which objects sug-gested by other children are members of theset. See if the children can figure out what theone child knows that allows him to decidewhich objects can be members of the set.

Using a collection such as the one shownin , take a single object such as a red car andstart a set with it. Let the children suggest aname forthe set and assemble the other mem-bers of that set to one side of the original col-lection. Call the children's attention to the origi-nal object again and have another appropriatename suggested. Assemble other members ofthis newly named set to the other side of theoriginal set. Talk with children about the origi-nal object and which set or sets it is a memberof.

A further refinement of the naming of setsmight include the distinction between "a set of

and the set of all the." For these activities, theteacher must be sure that the children under-stand the restrictions on the objects for the ac-tivityfor example, only those objects on therug, or the table, or the display board.

Begin any set (for example, a set of scis-sors) and have a child name it. Ask if there isanything else.that could be a member of the setand when there are no more possible mem-bers, ask if that is all the scissors. When thechildren decide that it is, say, "Yes, it is not justa set of scissors, it is the set of all scissors in theroom."

Have children name a given set and thenmake it "the set of all the _s." Or ask them toname a set and when all possible membershave been included, ask them to rename theset.

later activitiesActivities with sets of objects offer an oppor-

tunity for perceptual experiences that includesome ideas Of logic, proof, and Venn dia-gramming. The following activities combine theuse of negation and the idea of the complementof a set.

FHave the children work with a sorting de-

vice. (See page 82 of chapter 4.)

Bring out a variety of objects. Begin to putthe objects into two sets, perhaps on a piece oftagboard marked into two sections. Into one setput things with a common property or iden-tityanimals, perhapsand into the other setobjects that are not animals. As soon as theythink they know it, 41ave the children tell thename of either set. They will easily name the setof animals; the difficulty comes in finding aname for the other set. They may suggestnames like "things," "toys," or "objects," but ifany of these were the name of the set, then allthe animals would be members of that set, too,and they are not. If no satisfactory name can besuggested, put that display aside, still in view,and begin another partitioning, such as "yel-low" and "not yellow." Again seek names forthe two sets. If several such examples do notelicit the desired response from the children,return to the set of animals and say, "Listenwhile I think how I'll decide where this one goes.Is it an animal? Yes; so I'll put i% ere with theset of animals. Is this an animal? No; so I'll put itwith that set. This is the set of animals. Whatcould be a name for that set?"

The following activities approach Venn dia-gramming of the intersection of sets by show-ing with circular enclosures that an object canbe a member of two sets.

; Position two pieces of yarn or string innonintersecting, circular shapes on the floor ortable so that all the children can see. From acollection of objectsa universebegin toform two sets of objects (blocks and redthings). building each set within one of the cir-cular areas. (Perhaps labels reading "red" and"blocks" could be used.) As each object istaken from the universe, have the children as-


sign it to one of the sets. Be sure the universecontains some objects that could be membersof both sets. When the children clearly under-stand this procedure, produce a red block andask them where it should go. They will agreeboth that it is red and therefore belongs in theset of red things and that it is a block and there-fore belongs with the set of blocks. The prob-lem is how to get it physically into both sets,that is, within both circular boundaries.

Solutions that 4 1/2-to-5-year-old childrenhave suggested include putting the block equi-distant between the two yarn circles, cutting theblock in half and putting a piece in each set;getting another red block so that there will beone for each set; putting the red block asideand going on to the next object; moving thepieces of yarn together and balancing the blockon them ( ); and putting the two pieces of yarn

close together in a zigzag fashion and insertingthe block into a "pocket," as shown ine. Occa-

102 CHAPTER FIVE

sionally, a rare five-year-old will arrange thepieces of yarn so that they overlap, thus form-ing an area that is enclosed by the boundariesof both set areas as shown in C.1

If no child solves the problem, have the ma-terials available at free-activity time so thatthose interested can continue to explore anddiscuss the possibilities. While this activity pro-vides an opportunity for children to use theidea that an object can be a member of morethan one set, it tests for other cognitive behav-iors, including a certain flexibility of action andthought. It is quite possible for a child to ac-cept, and even be able to state, that an objectcan have membership in two sets and yet notbe able to solve the preceding problem.

Set off two circular areas with string. Over-lap the strings to indicate an intersection andbegin two, setsblue things and cars. Whenseveral objects have been placed in each setand in the intersection, have the children de-scribe the objects that are in the overlappingarea (blue cars). Point out that they are thingsthat share the identifying quality of both sets.

After children are able to place objects cor-rectly within the boundaries of the intersection.start two new sets and give them an opportunityto speculate about what would go into the inter-section of the two sets. Later, provide an op-portunity for them to speculate about whatwould go into the intersection of two sets evenbefore the sets have been started: "If we madethis a set of oranges and that a set of plasticthings, what would go here? Why?" Also in-

elude some instances for which the intersectionis the empty set.

comparing

Comparing is the process by which the childestablishes a relation between two objects onthe basis of some specific attribute. In suchsimple activities as comparing the length of twosticks and stating, "This stick is longer than thatstick," the child establishes the relation "islonger than." Also, comparing is the forerunnerof ordering and measuring.

Just as a child can classify only in terms ofthe attributes and identities that are known tohim, so can he make comparisons only on thebasis of those attributes that have already beenclearly discerned by him. Thus, the devel-opment of concepts for attributes, of objectscontributes to the ability to compare. It is es-sential also to develop (1) the techniques thatallow the child to make perceptual com-parisons and .(2) the specific language that isrequired for describing particular com-parisons. Comparing heights, for example, re-quires techniques and language different fromthose used in comparing sizes. The techniqueof attempting to establish a one-to-one corre-spondence allows the child to compare visuallythe numerousness of two sets. Juxtaposing.hefting, and using a common baseline areactivities that allow the child to compare area,weight, and height perceptually.

early activitiesBecause children are so intimately con-

cerned with their own height, the activities inthis section will Involve comparisons of height.(Height and length are both forms of line meas-ure; it is only physical position that makesheight measurement appropriate in one situ-ation and length measurement appropriate inanother.)

Ask two children who distinctly differ inheight to stand. Ask the other children whichchild is taller and how they know.

Give the children two objects that differonly in height. Have them decide which is taller,or shorter. Ask them what they had to do to besure.

Give children two objects of differentheights and in some instances of contrastingbulk, that is, tall and narrow, short and broad.Be sure they compare height and not bulk. Askwhat they had to do to compare the two ob-jects.

Have the children attempt to compare theheights of two objects placed some distanceapart. Ask them what they must do to confirmtheir judgment.

Have the children attempt to compare theheights of objects that are on different base-linesfor example, one child standing on achair and another standing on the floor. Askthem what they must do to be sure which istaller or shorter or if they are of the sameheight.

later activitiesMoving from comparisons of amount to

comparisons of numerousness requires thechild to pair the members of two sets, attempt-ing to set up a one-to-one correspondence be-tween the two sets.

Give three red spoons to one child and twoblue spoons to another. Ask who has morespoons. Have each child put a spoon on thetable, thus establishing a pair of spoons, a redone and a blue one. With each pairing, com-ment on what can be seen. "There is a bluespoon for that red spoon. There Is a blue spoonfor that red spoon. But there is no blue spoonfor this red spoon. There is a red spoon leftover. There are more red spoons than blueones."

Give one child three jars of paste and an-other child three paste brushes. "Is there abrush for each jar of paste? A jar for each pastebrush? How can we find out?"

Using sets of more than five members.have the children compare the number of dolls


and hats, cowboys and horses, and so on. Setup situations and ask questions similar to thosedescribed above.

Show a set of objects in a linear arrange-ment. With other sets in random arrangement,ask the children to find which sets have moremembers and which sets have fewer membersthan the set being used as a model.

Show a model set of objects. Have the chil-dren sort other sets into those that are equiva-lent to hie model set and those that are not.

For some kindergarten children, later activi-ties can include the use of the wards "fewerthan" and "equivalent." It is acceptable, how-ever, to continue to use only the terminology"more than" and as many as."

Daily classroom routines provide many op-portunities for children to use pairing to find ifthere are "as many as" or "enough" or morethan." These opportunities can be used topractice the actual pairing, the use of the infor-mation thus gained, and the language to reportthat information.

ordering

, Ordering builds on comparing. For theyoung child, ordering involves a physical ar-rangement of objects or sets of objects. The ar-rangement must have an origin and a directionand must reflect some rule. For example,things can be ordered from shortest to longest,the rule being that each thing after the firstmust be longer than the one it follows; sets ofobjects can be ordered from those with themost members to those with the fewest mem-bers, cr vice versa. Obviously, children can or-der only on the basis of attributes that areknown to them and that they can compare.

early activities

Materials that lend themsefoes to orderingon some clearly perceptible property should beavailable to children for their independent use.Examples of such materials include Cuisenaire

(18

It**

104 CHAPTER FIVE

rods, the Great Wall, blocks of progressivesize, unit blocks of graduated length, stackingbarrels, a collection of dolls or pencils or TinkerToy sticks of graduated lengths, objects ofvarying weight,.color chips in varying shades ofone color, tone blocks of varying pitch, and soon. Beginning activities in ordering should useno more than five objects, and their differencesshould be clearly discernible.

Show the children three objects differingin height. Locate the one of medium height.Have a child choose one of the remaining ob-jects, compare it to the medium one, describethat comparison, and place it to one side of themedium. one. Have another child compare theother remaining object to the medium one, de-scribe that comparison, and place the objecton the other 2icle of the medium one. Have thechildren describe the resulting arrangement.

Show three objects that have been ar-ranged according to height. Indicate one endas a starting point and ask the children to de-scribe how eadti object is different from the oneit follows. Indicate the other end as the begin-ning and have the children describe how eachobject is different from the one it follows.

Show three objects ordered by height. In-dicate and identify the shortest of three otherobjects. Have the children arrange the othertwo so that th&set will look just like the first set.Tell them thaMhe objects are ordered so thateach object aìter the first is taller than the one itfollows.

Show the children five objects that havebeen ordered according to height. Then showthem five more objects. Arrange the first, third,and fifth of these objects in order of height.Have the children insert the other two objectsso that the set is ordered like the first set.

Give the children sets of five objects andhave the.children order the sets according tosome physical property with which they are fa-miliar. Have them describe how they arrangedtheir 'objects.

P. Show live objects that have been orderedby height and, have the children describe thearrangement. Ask them to tell where a particu-

on

lar object is and why it belongs there: "The blueone is between the red and green ones. The or-ange one is last. It comes after the green onebecause it is the tallest of all...."

later activitiesAfter comparing and ordering objects and

comparing sets of objects for numerousness.children can use numerousness as a basis forordering sets of objects. Their first orderingsshould involve no more than three sets, with thenumerousness clearly disparate. When chil-dren are able to order three sets so that eachset after the first has more members than theset it follows, they can order as many as fivesets. In a similar way, sets can be ordered sothat each one has fewer members than the oneit follows. As with the preceding sequence ofactivities, these activities should move, not justtoward more sets to be ordered.. but also to-ward the acquisition of the language, both re-ceptive and productive, with which to describethe sets and the arrangements.

The preceding activities on classifyina, com-paring, and ordering are a prelude to the con-cept of number. The following activities neednot necessarily precede the naming of number,but neither do they require the use of numbernames. Many of them build on ideas alreadybegun in the activities using classifying, com-paring, and ordering.

measuring

Children's initial experiences with measure-ment should be leisurely enough to allow themto build concepts both of the properties to bemeasured and of the appropriate units. Later.there should be kinesthetic and physical expe-riences with standard units before any stand-ard measuring device is used. The teachershould be in no hurry to use standard units;rather, she should make sure that each experi-ence related to measurement contributes to ageneral concept of the process of measuringand to the concept of measurement.

Activities related to measurement begin withtasks of comparing and ordering. The earliersections on comparing and ordering ccncen-trated on height as a variable dimension. This

section on measuring concentrates on weightand capacity ways for developing these con-cepts. and the language and techniquesneeded.

Weight. Obviously, it is not possible to meas-ure a property unless the property itself hasconsistent meaning. The child must be able todistinguish weight from size, texture, or otherelements of appearance before he can com-pare and measure it.

early activitiesIt is easy to plan experiences that will lead

the child to realize "I cannot say which is heav-ier by lookingI must lift the objects to besure." In this way he realizes the power of, andthe limitations on. his perception.

Using a collection of objects including ballbearings, Ping-Pong balls, golf balls. solid rub-ber and foam rubber balls, balsam wood,boxes of cotton, and other objects whoseweight and size seem disproportionate, showthe children two objects and ask them to findout which is heavier. The first discriminationsshould be obvious and not necessarily offerany contradiction in size and weight. Ask thechildren to describe the different feelings of lift-ing very heavy and very light objects. Activitieswith finer discriminations and with pairs that in-clude large, light objects and small, heavy ob-jects can follow.

Have collections of objects available for aself-chosen activity. Let the children collect or-dered pairs of objects and arrange them so thatthe relation "is heavier than" or "is lighter than"exists between the two objects. A large piece oftagboard placed on a table or the floor with the

r- is heavier Phan


relation inscribed at the top and spaces ruledoff can help in the physical arrangement of thepairs of objects (U).

Expand the preceding activity. both as adirected and as a spontaneous experience, toinclude ordering three or more objects. Therule might be that each object after the first isheavier than the object it follows. Thus, the firstobject is the lightest and the last is the heaviest.

The preceding activities have used physicalsensation in comparing the weight of one ob-ject to the weight of another object. Building onthe knowledge of muscular comparison. thefollowing activities use a balance beam in com-paring weights.

Have children lift heavy things that requiregrasping by two arms and have them describewhat it feels like to lift them. Then have them liftbuckets of sand or rocks by using only onehand to grasp the handle. Ask them to describehow that feels, how it makes their arm feel, andwhat they had to do with their lifting arm. Haveother children observe and describe the liftingposture.

Show the children a balance beam like theone in and ask them to predict what wouldhappen if they put something heavy in one pan.Then have them do it to check what does hap-pen.

3 9

106 CHAPTER FIVE

With a balance beam, have the childrencompare the weight of two objects of grosslydifferent weights, establish wh.cn is heavierand which is lighter, and predict what wouldhappen if one object were put in one pan andone object in the other. Have them check theirpredictions. When the children agree that theheavier object causes the arm from which it issuspended to come down, let them compareseveral dinerent pairs of objects, this timereading the balance beam rather than relyingon their physical, kinesthetic sensations to saythat one object is "heaviel than," ;almost asheavy as," or "lighter than" another object.

Provide a balance beam and from two tosix look-alike Plasticine balls, with one ballheavier than the others (a metal ball can beplaced inside). Have the citi:dren use the bal-ance beam to find the heavier ball.

Capacity. Although capacity and volume arerelated properties. a visual comparison be-tween capacities can easily be made. whereasa visual or perceptible comparison betweenvolumes, in the sense of space occupied. can-not be so clearly made. For example, we canpour from one container to another and haveevidence that the first container holds more be-cause material is left in it when the second con-tainer is full (or that it holds less because wehave emptied its contents into the other con-tainer and the other container is not full).

Before introducing standard units of capac-ityliters, cups. pints, quartshave the chil-dren compare the capacities of containers ofdifferent size and shape.

Provide liquids and other "pourables,"such as rice, sand, lentils. or beans, and a vari-ety of containers, some identical in shape andsize. some varying in shape but the same in ca-pacity, and some differing in capacity. Let thechildren pour from one container to another.They can see that containers of quite differentdimensions can hold the same amount.

Careful attention should be given, in guidedactivities, to the acquisition of the languageneeded to describe the result of pouring thecontents of one container into another.

Have the children compare the capacitiesof two containers: "This holds more than (lessthan, almost as much as) that." Have them or-der three or more containers by capacity. Letthem label the ones that hold the most and theleast.

organizing and presenting data

Presenting information in graphic form andreading the graphs are interesting activities foryoung children. Real objects can be arrangedin three-dimensional bar graphs. With just theability to compare height and use one-to-onecorrespondence, the young child can "read"and interpret such an arrangement. providedthe objects are stackable and of the same size.

Provide a shoe box full of colored cubicblocks and ask the children to name the colorsof the blocks in the box. Ask them to find outwhich colored blocks are more numerous. Letthe children sort the blocks, stack them, andarrange the stacks in a side-by-side arrange-ment. They can tell perceptually which stack istaller, establish the one-to-one relationshipacross the columns, and then tell which stackhas more blocks.

Translate the preceding activity to pictorialform by using drawings or pictures of blocks,colored like the real blocks. Have the childrenfind an appropriately colored drawing for eachcolored block and place the drawings by coloron a paper marked off in columns. Again, with-out naming the number in any column, they cantell which color of block is most or least) nu-merous.

Have the children choose pictures to rep-resent each one of their brothers and sistersapicture of a boy represents a brother; a pictureof a girl represents a sister. Have each child ar-range his pictures in a column above his name.The tallest columns show the children with themost brothers and sisters. Comparing the

:.2:heiglits of the -columns allows the children tosay that this child has as many brothers andsisters as that child, these children have morebrothers and sisters, and so on.

Children can organize into a graphic pre-sentation pictorial data related to attendance,shoe styles, pets, birthday months, dessertpreferences, and so on.

Many science and social studies projects of-fer the possibility of a pictorial or symbolic pre-sentation of data. One simple example devel-oped from a study of peanuts. Each child wasgiven a .peanut to open and was asked to notehow many kernels were inside. The teacherpreviously had prepared cards with drawingsshowing peanuts containing from one to fivekernels and a chart on which these cards couldbe displayed. The cards looked like this:

Each child found a card showing a peanutshell with as many kernels as his peanut. Heplaced the card on the chart in the columnmarked with a similar picture. The completedchart is shown in NI . Again, the data were orga-nized and presented with no need for counting.

shape and space

The geometric idea of shape has been used,overused. and misused. Children are asked


00r

frequently to handle, name, and describe cut-outs shaped like triangles, circles, squares, andso on. From some such sequences of instruc-tion, children could conclude logicallybut in-correctly(1) that a circle can be held in thehand, (2) that a shape is made of posterboard,(3) that a shape is red, green, blue, or yellow,(4) that "square" and "rectangle" are mutuallyexclusive categories, and (5) that triangles haveat least two equivalent sides and their basesare firmly horizontal. To prevent st h incorrectconclusions, activities with object should beplanned to develop a general concept of shape,to give the children the opportunity to observeand describe likenesses and differences in theshapes of objects, and, finally, to enable thechildren to learn the names for the most com-mon shapes.

Explorations of shape begin in infancy.Babies and small children feel shape as theygrasp rattles, clutch bottles, and roll balls, asthey crawl around chairs and through cartons,climb up and down stairs, and finger the edgeof a coffee table while walking around it; as theyplay with blocks, dishes, stuffed animals, wheeltoys, and dolls; and as they learn to feed them-selves by holding their glasses of juice or milk.Although small children may not consciously

108 CHAPTER FIVE

communicate what they see and feel, these ex-periences during the early years of their livesdevelop in them an awareness of the likenessesand differences in the shapes of a variety of ob-jects. With guided learning experiences, chil-dren learn to describe what they see and feel.

Gather a collection of objects, such ascheckers, buttons, bottle caps, small wheels,sheets of paper, corks, crayon boxe& stamps,pennants, toy irons, cards or sheets of papercut diagonally, or other objects that can beclassified by an attribute of shape. Select oneobject, such as the cork, and place it on onetable and select another object with distinctlydifferent shape, such as the crayon box, andplace it on another table. Place a stamp withthe crayon box and a checker with the cork.Continue to select objects or ask children to se-lect objects, one at a time, and have the chil-dren indicate the table on which the objectshould be placed. Verify that their suggestion iscorrect, or if it is not, say, "No, let's put the____ on the other table." After all the objectshave been classified into one or the other of thetwo sets, ask what is alike about the objects ineach set. Check each suggestion to see if it ap-plies to all the objects in that set. Although youhave imposed a means of classification, thechildren may not perceive it. Try to lookthrough their eyes, not yours.

Further activities can be designed to usechildren's sight, touch, and kinesthetic senses.

Blindfold a child and have him trace withhis fingers the outline of a cutout (circular, rec-tangular, or triangular) and then give the cut-out to you to conceal before taking off his blind-fold. Have him rely on his kinesthetic memoryto trace the shape in the air with his eyes openand describe it as being round or as havingsides or corners.

MSeat the children in a giggle facing a pile of

blocks of various shapes and ask them to close

their eyes and put their hands behind theirbacks. Pick up one object from the pile andplace it in a child's hands. Ask him to find, with-out seeing the one he has handled, one exactlylike it or similar to it.

Vary the preceding activity, using blocksthat vary in only one dimension with their shapestaying the same; for example, use dowels ofvarying diameter or sticks of varying length(Cuisenaire rods).

Provide a collection of felt or cardboardpieces. Include some with sides and cornersand some with no corners, some that are exam-ples of common shapes and some that are not,some with the same shape but of varying sizes,and some with the same shape but with differ-ent proportions. Let the children sort the piecesaccording to whatever likenesses or differ-ences they can see. Successive sortings canhelp the children to see and describe the char-acteristics of the pieces that have, for example,a rectangular shape. It is, however, quite pos-sible to know and state that two things have acommon shape or common elements in theirshape without being able to name the shape.Do not be in a- hurry to use the words circle,square, triangle, and the like.

From a collection of three-dimensional ob-jects, select one object that can be a model of acylinder, another that can be a model of a rec-tangular box, and another that can be a modelof a sphere. Choose other objects that will gowith each. The first grouping is called cans, thesecond boxes, and the third balls. Have thechildren continue the sorting. Include some ob-jects whose names offer competing ideasforexample, a sardine can, which is shaped like abox, or a salt box, which is shaped like a can.Invite the children to bring in other objects thatwould fit into one of the groupings.

Provide a collection of cutouts that vary inshape, color, or size; a spinner showing thesame assortment of figures; and, for each child,a Bingo type of card with nine such figures ().For his turn a child spins and if he finds a figureon his card like the one indicated by the spin-ner, he takes such a figure from the collectionand covers the appropriate space on his card.The winner must have three objects in a row ora full card.

100 ELz\Z\

e11

Provide each child with the needed papercutouts. (See table 1 for suggestions.) Hold upa square and say, "Find two of these and fitthem together to make this." (Hold up a rec-tangle.) After the children have become familiarwith the task, the instructions-may be given asshown in the table.

TABLE 1

Find Make


Vary the preceding activity by having thechildren use scissors to convert one shape intoanother. Some suggestions are shown in table2.

TABLE 2

Find MakeIWhen the names for some common

shapes have been learned, give the childrensuch instructions as,these:

"Point out some ofthe triangle's youcan see in this pic-ture"

"Point out some ofthe rectangles youcan see in this pic-ture"

"Point out some ofthe squares youcan see in this pic-ture"

"Draw a picture using just triangles and an-other using just rectangles. Which picture doyou like better? Why?"

110 CHAPTER FIVE

Studies by Piaget have suggested that thechild's first geometric discoveries do not nec-essarily involve shape but are concerned withsuch ideas as separation, proximity, closure,and order. Children have been observed to re-spond to the topological aspects of figures be-fore perceiving sides and corners. For ex-ample, shows how one child reproduced theshapes shown in 0. Responses like this implythat children see and represent "closedness"rather than configuration. However, one alsomight speculate that because of their age, theirresponses are governed by a lack of musclecontrol rather than by an error in perception.

0flOther ideas that are geometric in nature

and that depend on these intuitive experiencesare closed and not-closed paths. The idea of aclosed path is related to the various geometricfigures that children have used in observingand comparing shapes. Consider the path thatthey take in walking around the room from onecorner along all four walls and back to the start-ing place. This is a closed path. Tracing withtheir fingers the closed path made by the edgeof a table, the rim of a wastepaper basket, orthe edge of a book provides other sensoryexperiences with which they associate thewords dosed path. By contrast, let them tracewith their fingers the not-closed path made by apiece of string or a stick lying on the table. Thisis different from a closed path, for they cannotget back to the starting point unless they re-trace their steps.

Have the children use ropes or cords toform paths on the floor so that they can walk onthem from one point back 'to-that same pointwithout having to turn around. Have them formother paths that cannot be so, traced.

Have the children tell if the path formed bya lope is closed or not closed and then let themtest their conclusion by walking on the ropepath or tracing it with toy cars.

Give The children pipe cleaners formedinto closed and not-closed curves and havethem sort them into two piles sothat those ineach pile are somehow alike.

Have the children use pipe cleaners toform a variety of closed and not-closed curves.

It can be expected that children have ac-quired definitions for "inside" and "outside"from hearing such everyday statements as "It istime to come inside" or "You may go outside toplay." Learning to associate these ideas withclosed and not-closed paths can bring an-other dimension to their learning. For example:

Have the children make closed paths onthe floor with pieces of string. Have them placeobjects or other children (or everithemselves)inside the path, outside the path, and on thepath. Show them that they must cross a closedpath to go- from the inside to the, outside butthat they do,not have to cross a not-closed pathto go from one side to the other.

These activities all -help children to observeand classify objects according to shape, tocompare objects for similarities and differ-ences in shape, and to perceive differences be-tween closed and not-closed paths. Their pur-pose is to illustrate the spirit of involvementnecessary for the young child to develop anawareness of geometric ideas and to acquirethe appropriate language for communicatingthese ideas. In this spirit of involvement, otheractivities can be devised.

wI

patterns

Basic to mathematical insight is the ability torecognize pattern. With the use of real objectsand pictures and drawings, children can bemade aware of patterns and can practiceseeing, describing, extending and completing,Jr repeating patterns. They can be requiredmerely to perceive and repeat or extend a pat-tern; they can be asked to describe a pattern orto explain why they think a particular thingcomes next; or they can be asked to make apattern themselves and describe it. Theteacher must be aware of the burden of lan-guage that a given task places on the child andmust make a considered judgment about theappropriateness of that burden for each groupof pupils.

Patterns may be made up of concrete ob-jects, pictures, designs in weaving and stitch-ery, symbols, and so on. The first patternsshould be strong_and uncomplicated.

Using blocks, shells, dry pasta, or anythingof distinctive -shape, color, or other attribute,begin a pattern, say, of two red blocks followedby two green blocks. Have the children make apattern directly below the one shown and justlike it, and another directly below: that. Theyhave replicated the pattern.

Show another pattern, such as two oystershells and one snail shell. Have the children re-peat that pattern in..a.,linear. arrangement fol-lowing the original. They have continued, or ex-tended, the pattern.

Show another arrangementfor example,two large blocks, two small bloVs, two largeblocks. Have a child choose the next block andthen the next one. Ask why each of those thingscomes next. (Note that what he chooses nextdepends on the pattern he sees.)

Show another pattern (a .piece of bowknotpasta, a piece of elbow macaroni, a piece ofrigatoni, a piece of bowknot pasta, a piece ofelbow macaroni). Ask the child to show or tellwhat the pattern is and to choose the next itemin the pattern.


Patterns need not be in a strictly linear ar-rangement. The following activities will allowthe child to practice seeing nonlinear patterns.

Show the child a row of evenly spaced dr sand have him put a counter on each of the do-3.Ask him to take these counters and make a pat-tern or arrangement. Then have him match an-other set of counters to the dots solhat he nowhas another set with just as many counters in it.Say, "Try to-make these look different." Hereare three possible patterns using six counters:

Show the child different arrangements ofsix or more blocks. Ask how the different ar-rangements are alike or different. Challengethe child to build a different.arrangement withthe same number-of -blocks.

Show a linear pattern Of.red and blue cut-outs such as-this one

Cover one of the cutouts like this:

Ask, "What is covered? How do you know?"

Cover more than one cutout. "What color arethe ones that are covered? How do you know?"

112 CHAPTER FIVE

Eu -Pq 1160,40___000-0-otShow pieces of paper in a linear arrange-

ment and begin a pattern on the papers withblocks:

zg-7

Ask a child to put blocks into the next "car," oronto the next paper. "How do you know howmany to put there? Why did you use that manyblocks?"

Show the children eight block "buildings"and two trains like the ones shown in E. Thengive them the following problem: "All thehouses have to be moved at the same time. Thehouses on each train must be the same in someway. How will you load the trains?" The childrencan choose to sort on the basis of color, shape,or numerousness.

Begin making an arrangement on ageoboard with differently shaped colored'beads or spools (0). "Which bead comes next?Where do we put it? How do you know?" Thesequence used can involve spatial arrange-ment only or color and shape.

° °

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summary

AIM

Considerable attention has been given in thissection to activities in which children can en-gage before they are able to recognize andname number. None of these ideas or activitiesis trivial. They all contribute to the child's first-hand experience and allow him to store im-ages, meanings, and language that he canbring to later, more symbolic learning experi-ences.

Teachers of young children should be able todefend a mathematics program that does notbegin with the naming of number, counting,and the recognition of numerals. Work withnumbers and number names can be mean-ingful only to the extent that the child is allowedto accumulate meanings for the symbols used.

developing the conceptof number

number awareness

The young child's awareness of numberseems to parallel the historical development of

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mankind's concept of number. In response tothe question of "how many," primitive mancounted only to twoany set beyond this levelwas dismissed as "many." In a spurt of explora-tion, a two-year-old seeks out a balloon andthen scurries off to find another balloon. Heproudly reports "two balloons" as he exhibitshis collection. He announces that there is "onecar" as he peers out the window and sees a carin the driveway. If he is playing surrounded bytoys and the toys are removed one at a time, heseems to show no concern until only two orthree remain. At that point, however, anychange in the location of one of the remainingtoys causes him to investigate what has hap-pened to it. A year later he continues to reporthis age by holding up his fingers, changingfrom two fingers to three. Also, he may reportthat there are two toys in his tent without havingthe set in sight as he relates the information.Throughout these early years, the child asso-ciates number names with collections of ob-jects just as he associates the word milk with aglass of milk, cookie with a certain food object,red with a certain item of clothing, and so on.

A premature effort to teach the child to countmay have the undesired effect of teaching himto associate number names with single objectsrather than with sets of a certain number. Forexample, in response to the question, Howmany cookies? a child may count "one, two"and report two cookies. (We are pleased withhis response.) Later, when he is directed to puttwo toys into his toy box, he counts "one, two"but carries only the second toy to the box. Al-though the first experience does not clearly in-dicate what, he is thinking, the second indicatesthat he is associating a number name with eachobject much as he associates a name with hismother, pet, toy, or food. If early experienceshave caused the child to attend to the numbersone, two, and three superficially at the level oflanguage and counting, then there is need foractivities to associate meaningfully and con-ceptually the language with perceived sets forthe purpose of naming the numbers of thosesets.

Readiness for the systematic development ofthe concept of number is provided by devel-oping the child's ability to perceive a collectionof objects as an entity, that is, as a set of ob-

EXPER!ENCES FOR YOUNG CHILDREN 113

jects. This is followed by experiences to ascer-tain the number relationas many as, morethan, fewer than between two or more sets.These relationships, as described in the pre-vious section, are established by pairing mem-bers in the sets.

The numbers 0-4. Sets with zero throughfour members are called "elemental" sets be-cause their number can be seen, and named,without counting or partitioning. This tie withperception makes them special sets and thenumbers zero through four special numbers inthe young child's learning. The imagery andknowledge accumulated for these numberscontribute to the concepts that the child is ableto build for numbers associated with sets of fiveand more members. It is important that thechild have a variety of experiences with setswith zero to four members.

A concept of cardinal number evolves from aknowledge of specific numbers. Because car-dinal number is the property of equivalent setsand because the body provides so many modelsets for twotwo hands, two eyes, two ears,two feet, two knees, two elbowsit is sug-gested that the first number considered be two.

Arrange several sets of two objects eachon a display board (magnetic board or flannelboard), table, or floor. Ask the children to de-scribe each set. Then ask what is alike about allthe sets. (There are just as many in one set asin the others.) "Flow maro, ?" The children mustlearn to name the "manyness" two. The num-ber is not identified, however, by counting"one, two." The number of each set is two.

Have the children identify or-collect sets oftwo objects in their classroom or pictured instory books or magazines.

Have -the children draw pictures of sets oftwo objects.

Have the children paste on sheets of papertheir collections of pictures showing two ob-jects.

From a collection of objects, have the chil-dren form sets of two. Ask them to describeeach set formed and tell the number of the set.

114 CHAPTER FIVE

Children learn to recognize the spoken name"two" and to associate It with sets of two just asthey learn to recognize other spoken words andto associate these words with certain objectsand actions. However, written symbols, eithernumerals or words. should be used cautiouslywith the preschool child. Recognition and nam-ing numerals should be restricted to those nu-merals for which meaning specifically has beenbuilt. When children do have the imagery andlanguage for twoness, the numeral "2" can beintroduced Children learn to associate mean-ing with the symbol. Certainly they should notbe expected to write numerals themselves dur-ing these early years.

When the children can identify the number ofsets of two objects, sets of three can be consid-ered.

Show two sets of two objects. Have thenumber o! each set named. Change one set byputting in another object and ask, Is the num-ber of this set (the changed set) two?" By com-paring this set with the other set they can seethat the new set has more members and thushas a different number. Form sets of three.

Extend the activities for twoness to de-velop the meaning of threeness. Also, sincenumber is independent of arrangement, besure to vary the arrangement of the objects.

The knowledge of twoness and threenesscan be used to introduce one and four. Al-though work with sets of one member mayseem trivial, it is important that children think ofa single object as being a set whose number isone and that they relate the cardinal character-istic (numerousness) of a set of one to thosesets whose numbers are two, three, and so on.

Display a set of three objects and ask thechildren to name the number of the set. Thenchange the set by removing an object and ask

q

the children to report the number of the set.Change the set again by removing another ob-ject.

Show a set with a single object. Name theset and identify the number of the set as one.Have the children find other sets with onemember from a collection of sets having one tofour members. Have them name each set andits number. "This is a set of apples. The num-ber of the set of apples is one."

Ask, "What is the number of the set of pi-anos in our room? What is the number of sinksin our room?" Have children look for anotherset with one member, such as the set of clocks,paper cutters, soap dispensers.

Changing sets from one to two to three tofour objects by putting in another object leadsto other activities related to fouriless.

Give each child three objects. Have sev-eral children tell how many they have. Giveeach child one more object. Establish that eachchild now has fou' objects. Have the childrenpick up the four objects in their hands and tellhow many they have in each hand. Have themput the objects back together and tell howmany there are in all.

Give each child two small containers andfour small objects. Have them put their objectsinto the two containers and report how manythere are in each container.

Give each child four objects and havethem make as many arrangements as possible.Have children describe their arrangements as,for example, "I put one and then two more andthen another one" or "I put three down and oneon top."

The idea of a set having no members is notforeign to the young child. He has seen and ac-cepted the significance of an empty plate whensomeone has said regretfully. "No morecookies," and he has seen his empty cerealbowl as his mother has approvingly said, ''Allgone." What is not familiar to him is the use ofthe word zero to tell how many cookies there

are when there are "no more or to tell howmany cereal flakes are in the bowl when he haseaten them allthat is, to name the number ofthe empty set.

Begin with a set whose number the ,:hildcan nametwo, three, or four cutouts of a rab-bit on a flannel or magnetic board. Follow thesame procedure that was used when focusingon the set with a single member. Each time, af-ter removing a rabbit from the set, ask the chil-dren to tell the number of the new set. Repeatuntil all the rabbits have beer removed. "Howmany rabbits are in the set on the board now?Yes, you are right, there are no rabbits left.There are no members in the set. The numberof rabbits on the board is zero. There are zerorabbits on the board." Repeat with other sets.Then ask, "What is the number of the set ofreal, live elephants in our room?" Ask childrento name other sets whose number is zero.

After developing the names for the numberszero, one, two, three, and four, continue withactivities that reinforce these ideas and enablethe children to establish firmly the concepts ofthese numbers and the recognition of numeralsif the symbols have been provided. Little is tobe gained from teaching numbers greater thanfour too soon, before meaning and languagefor zero through four are firmly established. In-formal activities like the following can be usedto attain this goal:

Have the children sort picture cards intoappropriate boxes that show model sets. (La-ter, the boxes can be labeled with numerals.)

p--,i

0o

oo0

0 ,la


Attach model sets or numeral cards to.pockets or boxes and have the children depositthe number of cards (or beans, stichs, etc.) in-dicated .by the numeral.

Use interlocking blocks to provide practicewith matching model sets and their numbernames Ask questions about the numerousnessof the sets and the numerals fitted together todirect the children's attention to the purpose ofthe task.

The number 5. Some children may recognizesets of five as elemental sets, but others mustbuild the meaning of five from their perceptionof the subsets of two and three or one and fouras they partition a set of five. A model set canbe introduced by starting with a set of four, asillustrated in the following activity:

Display a set with four members. Have thechildren name the set and its number. Thenchange the set by putting another object in it.Ask if the number of the new set is four or ifanyone knows the number of this set. Their re-sponses to such questions show how familiarthey may be with the number five. Reaffirm ortell them that the number of this set is five. Havethem use it as a model set to find other setswhose number is five.

Give- each child a set of five objects andidentify the number of the set. Ask the childrento pick up a subset of a given number and toobserve how many are in the remaining set. Dothis until they are able to state from some givendisplay that a set of five can be a set of threeobjects and a set of two objects, and so on. Af-ter the objects they picked up are returned tothe original set and the number of the totalgroup is again identified, continue until the chil-dren, using the imagery built earlier, can cor-rectly respond to the question, "If you pickedup three blocks, how many would be left?"

Give each child a collection of six to eightobjects and ask the children to form a set of

1 1

116 CHAPTER FIVE

two objects. Then ask them to put more objectswith that set to make a set whose number isfive. Repeat the activity, starting with a set ofthree, a set of four, and a set of one.

If the children have shown any confusion orhesitancy about the numbers zero through fiveand their symbols, the teacher would be welladvised to go no further than five in the devel-opment of numbers at this time. Using addi-tional activities can provide greater experiencewith sets having zero to five membeis, the as-sociations of numbers with those sets, and therecognition of the numerals if they have beenintroduced. It should be remembered that achild's perception is a potent factor in the nam-ing of number in sets with five or fewer mem-bers. For sets having more than five members,the child must rely heavily on his cognition, thatis, on what he has learned about the elementalsets as he learns to associate numbers withsets of more than five members. Many five-year-olds need to engage in a variety of activi-ties involving zero through five before being in-troduced to numbers greater than five.

The following activity not only will allow thechild to use the numbers he already knows butalso will contribute to an experiential back-ground on which he can build the idea of base,a concept fundamental to our numeration sys-tem.

If the child can the number for setshaving at least three objects, show a set havingfrom three to eleven membersfor instance, aset of ten pencils. "Let's find out how manypencils we have. Can you show fie a set ofthree pencils? Put it here. Is there another setof three pencils? Put it here. Can we make an-other set of three pencils? Put it here. Look, wehave three sets of three pencils and one pencilleft over--three sets of three, and one left over.That's how many pencils we have."

Repeat with other sets of three to elevenmembers until the child can perform the group-ing and can describe it. Give each child a set ofeight objects. "Show me how many things arehere by making sets of three. Tell me how many

there are. Good, there are two sets of three,and two left over."

Some children may need more guidance."Do you have enough to make one set of three?Do it. Do you have enough to make another setof three? Do it. Do you have enough to makeanother set of three? How many do you haveleft over? How many sets of three do you have?How many left over? Yes, you have two sets ofthree, and two left over."

For example, a child's arrangement of eightdolls might look like this:

j14

Using equivalent sets and "leftovers," a childwho has number concepts only to five canname as many as twenty-nine objects as fivesets of five, and four more. Even if he can rec-ognize the number of only those sets having nomore than three members, he can name elevenas three sets of three, and two more. The use ofequivalent sets and leftovers is analogous tothe use of base in our numeration system.When the child can communicate number insuch terms, he will be prepared for communi-cating number in terms of tens and ones."Place value" in two-digit numerals would thenbe a way of representing something the childalready sees as meaningful.

The numbers 6-10. Sets associated witheach of the numbers six through ten should bemanipulated and partitioned in ways similar tothose suggested for the number five. The

teacher should move slowly in introducingthese numbers. It is difficult for five-year-oldsto establish strong images of these numbersbecause they cannot name the number withoutcounting or pE-:itioning. Then, too, the num-bers zero through ten are the building blocks ofour numeration system, and only as they aremeaningful can numbers greater than ten bemade meaningful. The children should, there-fore, be given ample opportunity to establishmeaning for each number before being in-troduced to the next number.

counting

In order to develop counting as a meaningfulprocess, the child must be asked to count onlywith number names that are meaningful to him,the order of the counting numbers must beseen as having some logical basis, and theprocess of counting must be introduced in sucha way that it is seen as naming the number ofsuccessive sets.

Because sets with five or more members re-quire counting or partitioning for their numberto be named, the process of counting shouldfollow the establishment of meaning for one,two, three, four, and five. It is assumed that or-dering as a process has been previously devel-oped, and that the children have had experi-ence with ordering sets so that eachsucceeding set has "more members" and also"one member more" than the set it follows.

The following activities give the child an op-portunity to conclude that numbers appear inthe counting order in such a way that each suc-cessive number is one more than the precedingone.

Display in random order sets with one,two, three, four, and five members, as shown in111

Have the children begin with the set of oneand order the sets so that each set has onemember more than the one it follows. Have thechildren tell the number of each set. If the nu-


eeN/1

merals have been introduced earlier, havethem displayed by each set (0).

Ask the children to look at the display andtell how many are in the set having one membermore than the set of three. Continueivith otherquestions that call attention to the "one morethan" and "one fewer than" relationships be-tween two adjoining sets.

z3LI

1

Give children boxes containing pairs ofsets with no more than five in any set, with oneset having one member more than the otherset. Have each child arrange his sets to see ifthey have the same number of members, or ifone set has more members than the other. Achild's arrangement might look like° .

Ask, "How many are in each of your sets?Which set has more members? How manymore?" After one child has reported that he hassets with four members and three membersand that the set with four has one membermore than the set with three, ask each of sev-eral other children who have sets of four and

1-'1)frv,

118 CHAPTER FIVE

41 0

0three if his set of four also has one membermore than his set of three. Perhaps find othersets 4.h four members and three members inthe room and verify that all sets with four mem-bers have one member more than the sets withthree members:

Releated orderings of sets with one throughfive r. embers so that each set has one membermore than the set it follows will always producethe same order of sets: a set of one is alwaysfollowedlby a set of two, which is always fol-lowed by a set of three, and so on. Since thenumber of each set is named in succession, theword order is always "one, two, three, four,five."

Again display, in a random arrangement,sets with one, two, three, four, and five mem-bers. Have the children order-the sets so thateach set has one member more than the set itfollows. Then have the children name the num-ber of the sets in order. Repeat with other sim-ilar orderings, each time asking the children tolisten to the order of the words.

Such activities provide the ch:ld with the or-dered set of number names. The next activities(1) apply the ordered set of number names tothe members of one set, (2) emphasize thatcounting names the number for the sets so farcounted, and (3) establish that the order in

which the objects are counted does not changethe accuracy of the count.

Conceal in your hand or in a container aset with five members. "I have a set of tops.Let's find out how many there are." Take onetop out and ask, "How many? Good, there isone." Take out another. "How many are therenow? Good, there are two." Continue until thecontainer is empty. "There are five. We countedand found that there are five."

Repeat with other sets of four or five, havingthe children take out the objects one by one.Each time, after a set has been counted, de-scribe what has happened, noting the last num-ber name used and its significance.

There is a reason to begin counting with a setwhose members are not visible to the children.When the objects are taken out one by one, thechild each time is naming the number of the setcounted. When he says "one," he sees one.When he says "two," he sees two. By contrast,when the first counting experiences are with aset whose entire membership is fully visibleand the child points to the objects as the count-ing progresses, it may seem to him that thenumber names are applied to individual ob-jects and that a particular object is, for in-stance, named three.

The child's first experience with countingfully visible sets should be with sets whosenumber he already knows, and it should assurehim that if he uses one number name in thecounting order for one object, he-can count theobjects in any order.

Give the child a set whose number he cansee and name and whose members are distin-guishable (perhaps four blocksa red one, ablue one, a green one, and a yellow one). Lethim experiment freely with counting his blocksseveral timeshe can put them in different ar-rangements as he counts, he can begin withdifferent blocks, he can proceed with a differ-ent order of blocks. He knows there are four, so

this experience is not counting to find out "howmany" but rather to experiment with differentways of counting the same set.

Let the children see the teacher "make amistake"either failing to count objects orusing more than one number 'name for an ob-ject so that the last number does not name thenumber in the set. When the number in the setis readily perceptible, they can confidently tellher she is wrong and how and where she went

.wrong.After experiencing a variety of activities that

use counting to name the number for setswhose number is already known, children de-velop confidence in using the process of count-ing to answer the question, How many?

As counting is used with sets with moremembers, the teacher can reinforce that anumber name is associated with successivesets by moving the objects apart as they arecounted or indicating the sets,with her cuppedhand, as in

It is easy enough to .find opportunities forcounting, but they should be restricted to thosethat require using only the number names forwhich meanings have been built. Counting tofind how many children are in attendance in aclass of twenty would require most pre-first-grade children to use words for which theyhave no meanings.

A child should be encouraged to vary hiscounting procedure by naming the number ofan elemental set or some set whose number isknown to him and then counting on to find thenumber of the set.

Show a child a set of nine members ar-ranged in groups. Seeing, perhaps, a set ofthree members together, the child could namethe number of that set, "three," and count on."four, five, six, seven, eight, nine." If he knowsthat a set of three and a set of two is one ar-rangement o' a set of five, he could start bynaming the number of the set of five and thencount on: "six, seven, eight, nine." Childrenshould note that they can name the number inthe set, nine, by using either procedure.

'EXPERIENCES FOR YOUNG CHILDREN 119

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Many older children join sets and count fromone to find the solution to any addition equa-tion. With practice in "counting on." these chil-dren could name the number of the starting setand count on through the set that "came tojoin" to find the number of the union set. Thisprocedure would help them retain the numberof the starting set, attend to the whole equation,and, perhaps, remember the addition fact.rather than simply stating the number theyfound from a process of counting.

k 4

120 CHAPTER FIVE

summary

The focus in this section has been (1) on de-veloping meaning and names for some wholenumbers and (2) on counting. These devel-opments should come only after much experi-ence with other mathematical situations andideas.

The suggestion was made that many five-year-olds should be introduced only to thenumbers zero through five. Because the num-bers zero through ten are basic elements of thewhole numbers, it is important that childrenhave special and individual concepts for eachof these numbers. Additional suggestions foractivities with zero through ten and for begin-ning concepts related to the numeration sys-tem can be found in later chapters.

activities with numbers

When children have acquired stable con-cepts and language for some numbers, suchactivities as classifying, ordering, and measur-ing can include the use of number. Using num-ber in these activities not only allows for theirextension but also provides practice with thosenumbers with which the children are familiar.

classifying, comparing, andordering

Because activities with classifying, com-paring, and ordering are not changed qualita-tively with the introduction of number, thesethree areas are considered together in this sec-tion. When children can recognize and namem..mber, they can use it to describe objectsbeing classified, compared, or ordered. Theyalso can use number in the elescription of therules for these activities. (Only numbers withwhich the children are familiar should be used.)

Have the children put together all the setsthat have the same number.

Display pictures and have the children puttogether thosepictures that show sets with thesame number.

Have the children sort sets.of objects sothat in one location are sets with, for example,eight members, in another location are setswith fewer than eight members, ancrin anotherlocation are _sets with more than eight mem,bers.

Have the children put together objects thathave the same number of sides (corners, faces,edges).

Show the children a set and,have them findanother set with one (two, three) member(s)more. "How do you kniii0 it has one membermore? Show me."

Show sets of different .number and ask,"Which set has more members? How manymore are in this set than in that one?"

Again show sets of different number andhave the children order them.so that each setafter the first has three (two,. one) membersmore than the sett follows.

Show the child two numeral cards at atime. Have him read them and arrange them inorder with the one with the greater number onthe right. Be sure to use only numerals that thechild knows.

Repeat the preceding activity using morethan two cards or varying tile order.

Teachers should be aware of the numberswith which the children have h,ad experienceand take every opportunity to use those num-bers in the everyday classroom routines.

measuring

Being able to use numbers makes a signifi-cant difference in what children can do withmeasurement. When children can recognizeand name number, they can use, and describethe use of, a repeated unit. They are no longerlimited to saying, This thing is longer than thatone." With number, they can say, This thing isalmost as long as three of those." The use of a

repeated unit is basic to the process of measur-ing, and young children can be ingenious in de-vising such units. (Exploring what constitutesan appropriate unit might help prevent the con-fusion frequently seen in older children whenthey must choose appropriate units for areaand volume and when they try to understandour standard units. However, neither the use ofstandard units nor formal information aboutequivalencies among those units seems appro-priate for the preschool child.)

Linear measurement. In the sections oncomparing and ordering, only height was con-sidered. As children move into activities relatedto measuring length, they should be introducedto the more general process of measuring aline segment. Both height and length involvelinear measurements. One speaks of how longa block is when it is placed in a horizontal posi-tion; it is measured for height when it is in a ver-tical position.

As children build with cubical blocks,stand another, longer block by the stack andhelp them make the statement that is appro-priate: "This block is not as long as the six youhave stacked together," or "It is a little longerthan four of them."

Have one child stack blocks to see ap-proximately how tall another child is. Then havethat child lie down and the same blocks laidend to end to find out how "long" he is. In-troduce the language for describing that he islonger than, about as long as, or shorter than, aspecified number of blocks.

Have the children place erasers, blocks, orany other objects of reason-able size along theedge of a table to see if it is longer than someother piece of furniture in the room.

Show a picture such as the segments in A(U). Ask, "On which 'line' could you put themost candies (blocks, beans)?" Repeat whenone "line" is broken, as in B.

Ask the children to draw a line that is twoblocks long (or as long as three of these can-dies, etc.) and to check to see how close theycame.


A.

B.

Weight

Supply a balance beam. Have the childrenput an object in one pan and balance it asclosely as possible with some other, lighter ob-jectsperhaps one-centimeter cubes, metalnuts and bolts, marbles, or ball bearingsanddescribe what they see: "The is almostas heavy as five of these."

If possible, have' available a spring scalewith no markings on it and with the spring en-cased in a clear plastic cylinder so that the.chil-dren can see how far down an object pulls themarker. Have them decide how many lighterobjects it takes to pull the marker that far. Theheavier object is "almost as heavy as" or "alittle heavier than", or "just-about as heavy as"some number of the, lighterobjects.

Have the children record weight com-parisons using repeated units as shown in O.Let them place actual objects on the chart. Indi-vidual children or teams of children can workindependently, or the.work can be done as a di-rected activity.

is about as heavy as

0 00 0

0 00 0

; 0

122 CHAPTER FIVE

CapacityWith some pourable material, such as liq-

uid or sand, and a collection of containers, in-cluding several of like size and shape, give chil-dren opportunities to pour from a largercontainer into several smaller, similar contain-ers and describe what they have discovered.The big jar holds "as much as," "a little morethan," or "a little less than" some number of thesmaller jars.

Have the children order objects on thebasis of their capacity after they have filledthem with repeated use of a smaller container."The tall jar holds as much as two cups. Theblue jar holds as much as three cups. The canholds as much as four cups. The tan jar holdsthe least, the blue jar holds a little more, andthe can holds the most."

Have the children record comparisons ofcapacity in the same way they did comparisonsof weight, using real objects or drawings, or letthe children express the comparisons in narra-tive form.

organizing and presenting data

With the ability to recognize and name num-bers, the child no longer needs to rely on one-to-one correspondence to organize andpresent data. Quantities can now be repre-sented numerically rather than pictorially. Incontrast to the earlier suggestions for showingthe number of kernels in a peanut shell, con-sider how that data might be organized if thechildren can count to five or six and can readthose numerals.

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ofpeanuts

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Nupnber of kernels

0

Have each child carefully open a peanutshell and count out how many kernels are in it.Have them assemble all the peanuts with thesame number of kernels and count to see howmany are in each group. Record the data on achart in narrative form, as shown here.

3 peanuts had 1 kernel.5 peanuts had 2 kernels.5 peanuts had 3 kernels.2 peanuts had 4 kernels.1 peanut had 5 kernels.Most peanuts had 2 or 3 kernels.Fewest peanuts had 5 kernels.No peanut had more than 5 kernels.

A graphic representation of the same,informa-tion is shown in 0

shape and pattern

The use of number allows for different de-scriptions of shapes and patterns. For instance,the teacher and children can now talk about"the objects with three sides" or "the patternwith two blue blocks and three green blocks."Children can be asked about shapes and pat-terns with questions that require them to recog-nize and name number.

Show the child a rectangular array of dots,staying within the limits of numbers that aremeaningful to him and that he can count. Havehim put a counter on each dot. "How manycounters are.therel-How do you know?"

Using the same arrangement of dots, put acolored rubber band around some counters orused colored counters. Give a number name tothe red counter and have the children guessfrom which corner you started to count. Con-tinue to ask questions such as, "If the red one is5, what should we call the black one? If thepink one is 3, what should we call the blackone now?"

00

0 0 0

0

"What would the names for the black one beif we use each of the following patterns?"

Yw .

Show the child a pattern of numerals andhave him decide what comes next. (Use onlythose numerals that are meaningful to him.)

L!T21 3 2 is '4 3 2F-1

Show one of the following patterns, or asimilar one:

-

Have the children put a counter or block oneach dot. "How are the three parts alike or thesame? How are they different? Can you makeanother part that will look like these but be dif-ferent? Another?"

Show the child one of the following se-q uences:

A

L1


0

"Can.you make the next one?"

Begin a pattern with blocks so that therows and columns vary in size, shape, or color:Have the child find the appropriate blocks forthe empty places and tell how ha knew whatshould be used. After the pattern is"completed,cover or remove one object and ask the childwhat has been removed from view.

IA* 11111III MEI"

summary

The activities suggested in this section areonly a few of the many kinds of things that canbe done with young children who have ac-quired concepts and language for some wholenumbers and who can count meaningfully. Inlater chapters, the reader will find extensions ofthe areas considered here and additional activ-ities for children who are ready for them.

concluding remarks

The activities in this chapter are designed forchildren who are ready for directed study. Al-though this directed study need not take placein a group setting, most children work well ingroups. For children who are at ease in theschool situation, who are self-directed, whotrust the adults in charge, and who have suf-ficient language to carry on dialogues and con-versations about the planned experiences,

124 CHAPTER FIVE

large-group instruction has some advantages.The social setting of large groups can be used(1) to foster group consciousness, social skills,and communications skills, such as those ofrepartee and argument; (2) to increase pupilexchange and minimize teacher domination;and (3) to build a sense of responsibility andappreciation for each other. If a teacher plansto work with groups as large as twelve to twentychildren, she must develop her skills of differ-ential questioning and task assignment to alloweach child to participate successfully at his ownlevel; learn to take full advantage of.the inter-action between children having different abili-ties, experiential backgrounds, and motivation;and make a careful selection of activities andmaterials to avoid having many children lookon while only one or two perform a task.

For younger children, and for five-year-oldswho need more language practice and who areless secure in a group and with the adults in theschool, small-group instruction can be moreprofitable. Small groups allow more interactionwith the adult, thus providing a more depend-able language model; offer each child more op-portunity for language practice; and offer theteacher an opportunity to observe more care-fully each child's response to the tasks givenhim.

Since it is possible for a child to manipulateand observe materials without perceiving themathematical significance of his actions, it isessential to have in any program verbal inter-change between the learner and some know-ing, verbal person. Language can be acquiredonly in dialogue with someone who possesseslanguage.

When directed instruction in mathematics isa part of the school program, it needs to be bal-anced with self-initiated activity, that is, timeswhen children may independently and'individ-ually manipulate and observe materials andperhaps record their observations. Even whena carefully planned, sequential program in-volving directed group instruction is used, theteacher should not overlook the tremendousvalue of the child's play with materials in self-chosen activities.

No suggestion has been made for using be-haviorally defined objectives or for developing

a sequence of carefully determined and mi-nutely ordered abilities. It is hoped that whenprovision is made for learning that is insightfulrather than totally incremental, at least somechildren can experience the thrill of a "flight ofintuition." By not employing mastery tasks withthe very young, teachers can prolong their op-portunity to learn from their involvement andfor the fun of it. When the ability to do and tosay is considered an integral part of eachlesson or teaching episode, the teacher hasample opportunity to make judgments abouteach child's understanding and his acquisitionof new knowledge and language. Progress be-tween children and even within each child willbe uneven, but this will not prevent children ofvarying abilities and backgrounds from re-sponding satisfactorily to a common experi-ence.

It is suggested (1) that mathematics instruc-tion for young children emphasize concept de-velopment and language acquisition, (2) thatinstruction be planned to allow the child maxi-mum use of oral language, (3) that a primarygoal of instruction be to build the child's con-fidence in his perception and cognition and in.his powers to seek out information and drawconclusions from it, (4) that movement to newlanguage, and especially to written symbols, beunhurried, and (5) that instruction be plannedto allow and encourage the perception, de-scription, and extension of patterns, for "themathematician, like the poet and painter, is amaker of patterns."

reference

Bruner, Jerome S. "The Growth of Mind." American Psy-chologist 20 (1965):1007 -17.

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How old am I? When is my next birthday?How much do I need to grow to be as big as

you are?How many are coming to the party?Who has the most -=you or me?What lime is my TV program? What channel?

These questions reveal personal needs chil-dren have for number ideas, for number lan-guage, and for number symbolism. Childrenencounter problem situations that requirequantitative solutions and number responses.Parents ask children quantitative questions.Number ideas and language permeate chil-dren's television programs.

At times the use of number ideas and lan-guage is indirect and unplanned. At other timesthe content is systematically planned andtaught to children. In either event, the need fornumbers and the language and symbolism ofnumbers is universal.

early stages with number

Children very probably begin their numberand quantitative notions with comparisons ofsize. Early in life the child is aware that he is notas big as his mother or his older brother. Heknows he cannot reach the top of the cabinet toget a drink of water. These size comparisonsare not very precisely delineated for the child.At times the child gives attention

to length"You are higher up";to area"My hand can't cover yours";

ii in A;il 4 0...,

How many pieces of candy?

NUMBER AND NUMERATION 127

How old are you? Du I have enough& buy the toy?

to weight"I can lift the cat but not thedog";to volume"That pile of dirt is bigger."

These early perceptual comparisons help chil-dren develop the vocabulary needed for ex-pressing quantitative ideas. For example,words such as same, more, not as many, just asmany, higher, lower, bigger, and smaller arisenaturally in such situations.

Whatever the features chosen, some of theminvolve comparisons between "mounts that areabout the same. "We have the same amount ofcookies," a child might say, thinking either ofthe quantity or of two sets that match one toone. It is from the idea of matching one to onethat number idea emerges.

The main component of the concept of wholenumber is the classification of sets that are inone-to-one correspondence. If the elements oftwo sets match one to one, then the two setshave the same number. One set has just asmany as the other. When a child holds up thecorrect number of fingers in response to thequestion, "How many pieces of candy arethere?" he is showing the essential feature ofthe attribute called number.

The concept of one-to-one correspondencerequires that other features, or attributes, beexcluded. The texture, the feel, the softness orhardness, the color, the length, and the posi-tion must receive no attention. Consequently,initial developmental activities should aim tohelp children exclude such irrelevant features.Chapter 5 provides many suggestions for thisinitial experience.

If two sets match one to one, they do havethe same number of members. The specificnumber of members, however, is of use mostoften, and it needs a name, both oral and writ-ten.

The oral name "two" or the written symbol"2" may get attached simply by association toany set that corresponds to the eyes of a nor-mal person. Enough examples may have oc-curred for the name to come to tell how manyare in a set. Likewise, the name "three" and thesymbol "3" may get attached to the wheels of atricycle. With "two" and "three," counting maynot have been necessary for the child. Count-ing will, however, likely accompany the learningof the names.

Counting is itself an example of one-to-onecorrespondence, though more abstract thanmatching two sets of objects. In counting, acorrespondence is made between objects andthe number names. Wilder, a mathematicianwell recognized for his work in foundations ofmathematics, states this idea clearly (1968, p.34):

True counting is a process whereby a corre-spondence is set up between objects of the col-lection to be counted and certain symbols. ver-bal or written. As practiced today, the symbolsused are the natural number symbols 1. 2, 3,and so on. But any other symbols will do. tallymarks on a stick, knots on a string, or marks on

paper such as /i are all sufficient

for elementary counting purposes. Counting is,then, a symbolic process employed only byman, the sole symbol-creating animal.

128 CHAPTER SIX

In counting a set of five members, one objectis matched as each name is said. one, two,three, four, five.' The last name said tells thenumber of members in the set counted. Thusfwe identifies the last object counted as well

as the total number in the set. More is said onthis later.

Learning to distinguish those transforma-tions of a set that leave the number of its mem-bers unchanged from those that change thenumber of members is of major importance inthe learning of number in the early stages. Thetransformation most often reported +. a changein position of the objects in a set. When a childviews a change of position as not affecting thenumber, the child is conserving that number.

conservation

More has been written on the topic of con-servation than on any other aspect of the youngchild's mathematical development. Piaget andhis associates have written extensively aboutthe topic. In turn, their initial findings and con-

lectures have led to extensive research and vo-luminous writing by psychologists and educa-tors in the United States and other countries. Itis difficult to interpret the total effect that all thiswork has had on educational practice. It doesseem certain, however, that one result is betterunderstanding of what children know at variousages.

The age at which conservation occurs, thereasons for lack of conservation, and the ef-fects of training have been studied extensively.The classic conservation task for number be-gins with two sets of objects, the same numberin each set. Through questioning, it is deter-mined that the child recognizes that the setsmatch one to one. In the initial setting, lan-guage such as "same." "just as many," "onehere for each one there" is used. Then the ob-jects in one of the sets are rearranged, and thechild is asked, "Do both sets have the samenumber of things, or does one have more thanthe other?" When the child recognizes that therearrangement of objects does not change thenumber. the child is said to conserve. Piagethas said that conservation occurs at about agesix or seven.

Task I. Before irciniforniailion

123

In a study of conservation, Miller, Held-meyer, and Miller (1973) found that perceptualclues affected conservation, and they pointedto the developmental nature of conservation.They studied 64 children ranging in age fromthree years and one month to five years and tenmonths, with a mean age of four years and fourmonths. (Sixteen children were excluded fromthe study because they did not have facility withthe language of "same number," "more," "notas many," and "just as many.") On task 1, theexperimenter conversed with the child to em-phasize that the animals were the same kindand that a cage of one color was for the childand a cage of another color was for the experi-menter. The purpose was to provide the childwith perceptual clues that emphasized one-to-one correspondence. The questions askedwere, "Do we both have the same number ofanimals or does one of us have more animals?"After the child replied, he was asked. "How didyou figure that out?"

On task 1, 77 percent of the children an-swered the conservation question correctly,and 50 percent of the children could give anadequate explanation of conservation.

In task 2. flat beads were arranged in a verti-cal line before and after the transformation. Noperceptual clues were provided. Only 41 per-cent of the children were correct on this typicalconservation task, and only 22 percent wereable to give an adequate explanation of con-servation. It seems clear that perceptual clueshelped more children perform the conservationtask correctly.

Task 2


§(q.O 0O 0

8°0O 0

Before Iraq'dim

000

00008°0000

cTask 2 After iransfortPallon

Six other conservation tasks, having diffi-culty levels between the two tasks described,were included in this study. The tasks variedthe use of color, the kind of objects, the num-ber of objects, and the linear or nonlinearplacement of the objects.

The experimenters found that any responsemade after the arrangement of objects wastransformed by length substantially reducedthe number of children who conserved. Lengthwas a major perceptual distraction. The kind ofobjects, however, seemed to have no effect,

Task 1. After fraits-jormafion;

I 04

130 CHAPTER SIX

and the effect of the number of objects was notconclusive, although for objects that seemed togo together, more children conserved with fourobjects than with eight objects. When objectswere arranged linearly, about the same percentconserved using four objects as eight objects.which suggests the overriding influence of thelinear arrangement.

The children in this study demonstrated a be-ginning understanding of conservation of num-ber at an earlier age than that generally sug-gested by the literature. For example. 17 of the20 three-year-olds made at least one con-servation judgment, and 15 of these childrencould also supply at least one adequate ex-planation. (Miller, Heldmeyer. and Miller 1973.P. 91

Gelman. in an earlier study. found that"young children fail to conserve because of in-attention to relevant quantitative relationshipsand attention to irrelevant features in classicalconservation tests" (1969. p. 167). She alsofound that discrimination training with feed-back helped children with a median age of fiveyears and four and a half months who did notconserve. She provided tasks with objectsvarying in color, size and shape (large or smallrectangular or circular chips). starting arrange-ments (horizontal, vertical, and geometricallyarranged rows of chips or sticks), and numberof objects. When a task was presented to thechild, he was provided immediate feedback onwhether or not he was correct. The training andtesting took place in a three-day period.

At the end of the training. the children hadattained a performance level of approximately95 percent correct on number problems and onproblems involving length. On a test two tothree weeks later, the same results were found.

The importance of feedback was evident inher study. Children who were given the taskswith no feedback performed only slightly betterthan children in a control group that used"junk" stimuli, essentially a placebo treatment.

It seems clear that a key component in at-taining conservation of number is the attentionthe learner gives to one-to-one correspon-dence under different arrangements and theextent to which he can ignore irrelevant attri-butes.

knowledge of enteringkindergartners

Any planned mathematics program musttake into account the knowledge already pos-sesset oy the children. Helpful data have beenobtained by Rea and Reys. They took a crosssection of 727 urban children entering kinder-garten and assessed their knowledge of mathe-matics. Some excerpts from their results re-lated to number follow (1971, pp. 391-92):

Oral Names for Numerals

Numeral1

3

4

5

10

16

21

% of CorrectResponses

8371

71

71

3817

7

Sequence

What comes next? (Count orally: "one,two. three")

What comes next? (Count orally: "five,six, seven")

What comes after three?What comes before three?

Cardinal NumberGive me three disks.Give me seven disks.Give me thirteen disks.How many pins are here? (Show card

with three pins)How many stars are here? (Show card

with eight stars)

Ordinal Number

Point to the second ball.Point to the first tree.Point to the third ball.

Group ComparisonWhich group has more (Show four-.)spades and three buckets)

Is there a disk for each child? (Show fourdisks and four children)

90

887543

825534

92

72

487025

72

88

Among the conclusions drawn by Rea andReys are these (pp. 393-94):

Skills in counting and recognition of smallgroups were well developed by most of the en-tering kindergarteners.

125

Development in ordinal relationships has riotreached the level of competence displayed incardinality.

Most children were able to make accuratesmall group comparisons.

Another part of the study was reported in theArithmetic Teacher. There Rea and Reys re-ported that almost 40 percent of entering kin-dergarten children could count by rote totwenty and about the same percent could countrationally to twenty. more than 75 percent couldcount by rote beyond ten and about the samepercent could count rationally beyond ten(1970. p. 68).

activities for numbers0 to 10

Most children have some well-developedideas about number at the time they enter kin-dergarten. The kind of meaningful activities forthese children will depend on the backgroundof particular groups of children and particularchildren in a group. Overall, the activitiesshould be directed at the following five majorobjectives:

1 Matching sets. This includes matching ob-jects of two sets one to one to determine if thetwo sets have the same number. If the two setsdo not match one to one. then one set is identi-fied as having "fewer' and the other "more."

Rearranging objects should be perceived asa transformation that does not change thenumber of members. Removing or adding anobject should be recognized as a transforma-tion that does change the number of members.

2. Relating a set of objects. the oral name.and the written symbol to show "how many"(cardinal number). These relations can beviewed in this triangular arrangement:

A

Oral nameaihree"

Wriften SVP/160(

"3'/


Each of the three paired relationsset oralname, set -- written symbol, and oral name-- written symbolrequires two questions toassess the objective: (1) for example, given aset, the child is asked to give the oral name; (2)given the oral name, the child is asked tochoose the set. The child's production of thewritten symbol, 3." should come substantiallyafter the recognition stage for the numeral.

3. Ordering numbers 0 to 10. This includesarranging sets of from zero to ten members inorder, counting in order to ten, and giving thenames "one, two. three, ..." and "first, second,third...." to positions of objects in an orderedset (ordinal number).

4. Recognizing sets of from two to five mem-bers without counting. This is facilitated by pat-terned arrangements of objects.

5. Separating sets of up to ten members intosubsets and naming the numbers of membersin each. This is in preparation for addition andsubtraction and also for place value.

matching sets and the numbers 0to 10

Activities for matching should include thephysical pairing of real objects and subsequentquestions about the number of members. Atthe same time, the objects can be counted tofind out what the number is. This gives the childtwo dependable and varied ways to work withnumber. By making this variety available, theteacher provides the child with the flexibility ofthought needed in his subsequent use of num-ber.

Activities involving matching should beginwith natural situations where there is someneed for the matching. The language shouldsuggest the reversibility of one-to-one corre-spondence.

Match parts of the body. "Look at yourshoes and think about your feet. Is there a shoefor each foot? A foot for each shoe?" Then askabout the numbers of the two sets. "how manyshoes do you have? How many feet?"

132 CHAPTER SIX

Repeat with parts that do not match one toone. "Do you have a nose for each hand? Doyou have the same number of noses as hands?How many hands do you have? How manynoses?"

Use situations where matching seems nat-ural in the classroom setting. "From this box ofpencils, take one for each hand. Now is a pencilin this hand? In the,other hand? Is a hand witheach pencil? _Flow ,many hands do you have?How many pencils do you have in your hands?"

Vary the procedure and the way questionsare asked. "Which set of bracelets gives youone for each arm? How many arms do youhave? How many bracelets do you need?" ( .)

Vary the size of the objects used. Place asmall block of wood in each hand of a child. "Isthere a block of wood for each hand? Doeseach hand have a block? How many handshave you? How many blocks of wood are inyour hands?"

as small blocks, and hands have the samenumber as large blocks; so small blocks havethe same number as large blocks).

Use rows of equal-sized chips- (pokerchips are excellent). "Look at the two rows ofchips. What do you see that is the same?" Chil-dren may say something about color, about thesize of the chips, and almost certainly about thefact that they stretch out the same, that is, thatthey are the same length. Pursue the matchingand number questions. "Is there a chip in thetop row for each chip in the bottom row? Isthere a chip in the bottom row for each chip inthe top row? Count the row of chips on top.Now without counting, can you tell me howmany chips are in the bottom row?- Do I stillhave five chips in this row? How can you findout'?" To the last question, the child may matchto find out or he may count.

000000 0 0 00

To helpcue for number-instead of length,link each member of enetet to each memberof another set with-a piece (*masking tape orstring (buttons are durable and easy to use).

Use the larger bloc!'s of wood and repeatthe questions. Then ask further questions."Which blocks are bigger? Longer? Do theblocks match one to one? How many smallblocks are there? How mar/ large blocks?"

Activities like these help a child distinguishnumber from size, and they provide experiencewith transitivity (hands have the same number

"Look at the top row of buttons. Is a,string ineach button? Does each string go to anotherbutton? Is there a button on top for each button

on the bottom? Is there a button on the bottomfor each button on top? How many buttons areon top? Can you-tell- without counting howmany buttons are on the bottom?"

Spread out the buttons on the bottom andask similar questions to bring out the idea thatthe number does not change with, different ar-rangements.

As an extension of this activity, hold in yourhand three of the buttons in the bottom row."Can you begin herewith threeand countthe rest of the buttons?" Repeat with a differentnumber of buttons in your hand. This "countingon" strategy is a valuable one for children todevelop.

Contrast sets that match one to one andthose that do not. Develop the language of"more" and "fewer." "Look at the two sets at theleft Is there a pear for each apple? An apple foreach pear? How many apples? How manypears?" Then point to the sets on the right. "Dothe sets match one to one? Is there the samenumber of pears as apples? Which set hasmore members? Which set has fewer mem-bers?"

Ask similar questions using two rows ofchips with the chips in one row larger thanthose in the other row. This again brings out theneed to focus on matching instead of on thesize of the objects.

000 00CO 00

Make "Match Me" cards. For each numberone through ten, make a set of five cards withfive different arrangements of dots represent-


ing that number. Substitute a card from one setfor a card in another set. Ask the child, "Whichcard does not belong?"

1000 0

0 0 00

0 000

0 o

0 0

A child can answer the question by matchingthe dots visually or by counting the dots. Markthe back of the odd card with colored tape toallow for individualized work.

Use rhythmic activities for matching. "Let'sclap our hands together. Each time we do it.say, 'Clap.' Do we say something each time weclap our hands?" Variations include beats on adrum or other musical instrument or taps of thefinger on a desk. The sounds are matched withan oral response.

Matching members of one setwhetherclaps of the hands, sounds on a musical in-strument, or objects themselveswith verbalsounds is one essential part of oral counting.Oral counting can also be done as hands areclapped or as beats are made with fingers or amusical instrument. As an extension of this ac-tivity, tap your fingers three times and have thechildren count silently. Then tell the number.Repeat with other numbers and have the chil-dren say the number. Challenge the children togo as far as they can.

Use matching activities for larger numbersfor which the children do not have the names."Look at the chairs in the room. Is there a per-son on each chair? Is there a chair left over?Are there more people, more chairs, or thesame number of each?" Pose similar questionsas cups are passed out for juice. Stress match-ing as a way.to tell if two sets do or-do not havethe same number and which set has more orfewer members.

134 CHAPTER SIX

The previous activities emphasize matchingand oral language. In such activities, quan-titative language"same," "more," "fewer,""just as many," "one here for each one there,""match." "big." "small." and so onshould bedeliberately used.

Make a set of "Show Me" numeral cardsfor each child. Use cards about seven centime-ters by ten centimeters. Put one of the numer-als 0 through 9 on each card. Tell the childrento hold up the correct numeral card when a setis displayed or when the oral name is given. "Iam holding up some fingers. You pick the cardthat shows how many I am hblding up. When Isay 'Show me,'hold up your card." It is easy tocheck the entire group this way, and each childis responding to the question.

Repeat with questions that use the oralname. "I am going to say the name of a num-ber. You hold up the card that goes with thename when I say 'Show me.'

Make a set of "Show Me" dot cards foreach child. Use cards about seven centimetersby ten centimeters. Put a different set of dotson each card to represent the numbers 0through 9. Arrange the dots in some regularpattern for ease in determining the number andfor practice in recognizing the number withoutcounting. Dots can be drawn on the cards orgummed stars may be affixed. Have the chil-dren respond to the written symbol for num-bers and to the oral names of numbers by hold-ing up a card. "I am going to say a numbername When I do, you pick the correct dot card.When I say 'Show me,' hold it up." Repeat thequestions using a numeral written on thechalkboard.

Recognition of the numerals is helped bytalking about the essential features of the sym-bols. Establish the meaning of "straight" and"curved." Then write the digits on thechalkboard. "Which numerals have straightmarks?" (1, 2, 4, 5, 6, 7, 9) "Which ones havecurves?" (2. 3, 5, 6. 8, 9, 0) "Hold your finger toshow the way the straight marks go in each."(Have them show the finger in vertical and hori-zontal positions, depending on the numeral.)

"Which ones have-loops that are closed up?"(6, 8, 9, 0) "Which ones have both curves andstraight marks?" (2, 5, 6, 9)

Make sets of "Match Me" numeral cardsfor the numerals 0 through 9, with.five cards ineach set. Then rearrange so that four cardshave the same numeral on them and the fifthcard has a different, but similar, numeral. Forexample, four cards might have .a "9" and theother card'a "6." Have the child choose the onethat is different. Mark the different card on theback so that the children can check them-selves.

comparing, ordering, and ordinalnumbers

Comparing and ordering using sets of ob-jects, oral names, and written symbols are bothimplicit and explicit in many of the prior activi-ties. Children should be able (1) to comparetwo numbers and tell which is greater andwhich is less. (2) to arrange any three numbersso that the first one is least and the last one isgreatest, (3) to give the number "before" andthe number "after" a given number, (4) to ar-range the numbers 0 to 10 in order, and (5) togive the ordinal names "one, two, three, ..."and "first, second. third_ ." As the numbers 0to 10 are learned, comparing and orderingquestions arise naturally. Some other activitiesfollow.

Use the "Show Me" numeral cards and the"Show Me" dot cards. Choose two cards fromone of the sets. Put one on the chalkboard tray."Does the other card go 'before' or 'after' thecard in the tray?" Now choose three cards. Puttwo of the cards in the tray. "Does the othercard go 'before,' 'in between,' or 'after' the twoon the tray? Can you arrange all the numeralcards in order? Can you arrange the dot cardsin order?"

Counting itself gives one ordinal name foreach object. When counting children in line,you say "three" as you point to a given child.

one two

first second

-84S

three fot4r

third fourth


That child is "number three"an ordinal use ofnumbers. It would seem reasonable that chil-dren do indeed use ordinal numbers just aswell as cardinal numbers. The difficulty is morelikely to be with the special ordinal names of"first, second, third, ..." and not with ordinalideas themselves. The names for "third"through "tenth" are easily related to the names"three" through "ten." The special name "first"is learned early. The name "second" must belearned separately. (See M.)

Have children count oft-aS-they-line up togo to lunch or recess. Count "one, "two, three,..." Then count-off "firel_second, 'third, ...""Will the 'number five'Thold up 'pis "hand? Willthe fifth person hold up his hand?"

Show hoW the ordinal namtells_riot only"which%One",but "how many" thefourth:person hold up his hand?AreyOU,'num-ber four' "also ?" (Yes) "How.merV.ctiitciren arein line. counting youFoUr)--Aek simpaques-tions using Pages ine:book.'"Turri,1cipage 10Ian ordinal idea ). How many pages are there tohere, counting the tenth one?" (Ten)

Knowing how many numbers are in a set tellsnothing about how the members are ordered.However, knowing the ordinal name of an ob-ject does tell the cardinal number of the setcontaining that member and all before it.

reproducing symbols

Although it is not unusual for children of agesthree through five to begin trying to write theirnames and certain numerals, most educatorsprefer to delay formal writing until age five or

six

sixth

seven eight

seventh eighth

nine ten

ninth tenth

six. The main reason for this delay is the lack ofdevelopment of the small muscles needed towrite. However, it is doubtful that any harm isdone if children, on their own initiative, beginearly to write the numerals; indeed, many chil-dren get much satisfaction from it.

The following activities are suggested for thetime when a child is learning to write the nu-merals.

Trace the numerals in the..alr-, :tieing thelarge muscles. Then havef,the,:child'traCe thesame motion, making sure enough epacels be-tween successWel.childrart:lo avoid',0110Cp. Asthe tracing is darn ;in the air, talk..abbut theshapes used:;.ibmake the numeralk'Shor de-scriptions khe,(5: 1"straight cfpwrIF!;"around :and` ;back "; 3"around irid::arbUndagain"; 4:-.4.'4,C-Wnand out then,a4§4Atottirn";5"straigt*doWn;:and-arduifa:;.;p:400..".flat,pntop", 6"iloii-,"On&-o'F000.,-;7,4!?64f anddown", 8"doWnzerouriclfahe way arid14aok upthe other"; 9"arouiic afid7down"; theway around."

Make thallumerals from sandpaper or feltand pasteiliembn cards. Put eA7ed'Imark toshow where the child is to begiriihe:rtumeral.Allow the child to trace.',Ihi-riumeral manytimes, using the,roil§hIgictUrefor: -tduCh. An-other child, eeacher'elde, or thetteaCher cancheck to sdeliffleihild can makettrienumeralin the air first:by-looking at what he traced andthen without:looking. It is difficult forachild tocheck himself because he can be reversing andnot be aware of it.

Dots carilpe.iised-to--show paderna for thechildren to followiri.Writing numerals. A crossmark might slidw where to begin.

136 c CHAPTER SIX

recognizing sets of two throughfive without counting

A major goal of the extensive activities andexperiences with numbers in the age rangethree through six is to enable children to recog-nize without counting the number of membersin sets containing zero through five members.This ability is especially important as the num-bers six through ten are studied, as grouping isdone to teach the base ten of our numerationsystem, and as addition and subtraction arelearned. For numbers beyond five, it is difficultto recognize the number without counting orwithout making subsets whose numbers areeasily recognized.

Initial instruction should involve arrange-ments of objects and dots that are easy to see,such as the ones shown in this diagram.

1 .1 a

'

Hold up a dot card, allow no more than twoseconds for the child to see it, and then ask."How many dots?" After the children see whatthey are to do, let them practice in pairs. (Thisactivity could easily be carried out by an olderchild or by a teacher aide.)

As the child gains proficiency in recognizingthe number in easy arrangements, use a widervariety of arrangements. Several arrangementsof four are suggested in this diagram:

The number zero. Usually the number zero isnot introduced until some other numbers arestudied, the concepts well established, thenumbers ordered, and the written and oralsymbols learned Frequently the study of zero,then, comes after the numbers one, two, three.and four have been studied. It should be

treated as any other number and should begiven its correct name, "zero," from the start.

The number property of the empty set iszero. The word name "zero" and the writtenname "0" show the number. The number zeroanswers the question, How many members inthe box?

Using a paper bag or theta' can, place ob-jects inside the container and'mdve it back andforth "Do I have objects inside? Can you hearthe objects, move?" Then remove,,the objectsand ask similar questions 'to illuStrate theempty set. Introduce. the oral and writtennames as the way to answer thequestion, "Howmany objects-are there?"

Another effective activity is to have all pupilsextend their fiVe fingers. "How rnanylingers?"Then turn-one down and ask, "How many fin-gers now?" As each finger is turned"down, askthe pUpilS to tell how many. When all the fingersare turned down, it will then seem to an-swer the question "How ,,many ?" by replying,"Zero."

separating sets into subsets andnaming the numbers

As numbers beyond two are studied, setsshould be partitioned and the numbers named.For example, the chart shows the numbers ofthe two subsets of a set of six partitioned justonce (111).

The partition can be shown on the chalk-board or on a felt board using a piece of yarn.For an individual activity, a child can makeseveral sets of six and do his own partitions (4)).

Words that children could use to describethis action include split, part, partition, or sepa-rate. Using the words whole and parts empha-sizes the essential relation to be learned. Setsshould be partitioned more than once. Six, for

D

L

\''0.50 I

t,:° /

example, might be partitioned as 2, 3, and 1;equal partitions, such as 2, 2, and 2, shouldalso be made.

Activities that focus on partitioning a set andnaming the numbers of the subsets have twoimportant goals. (1) the recognition of the sub-set or inclusion relation, and (2) preparation foraddition and subtraction.

To focus on goal 1, such questions as thesecan be asked: "Where is the whole set? Whereare the parts? Can you cover the whole set withone hand? Can you use one hand to cover onepart? The other hand to cover the other part?"

The preparation for addition and subtractionis very direct. The concept of addition involvesnaming the number of the "whole" when two"parts" that do not overlap are joined. For sub-traction, the concept is giving the number ofone -part" when the numbers for the "whole"and the other "parr are known.

Partitions also provide a good opportunityfor "counting on Cover four objectsperhapsjust recognizing them as "four" rather thancounting themand then "count on" to get six.

Competence with the numbers zero to ten isessential for the meaningful learning of largernumbers It is essential that the relation of thesets of objects, the written symbols, and the nu-merals be well understood, as well as the orderrelations. The progress chart at the end of thechapter suggests a way that information onthese topics may be collected by the teacher.

142


numeration

The term numeration is used here to denotethose concepts, skills, and understandingsnecessary for naming and processing numbersten or greater. The purposes of this section areto describe the components of numeration andto present a sequence of learning experiencesthat will help children deal meaningfully withnumber.

Most children are introduced to numbersgreater than ten by counting. Hence, it is onlynatural for them to associate cardinality withthese numbers. They think of eleven simply asone more than ten and twelve as one more thaneleven.

Although counting is essential, children havemuch greater use for thinking of numbers be-tween 10 and 100 as tens and ones. Thinkingusing tens and ones, called base representa-tion, gives the child flexibility and facility indealing with a great variety of tasks. For ex-ample, base representation helps with oralnaming and counting, with writing numerals,and with comparing numbers; moreover, it isessential for all algorithmic work with wholenumbers.

Learning about numeration is not a singulartask. Children need to develop many conceptsand skills before they can deal with numbersflexibly The major components of this complexof ideas called numeration are listed below. Abrief description is included here, with greaterelaboration in the following sections.

1. Grouping once and naming the numberof groups. The essential notion is grouping ob-jects into equivalent sets. The number ofgroups and the number of ones left over arethen recorded orally and in written form. Forexample, a set of sticks may be grouped bytens. If there are two groups of ten and threesingle sticks left over, the number is noted as "2tens 3 ones."

2. Scheme for grouping more than once. Notonly are ten ones grouped to form one ten, butalso ten tens to form one hundred, ten hun-dreds to form one thousand, and so on. A gen-eralized scheme for grouping needs to be con-sidered prior to the introduction of hundreds.

138 CHAPTER SIX

3. Scheme for recording groups. Aftergrouping and naming the number of groups, apositional scheme is developed for writing thenumerals. This is often referred to simply as"place value." It should be noted that placevalue requires grouping as a prerequisite.

4. Representing numbers three ways. Thenumber of objects in a set can be representedby (a) the oral number name, (b) the written nu-meral, and (c) a base representation. Base rep-resentation means some form that directly in-dicates the number of groups, for example,tens and ones.

5. Translating from one representation to an-other. Students should learn to translate fromany representation of a number to the other twoforms of representation. as indicated by thethree pairs of translations shown.

Baserepresetliedion

V

Oral number 74/o-digit numeralnakyie

If the child can easily shift from one repre-sentation to another, he will be able to deal withtasks more flexibly For example, to add 32 and45, a child thinks "3 tens 2 ones" and "4 tens 5ones." Then he addVhe tens and the ones,getting 7 tens 7 ones, and converts back to 77.Similarly, ear in translation is of great use inordering, in approximating and estimating, andin regrouping and renaming.

1/13

a sequence for learning two-digitnumerals

Four major units on numeration are sug-gested in the following sequence. Suggestionsfor activities useful in teaching the units are in-cluded.

At each stage, the main concern has beenthe development of a meaningful relation be-tween the various numeration components.Each stage begins with few prerequisites andbuilds competence in such a way that rotelearning is minimized.

unit 1: grouping once and namingthe groups

The primary objective is to teach children togroup objects into equivalent sets and nameorally the number of sets and single objects leftover. Preliminary work on grouping objects hasbeen suggested for inclusion in the learning ofthe numbers zero to ten. This unit builds onthese experiences and extends through a mas-tery of grouping by tens and ones.

Group objects by four, using plastic bags."Here are some blocks and.some plastic bags.Put four blocks in a bag. Are there enoughblocks left over to put another set of four in a

1,1 ter.irik,

vat

toov

,t-'

Vr

bag?" (Yes) "How many bags of blocks arethere?" (Two) "Are there any blocks left over?"(No) Use the bags and group the blocks byfives and by sixes. Have the children tell thenumber of groups orally.

Use different kinds of objects, such assticks and counters. Display sixteen sticks."Make a group of five sticks. Are there enoughsticks left over to make another group of five?"(Yes) "Make as many groups of five as you can.How many groups of five are there?" (Three)"How many sticks are left over?" (One) Arethere enough sticks left over to make anothergroup of five?" (No)

"Group these counters by sixes. How manysixes are there?" (Three) "How many . s leftover?" (Four) "Are there enough counters leftover to make another group of six?" (No)

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Not only are situations like the ones abovepreparation for grouping objects by tens, butthey also help children to consider a group as aunit just as individual objects are consideredunits. They help children avoid saying "thirty" in

'Am

140 CHAPTER SIX

response to the question (when three tens areshown), "How many tens?"

Next, grouping by tens can be emphasized.

0. Display thirty-four counters in a randomarrangement on a flannel or magnetic board."Let's group the counters by tens. Are thereenough counters to make a set of ten?" (Yes)Have achild line up a set of ten counters on theleft side of the board. "Are there enough count-ers left over to make another set of ten?" (Yes)Let another child show asecond set of ten fromthe remaining counters. Continue this processuntil the grouping is completed. "Are there

n

enough counters left over to make another setof ten?" (No) "How many sets of ten are there?"(Three) "How many ones are left over?" (Four)"That's correct, there are three tens and fourones."

For a similar class activity, group sticksthat have been placed in a box. "Ed, come andtake ten sticks from the box. Now, Julie, will youcome and take ten." Repeat until all tens havebeen taken. When fewer than ten sticks remain,have the child taking those sticks stand slightlyapart from the others. "Are there enough sticksleft over to-make another set of ten?" (No) "Howmany sets of ten are there?" (Five) "How manysticks are left over?" (Two) "Yes, there are fivetens and two ones."

iz15

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With all experiences in grouping, the essen-tial questions are these: "Are there enough ob-jects left over to make another set of ten? Howmany sets of ten are there? How many objectsare left over?" The number of tens and ones isthen reported.

The reverse task is also important. Whenasked, children should be able to show the ap-propriate number of tens and ones. After de-veloping competence in constructing five tens,two tens, four tens, and so on, the childrencould construct representations of tens andones. The girl in the photograph has shown"three tens two ones."

-

For individual work, each child can be givena set of materials. Coffee stirrers, tongue de-pressors, Popsicle sticks, or squares cut fromheavy paper or cardboard work well. Thesquares, shown in the photograph above, caneasily be drawn by the children. Tall, skinnyrectangles represent tens, and little squaresrepresent ones.

Evaluation should be based on the ability togroup by tens, to describe groupings of tensand ones orally, and to construct a concreterepresentation from an oral description of agrouping. Repeated use of the question "Arethere enough objects left over to make anotherset of ten?" should help children focus 'on theidea of forming as many sets of ten as possiblewhen they are grouping by tens.


unit 2: oral number names fortens and ones

The major objective of unit 2 is to help chil-dren relate groups of ten to the usual oralnames. It is here that the child relates the nu-.

merosity of a set to some base represent .tionof the set.

They first task of the child is to learn thenames for multiples of ten"ten, twenty, thirty,..." One reason for beginning here is thatthese names are prerequisite to naming num-bers such as "3 tens 2 ones" as "thirty-two."Another reason is related to the counting skillsthat children need to acquire. By learning thenames "ten, twenty thirty, ..., ninety," childrenhave an effective but simple scheme for nam-ing the number or counting to one hundred.

The knowledge that some children alreadyhave in counting by ones can be used to showtwo ways to count to fifty: counting'bY ones andcounting by tens. "-

Display fifty.cOunters in a reridoM order."How many counters are there?"'Theleacherand children.count them together by ones. Asthey are,countect, the teacher gtouPS-them bytens. "Yes,,there are fifty counters.'llaw manycounters are in eicb:§rOup' (Ten) "Whdn ob-jects are grouped bylens;,there is dr differentway to count them.'We say, 'Ten (point to thefirst group), twenty,' and so on. This is calledcounting by tens."

?pp

nn/r-.

142 CHAPTER SIX

Remove a set of ten. "How many countersare there now? Count them by ones. What doyou get?' (Forty) Now count them by tens.What do you get? (Forty) 'Do you get the samename when you count both ways?" (Yes)

Remove another set of ten. "Count them bytens. How many are there now?" (Thirty)"What number name would you get "! youcounted by ones?" (Thirty)

Most children have considerable confidencein their ability to count by ones. The purpose ofthese activities is to develop a similar con-fidence in naming numbers when counting bytens. Children need to be thoroughly convincedthat counting by tens and counting by onesyield the same result.

Show a given number. of objects groupedby ten and the same number of objects ar-ranged in a random order. Have- a studentplace two sets of ten on one side. Have anotherstudent count out twenty counters by ones andplace them at the other end. Point to the twotens. "How many counters are at this end?"(Twenty) "How many cotinters are at the otherend?" (Twenty) "Are more counters at one endthan the other end, or is the same number atboth ends?"

AL

An interesting discussion should follow.There will probably be some disagreement.Guidance should be provided to help comparethe sets. Previous experience with comparisonby constructing a one-to-one correspondencecan be used. The counters could be lined up ina one-to-one correspondence, or yarn used tomatch counters, or perhaps some childrencould pick up one counter from each end untilall the counters have been removed to see if it"comes out even." Some children may suggestcounting the two sets of ten by ones. This pro-cedure should be supplemented with a match-ing comparison.

With the understandings and skills de-scribed above, children are ready to relate thenumber of tens meaningfully with the usualnumber names. This task is not difficult. Theyquickly learn to name concrete representationstwo different ways.

The photograph on the next page showsphysical representations for one ten, for twotens, and for three tens. By using such repre-sentations, count two ways"one ten, two tens,three tens" and "ten. twenty, thirty."

Learning the names can be facilitated bycomparing the names of the number of tenswith the names of the numbers one throughnine.

one ten two tens three tens four tensten twenty thirty forty.

five tens six tens seven tensfifty sixty seventy

eight tens nine tenseighty ninety

Note that all names beyond ten end in -ty. Thety means ten. Discuss the similarity and differ-ences in names:

"twen-"sounds like "twins", twins meantwo

"thir-"sounds like "third""for-"sounds just like "four""fif-"sounds like "fifth""sixty through ninety"just like the names

you know

The child should be able to move easily fromthe name using "tens" to the usual name, andconversely Varied experiences are needed incounting objects by ones, by tens using thename "tens," and by using the usual names fortens Since the same concrete representation isused for all the ways to name, the child has asimple but powerful way to think about num-bers. regardless of the form in which a task isposed.

Next, numbers other than multiples of tencan be considered. The usual oral numbernames should be developed for groupings oftens and ones.

Because the students already know that twotens is twenty, that three ..ens is thirty, and soon, they should have :ittie difficulty developingthe name "forty-three" from a grouping of fourtens and three ones. The translation is nearlydirect The reverse task of constructing a con-crete representation that corresponds to anoral number name is also relatively easy.

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If tenunit strips and unit squares are used asmaterials, tens and ones can be easily repre-sented by children pictorially. Since they enjoydrawing, this variation of the task is a pleasantchange for th 71. An example of the materialsand a first gr,oer's drawing of them is shown.

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In counting the number shown, childrenshould learn to count by tens as far as they canand then count the ones. They would count,"Ten, twenty, thirty, forty, forty-one, forty-two,forty-three."

41IP. ,,

144 CHAPTER SIX

Next the teens can be considered. It shouldbe pointed out that "eleven" and "twelve" arespecial names for "1 ten 1 one" and "1 ten 2ones." The other names are all backwards. Forexample, in "fourteen,- the part referring tofour is named first and indicates ones. It shouldbe made explicit that the names are backwardsand that teen indicates "one ten."

The naming irregularities cause many errorsin naming the teens. Frequent reminders tostop and think about the naming for the teenswill help. Many experiences discriminating be-tween numbers such as sixteen and sixty-one,eighteen and eighty-one. and so on are alsonecessary.

Evaluation should be based on the children'sability both to group objects by ten and give thecorrect name and to construct groups of tensand ones when they are given the name.

The children's depth of understanding canbe assessed with tasks such as these:

Show some bundles of ten sticks and sev-eral single sticks.

1. Say, "Pick up forty sticks." See if the childneeds to count by ones or if he choosesfour bundles immediately. Repeat withother numbers, such as thirty-two.

2. Point to the tens and then to a large groupof single sticks. Ask, "Which set has more,or do they have the same?" See if thechild arranges the single sticks in tens forease in comparison.

fi

3. "Choose as many sticks as there are fin-gers on all our hands." See if the childchooses by groups or by single sticks."Give two names for the number of fin-gers." See if the child can give the twonames.

unit 3: two-digit numerals

The major objective is to enable the child towrite and interpret two-digit numerals and torelate two-digit numerals to the usual oralnames This completes the initial instruction forthe translation tasks suggested in component5.

Baserepresentation

Oral number Tvo-digitname numeral

The prerequisites for the instruction up tothis point in the sequence are minimal; how-ever, before moving into unit 3, children shouldbe able to read and write the numerals 0through 9. They will also need to recognize thewritten words tens and ones.

Show a grouping of tens and-ones. "Howmany tens and ones are there?" (Five tens andtwo ones) "We can keep a record of how manytens and ones there are by writing numeralsunder the words tens and ones. How many tensare there?" (Five) Write the numeral 5 underthe word tens to show five tens. "How manyones are left over?" (Two) Write the numeral 2

tens ones

under the word ones to show two ones. "Thischart tells us that there are five tens and twoones."

tens ones

.)

After discussing several examples, distributeworksheets on which the students can recordthe number of tens and ones. Drawings of tensand ones could be used when the children areworking individually. Concrete objects could beused when the class is recording the numeralstogether.

tens ones

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n.

tens ones

II


Include a review of grouping by tens by vary-ing the task as shown at the top of page 146.

Next have the children draw pictures of tensand ones for the numerals provided on a chart.

tens_ones

5 2tens ones

tens ones

4)\(-

Finally, all the missing parts could be filledin. as shown in the chart at the bottom of page146.

Children will generally attack these work-sheets with great enthusiasm. After the task isclear, the only serious problem they have isdealing with zero. Special attention must begiven to the fact that "no ones- is recorded bywriting the numeral 0 under the word ones,

Similar tasks could be performed by lettingchildren form their own sets of tens and onesfrom graph paper. They can cut strips ten unitslong for tens and single squares for ones.These cutouts can then be matched to the ap-propriate tens and ones charts.

Soon after beginning this unit, the studentscan learn to write two-digit numerals. First havethem record the numerals in charts without aseparating line. Then after a few experienceslike this, they will be able to write two-digit nu-merals without reference to the words tens andones.

tens ones

3is

tens ones5

146 CHAPTER SIX

ir),DODOODOCI O 1L1L.I( iL [inn,

Worksheets_ like_ _the. one. on the .following.page provide opportunities for children to re-late two-digit numerals to groupings of tensand ones. Similar problems can be used toassess their ability to relate base representa-tions to two-digit numerals.

Reading and writing numerals is generallytaught earlier in a numeration sequence thanindicated here. The reason for the post-ponement in this sequence is to insure that thestudents have a sound basis on which to relatethe number names with the two-digit numerals.

r 1

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illyfens ones

tens ones

1

Since groupings havateen_closely_tied_both_tothe numerals and to the usual oral numbernames. reading and writing numerals can nowbe made more meaningful by reference to thebase ideas. Without this reference to base rep-resentation. connections between the oralnames and the two-digit numerals must neces-sarily border on rote learning.

The students are able to produce numbernames for symbols such as 68 by thinking. '68is 6 tens 8 ones. Since 6 tens is sixty. 68 is sixty-eight." By reversing the procedure, the children

fens

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148 CHAPTER SIX

3 ones

can develop two-digit symbols just as easily. All work that children have developed for number,the prerequisites for making this a meaningful emphasis on sequence too early may distractprocess are present. attention from groupings of tens and ones and

lead children to make responses based on pat-terns they have learned for counting.

The numbers zero to ten are ordered usingthe "one more" idea. This idea is also used inwriting the numerals in order and in readingthem as they are written. The chart at the bot-tom of the page shows how this idea can beused in ordering the two-digit numbers.

Many.other activities can help students mas-ter the two-digit number sequence. For ex-ample, they could fill in the missing numerals ina hundreds chart. Some sample lines areshown.

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43 43 is foriy-ihree

unit 4: sequence of numbers andnumerals 10 through 100

The major objective of this unit is to teach thechild to count and write the numerals in orderup to 100. The sequence of the numerals wasnot presented in earlier units so that the centralnotion of base representation could receivegreater emphasis. Since counting is such aprominent component of the cognitive frame-

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A slightly more difficult task is to have thechildren fill in missing numerals in a portion ofa sequence.

68 78

Another related task is to ask the children towrite numerals for the numbers one less andone more than a given numeral.

38

79

60

A hundred board with circular tags contain-ing the numerals 1 through 100 is useful forcounting in order and recognizing the numeralsin order. By turning all the tags over, a child canchoose one and give the number before heturns it over to check the numeral. For ex-ample, if a child turns over the fourth tag in thethird row and says, "Twenty-four," he is right.

comments on the sequence fortwo-digit numerals

That children have difficulty learning aboutnumeration has been well documented by in-formal observations of teachers as well as bymore careful evaluations of researchers. Threepossible reasons for this difficulty, along withprecautions that should be taken to minimizethem in this sequence, follow.

1 Children often have difficulty with numer-ation because too many different but closely re-lated tasks are presented at nearly the sametime It is not uncommon for children to be ex-pected to count by ones, count by tens, groupby tens. make tens and ones charts, namenumbers orally, read and write two-digit nu-merals. recognize the significance of ten in ourbase-ten system of numeration, recognize the

1S4


equivalence of numbers named orally andnamed by their respective base representa-tions, and perhaps grasp several other ideasall during their initial exposure to tens andones. Children need time and assistance to sortout these base and place-value ideas and to re-late all the components of numeration. In thissequence, all the major components are re-lated so that children will be able to deal withnumbers-flexibly.

2. Another problem that children have withnumeration involves the use of oral numbernames. Besides the normal confusion due tothe irregular number names, such as the teensand some multiples of ten, there is a moresubtle and more significant difficulty. Childrengenerally do not have a way to think about hownumbers are named. Initially, they do not learnto relate the names to groupings of tens andones. For example, they do not recognize thatthe name "sixty-four" is derived from, or iseven related to, six tens and four ones. Rather,they learr *he number names in sequence fromcounting patterns. "Sixty-four" is simply thenumber after "sixty-three." This type of thinkingseverely restricts the use of number. It ac-counts for some of the difficulties that childrenhave with numbers taken out. of the context ofsequence. It also accounts for the slow devel-opment of meanings and skills related to oralnumber names, particularly the tasks of read-ing and writing numerals (Rathmell 1972).

The sequence presented here has beencarefully structured to insure that naming num-bers can be meaningful. Simple counting skillsare used to reiate the oral names to the baserepresentations from which they are derived.This provides students with a thinking patternfor naming numbers. Base representations canthen be used as a mediator between namingand symboling numbers. Names or numeralsthat have been forgotten can easily be recon-structed by the patterns. Since the teens arethe exception, they are presented after the gen-eral scheme is well established. Reversal errorsare less likely because the patterns stem frombase ideas. Unit 2, which is devoted to namingnumbers, also provides the opportunity forchildren to have many experiences with group-ing and naming before symboling is necessary.

150 CHAPTER SIX

After these developmental experiences, writingnumerals is a natural way to record the ideasalready established.

3 Many children do not recognize the signif-icance of grouping by tens. One contributingfactor to this lack of understanding is thatgrouping activities are often performed in con-nection with counting by ones. When childrencount, they may not recognize that each newdecade is really a set of ten.

The sequence presented here features aspecial emphasis on grouping objects to stressthe base idea. Base representation is central tothe entire development. Besides the concentra-tion on grouping and the link between groupingand the oral number names, tens and ones areused as the initial model for two-digit numerals.Since references to counting may distract fromthe significance of grouping and base ten. theusual number sequence is not emphasized un-til base ideas are thoroughly developed. Chil-dren are also given the opportunity to comparebase representations of a number to the idea ofnumerosity.

other related topics

Related to numeration is a need to ordernumbers. to use estimation and approximation.and to begin regrouping and renaming. Activi-ties for these topics are suggested in this sec-tion. Many of the activities could be intertwinedwith those for two-digit numerals, but masteryshould probably be expected later.

order

Initially, children probably learn to ordernumbers greater than ten by using the numbersequence they have learned from counting. Forexample, a child might say, "Twenty five comesbefore twenty-eight, so it is less than twenty-eight."

Although such thinking is not incorrect, or-dering from a number sequence is inadequate.

If two numbers being compared are not closein the sequence, counting is not practical, and ifcounting is not done, a child may focus on thesize of the greatest digit. For example, studentsoften say that 39 is greater than 51 because 9 isgreater than either 5 or 1.

A more effective method is to use base rep-resentations. Children who can compare 3 and5 have little trouble comparing three tens andfive tens.

Relate initial comparisons of tens to com-parisons of ones. "Which is greater, three orfive?" (Five) "Which is less?" (Three) "How canyou tell? What would happen if you tried tomatch the sets?" (There would be some leftover.) "Which is greater, three tens or fivetens?" (Five tens) "Which is less?" (Three tens)"How can you tell? What would happen if youtried to match the tens?" (There would be sometens left over.)

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40

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The students will soon learn that numbers oftens are ordered the same way as numbers ofones. After the oral number names have beenlearned, they may also be used in comparingnumbers.

"Which is greater, thirty or fifty? Which is

c-r.c)0.

less? How can you tell? Yes, thirty is only threetens, and fifty is five tens."

Sets of tens and ones can then be comparedwith or without the oral names. Teach the chil-dren to look at the "big things"the tens. Onlyif the tens are the same will they need to look atthe ones. Later, after the two-digit numeralshave been learned, the children might respondas follows:

"How many tens are in thirty-nine?"(Three) -How many tens are in fifty-one?" (Five)"Which is greater. thirty-nine or fifty-one?"(Fifty-one) "Which is less?" (Thirty-nine) "Howcan you tell? Yes, thirty-nine has only threetens, but fifty-one has five tens." (See .)

Besides comparing two given numbers,many other appropriate order experiences withnumbers are possible. Concrete representationwill probably be necessary during the first en-counters with such tasks as these:

Name a number greater than forty-nine.Name a number less than thirty-five.


Name a number greater than twenty-three,but less than forty-one. Can you find all thepossibilities?Order forty-three, seventy-one, and sev-enteen from smallest to largest.Here are two boxes, A and B. Box A has morethan forty beads. Box B has less than fortybeads. Which box has more beads? How canyou tell?Here are two boxes, A and B. Box A has morethan thirty beads. Box B has more thantwenty-five beads. Can you tell which box hasmore beads? (No)

estimation and approximation

Skills in both estimation and approximationdepend heavily on the ability to deal with tensand multiples of ten. Developmental experi-ences for these estimation and approximationtasks must be provided in the primary grades.

Initially, an intuition should be developed forthe approximate location of points represent-ing given numbers on the number line. Stu-dents should not have to scan the teens to find83.

152. CHAPTER SIX

Locating points on the number line can bedone in conjunction with ten-unit strips and unitsquares matched to the points on the numberline. This activity has a built-in self-check.

Predict where six tens and two ones will beon the number line.

The number line is also useful for roundingto the nearest ten. Thu difference between twonumbers is represented visJally by a line seg-ment on the number line, and so rounding tothe nearest ten involves the comparison of twoof these line segments. For example:

"Find 69 on the number line. Is it closer to60 or 70?" (70) "Which is the nearest ten, sixtens or seven tens?" (seven tens)

"Find 32 on the number line. Is it closer to30 or 40?" (30) "Which is the nearest ten. threetens or four tens?" (three tens)

Now show only a portion of the numberline with cards covering up both ends. "Here is48. Is it closer to 40 or 50?" (50) "Which is thenearest ten, four tens or five tens?" (five tens)

"Think of 42. What is the nearest ten?" (40)Show this on the number line. "Think of 72.What is the nearest ten?" (70) Show this on thenumber line. Do the same for 12, 32, 92, 52. 22,82, and 62. Do the same for other numbers inthe ones place.

These questions should lead to a gener-alization about rounding. to the nearest ten. Ifthe ones digit is 0, 1, 2, 3, or 4, the nearest ten isthe number of tens indicated. If the ones digit is6, 7, 8, or 9, the nearest ten is one more thanthe number of tens indicated. An opportunityfor creative expression comes when dealingwith a 5 in the ones place. Usually 5 is roundedup. Later, this generalization is an aid in esti-mation for computation. (See a .)

regrouping and renaming

Although regrouping objects and renamingnumbers are ideas that are not generallythought to be of major concern for the early pri-mary grades. it is essential for students to un-derstand these ideas if they are to understandthe computational algorithms. Unfortunately,the ideas of renaming cause a great deal of dif-ficulty. More developmental activities dealingwith regrouping and renaming may prove to bequite beneficial.

Several renaming ideas have already beenpresented. They include the naming of ten onesas one ten and the tasks comparing the resultsof counting by ones with counting by tens. Thepossibility of a number's having more than onename should be made explicit during this in-struction.


3 tens + 2 tens =5 tens

so 18 + 29 is

about 5018 is about

2 tens

000

What is the approximate cost of thecar and the balloon together?

Other developmental tasks could also bepresented. After the students can representnumbers concretely by tens and ones and re-late this to counting objects by ones, they couldbe' asked to find different ways to show thenumber concretely. For example, after showingthirty both as thirty ones and as three tens, theteacher might ask. "Can you show thirty in astill different way?" Some direction will have tobe provided at first.

The preceding task is no doubt easier if thematerials used do not require physical ex-changes. Bundling sticks by tens is quite ap-propriate, since the removal of a rubber bandregroups the sticks.

The comparison of different representationsof numbers is also an effective way to providedevelopmental experiences for renaming. Con-sider the following:

Give one student three bundles of tensticks and two single sticks. "How many tensand ones do you have?" (Three tens and twoones) Give another student two bundles of tensticks and twelve single sticks. "How many tensand ones do you have?" (Two tens and twelveones) "Who has more sticks, or do you have thesame?" (Same)

154 CHAPTER SIX

By establishing a bank, where exchanges ofone dime for ten pennies can be made, smallgroups of students could compare differentrepresentations of numbers. The group firstmight decide what exchanges they would needto make, and then one student could go to thebank to make these exchanges. On his return,a direct comparison of the representationscould be made.

Other situations that force physical ex-changes could also be presented. For example,three students could be given four dimes andtwo pennies to share equally. A bank couldagain be used for making the appropriate ex-changes.

Nearly all the renaming experiences in theprimary grades should include concrete refer-ents, Money. bundles of sticks, unit squaresand strips, and many other materials can beused to provide excellent problem-solving ac-tivities that promote the ideas of regroupingand renaming.

These developmental experiences shouldprovide an intuitive understanding of renamingbefore it is requiR,d for the computational al-gorithms. This should facilitate both an under-standing of the algorithms and computationalability.

a sequence for learningthree-digit numerals

Base ideas, or groupings, must continue tobe the central theme in the teaching of three-digit numerals. This not only helps promote thebasic concepts of our numeration system butalso helps students maintain a concrete refer-ent for the symbolism.

The major steps in the sequence suggestedhere include (1) using a general scheme forgrouping from which 100 emerges as ten setsof ten, (2) naming the number by using a place-value numeral, (3) giving the usual oral namefor groupings and the three-digit numerals, and(4) sequencing and ordering the numbers.

generalscheme for grouping

The generalized scheme for grouping can bedeveloped by grouping more than once usingnumbers less than ten. In this way, pupils canexperience grouping activities without havingto work with so many objects.

"Today we are going to work in Six land. InSixiand, things are grouped by sixes: six ob-jects to a box, six boxes to a carton, and so on.

Here are some blocks. Let's group them asthey would in Sixland. How many blocks to abox?" (Six) ''Now let's make as many boxes aswe can." Have the children help complete thisfirst grouping. "How many blocks are leftover?" (Three) Are there enough left over tomake another box?" (No) "How many boxes toa carton?" (Six) "Are there enough boxes tomake a carton?" (Yee) "Let's make a cartonAre there enough boxes left over to make an-

other carton?" (No) "What did we end up with?Yes, one carton, one box, and three singleblocks."

cartons boxes singles

3


Grouping activities like the proceding onecan be completed by using strips of p. per sixunits long and separate square units. Childrencan line up six squares to form a "box" and sixboxes to form a "carton."

Similar grouping activities could be providedin other bases; for example, in Threeland therewould be three objects to a box and threeboxes to a carton. In Fourland, four objectsmake a box and four boxet' make a carton.Poker chips may also be used to establish thegrouping. In Fourland. four white chips can betraded for one red chip; four red chips for oneblue chip.

A few activities like the preceding ones willhelp children anticipate the possibility ofgrouping groups of objects. Now special atten-tion can be given to grouping ten tens to formone hundred.

"Here is a container with lots of sticks.Let's group them by tens, ten sticks to a box,ten boxes to a carton." Let several children inturn form sets of ten. Have them remove eachset of ten from the container. and place it in anappropriate box or bundle it with a rubber bandto represent a box. Continue this process untilthere are no longer enough sticks left over toform another set of ten. "How many sticks areleft over?" (Four) "Are enough left over to makeanother box?" (No) Place the sticks that are leftover on a desk top or table. "How many boxesare needed to make a carton?" (Ten)."Are thereenough boxes to make a carton?" (Yes) Havethe students form as many cartons as possible."Are enough boxes left over to make anothercarton?" (No) "What did we end up with? Yes,two cartons, two boxes, and four single sticks.How could we record this in the chart?"

Since there are ten sticks in a box, count bytens to see how many sticks are in a carton."The name for ten tens is one hundred. A car-ton in Tenland is one hundred." Record the an-swers now in a hundreds-tens-ones box (11).

156 CHAPTER SIX

r-T

three-digit numerals

It is important that all the components of nu-meration be taught and related well. Familiaritywith the different forms of representing num-bers will grow as groupings of hundreds, tens,

_ hundreds

I hundreds 2-1 tens

hundreds tens

_ tens

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cartons boxes singles

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tens ones

2 Li

and ones are recorded. Recording physicaland pictorial groupings in a chart should re-quire little practice. The reverse taskcon-structing a grouping from a chartshould alsobe completed. Worksheets such as the oneshown provide opportunities for both tasks.

ones

hundreds tens

hundreds tens

ones

hundreds tens

one 2.

ones

1 1

ones

ones

t51

As with two-digit numerals, three-digit nu-merals may be constructed from the charts intwo steps. First, the separation lines can be re-moved: then later, the words can be removed.

hundreds tens one hundreds tens ones

fl 3 7 fl 13 7 i

reading three-digit numerals

Some guidance for reading three-digit nu-merals can be quite helpful, Children need tobe given a way to decide which numerals in-volve hundreds and also a process that will en-able them to be successful at reading the nu-merals.

Discrimination tasks can be used to focus at-tention on the number of digits in the numeral.For example:

Circle the numerals that involve hundreds42 375 506 98 111 694 63

After the child realizes that a numeral in-volves hundreds, he needs to remember onlythat the first digit ,,(on the left) indicates thenumber of hundreds and that the other two dig-its are read just as other two-digit numerals areread. Consider 632. A student should think,"There are three digits: so the numeral involveshundreds. The first digit indicates hundreds; sothe number name is six hundred thiity-two." Itis sometimes helpful to separate the numeralwith a long slash. Writing 6/32 emphasizes thedistinction that needs to be made among thedigits when numerals are read. When a studentis unable to name a number correctly. this longslash is quite often enough of a cue for the stu-dent to reconstruct it without calling it sixty-three-two.

number sequence

After becoming fairly proficient at the six ba-sic translation tasks, children need to considerthe number sequence for three-digit numerals.Activities like those for the number sequencewith two-digit numerals are quite appropriate.


The children could fill in the missing numeralsin a chart:

1291 299 2% 299

.253 257

Special attention will need to be given othose numbers between 100 ,did 110, 200 and210, and so on.

Writing numerals for numbers one more orone less than the number represented by agiven numeral is also appropriate:

735-

560In preparation for the computational al-

gorithms, base representations can be used toname the numbers ten more or less and onehundred more or less than a given number. For347, the students might think, "347 is 3 hun-dreds 4 tens 7 ones, and so ten more would be3 hundreds 5 tens 7 ones, or 357." Similarthought patterns could be used for the otherproblems.

oth otational schemesIn the past fevA, years. nondecimal numer-

ation has become a popular topic in the elementary mathermit,cs curriculum. Originally.the rationale for including this top c was that itwould improve our understanding of our deci-mal numeration system. Research studies, pri-marily at the middle- and upper-grade levels.have generally supported the hypothesis thatan equivalent amount of instruction in base tenis more effective.

Although a study of nondecimal systems ofnumeration has not proved to be particularlybeneficial, a study of the Egyptian numerationsystem by third graders has been shown to be

162

158 CHAPTER SIX

effective in promoting the concepts and usageof zero (Scrivens 1968). Evidently the lack of asymbol for zero, as in the Egyptian system,draws attention to the way in which zero is usedin our own system.

Roman numerals should also be taught. Al-though they are not used excessively, they doarise often enough to be included in the cur-riculum. However. addition and subtraction areprerequisite for a comprehension of this sys-tem.

record of progressThe key to a successful mathematics pro-

gram is its ability to provide experiences to a

child when he is most likely to benefit fromthem. Regardless of whether the program isgeared toward the whole group, is flexiblygrouped, or is designed for individualized in-struction, the teacher must know what a childund ;ands to be able to provide these experi-ences.

One of the best ways to obtain the informa-tion on which curriculum decisions can bebased is to conduct short interviews to supple-ment paper-and-pencil assessments. Recordsof these interviews can be kept on charts suchas the ones that follow. Each teacher can de-vise his own scheme, but by using checks ormarking the date of achievement, he will alwayshave available a current profile of studentachievement.

i,LSW xrd 21

OldISI at r

0101t t. tl

ovttt 1'1: u1 II,t ,,0,11::11), dot p.iit t

- 1. ,1. , ' 'tot ," ltek cork, work with

"Ikss" and "tower"

RECORD OF PROORKSS

Jell

n,, a, litt)1" r,ore work

on "child" and "Huh"

OK

mod, tt01

oR LO tour

id,ntityine "parta"Ind %holt."

re.ogniting that rearrange-ea t dok 'a't flange

nunibers (..owiervat ion)

tra..sitlon S wge for

larger numbers

Kara.

oR oR

oi, r I I nu

onk to 1

only mo

Or

Fl}

OR

oy

rv. I

needs more work with"less" and "tower"

tptua I I y OR

1 6 3


Two -St tt lodv

grouping needs r.ore work with

grouping onee

representing numbers needs sore wort` with

oral names

,ounttng OK tor ones: needs somehelp for tens

ordering uunhers uses misleading cueshometLtes

rounding to nearest ten guesses

legrouping tens and ones not ready

three-di:it

rouptng

re q-. pent fug nu.bers

eunt t ng

lug numbers

OK

Neal Patri,k

OK

OK

OK

OK

OK

a few reversals, needsmore work with base

OK

OK

depends on number line

ready for initial depends on eonereteexposure materials

ready for grouping molethan once

beginning to try

trying, but has tfrom 100 to 110

needs review tar groupingmore than once

oral names eause smoproblem, also evro-ten:

OK

has some trouble whenhundreds are the same

conclusion

Throughout the learning of the numbers 0 to10. 10 to 100. and 100 to 1.000. concrete refer-ents for thinking are critical. The concepts thatgrow from the concrete referents with such

bibliography

Bruns, James V and Helene Silverman. -Developing theConcept of Grouping." Arithmetic Teacher 21 (October19741:474 79.

Davidson. Patricia S Grace K. Gallon. and Arlene W. Fair.Chip Trading Activities Arvada, Colo. Scott Scientific,1972

Dienes. Zoltan P 'Multi-Base Arithmetic." Grade Teacher79 (April 19621:56.97 100.

Dienes. Zoltan P and E W Golding. Modern Mathemifor Young Children. New York: Herder & Herder, 1970.

heavy emphasis on base representation mustbe related carefully and thoroughly to the oralnames and written symbols. If a child learns allThree of these and can move easily from anyone to the other two, we shall have providedhim with the best foundation possible for allsubsequent work with whole numbers.

Ellond, David. "Con4ervation and Concept Formation " InStudies in Cognitive Development: Essays in Honor o/Jean Piaget, edited by David Elkino and John H. Flavell,PP 171 89 New York. Oxford University Press. 1969.

Gelman. Roche!. "Conservation Acquisition. A Problem ofLearning to Attend to Relevant Attributds." Journal of Ex-perimental Child Psychology 7 (April 1969):167 87.

'The Nature and Development of Early NumberConcepts." In Advances in Child Development and Be-havior, edited by Hayne Reese. vol. 7. pp. 115 67. NewYork: Academic Press. 1972.

160 CHAPTER SIX

Grossnickle. Foster E.. Leo J. Brueckner. and John Reck-zeh. Discovering Meaning in Elementary School Mathe-matics. 5th ed. New York Holt. Rinehart & Winston. 1968.

Miller. Patricia H.. Karen H. Heldmeyer. and Scott A MillerFacilitation of Conservation of Number in Young Chil-dren. Report no. 20. Developmental Program, Depart-ment of Psychology. The University of Michigan. 1973.

Piaget, Jean. The Child s Conception of Number. NewYork: W. W. Norton & Co.. 1965.

Rathmell. Edward D. The Effects of Multibase Groupingand Early or Late Introduction of Base Representationson the Mastery Learning of Base and Place Value Nu-meration in Grade One. Doctoral dissertation. The Uni-versity of Michigan. 1972,

Rea. Robert E.. and Robert E. Reys. -Competencies of En-tering Kindergarteners in Geometry. Number. Money.and Measurement. School Science and Mathematics 71(May 1971):389-402.

115

Mathematical Competencies of Entering Kinder-garteners. Arithmetic Teacher 17 (January 1970).65 74.

Scrivens, Robert W. A Comparative Study of Different Ap-proaches to Teaching the Hindu-Arabic Numeration Sys-iem to Third Graders." Doctoral dissertation. The Univer-sity of Michigan. 1968.

Steinberg. Esther R., and Bonnie C. Anderson "TeachingTens to Timmy. or a Caution in Teaching with PhysicalModels. Arithmetic Teacher 20 (December 1973).62025.

Stern. Catherine. Children Discover Arithmetic New York:Harper & Brothers. 1949.

Van Engen. Henry. "Place Value and the Number SystemIn Arithmetic 1947. edited by G. T. Buswell. pp, 59 73Supplementary Educational Monographs. no. 63. Chi-cago. University of Chicago Press. 1947.

Wilder. Raymond :.. Evolution of Mathematical Concepts.An Elementary Study. New York. John Wiley & Sons.1968.

r...0C;,Z''

mary folsom

OPERATIONS on whole numbers represent amajor area of learning for the primary schoolchild. Understanding and competency in thisarea are important both in the everyday worldof the child and for his future mathematicallearning. The four operations on whole num-bers are frequently equated with techniques forcomputing answers to examples; yet learningabout operations on numbers encompasses farmore than this. In the early stages of learning, aprimary goal is to help the child move fromphysical-world situations to the abstract ideasof operating on numbers. Consequently, it isimportant to provide the child with many expe-riences involving both physical representationsof the operations and the properties of a givenoperation so that he has the basis needed forabstracting the mathematical ideas. For ex-ample, from a multitude of situations involving':ti, sets being joined, the idea of finding thesum of a pair of whole numbers may be slowlyabstracted. Although the child does determinesums, proficiency in computing is not the goalof early instruction.

the meaning of operationsand their properties

An integral part of the work with operationsis guiding children in their investigation of theproperties associated with each operation. Thisexamination helps the child focus on the idea of

operations rather than on the ccmputationaltechniques alone. Because the emphasis shiftslater to techniques for computing numbers, anearly emphasis on conceptualizing the opera-tions and investigating their properties has sig-nificant value. Indeed, it complements and fa-cilitates the learning of basic facts and thedevelopment of algorithms.

After having many experiences with com-bining sets, the child can learn, for example, toassociate the number five with the number pair(3, 2) under the operation of addition. Fromcarefully planned lessons, the following prop-erties can be developed:

1. The order in which the numbers areadded does not affect the sum; that is, a +b = b t a. (Commutative property of ad-dition)

2. If one of the addends is zero, the sum isthe other addend; that is, a + 0 = 0 + aa. (Identity element of addition)

3. Given three numbers, the way in whichthey are grouped does not affect the sum;that is, a + (b .c) = (a ÷ b) c. (Asso-ciative property of addition)

4. If two whole numbers are added, the sumis a whole number; that is, a + b is a wholenumber if a and b are whole numbers.(Closure property of the whole numbersunder addition)

Since none of the properties of additionholds under the operation of subtraction, careshould be taken to prevent children from mak-ing such assumptions. Because the identityelement for addition is zero, any number minusitself iszero, and any number minus zero is it-self; that is, n - n = 0 and n - 0 = n. Both theiegeneralizations can be developed through theuse of set removal and set partitions.

The idea of subtraction as the inverse opera-tion of addition is difficult for a primary-gradechild to grasp. Although .( is used when he isready to master basic subtraction facts, his un-derstanding of it is probably only intuitive. Tobe precise, the inverse operation for the equa-tion 3 + 2 = 5 must be either 5 - 2 = 3 if 2 wasadded to 3 or 5 3 2 if 3 was added to 2.When a child knows that 4 + 5 = 9 and sees 95 ?, he can think, "What number plus 5equals 9?" This does not necessarily mean that

OPERATIONS ON WHOLE NUMBERS 163

he grasps the full significance of an inverseoperation. Teachers might well question theadvisability of asking young children to write anequation to show the inverse operations forgiven addition equations. Time might be usedto better advantage by using a concrete or pic-torial situation in structuring the addition andsubtraction equations associated with a givenset partition.

Multiplication of whole numbers, like addi-tion, is a binary operation that associates with agiven pair of numbers a unique third number.For example, (3, 2) and 6 are associated underthe operation of multiplication. The numbers inthe pair are called factors, and the third num-ber is called the product. Multiplication is re-lated to the product set of two sets just as addi-tion is related to the union of two disjoint sets.The product set of two sets, A and B, resultsfrom the pairing of each element of set A witheach element of set B. If A and B are two sets,

A= lx, y, z1 and B = I1,2I,we may find the product set of A x B by pairingthe elements as shown:

Thus the product set is the set of ordered pairs,A X B = 1(x, 1), (x, 2), (y, 1), (y, 2), (z, 1), (z, 2)IThese six pairs can be arranged into three rowsand two columns, called a three-by-two array,as shown in it

xyz

2

3;

2(x, 1)

(y, 1)

(z,

(x, 2)(y, 2)

(z, 2)

six dfferetit pairs 3 X 2= 6

Since primary-grade children tend to see,not six pairings as shown in the array on the leftin n, but rather twelve members in the array,only arrays such as those shown on the rightwill be used with children of this age. Every

164 CHAPTER SEVEN

product involving counting numbers may berepresented by an array. The product a x b canbe visualized as a rows with b objects in eachrow. Consequently, it is possible to guide thechild to abstract from his experiences involvingarrays the properties of the operation of multi-plication:

1. The order in which the numbers are multi-plied does not affect the product; that is, ax b = b x a. (Commutative property ofmultiplication)

2. If one of the factors is one, the product isthe other factor; that is, ax1=1xa = a.(Identity element of multiplication)

3. Given three numbers, the way in whichthey are grouped does not affect theproduct; that is, a x (b x c) = (a x b) x c.(Associative property of multiplication)

4. If two whole numbers are multiplied, theproduct is a whole number; that is, a x bis a whole number if a and b are wholenumbers. (Closure property of the wholenumbers under multiplication)

5. Given two factors a and (b c), theirproduct is the sum of the two products ax b and a x c; that is, a x (b c) = (a xb) F (a x c). (Distributive property ofmultiplication)

Trying to make an array that is 3 by 0 or 0 by3 helps the child develop the generalizationthat if one of the factors is zero, the product iszero.

Since a similarity exists between the opera-tions of addition and multiplication (both oper-ations are commutative, associative, andclosed in the set of whole numbers), it seemsnatural to ask if there is an operation that bearsthe same relation to multiplication that sub-traction bears to addition. Of course, the an-swer is yes. The inverse operation for multi-plication is division.

To find the product 3 y 4, we count the mem-bers in a 3-by-4 array, that is, in three disjointsets with four members each. An associatedquestion is to start with twelve objects and askhow many disjoint subsets are in this set if eachsubset has four members. In terms of arrays,the question is, If a set of twelve is arranged in

three rows,, how many columns will there be?The answer is four:

SO IIe

teesTwelve objects-arranged four to a row.

Often there is no whole-number answer tothe question, How many rows? For exam ple,eighteen objects arranged in seven rows doesnot yield a whole number of equivalent col-umns. Although we do carry out such a divisionprocess as eighteen divided by seven, obtain-ing a quotient and a remainder, we cannotspeak of this process as the operation of divi-sion, since the expression is meaningless in theset of whole numbersthere is no whole-num-ber quotient. The process, however, can be de-veloped.

Division by zero must be excluded becausethere are many answers in one case and no an-swers in all other cases. For example, there aremany answers for 0 -:- 0, and no answers for6 i 0. Both kinds of examples are explainedusing multiplication.

0 0 = 0 means 0 X 0 = 0Since any number times 0 is 0, any numbersolves the equation.

6 0 = 0 means 0 X 0 = 6Since no number times 0 is 6, there is no solu-tion. Consequently, any multiplication chartthat children use for practicing basic divisionfacts should eliminate zero as a divisor.

The role of division as the inverse operationfor multiplication suffers from the same lack ofunderstanding as that of subtraction. But withdivision another hazard is involved. Whereas itis true that 15 : 5 3 and the statement for theinverse operation is 5 x 3 15, 8 3 is not de-fined in the set of whole numbers, since 8 is nota whole-number multiple of 3. Even if the childobtained the true quotient 2 2/3 from the set ofrational numbers, it is doubtful that he wouldcheck his answer by using the inverse opera-tion 3 2 2/3. The usual procedure would be(3 x 2) I. 2 = 8.

Because the identity element for multi-plication is one, the following generalizations in

division may be developed: n 1 = n and n n1, provided n does not equal zero. Division

distributes over addition when division is on theright. For example, 91 + 7 = (70 21) + 7 =(70 + 7) + (21 + 7). Of course 7 + fi1 (7 + 70)t (7 . 21), but this is not usually attempted byprimary school children, although many ofthem, unless carefully instructed, are temptedto subtract 9 from 2! The point is that propertiesof addition and multiplication must not bethoughtlessly carried over to subtraction anddivision.

addition and subtractionwith one-digit addends

the role of counting

Since meaningful counting is the main strat-egy used by children to find the sum in additionand the difference in subtraction, countingshould play an important role in the number ex-periences of primary school children. Fingersare often used in counting because they areconvenient. The teacher can develop alterna-tives to finger counting by providing morecounting experiences and by using concreteobjects in counting.

By the end of the first grade, the majority ofchildren Mould have mastered counting to 100by ones, tens, and fives and to 20 by twos. Atthe next grade level these counts should be re-viewed so that those who have not masteredthem will be able to do so, and counting bythrees and fours should be introduced, usingthe number line. The number line should beplaced so that children can touch the symbols.A mastery of counting by fives, twos, threes,and fours is a prerequisite for learning multi-plication facts once an understanding of theconcept of multiplication has been developed.Hence, counting activities should occur fre-quently in the early grades.

170


procedure and sequence foraddition

The physical representation for addition isthe union of two disjoint sets. The words unionand disjoint need not be used with children.The meaning of set should be developed by us-age rather than by definition.

Place a number of .dissimilar objects on atable (dissimilar :So that children. will not de-velop the:ideaThat the membefS of a set musthave soffit- characteristic in :Common otherthan that they are members'.o.f.the_same set).Ask the children to nartie,inernernbers of theset (a block, a penCil, a keY) aS'each'iS pointedto. The set will be well definedaltie;childrencan answer 'such questions as Ihesk, Is theblock a rnember of the set?" (YeS),"Wby?" (It'sthere.) "IS:the elephant a member.,.attie set?"(No) "W00-" (it's not there.)

USe.the:tlannel board to:del/etc:5'00e ideaof joining sets Ao..firld',the."ridtribet.of '0,),,c OW setand begin work on the coilimillative,pipi:iperty ofaddition. Ask ttle-c-hitdien, "What things do yousee in each set?- How tpaliyirternbers are in thefirst set? How many in the other set? Move theset with the apple and the duck tdjoin the otherset and ask, "If you join a set of two with a set ofthree, how many will be inffie-new set ?" Sepa-rate the sets and then jointhe setofthree withthe set of two .and.-ask the same -question.Present several different joinings, using num-bers other than zero. The children may need tocount to determine the number in the new set.

166 CHAPTER SEVEN

Every child should be given a box with atleast ten objects, such as blocks, disks, andother materials. Much practice can be given byasking children to show, for example, a set offour and a set of two and then to join them andname the number of the new set. There areseveral advantages to such a procedure: all thechildren are busy working; they have many ex-periences in seeing sets of three to five mem-bers, which they ordinarily cannot recognizewithout practice, and after the novelty of theobjects has worn off, the children will regardthem simply as things used to determine num-ber, in the same way that an adult regards pa-.per and pencil.

Partial counting, that is, counting on from agiven number, is a difficult task for almost allchildren. Much practice is needed before chil-dren master the task.

Place several flannel cutouts on the flannelboard. Say to one child, "Come up and placeyour hands around the number you can seewithout counting and tell me how many that is."Then have the class help the child count therest of the members in the set. The:ability to dopartial counting will heio increase the children'ssophistication as they learn the basic additionfacts. This will be shown later.

Siort, Clra

(P. 2 3, 4, 5, 6, 7

kojc

Avemge Child

'.21, 3, 4, 5, 6,7

Very Cor;:pLIt. I:1

51, 6,7

171

2 5

Building equations should be a slow proc-ess. Symbolism should be introduced only af-ter the concept being taught is well developed.The following sequence is su,jested for addi-tion.

Use objects and numerals. (Cutouts on aflannel board are easy to manipulate.) Point tothe first set and ask, "How many are there inthis set? Put the rumeral under it. How many inthe other set? Put the.numeral° under it. If wejoin the two sets, how many will be in the newset?" Place ,a 5 to the right of the joined sets. Dothis with several different sets, but have nomore than nine objects in the joined sets. (II .)

Transfer to the chalkboard and representthe objects' there. Extilain, "We bought threecans of orarige,juice." Draw.rings to representthe cans of orange juice. "Where should the 3be written?" Put the in the proper space."Then we bought two cans of applesauce."Make the rings to represent the two cans."Where should the 2 be written? How manythings did we buy? Where,should the 5 be writ-ten?" (It is not necessary to explain the use ofrings for representing the cans. If the rings aredrawn as the words -are spoken, the childrei,will understand. The reasoning behind the useof rings is simple. It prevents wasting time todraw objects and enables the teacher to devotemore time to the mathematics.)

1°300

3 i 2

Transfer to a worksheet that will holdabout six boxes like those shown in the preced-ing activity. Make up word problems similar tothose in the previous activity and have the chil-dren draw the rings and write the numerals asthe questions are posed. Thus you can walkabout observing while telling the story andmake certain that everyone understands whatis to be done. Six such exercises may take theentire mathematics period.

On another day, after it is clear that every-one has understood, return to the flannel boardand use cutouts to represent the sets. After thenumbers of the sets and the number of thejoined set have been identified and the numer-als placed under the objects, -say, "We joinsets, but we add the - numbers. Wendve a spe-cial sign that says-'add-the'numbers.' It is calledthe plus sign." Show the children the flannelplus sign and place it between the numbers ofthe sets. "We can read this sentence like this:Three plus two is the same-as five."Repeat withother sentences,, and then transfer-the activityto the chalkboard, and finally tdthe Worksheet,using story probleths. The figtires-dritlie-work-sheet will-now, lOok like this:

00

o 0

(It is not necessary-td explain whata-sentenceis. Simply read itlor-the Children as you point toit. Ask -them 'to i read the Sentence after eachstory is completed. To reinfOrde:the language,ask where each numeral shotild,be, written, thatis. before or after the pluS'sign:

Return to the.flahn-eftoded*cf use the setobjects, flannel 'filitiieralS,' and.prus sign. In-troduce:the equals sign. Say, "We-have a signthat means 'the same as.' " ShOw the childrenthe equals sign and place it in:the prOper posi-tion on the flannel board. "We can now read theequation as three plus two _equals five." Notethat the word equation -has now been in-troduced. Again, there is no need to give thechildren an explanation. Once 3 + 2 = 5 is writ-ten, an equation is represented; in the previousactivity, the children were just supplying an an-swer.

172


Follow the same- sequence from flannelboard to chalkboard to worksheets. The work-sheet now looks like this:

When the children are readY, teach themto find the sum for simple equationS, such-as 4+ 2 = ........-without using story problerris. Havethem repreSent the number with concrete ob-jects. Let-them -Show.fotir blocics, join ,a set oftwo, and dounttO find -the nUmber'of.:_the newset.

Children will not use the blocks as a crutchunless they really need them. When some ofthem ask if they can use marks instead of theblocks, encourage them to do so. Usually theywill follow a pattern like this:

(a) ///// // to (b) // to (c)5 + 2 = 7 5 + 2 = 7 5 + 2 = 7

Advancing from (a) to (b) indicates that thechild has mastered partial counting. Goingfrom (b) to (c) tells us that the child "justknows."

Worksheets like the following may be used todetermine (by watching the children at work)which children have generalized the commu-tative property of addition and the idea thatadding one produces the next whole numberafter the first addend.

2 + 3 =1 + 1 =

3 + 2 = 2 + 1 ='

4 + 1 = 3 + 1 -=

1 + 4 = 4 + 1 =

If children are still using blocks or marks to findthe sum of 3 + 2 after using them to find thesum of 2 f 3 or are still using material to findthe sum when one is added, it is evident thatthey have not generalized these concepts. Fur-ther attention will be given to the commutativeproperty when set partitions are used, but morecounting experience is needed to generalizethe addition of one.

168 CHAPTER SEVEN

Use the number line to show addition.(This is a rather sophisticated idea, which maybe postponed until later in the year, dependingon the ability level of the class. To be success-ful, the child must be able to move his fingerfrom 0 to 1 to 2, and so on, which he shouldhave mastered in counting exercises. He mustalso be able to do -partial counting.) Show thechild how'to. find, for example, 5 + 2 = 7. It isoften necessary to teach him to 'Cover the firstfive spaces with his hand as -he counts twomore to.reach the 7.

procedure and sequence forsubtraction

The physical representation for subtractionis the removal of a subset. It is not necessary todefine subset, nor is it necessary to dwell onthe empty set's being a subset of every set, onevery set's being a subset of itself, or on thedistinction between a single element and thesubset consisting of that one element.

Place five objects on the flannel board.Ask a child to remove a subset. "How many arenow in the remaining set?" Eventually someonewill be brave enough to remove the entire setso that the number of the remaining set is zero.

Someone, perhaps the teacher, might even re-move the subset with no elements so that thenumber of the remaining set is five.

t Place objects and numerals on a flannelboard like this:

1 73

Set

5

Then-remove a subset

3

*12aPie 11,4;

Show the remaining set

2

Many examples similar to this are needed.

Move to a chalkboard and represent theobjects on it. "John had five balloons. Hisbrother broke two. How many were left?"

I> Transfer to a worksheet like the oneshown. Tell a story and have the children worktogether using rings to represent the objects.

Introduce the minus sign. Explain that it isrelated to the idea of removing a subset butthat it means to subtract the number. Read theequation as "six minus two equals four." Stressthe fact that in subtraction the numeral for thenumber of the entire set is always Written first.Do this at every opportunity. Since subtractionis not commutative, every effort, should bemade to see.that no one writes 2 6.

Have the children complete subtractionequations such as 6 - 3 Use concreteobjects first. Ask the children ,to show sixblocks and remove three to find the number ofthe remaining set. On paper, their work willlook like this:


Next, show su htractiAir,On'the,n um ber line.

2=6

Teach the afilidti.0ieftiSlifiger okth0,09neralfor the number of the entire set. Use a pencil tomake the swing back to 7 and then to 6 as hecounts, "One, two."

teaching the related facts

Partitioning a set into two subsets will beused to teach the commutative property of ad-dition and later the connection between the in-verse operations of addition and subtraction. Inattempting to relate addition and subtractionfacts, the teacher should work on the additionfacts for a given p,artitipp,first. Again, there isno need to defiriepkiiiiPrissittipty do it so thatthe childrercuPdeiitand what iS:Meant.

Partition e:eeti,endgiCre-eqdatioris orally.Place a number pt objects on the kennel boardand say, lerit'gping to partition 0100:Onto twosubsets:'';Use colored yarn to nia'ke-the parti-tion.

A el )1

0 0 1 o 00

170 CHAPTER SEVEN

Stand on the side of the flannel board next tothe subset of four. "If I stand on this side of theflannel board, how many are in the first subset Isee?" (Four) "How many are in the other sub-set?" (Two) "What addition equation can I

write?" (Four plus two equals six.) Move to theother side of the board and ask the same ques-tion, which will result in the equation two plusfour equals six. This activity illustrates the com-mutative property of addition and should helpmost, if not all, of those children who have notmade the generalization.

Move to the chalkboard and do severalother partitions so that all the children under-stand. Ask the same questions as before andwrite the equations suggested by the children.

/ /

Provide worksheets with enough partitionsto fill the page but not crowd it and have thechildren work on their own. It may be necessaryto use two different worksheets for childrenhaving different levels of knowledgeonemight be with partitions not greater than fiveand another with partitions as great as ten. It isbetter to divide the group and have the childrenworking on their own level than to give themwork that is too difficult, which will develop feel-ings of frustration and failure.

Once the children are able to do the pre-ceding activity, move to subtraction facts forthe partition Say, "Today we are going to writesubtraction equations for a partition. What isthe first thing we do? That's right; we write thenumeral for the nur of the entire set first."(They may say, "Wri_ le number of the set.")Give the children flannel numerals and minussigns to show 9 and - twice.

Cover the subset of four and ask a child to fin-ish the first equation. (9 4 5) Uncover thesubset of four and cover the subset of five."Finish the other equation." (9 - 5 = 4)

Do several activities on the chalkboard,but instead of covering a subset, just point to itwith your finger.

Do similar activities with worksheets.

//

While the children are working, the teachershould walk about, observing (1) those childrenwho must count to find the number of the set.whether or not they write that numeral for bothequations before completing the first, (2) thosewho cover the subset to be removed for eachequation, and (3) those who go through someof these steps to obtain the first equation andthen automatically write the second. (Ofcourse, the same types of observations shouldhave been made while the children were doingthe addition equations for a given partition.)Obviously, the child who must count the num-ber in the first subset and the number in thesecond subset and then count the members ofthe. partition by ones is far below the child whocan recognize a subset of four and a subset ofthree, count "five, six, seven" to find the sum,and then simply write the second additionequation. It is possible to discern various levelsof maturity by watching what the children aredoing and how they are doing it.

Use the partitions to write all four additionand subtraction facts suggested for

175

. For the partition , where

the subsets are equivalent, only two facts arepossible. When this occurs, it is not necessaryto explain it to the childrenprovide only themodel for one addition and one subtractionequation. If a child asks why this is so, simplyask him to write the equations; he will d;scoverthat he has written the same thing twice. A childbright enough to ask the question can under-stand if you explain that this occurs when thesubsets are equivalent.

No one should take it for granted that writingthe addition and subtraction equations for agiven partition is an easy thing to teach all chil-dren. It is difficult and requires great patienceand a slow pace from one idea to the next. Atany given moment it is possible to observe chil-dren doing the task eight or nine different ways.

The child who looks at and

writes 4 t 5 = 9 is indicating that he knows thatfact, when he automatically writes the otherthree, he is showing that he has the structurewell in hand. He has arasped four basic facts atone time. Although others may labor at lowerlevels of sophistication, they will move from onelevel to the next if they have the freedom tomove at their own speed.

It may be useful at this point to consider howthe physical representation of subtraction asthe removal of a subset has been expanded bypartitions so that it encompasses far more thanthis simple idea. The word problems and equa-tions with pictures, such as , al-lowed the subset that was to be removed to re-main in full view. The use of the partitions tookthe child even further. After the initial coveringof the subset to be removed, he not only doesnot cover, remove, or mark it out but visualizesthe separation of the set into two subsets with-out even moving the subset. Relating the ap-propriate addition and subtraction equations toa given partition is a flowering of the ideas ofpart-part-whole and whole-part-part. if thechild can write the equations for a given parti-


tion, he has attained not only conservation ofnumber but also the concept of reversibility.

Frames and partitions. Once the children un-derstand the work with partitions, it is possibleto use the partitions to solve other equations.Some teachers use frames. Instead of writing6 + 4 = they write 6 + 4 = 0 . Ask thechildren if ;I would make any difference if wewrote = 6 -1- 4. Let the children discuss thematter until they agree that it makes no differ-ence, that the sum can be written on eitherside. Moving the position of the frame to anaddend in addition or to a sum in subtractioncan cause difficulty for both teachers andpupils. In the equation .1. 2 = 7, for example,children do not understand that they are beingasked, "What number plus two equals seven?"They do not know that this can be determinedby subtracting 2 from 7. Use partitions to helpchildren develop these concepts.

Give the children the following problem:+ 2 = 7. "Do we know how many are in the

entire set?" (Yes, seven) "Make seven marks.") "Do we know how many are in

one subset?" (Yes, two) "Mark off two.") "The other subset, then, must

have fiYe members." Use the same procedurefor 3 1- = 8 and for 7 -0 =2. For 0 - 3 = 4,say, "Do we know how many are in the entireset?" (No) "Do we ,know how many are in eachsubset?" (Yes, three and four) "Make a drawingand count to find the number of the entire set."( ) A child might recognize threeand count on to seven, or recognize four andcount on to seven, or simply recognize groupsof three and four as seven.

Practice can be given in addition by usingthe partition to give different whole-numbername combinations for a given number. Thefollowing example shows names for 4 corre-sponding to the partitions.

172 CHAPTER SEVEN

4

4 + 0 / / //)3 + 1 / / /2 + 2 / ///1 + 3 /Z/ //0 + 4 / / /

Later this can be written as + = 4. Pointout that the two frames are of a different shape,making it possible to use either the same num-ber or different numbers in both frames. If theframes are the same shape, as in + = 4,the same number must be used in both. In-serting 2 makes this sentence true. (Using thesame frame is equivalent to x x = 4, and xmust be the same number both places. Usingtwo different frames is the equivalent of the al-gebraic equation x y 4, and there is no rea-son why x and y cannot be either the same ordifferent numbers.)

Subtraction word problems. In the primarygrades children encounter three types of sub-traction word problems:

1. How many are left?2. How many more?3. How many more are needed?

The first type does not prove difficult, since itis common in life situations and is used exten-sively in developing the subtraction equation.Looking at the drawing 0 0 0 0 0, childreneasily recognize that the operation of sub-traction is suggested and that the question is ofthis type: "Mary had 5 pieces of candy. She ate2 pieces. How many pieces does she haveleft?"

The second type, however, involves a com-parison, and many children do not recognizethe question as one solved by subtraction. Itmay help to ask the children to represent theword problem with blocks or other concretematerials. For example: "Mary has 5 books.Jane has 2. How many more does Mary have?"The following representation would result:

Mary's hooks EN [0 El 14Jane's books

From this arrangement, it now becomes appar-ent that Mary has three more than Jane. It is notequally apparent that the equation suggested is5 - 2 = 3. Many experiences with blocks or pic-tures are needed before the teacher can sug-gest that it is not necessary to represent thenumber of books Jane ha;. "Can't we removethe number of Jane's books from Mary's?" The

picture will look like th s: 0 0 0 0 O. The

children will then recognize the operation ofsubtraction.

Many adults tend to consider the third typeof problem the.same as the second type. Manychildren, however, do not agree. Using blocksto represent the problem should convince anyteacher that this type of problem needs atten-tion. When asked to represent the number ofMary's books and the number of Jane's books,some children will move in three more blocksto make the sets equivalent.

Maly's books 1#11 in

Jane's books I Pji ;k1 ;

The teacher must first convince them that it isnot necessary to move in the three blocks tomake the sets equivalent and then suggest thatthey remove the number of Jane's books fromMary's. In other words, it may be necessary tomove through three different physical repre-sentations before the children see that thequestion suggests subtraction.

From

to

1.1 i2:11 '174

C411

1,141 14

JAI 1(/)11

to),1^;--

.1

'11,

'-`77

ii \% 41\ 1,V,

using the associative property ofaddition

If ,:ldren know the addition facts up tosums of ten, the associative property of addi-tion can be used to obtain facts involving sumsof eleven to eighteen. They must learn to thinkof 8 + 4 as 10 + 2, for example. The followingsequence is suggested:

1. Present the problem at the flannel board,using objects.

Place a set of eight objects and a set offour on the flannel board. "Do Ill_ . enough tomake a set of ten?" (Yes) "How many must wemove"from the set of four to make a set of ten?"(Two) Move the two objects accordingly. "Howmany do we have now?" (Ten plus two, ortwelve)

This must, of course, be done with manyother combinations that will result in a sumgreater than ten.

2. Present th6 oroblem at the chalkboard,but let the children use the objects.

Give the children paste sticks, lollipopsticks, tongue depressors, or blocks. "Showme a set of seven and a set of four. Are thereenough to make a set of ten?" (Yes) "How manymust you move from the set of four to maketen?" (Three) After they have done this, ask,"What numbers are represented in the two setsnow?" (Ten and one) "What is their sum?"(Eleven)

After many such examples have been done,it may be helpful to write on the chalkboardwhat has taken place. For the preceding ex-ample, write

7 + 4 = 10 +1= 11

3. Transfer to a worksheet, using marks forthe objects in both sets.

Distribute worksheets to the children withproblems like this:

ID)

g-r 'I -10


4. After the children are successful at work-ing problems like those on the precedingworksheets, change the worksheets toshow marks for the second addend only.

Distribute worksheets like this:

I ' l l

g+ 9- 10 +

If a child does not make all the marksas onthe first worksheets, but makes marks asshown above, he either knows the sum of 8 + 2or he is doing partial counting: "eight, nine,ten."

5. Provide no marks at all on the work-sheets.

Give only combinations such as 8 + 5, 9 +3, 7 + 5, and so on. At this point, children ei-ther make all the marks, as in the third step,make only enough marks for the second ad-dend, as in the fourth step, or write only thesum, since th;.1.y have done enough of these to"just know." Permitting these different levels ofsolutions on the same worksheet is one way ofproviding for individual differences.

using partitions for sums of11 to 18

Partitions can be used to practice the relatedfacts with sums greater than ten. The usual se-quence from flannel board to chalkboard toworksheet is used. Any of the three forms in-dicated here may be used on the worksheet.

1.

will An v.)

174 CHAPTER SEVEN

2.

MIN 11I11111

3.

fill IIII Ill! in

If form 1 is used. the child must count to findthe number of each subset in order to get thecombination. Then he either knows the sumand writes it, counts on from eight to fifteen, orcounts the members of the entire set by ones.Once he has the first equation finished, he maywrite the others, indicating that he knows thecommutative property of addition and the in-verse operations. Others may combine in vari-ous ways counting and covering subsets, assuggested for sums of ten or less. Form 2 al-lows the child to complete the equations if heknows tnem without having to count to deter-mine the number of each subset. If he does notknow the structure, he v4111 use all the ways in-dicated in form 1. Form 3 allows the child whorecognizes two sets of four as 8 and a set offour and a set of three as 7 to obtain the num-bers of the subsets without counting.

lesson plans

The ideal lesson plan should be a blend ofseveral ingredients. At least ten minutes of theperiod might well be devoted to group activities

t' %9

in which everyone is involved and the feeling ofa class working together is maintained. Thegroup lesson provides an opportunity for ques-tions and discussion. Everyone can participateat hiS own levelthe fact that a student doesnot know all his basic facts does not precludehis learning other things. The development ofconcepts should continue while the studentsare learning the facts and ways of thinking

;about the facts. The combination of adequatetime and good teaching results in a greaterrange of ability and skill in solving problemsand handling computation. Therefore. theteacher must base the small-group assign-ments for a particular day on a careful eval-uation of the work done by the children the pre-vious day and allow enough time for every childto learn. The following lesson plan might befeasible by midyear in grade 1.

All pupils: Count by tens and fives to 100 on thenumber line. Any child who volunteers may be theleader and move his finger along a number line asthe class counts aloud. (When a child first startscounting. the teacher should be sure that he putshis finger on 0.) No child will ask to 'do the tens' or"do the fives" if he is not confident that he hasmastered the count.

Use a word problem for group solution. Ex-ample: There are 16 boys arid 15 girls presenttoday. How many children are here? Use the num-ber line and have a child find 16 on the line. Let theclass count 15 more to arrive at 31. (Note that thisword problem can be solved even though the chil-dren have not had experience in adding 16 and 15.Every class period in the primary grades shouldinclude a word problem so that children have ax-pertence in solving word problems and builci con-fidence in their ability to do so.)

Play a game. such as identifying a number by itsrepresentation on charts showing tens and ones,for examnle. 34:

Tens Onesl*t ((Lt

New work Introduce subtraction equations for parti-tions (sums 6-10).

Giouping: Eddie. Rosa. Leroy. and Susanwork-sheet with partitions involving sums not greaterthan five. They will write subtraction equations forthese partitions.

Joe. Bobby. and Leonworksheet with sub-traction equations (sums to ten). still using blocks.

Ethel and Jimworksheet with addition equa-tions (sums to ten). using blocks or marks.

Rest of classworksheet with subtraction equa-tions involving partitions with sets of five to ten.

addition and subtractionwith two-digit addends

Before adding and subtracting with two-digitaddends. the teacher must change the equa-tion from the. horizontal form to the verticalform. Tell the children a story problem: "Billhas five marbles. His brother gave him fourmore. How many ma oles does he have now?'Ask the children what addition equation can bewritten for this problem. When the children say.-Five plus four equals nine.- write it on thechalkboard like this:

5

..4

9

Continue with other. similar stories and writethe suggested equations in the vertical form. Itis not unusual for children to accept the formwithout question. If no questions are asked,they have probably assumed that the teacher,has simply decided to write the equation upand down rather than across. Subtractionequations from story problems can then bewritten in the vertical form. From this point on,practice worksheets should take this form. Ifprevious equations have been read as "fiveplus four equals nine" and "seven minus twoequals five," then the reading is the same fromthe top down.

The following sequence of steps is sug-gested for adding and subtracting with two-digit addends.


1. Add multiples of ten less than 100 by us-ing the tens and ones charts.

Show 20 ± 20 = 40 this way first:

tens Onespimi pig its14 pi;(unrl 'Ulm rirric lrrnra

Then like this:

2 tens 20+2 tens i-20

4 tens 40

Tens

it ..

Ones

2. Subtract multiples of ten less than 100 byusing the tensand ones charts.

Show 40 - 10 = 30. Begin by showing thefour tens:

Tens -Ones

Then show the:charts after one ten has beenremoved:

4 tens 40-1 ten -103 tens 30

Tens Ones

n ( rag- tiiiilif

176 CHAPTER SEVEN

3. Practice facts orally while the children arewaiting in line to fia to lunch or to physical edu-cation.

Ask what is 20 r 30? 40 + 10? 70 r 20?80 - 40? 70 - 20? 40 - 20?

The preceding steps are designed to helpchildren think in tens. For example, "Two tensplus two tens equals four tens, aridWity plustwenty equals forty." They should not.think me-chanically, "0 + 0 = 0 and 2 2 = 4."

4. Develop the expanded form for two-digitnumerals from the tens and ones charts.

Show the charts and ask, "What is thenumber?" (Forty-three) Stand between thecharts. "Today we are going to write forty-threein a new way. How many are here?" Indicate thetens chart. (Four tens, or forty) "Plus huw manymore?" Indicate the ones chart. (Three)

Tens Ones14.'1

gg'i IIIc tirrat

Write on the chalkboard:

43 = 4 tens + 343 40 + 3

Do several other exaMples. Tell the children,

that 40 + 3 isrcalled the expanded form. Give aworksheet that provides practice in writing theexpanded form 'f numerals.

5. Start with the expanded form and use theplace-value chart to get the simplest name forthe number.

Present the problem 50 + 6 = ____ andask the children, "What is the simplest namefor the ansv...or?" Show them the charts:

Tens Ones

Wears. We

The simplest name is 56.

After doing, several examples, distributeworksheets Jo.give_the childretWattice in go-ing from the .expanded form to the simplestname for the number.

6. Put together the join' skills previouslytaught.

Add:

43, 40 + 3+24 20 + 4

67 60 + 7The children have had practice with the basicaddition facts (3 + 4 - 7), adding multiples often (40 + 20 = 60), writing .the- exPanded form(40 + 3 acid 20 + 4), and uSirig the -expandedform to go to the simplestname for the number(60 + 7 = 67).

7. For subtraction, represent the number onthe place-value chart.

Subtract:

4§ 40 +-23: .20. +

.22 20 +2

Tens Ones. ago

igi.

Remove three from.the ones chart and twentyfrom the tens chart and sh-ow"the charts now:

Tens

II

.0nes

renaming in addition andsubtraction

Each step involving renaming in addition andsubtraction should be represented on the tensand ones charts.

Add 46 and 35.

46+35

Expand and show on the charts:

40 - 630 5

Tens Ones

Then show the addition on the charts:

70 - 11

Tens Ones

"Do wehave enough to make a set of ten on theones chart?" (Yes) Remove ten ones and placea bundle of ten ones on the tens chart and showthe charts:

80 + 1

Tens Ones


"What is the number?" (Eighty-one) The prob-lem can now be written vertically:

46+35

11

70

81

Present 73 - 29 to the children.

Tens60 13

73 J'ef ±-29 20 + 9

44 40 + 4

Ones

Remove a ten from the tens chart and put tenones on the ones chart:

tens Ones

I

Then remove nine from the ones chart andtwenty from the tens chart and show the an-swer, 44:

Tens Ones

Replacing the charts with a homemade aba-cus can save time. A useful abacus should havefour wires with eighteen disks in the ones place,

178 CHAPTER SEVEN

nineteen in the tens place, nineteen in the hun-dreds place, and nine in the thousands place.This type can be used throughout the primary

ides and is easy for both teachers and pupilsto manipulate.

The children should discuss the differencebetween how a number is represented on thecharts and on the abacus. They will say manythings. but the significant difference is that tenis represented on the chart by a bundle of tenones but on the abacus by one disk. Any trueanswers the children give. such as difference incolor or that one is horizontal and the other ver-tical. should. of course. be accepted until somechild suggests the significant difference.

A few minutes a day should be taken so thateventually each child will have had the opportu-nity to manipulate the abacus as he renames inaddition and subtraction. When they under-stand the concept. introduce the short form.which can also be adequately represerited bythe abacus.

Each variation in the algorithms for additionand subtraction will not be discussed here.Techniques similar to those used in the preced-ing activities can be used in renaming asneeded. such as ten ones as 10, ten tens as100, or 100 as nine tens and ten ones, Theusual sequence is available in commercial text-books.

learning addition and subtractionfacts

There are several properties and gener-alizations that children should know in order todecrease the number of facts in their memoryload. Although it is not expected that first-grade children will know all the basic additionand subtraction facts at the end of the year,

(.2'1.+ Vu

they will have made several generalizationsfrom their experiences:

1. The children will realize that when zero isan addend, the sum is the other addend. (Zerois the identity element for addition.) This elimi-nates nineteen facts from the memory load;these are indicated by color A on the additionchart.

2. The children will have generalized thatwhen one is an addend, the sum is one morethan the other addend. This eliminates anotherseventeen facts from the memory load; theseare colored B on the chart.

3. The children will have learned that tt e or-der in which the addition is performed does notchange the sum (commutative property of ad-dition). This eliminates twenty-eight more far ts,colored C on the chart.

4. The children will know the seven doubles,which are colored D on the chart, thus -Iirninat-ing seven more facts.

At the next grade level, an effort should bemade to master other facts so that they too canbe eliminated from the child's memory load.Three facts with sums less than ten. 2 + 3. 3 f4, and 4 + 5, can be worked around the dou-bles: if 3 + 3 = 6, then 3 + 4 - 7 and 3 -t 2 = 5.These are the facts colored Eon the chart. Thenines are easy once the children can see thatwhen nine is an addend. the sum is ten plus theother addend less one:

9 6 8+4 +9 t-9

13 15 17

The nines are colored F on the chart. All thisleaves nine facts with sums of ten or-less (thosecolored G) to pr Mice and to master.

Six facts with sums greater than ten must bemastered using the associative property of ad-dition. These are colored H on the chart. threemore, those colored I, can be worked aroundthe doubles as beforethat is, if 6 + 6 12,then 6+ 5- 11 and 6 + 7 13; if 7 t- 7- 14,then 7 18 - 15.

An understanding of inverse operationshelps Children relate subtraction to addition. Ifa child adds 8 to 6 and gets 14, then he shouldsee that under the inverse operation, 14 minus


A --gB =SIC = 0

A ddrlion chart.

D=E = illF = 11

m

8 is 6_ In the first procedure he adds the 8, andin the second he must subtract the 8. Com-mercial texts sometimes label these processes"doing" and "undoing." This is not an easy con-cept for children, who often do not grasp thatthe "doing" was adding the 8 and that there-fore. in order for the result to be the numberwith which they started (6). the "undoing" mustbe subtracting the 8.

Thinking "What number added to six equalsfourteen?" is a good way of using language anda knowledge of addition facts to eliminate sub-traction facts, for if this interpretation is usedand if the children know the addition facts, thenthere are no subtraction facts to be learned.

When correctly administered, timed testscan be used to ascertain what facts need fur-ther work_ Each problem should be timed, notmerely the test itself. If a timed test is given, theteacher must be sure that the children do nothave enough time to use immature ways of ar-riving at the sum or difference. The best way isto use a tape recorder that voices the com-binations with just sufficient time (about threeseconds) between combinations to allow thechildren to write the answers if they know themautomatically It is then possible to know ex-actly which facts are giving which child trouble.

The results also permit the teacher to groupchildren so that they work only on the facts theydo not know.

Holding up flash cards with combinationssuch as 4 : 5 or 9 - 2 is not a very productiveuse of time. Those children who know the an-swer will shout it out. Those who do not knowwill relax, since they know that the others willsatisfy the teacher by giving the correct re-sponse. Flash cards can be effectively used af-ter a timed test like the one described above.When the children have been grouped to workon those combinations they have not mas-teredperhaps, for example, into four groups.those working on addition sums of ten or less,those working on sums from ten to eighteen,and those working in similar groups on sub-traction factsone child in each group can beappointed leader for the day and hold the flashcards. As he holds up a card, the others showhim the answer by raising their appropriate nu-meral card. Then the leader turns over his com-bination card to show the answer. Anyone whohas the wrong answer is required to write downthat fact for future study. The leaders should bechanged every day so that they too have theopportunity to work intensely on those factsthey do not know.

-; 8.1

180 CHAPTER SEVEN

games that provide practice shown here, the children might generalize asshown by the arrows; that is,

Chalkboard,-Rocs.-Organize two (or more)teams and put-two sets ofnumbers, one foreach team, on the chalkboard. 13 14 15 16 17 18

7 8 9 10 11 121 2 3 4 5 6.

C{Lta.t, 2 7 5 1- cl Li 8 3 0 (0

13 kl 7 to I1- 9 6 12

acid, (0 tg 5W- 7

0 6) 2 _7 5 1 4 q(-.) 12 (3 13 ti 7 10 IT

Have individil members of the teams come tothe board. After giving such instructions as"Acid six, a-'d nine, add four," have the childrenwrite the answer below each number. The firstone finished with all correct sums-wins a pointfor his team. This genie can also be used forsubtraction-by putting the numbers 10 through18 on thilboard.

Hiddeh Plumbers. Think of some com-bination of numbers, and provide clues to en-able The children to guess the numbers. "Weshall use the set-of numbers zero through nine.I am thinking of two numbers. Their sum isnine." Let the children give all the possibilities:(0. 9), (1, 6). (2, 7), (3, 6), (4, 5). Since it is stillnot possible-Jar-the children to ascertain thecorrect combination, give another clue: "Onenumber is greater than seven :" This leaves (0,9) and (1, 8). When the children-determine thatthey still do not have enough information, theywill ask for another dile. Say, "The difference isseven." Now only one nossibilitY fit he infor-mation: (1, 8),

- Show Me. Provide each child with cardshaving, a :zero and two sets of-the digits onethrough nine. Hold up a combination such as4 4- 8. The children show the sum by holdingup the cards al [0 . Subtraotion com-binations can also be used.

Maneuvers on a Lattice. Make a lattice sothat the needed facts are used. For the lattice

2 9 /add 7) 9 (svibirad 6)

3 9 (add 61 15 := 5 'subtract 1C)

(Y.4 = 10(culd, 2) 18, .svibtrati 7)

After some !practice. give only the-,bottom rowof the lattice so that the children must-either vi-sualize the rest- of it- or make the -gener-alizations.

PatternS. 'Ask -the children to find_the miss-ing numerals in patterns-like these:

7,10,13.163,7, 19, 23, ,

1, 3, e 28

Card and Dice -Games. Most of thesegames are played by pairs of children.

Give each child numeral cards that include azero and two sets, of the digits one -throughnine. Have One-Child declare the Play-;4or ex-ample, to play nines. He then puts out one ofhis cards.-If-the other child has a-card-that hecan play that will result in a ,sumof nine, heplays it, takes the pair;-.0-nd leads another card.The child with the greater 1T-umber-of cards atthe end of the game wins. For subtraction, in-crease the digitson. the-cards to eighteen andask for a difference':

Have one child. toll a EPairof dice and havethe other child -name the sum. The:child getsone point for each correct answer. Theene withthe most :points: wins. For sums greater thantwelve, use two plain wooden blocks and num-ber the sides 4-9. For subtraction: ask for the

%difference between the two ,nurnbers. Oneblock could have the numerals8-13 and theother the numerals 4-9.

problems

As discussed in chapter 4, it is important thatchildren be asked to solve real problems. Storyproblems such as those used in commercialtexts and the problems used in the physicalrepresentations of addition and subtraction in-dicated earlier in this chapter are not reallyproblems. They are situations from which themodeling, such as 2 + 3 = 5 and 7 2 = 5, canbe derived. The following are some problemsthat may be found suitable for certain childrenor groups of children.

1. John had a box of chocolates. His mothersaid that he had more than 12 but fewer than32. When he counted his chocolates by fours,he had two left over. When he counted them byfives, he had one left over. How many choco-lates did he have? (26)

2. The first picture shows two rings. The left-hand ring contains 6 counters, and the right-hand ring contains 4 counters.

If we move a counter from the right-hand ringto the left-hand ring, there will be 3 counters inthe right-hand ring and 7 counters in the left-hand ring.

In how many different ways can you arrange 10counters in the two rings? (10)

3. Complete the following addition table:

12 11

26

1. 513

5 14


4. Ask a friend to think of a number between1 and 10. Find out what the answer is by askinghim no more than five questions that may beanswered only by yes or no. How many ques-tions would you need to ask in order to find anumber he has chosen between 1 and 20? Be-tween 1 and 50? Between 1 and 100?

5. Fill in the missing digits in these exam-ples:

3* *3 *7 *4 63-1-'7 +4* -1-** -2* -*7

62 92 90 35 3*

6. The first box shows how the numbers in-side the box were added across, down, anddiagonally. Find the missing numbers insidethe second box.

F7 X 15

.9 5 _ 14

17 16 13 12

14

15 13 17 15

multiplication

There are two approaches to multiplication,neither of which has been universally accepted:repeated addition and arrays. Both should be

0000_1

182 CHAPTER SEVEN

used when appropriate Repeated addition isan effective strategy when used for multi-plication with two, since children can be con-vinced that they know this table if they know thedoubles in addition. But is is not very effectiveto think of 8 x 7 as 7 +7+ 7 + 7 + 7 +7+ 7 +7, since children are not competent in countingby sevens or in column addition by the timemultiplication is introduced.

The rectangular array is a physical represen-tation of multiplication. It can be used effec-tively with the distributive property of multi-plication to provide a method for discoveringand learning multiplication facts and to dis-courage t le making of marks and other time-consuming counting procedures.

Before beginning multiplication, the teachershould review counting by threes, fours, andfives to enable the children to determine thenumber of members in a,; array with a mini-mum of difficulty. The following activity willserve as a good introduction .to arrays.

Arrange a number of cans in an array likethis:

00 CGC 0 0c 0C i. 0 0 0

"Mary's mother arranged cans of soup on thekitchen shelf in this way. How many cans ofsoup does she have?" (15) "How did you findthe number of cans?" (Counting by ones to 15.Counting by fives-5, 10, 15. Counting bythrees-3, 6, 9, 12, 15.) "When we arrange ob-jects in rows [indiCate that the rows are hori-zontal j and columns [vertical I with the samenumber of objects in a row, we call it an array."Present various arrays on thechalkboard (us-ing no more than five rows and five columns)and ask the children how many rows, howmany columns, and how many members are inthe array. Explain that the cans shown aboveare arranged in .a 3 -by-5 array, which has fif-teen members.

After several such activities, the children canbe given worksheets containing arrays andproblems like the following:

0 0 0 0 00 0 C G C0 0C

0 0 C

This _ by _ array has members.

') 0 Q 0OCCO

C C

This by_ array has members.

When the children feel comfortable with thisidea, change the questions: "How many rows?How many in each row?" These questions leadnaturally to the idea that, as expressed for thearrays shown above, four 5s = 20 and three4s = 12. Worksheets using these questionsmay be given next:

o

187

000 0 00 0 t,

3s =

, . .. . .

Gs = _ 3s = _

When the children are competent in namingthe number of rows, the number of members ineach row, and the number of members in an ar-ray, review the addition and subtraction signs,"the signs that tell us to 'add and subtract." In-dicate that to find the number of members in anarray, there is a special sign that tells them tomultiply the number of members in each rowby the number of rows. Then introduce thetimes sign. "For a 3-by-4 array with 12 mem-bers, we have been writing three 4s = 12. Wecan also write the equation as 3 x 4 12."

Show the children various arrays (nonegreater than 5 x 5 or less than 2 x 2) andask them to give the multiplication equation forthe array. Then use arrays like these:

These types of arrays may be used on work-sheets to help children master the following sixbasic facts:

3 x 3 = 9 4 x 4 = 16 5 x 5 = 253 X 4 = 12 4 X 5= 203 x 5 = 15

A mastery of these facts is needed before thedistributive property is used to discover the ba-sic multiplication facts greater than 5 X 5=25. Word problems similar to the followingmay be used to reinforce the modeling.

1. Mary's teacher brought 5 books to schooleach day for 3 days. How many books did shebring to school?

2. John broke 2 balloons each day of aschool week. How many balloons did he break?

3. Jane put 3 books on each of 4 shelves.How many books did she put on the shelves?

While working on the mastery of these facts,the children can be taught generalizations suchas the commutative property of multiplicationand the generalizations involving 0, 1, and 2 asfactors.

Prepare cards with various arrays. To de--velop the commutative property, display a 4-by-5 array. "How many in the array?" Then turnit so that it becomes a 5-by-4 array. "How manyin the array now? Does it make any differencewhether we multiply 4 x 5 or 5 x 4?" (No)

The array is an excellent way tq -show chil-dren that the operation of multiplication is com-


mutative, since they can physically see that thenurnber of members in the array is the sameeven if the number of rows has been inter-changed with the number of columns.

In most programs the language-'5f, -multi-plication is "factor times factor equals prod-uct," with the first factor being the number ofrows. This terminology is used with the chil-dren. Note that the physical representation of3 x 5 = 15 is different from 5 x 3 = 15, al-though the product remains the same.

Use various array cards having only onerow to help the children generalize that if one isa factor, the product is the other factor. Show1 X 4= 4 and 4 x 1 = 4, 1 x 3= 3 and3 x 1 = 3, and so on, as examples of boththe identity elementifor;multiplication (one) andthe commutative property of multiplication.Write all the equations on the chalkboard. "Ifone is a factor, what is the produCt?"

Use arrays with two tows or two columnsand ask the children,,ifthey:,aan write an addi-tion equation for .the -array. For example,

o o 0 0for 0 0 0 0 the,,Oliations 2 x 4, = 8 and

4 + 4 = 8-can both be written. US necessaryto convince the children that since they alreadyknow the addition doubles, they,knOw the prod-uct when one of the. factors, is two.

Once the children:have;learned the-siX basicfacts mentioned prev)desly,-an array card canbe folded to find the products of facts greaterthan 5 X 5.

Present 3 x 9 = for example, andask.a child to fold the card in two pieces so thathe can tell how many.members in the array areon each side. "What multiplication equation issuggested by the first side?" (3 X 5 = 15)"By the second side?" '(3 x 4 = 12) Writethese on the chalkboard and then add 1512 = 27 'in vertical form. "So, 3 X 9 = 27."

188.

: .1 r

I

184 CHAPTER SEVEN

After doing several of these, give the childrentheir own cards with 3-by-9, 4-by-9, and 5-by-9arrays. Given such products as 4 x 8, 3 X 7,and 5 x 8, the children can fold the cards,write tl-e multiplication equations, and add theproducts.

When the children have demonstrated theability to use this procedure, hold up a 3-by-9card and fold it so that the array pictured in thepreceding activity is evident. "How many col-umns are on the first side?" (5) "How many col-umns are on the second side?" (4) "What namehave you used for 9?" (5 + 4) Then write onthe chalkboard:

5+ 4 15 3 x 9 = 27x 3 +12

15 + 12 27

Once the children understand that the foldingof the card indicates the renaming of the num-ber of columns, they can dispense with the cardand select the name. for the number of col-umns. This enables them to obtain the productif they do not know it. Note that this procedureintroduces the distributive property of multi-plication at the simplest level possible.

It is not possible to show the zero facts, otherthan 0 x 0 = 0, on an array card. Askingthem to make the array is an effective way toconvince children that when zero is a factor, theproduct is zero. Giver. 3 x 0, for example, thechild may begin to write in the three rows, butwhen you ask him how many columns there willbe, he will have to erase the three rows. If given0 X 3, however, the child cannot start his pic-ture; since there are no rows. Any child makinga mistake on a zero fact should be asked tomake the array on his paper. By the time fac-tors of zero occur, children will be quite familiarwith the idea that 7 x 6 means seven sets of 6;then 3 Y 0 can be thought of as three sets of 0.Since three sets of 0 = 0, 3 x 0 = 0.

It is worthwhile to help children generalizenine as a factor. Present the products of thenine table-9, 18, 27, 36, 45, 54, 63, 72, 81invertical form and ask the children what theycan discover about these products. They wiltprobably mention many things, but the impor-tant discovery is that the sum of the digits is 9. Ifno one discovers this, ask what is the sum of

2 + 7, 3 + 6, 4 + 5, and so forth. Thenpresent the folio 'ing examples:

9 6 9 4 9x7 x9 X3 x9 X9

63 54 27 36 81

See if they can discover that if nine is a factor,the tens digit is one less than the other factor. Ifno one is able to see this, cover the 9 with yourhand and ask how the tens digit in the productcompares with the other factor. Combining thetwo discoveries, we see that for 7 x 9, for ex-ample, the tens digit is one less than 7, that is,6, and the ones digit must be 3, since the sumof the digits must be 9.

Also, since children can count by fives soeasily, it seems a shame not to allow them tocount by fives (using their fingers if necessary)to determine such facts as 5 x 7. Surely theywill know the fact if they count by fives for a suf-ficient length of time.

The teacher should strongly discourage anychild from finding the product of 4 x 8 bydrawing

3 x =and counting by ones from the first mark tofind 32. Instead, start him counting by twos,threes, fours, and fives as suggested previouslyand follow the procedures as outlined earlier inthe chapter.

drill on multiplication facts

By allowing.the children to count by fives andby teaching them all the generalizations andproperties of multiplication, the teacher caneliminate mai.y of the multiplication facts fromthe memory load of the children. Drill should beconcentrated primarily on those products out-lined in black on the multiplication chart (thefive perfect squares), those products left with-out color, and at least two facts from each ofthe colored products to make certain that the

189


Multiplication chart

A.1111 B=Egj C=111 D=10 E=111

generalization or understanding of a propertyis still well in hand. Of the one hundred basicmultiplication facts, the generalization on zeroeliminates nineteen (those indicated by colorA), understanding the identity element of multi-plication eliminates seventeen more (those col-ored B), and the generalization on two as a fac-tor (those colored C) cuts fifteen more, leavingforty-nine. The generalization on nine (thoseproducts colored D) eliminates thirteen facts.Of the thirty-six remaining, counting by fiveseliminates eleven more (those colored E). Ofthe twenty-five facts remaining, five perfectsquares must be learned. Only half of the othertwenty facts (those products not colored) mustbe learned, since the other ten are not neededbecause of the commutative property of multi-plication.

Using a tape recorder as suggested in thesection on addition can enable the teacher todetermine exactly which facts a child needs topractice. Children can then be grouped to workon the facts they need to learn, and a leaderwith flash cards can be used as suggested pre-viously in the section on addition. The teachercould then write on the back of a child's reportcard that he needs to practice 7 X 8, 6 X 6,and 8 x 8, for example, instead of writing thathe needs to practice his multiplication facts. If aparent knows exactly what must be done, he ismore likely to see that his child learns three,

ct.1;j1f

four, or five specific facts than if he is con-fronted with the problem of practice on a hun-dred facts. it is important to include facts thatinvolve a generalization or a property, since ifthe child misses two of these, it is likely that hisconcept of the whole generalization or propertyhas fallen apart and must be retaught.

games

The following games may be used for prac-ticing multiplication.

Chalkboard Race. Organize two teams andwrite two sets of numbers on the board.

X (9 3 8 / `175

13 '15 o 51 -34 111_30

C1 1 3 5 7 LI ec 2 S 0

5-1- 6 \?). 3° 11-Z .2.`v' 31D \2 Y.-`10 0

186 CHAPTER SEVEN

Have two players, one from each team, go tothe chalkboard. Say, "Multiply by six," andhave the two children record the products be-neath the numbers. The first one finished withall correct products wins a point for his team.Use a different factor with the next pair of play-ers. (Record for later drill any specific fact notknown by a player.)

Beanbag Throw. Organize teams andmake a design on the floor (or on paper tapedto the floor) consisting of nine squares with anumeral in each square. Provide beanbagsnumbered 0-9 and have the players stand be-hind a mark and throw a beanbag on thesquares. The player must give the product ofthe number on the beanbag and the number ofthe square into which he has thrown the bean-bag. One point is given for each correct prod-uct. The team having the most points wins.

Show Me. Provide cards like those usedwith this game in The section on addition. Usemultiplication combinations 'rather than addi-tion.

Hidden Numbers. Make up the game sothat the basic facts to be practiced are needed.(See the s,<;tion on addition.) For example, thefollowing clues might be given:

I am thinking of four numbers from one tonine.

1. The product of two of the numbers is24. (Requires either 3 and 8 or 4 and 6)

2. The product of two of the numbers is32. (Requires 4 and 8)



Answer: 4, 6, 8, 8

It is important to note that most games donot actually provide the practice the teacherwishes. A game like Bingo does not insure thatthose needing to know 6 X 7 = 42 will learnanything when the teacher calls out 6 x 7. Thechild who does not know the product will notplace a counter on 42, but those who do knowthe product will. There is no way that the

teacher can watch over thirty children and notethose who do not know that particular fact.Games like this provide a change of pace andfun for the children, but no adult should as-sume that they do anything else.

division facts

The inverse operation of the multiplicationfact 4 X 8 = 32 is shown by 32 ÷ 4 = 8.Practice on finding the missing factor can begiven as the children work with arrays. If the ar-ray is presented as

N

5 F25,the teacher can ask, "What is the missing fac-tor?" In this interpretation of division, the ques-tion is, How many 5s equal 25? However, it isnot sensible to ask such a question when thechild is confronted with a problem like1,875 25.

Children find division difficult not only be-cause they are required to use the other opera-tions of addition. subtraction, and multi-plication but because they do not reallyunderstand what the question is. When the ideais to develop not only the division facts but thealgorithm as well, it is profitable to provide chil-dren with concrete materials. Then they may beasked to show a set of eighteen objects, told tomake sets of three, and answer the question,"How many sets can you make?" After doingseveral of these examples, the equation42 6 = 7 may be shown. This is now inter-preted to mean having forty-two objects, mak-ing sets of six, and finding that there are sevensets. A worksheet such as the following may beused to emphasize this concept.

191

o00

0 00 (0

12 =3=

C0(° 0000 0 00 0

0 0 0

0 0 0 0

21= 7=7X =

00

00t 0o

0 0 GG

24 + 4 =4X

Explain that the symbol . is read "divided byand that in the examples above, : means tomake sets of 'three, seven, and four. Later,when the standard vertical form

63 Pr18

is introduced, the same interpretation is used."There are eighteen objects; we make sets ofthree and find that there are six sets." Whenconfronted with an example such as 56 ÷ 7,the children are instructed to make as manysets at- they are certain they can make in a"sure guess." The results might look like thefollowing:

8 2

1 8 2

2 3 25 5 2 8

7 ) 56 7 / 56 7 56 735 35 14 56

21 21 4214 21 14

7 287 14

14

14

Such a procedure has many advantages: (1)much practice is 'given on the multiplicationfacts; (2) the long-c-livlsion algorithm is in-troduced at the lowest ,evel possible; (3) thechildren understand what the question is; (4)practice is given on making a "sure guess,"which will later be called an estimation; (5) theconcept of division as a guessing game wherethe goal is to make the best guesSes possible isintroduced at a level where children can besuccessful; (6) by examining the various waysin which children arrive at an answer, theteacher can determine the level of efficiency atwnich a child is operating; and (7) as the advan-tages of "just knowing" the multiplication factbecome evident, the children become con-vinced that it is easier for them to "know" thanto do all the work 'required when they do notknow.


Once children know the multiplication facts,there are no division facts to be learned. Chil-dren should be taught with timed tests to thinkthat 72 8, for example, means "8 times whatnumber equals 72."

It is possible to use the multiplic4- lion chart(p. 165) for division facts if the chart is usedwithout zero as a factor. Since division by zerois not allowed (see p. 164), only ninety divisionfacts are possible. In the primary grades itwould be better to avoid an explanation by sim-ply using a chart that does not have zero as adivisor.

Word problems, similar to those below, maybe used to provide practic:: in thinking aboutdivision and to reinforce the modeling.

1. Mr. Jones has 12 cars at his automobilerepair shop. If he puts 3 cars in each garage,how many garages does he need?

2. David has 21 balloons. If he decides togive each of his friends 3 balV.ons, how manyfriends does he have?

3. There are 30 children in Charlie's class-room. They are going on a field trip and wilt ride5 in a car. How many cars will be needed?

multiplication and divisionalgorithms

Teaching division with dividends beyond thebasic multiplication facts cannot be done effec-tively ur efficiently if children are not competentin multiplying by tens, by hundreds, and laterby thousands. Much practice should be givenboth orally and in written form to ascertain thecompetence required before different types ofexamples are presented.

To teach the multiplication of those multiplesof ten that are less than 100, the place-valuechart may be used so that children can see that2 X 10 = 10 + 10 = 20, 4 X 20 = 20 + 20 +20 f 20 80, and so on. The algorithm shouldbe written on the chalkboard like this:

2 tens 20 4 tens 40X2 X2 X2 X2

4 tens 40 8 tens 80

After both oral and written practice has been

188 CHAPTER SEVEN

given on this type of multiplication, the follow-ing sequence should be developed:

1. 22 2 tens + 2 ones 20 + 2x4 x 4 x 4

8 tens + 8 ones 80 + 8 = 88

200 40 F 3x 4

or

- 972

243X4

(4 x 3)(4- x 40)(1 X 200)

800800900

1 160 12+ 170 + 2

+ 70 i 2

12160800

972

2.

Or

20 2 80 22x4 x4 +8 x480 8 88 88 and eventually

24322 x4x4

880

88

and eventually22X4

88

3. 20 + 8x 3

60 4- 2480 + 4 -84

Or

and eventually

28X3

2460

84

28x384

4. Before the children uan proceed with thissequence, they must be able to multiplyhundreds.

972

The following sequence for division is sug-gested:

193

1.

2.

10

7 FcT70

13

310

7 F9-E70

21

21

24

4

10

10

3. 6 1171

60

84

or

60

2424

24

420

6 11471

120

24

24

A good commercial textbook will carefullysequence the work in multiplication and divi-sion so that the children are not confronted withseveral types of difficulty-at one time. It is im-portant for the teacher to.pernit the children to"guess" in division. as shown in the first ex-ample of step 3 before moving them on to themore efficient type of algorithm, shown in thesecond example of step 3. If the children writetheir examples on the cha;kboard and discusswhat was done, they will conclude that in thesecond example given above in step 3, theguess was based on the basic fact 6 x 2 = 12,and so 6 x 20 = 120. It is recommendedthat children estimate, or guess. using thebasic fact and then use scratch paper to dethe second multiplication. When nonapparentquotients are introduced, an incorrect guess onthe basic fact is always too great, and so thechild will try a lesser number. Fancy estima-tions. such as rounding up and rounding down,are confusing to many children; children of av-erage or better intelligence will learn how to dothis for themselves with experience.

problems

As implied in chapter 4, problems that arereal should be included whenever possible.The following may be useful.

1. Use twenty cubes to make four piles sothat the first pile contains four more cubes thanthe second pile, the second pile contains onecube less than the third pile, and the fourth pilecontains twice as many cubes as the secondpile. (7. 3, 4, 6)

2. Four soldiers came to a wide river andfound that the only means they had of crossingit was rowing a boat owned by two small boys.


The boat would hold both boys; but it would nothold a boy and a soldier or two soldiers. Howmany trips were required for the soldiers tocross the river using only the boat and its oars?(17)

3. John and Rose were each given a numberof colored pencils for their birthday. In the dia-gram pictured, each has laid out the samenumber of pencils, either loose or in boxes.John has three full boxes, and Rose has two fullboxes. Each full box contains the same numberof pencils. How many pencils are there in abox? (8)

John

The following problems have more than onesolution:

1. Mary has 250. How many items costing 20,30, and 50 can she buy?

2. John has $1.00. How many items costing100, 200, and 250 can he buy?

summary

The number of mathematical concepts, op-erations, symbols, and algorithms introduced,developed, and supposedly mastered by pri-mary school children is of a large magnitude.Without the development of physical represen-

94

190 CHAPTER SEVEN

tations from which children can abstract ideasof number and operation, we have probablydoomed many to a below-average achievementin mathematics as well as built up in them a dis-like for the subject. The .dea that we can startwith symbols such as 2 t 3 5 and then tryto make some sense out of them for children bymouthing endless words of explanation is anoutmoded, even cruel, thing to do. Children are

deserving of our best knowledge, experience,thinking, and observation on what kind ofmathematics is suitable, how it can bepresented clearly, and what procedures in-crease logical thinking, sophistication, andmastery in it. It is hoped that teachers will findthe suggestions given in this chapter helpful asthey try them, change their approaches, andimprove on what has been developed here.

4 446 *it* 4 mi14 tts

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arthur coxfordlawrence ellerbruch

192 _

197

THE word fraction has as one of its dictionarymeanings a "fragment," a "bit," "a part as dis-tinct from a whole." In common language. frac-tion is used to designate some unspecified partof a whole. For example:

Only a fraction of the seeds germinated.

A small fraction of the children chose white milk.

We bought it on sale at a fraction of the cost.

The idea of part of something is the key ideahere. Children acquire this concept early. Theyoungster who has a piece of toast coveredwith jam, which becomes part of a piece afterseveral bites, has the idea. It is not unreason-able, then, to suggest that the first idea aboutthe fractional number concept that a youngstermay develop is that of a part of a whole, some-thing less than a whole, or a bit of something. Itis likely that when the youngster comes toschool, the "part of something" idea is his con-cept of fraction, if he associates anything at allwith the term. It is the purpose of the schoolprogram to take this idea as a starting point, toexpand it, to develop it, and to make it moreprecise. The result at the end of the total schoolexperience, K-12, should be a relatively so-phisticated and useful concept of fractionalnumbers and of rational numbers as well.

Making precise the idea of "part of some-thing" leads to discovering and to telling howmuch the part is. You may ask the child withpart of a candy bar "how much" there is. Thetypical five-year-old will say "half." regardlessof the amount.

In order to develop the correct fraction con-cept. a child needs to think of these three ques-tions:

1. What is the unit?2. How many pieces are in the unit?3. Are the pieces the same size?

A very young child will see a candy bar cutinto three pieces and will believe that there ismore candy after cutting it than before cuttingit. The child must learn to think of the whole

FRACTIONAL NUMBERS 1S3

object when talking about fractions. Practice inidentifying the unit and the whole object is ex-tremely important. It is wise to ask often."What's the unit?" and to have a copy of theunit available so that the children can refer to itas they look at part of the unit.

The number of pieces in the unit gives us in-formation for part of the fraction name. For ex-ample, having three equal size pieces in a unitleads to the name thirds,lour pieces to fourths,five pieces to fifths, and so on. Thus, the ordinalnames help in learning the fraction names. Al-though ordina! names are not helpful for twoequal size pieces, fortunately most childrenlearn the word half at an early age. and thisspecial name is not difficult to learn.

To have any consistent meaning for amountsother than a whole number of units, it is neces-sary to have "equal size" pieces in the unit.Children will see the importance of this if it isrelated to their experience in receiving a "fairshare." For example, if a candy bar is cut intotwo unequal size pieces, it makes a differencewhich piece you choose. The "you cut and Ichoose" method will help reinforce the need forcutting into equal size pieces to get a fair share.

Yore cut, I'll choose.

.1,

Much emphasis must be placed on equalsize pieces. When using fractions to answer the"how much" question, you must not depend onwhich pieces are chosen.

Children need to be able to recognize equalsize pieces and also to do the cutting into equalsize pieces. They need a lot of help in learning

198

194 CHAPTER EIGHT

how to cut a unit into equal size pieces. Most of-ten, the equal size pieces will be congruent.though this is not necessary.

One method for learning to "cut" a rectangu-lar region into equal size pieces is to fold asheet of paper to get halves, 'fourths, eights,and so on. You can also fold the paper to getthirds and ninths. Fifths are somewhat moredifficult, but still within reason if marks areplaced on the edge of the paper to aid judg-ment.

Segments 'and rectangular regions may be"cut" with a pencil by a halving process. Movethe pencil until it appears that you have half oneach side, then mark. By halving first the unit.then the resulting pieces, ycu can get halves.fourths, and eights fairly well. Thirds will re-quire two pencils. Place one pencil on eachside of the unit and slowly move them towardeach other until it appears that you have equalamounts in the three places. Then mark the po-sitions.

..----_,E-----___

-------,./

By combining the halving and "cutting intothrees" methods, you can cut a unit into thenumber of pieces most important for childrento use initially in working with fractions.

When a child can identify the unit, can tell thenumber of pieces in a unit, and can aecidewhether those pieces are equal size pieces, heis ready for naming fractions. The fractionnames should be related to ordinal names forease in remembering, noting the exceptions forone and two pieces.

Fractions are used to answer the question"how much?"

How much is shaded?The question is answered by working throughquestions such as these:

What is the unit? (Whole thing)How many pieces are in the unit? (Four)Are the pieces all the same size? (Yes)How much is each piece? (A fourth)How many pieces are shaded? (Three)How much is shaded? (Three-fourths)

When a child can successfully use fractionsto answer orally the question. "How muchr",hecan be shown how to use the written form. Aspecial effort is needed to teach the child theconnection between the oral form "three -

fourths"fourths" and the written form " ." Similarly,

the written form must be connected with theconcrete model.

Make sure the child can make all six con-nections between each of the following forms:oral, written, and concrete model.

concrete model form

three:lewd-1soral form

>34

written form

Number of equalsize pieces

1 piece 2 pieces 3 pieces 4 pieces 5 pieces

Ordinal names first second third fourth fifth

' Fraction names unit half third fourth fifth

C.

The following are descriptions of sample ac-tivities and questions to use to help the childlearn the connections between the concretemodel, oral, and written forms. A sheet of pa-per is used as the unit throughout.

Concrete -- Oral1 Show the child a sheet of paper folded in

thirds, with two-thirds marked in red andone-third marked in black. Ask, "Howmuch is marked in red?" (Two-thirds)"How much is marked in black?" (One-third)

2. Say, "Use a sheet of paper to show meone-third." Have the child mark a sheet ofpaper or point to the part from asheet al-ready folded or marked.

Concrete Written1 Show the child a sheet of paper folded in

fourths, with three-fourths marked in redand one-fourth marked in black. Say,"Write the fraction that tells how much is

1black." (-4 ) "Write the fraction that tells3

how much is red." (-4 )

12. Write2- on the chalkboard. Point to the

written fraction and say, "Use a sheet ofpaper to show this much." Have the childfold' a piece of paper or point to a partof a sheet already folded.

Oral ---- Written

1. Say, "Write three-fourths." (731)2 32. Write fractions on paper, such as -y, -g-,

and 5Say, "Read these fractions

6aloud." (Two-thirds, three-fifths, five -Sixths)

FRACTIONAL NUMBERS 195

Children should have a great deal of experi-ence working with concrete materials. Sheetsof paper can be used to represent a region.Sticks, toothpicks, or coffee stirrers can beused to represent length. These concrete ob-jects should be used to establish all six con-nections before using diagrams to representregions or lengths.

It is better to delay the use of sets of objectsuntil the children's understanding of fractionsused with regions and segments is very firm.

The following sequence is one that empha-sizes the most important parts of the fractionconcept:

1. Units

Identifying the number of units

Identifying more or less than a unit

2. Parts of a Unit Using Concrete Materials

Identifying the number of pieces in a unitIdentifying equal size pieces

Making a unit into equal size pieces

3. Oral Names for Parts of UnitsEstablishing fraction names

Using fractions to answer ''how much"Identifying fractions equal to 1

4. Written Fractions for Parts of UnitsOral to written

Concrete to writtenWritten to oralWritten to concrete

5. Representing Fractions with DrawingsTransition from objects to diagramsRepetition of parts 1-4 of the sequence,

using diagrams

6. Extending Notions of FractionsFractions greater than 1Mixed forms

Set modelusing sets to answer ques-

tions such as, "What is of 6?"3

Comparing fractionsEquivalent fractions

00I,

196 CHAPTER EIGHT

learning activitiesfor fractional numbers

This section is organized on the basis Ochedevelopmental sequence given in the preced-ing section. Each part of the sequence is given,followed by a short description of the majorlearning goals for that heading. This material isfollowed by one or more activities or situationsthat can be used to help youngsters developfractional number ideas.

The teacher should look for opportunities tohelp children develop fractional number ideasin natural settings. If such settings do notpresent themselves, then the teacher shouldplan "lessons" in which appropriate settings doarise. The following illustrations are to bethought of as examples that provide the basisfor further teacher-designed activities that havesimilar purposes.

units

Identifying the number of units. The idea of"fractional number" grows out of the idea of"part of something." The purpose here is to de-velop or reinforce the idea of "part" as distinctfrom "the whole" or "all of something."

Assemble a supply of bulbs to be plantedin paper cups of various colors. Say, "Plant partof the bulbs in the pink cups. Did you put all thebulbs in pink cups?"

Place some red beads and some bluebeads on the table. Ask, "Are all the beadsred?" (No) "Give me the part of the beads thatare red. Did you give me all the beads?"

Give the children drawings of flowers. Say,"Color rart of the flowers red; color anotherpart blue."

Ask, "Is all the class here today, or is only apart here?"

21.

Identifying more or less than a unit. Sincefractions depend on the particular unit beingused, children should be given experienceworking with different units and parts of units.

Place several sheets of paper on the desk.Hold up one sheet of the same size. Say, "Thepiece of paper in my hand is a whole unit. Howmany whole units are on the desk?" Repeatwith different-sized units.

Hold up a unit. Place a unit and anotherpart of a unit on the desk. Say, "I have onewhole unit in my hand. Is there more than a unitor less than a unit on the desk?" Repeat withdifferent-sized units.

Have children use sheets of paper to findhow many units will cover their desk tops.

Have children use pencils as units oflength to find the width of the door.

parts of a unit using concretematerials

Identifying the number of pieces in a unit.Children need experience in identifying thenumber of pieces relative to a given unit. Thisreinforces the notion of unit and its role in frac-tions.

Fold a sheet of paper into two parts. Ask,"If the whole sheet of paper is a unit, how manypieces are in the unit?" (2) Repeat, using differ-ent numbers of parts and different units. Havechildren "cut" a unit into various numbers ofpieces.

Identifying and making equal size pieces.Equal size pieces are fundamental to the frac-tion concept. Practice in both recognizing andmaking equal size pieces is important. The ac-tivities should use objects to represent botharea and length. They can be named "coverup" units and "how long" units.

Fold sheets of paper into equal sizepieces. Fold fltGrS into unequal size pieces.Have students pick the sheets folded into theequai size pieces.

Have each child mark and fold separatesheets of paper into two, three, four, five, six,eight, and ten parts. Be sure the paper is aneasy size to manage.

Show students how to fold a sheet of paperinto two equal size pieces, four equal sizepieces, and eight equal size pieces by match-ing the corners and edges.

Use pencils, as shown previously, to mirka sheet into three equal pieces. Mark a7,othersheet with six pieces by folding.

Use pencils, as shown before, to make fiveequal pieces. Then make .another sheet withten pieces by folding. Say, "How many equalsize pieces are in this sheet?" Hold up the sheetfolded into ten equal pieces. "Which sheetshows how you share a cake with ten people?"(See .)

oral names for parts of units

Establishing fraction names. As soon as chil-dren can recognize equal size pieces in a givenunit, they can start using fractions orally to talkabout how much of a unit is being considered.To do this, they first need the fraction nameshalf, third, fourth, and so on.


Count ordinally: "First, second, third,fourth, . ..." Pair the fraction names; with theordinals. Note the different name for "half" and"halves."

Using frabtions to answer "how much."

Discuss the names half, third, fourth, . . . .

Ask the children to tell how much each piece ofa unit is when they have the correspondingnumber of equal size pieces in the unit.

Discuss the fact that fractions tell us howmuch something is when compared to a unit.Say, "You know how much by telling how manypieces are'used and what part of the unit theequal size Oeces are."

Show tile children a sheetof paper foldedin six equal size pieces, with four marked in redand two in black. Say, "If this whole sheet of pa-per is a unit, how many pieces are red?" (Four)"How many equal size pieces are in the unit?"(Six) "What part of the unit is each piece?"(One-sixth) "Say the number of red pieces andthe name for each piece together." (Four-sixths) "This is I he fraction that tells how muchof the unit is red." Repeat with other units andmarkings.

Use the sheets of paper each child has andask similar qui)stions: "Place your finger ontwo-sixths; hold it up so that I can see. Pick thesheet with ten equal pieces. Put your fingers onthree of them. What part do you have your fin-gers on?" (Three-Zenths)

Repeat the previous activities with unitssuch as straws, sticks, or thin -strips of con-struction paper.

Identifying fractions equal to 1. A special sit-uation occurs when all the equal size pieces inthe unit are marked, that is, all the pieces makeone whole. Both the fraction name and thenumber one need to be related. For example,"six-sixths" and "one" are different ways to

. name the same amount.

n,n91.1N

198 CHAPTER EIGHT

Show a unit cut into fifths. As you mark the

pieces to show 1

5

2

5*35

45

5 , have the children5

say the fractions. After 5 ask, "Is there an-otherother name for this much?" (1)

Have Ma- children mark their own sheets of

paper to show 2 34

4. .. and ask, "How

2 3many sixths are there in a unit?" (Six)

written fractions for parts of units

After the children are successful at sayingfractions when shown a marked unit, they areready to start writing the fractions. The majorproblem in writ,ng a fraction is to get the orderof the numerals correct. Care must be used tomake sure that the children do not reverse thetwo numerals. some sensible scheme is valu-able. Following is a possibility:

3 ,10,,,;;;any pieces arc ;;;arAerl?

Melt size is each piece?

Concrete to written. Oral naming seems tobe helpful in moving from the concrete objectform to the written form, with fewer reversals."We say 'three- fourths,' and so we write 3 with abar underneath it. 'Fourths' means four equalsize pieces. and SO we write 4." Reinforce the

meaning by discussing what the 3 in 3 means4

and what the 4 in 3means.4

Use sheets of paper cut into eight equalsize parts. Point to four of the eight parts."How many parts are marked?" (Four) Howmany parts are there in all?" (Eight) "Say thefraction name." (Four- eighths) "Write 4 first.

Then write 8." Show 4 beside the sheet of pa-8

K03

per. Write the fraction on the chalkboard. Havethe children show the amount by holding up thesheet of paper with the amount marked.

Have the children write fractions. Firstpoint to parts of a sheet of paper you havemarked, "Write the fraction of the unit that isred (or blue. etc.)." Check to see that the chil-dren are writing the correct fraction for the rightreasons.

Written to oral. When the children can writefractions correctly. they should practice givingthe oral names for the fractions.

Make a pack of fraction cards. Use themas flash cards to have children read fractionsaloud.

Written to concrete. The most difficult con-nection to make is the one between the "writ-ten" and "concrete model" forms. Later workwith fractions will be more meaningful if thechildren can represent fractions in a correctconcrete manner. Some practice activities fol-low:

Give the child a unitcut into three equal2size pieces. Write --3-. Say, "Show me how

much of this unit this fraction is." Repeat withdifferent units and fractions.

Give the child a sheet of paper to repre-

sentsent a unit. Write . Say, "Use this unit to4

show this fraction." Repeat with different unitsand fractions.

1Write

2 and without providing a unit, ask.

"How would you show what this fractionmeans?" Have the child then follow his answerwith a demonstration. Repeat with differentfractions.

Give the children paper units "cut" into six-teen equal size pieces. as in the diagram. Say."Color three-sixteenths of the square red.Color seven-sixteenths yellow. Color six-six-teenths blue." Say, "Draw a square regionshoWing each fraction (for example, one-half.two-sixths, one-third. five-eighths, etc.).

S

Give the students paper units "cut" intoeight equal size pieces, as in the diagram. Say,"Frost the cake so that one-eighth 'pas whitefrosting. five-eighths has blue froSting, andtwo-eighths has red frosting Ask, "Does thewhole cake have frosting? Does eight-eighthsof the cake have frosting? How do you know?"

Give the students paper units ''cut" intosixteen equal size pieces, as in the diagram.Say, Color three-sixteenths of the square red.Color nine-sixteenths of the square blue. Colorthe rest yellow." Ask, "What part of the squareis yellow?" (Four-sixteenths) "How do youknow?"


representing fractions withdrawings

Transition from objects to diagrams. Up tothis point, the work with fractions has beenbased on re{. resenting fractions with concreteobjects. A too-early transition to drawings ordiagrams has been intentionally avoided. Atthis point in the sequence. however. it is appro-priate to move to a diagrammatic representa-tion of fractions.

Activities designed to make the transitionfrom objects to diagrams follow.

Have the children trace around a concreteunit (sheet of paper). Similarly, have them drawa segment along a stick unit to show a lengthunit. Encourage the children to think of the dia-gram as a "picture" of the concrete unit.Throughout this process. refer to the diagramsby the object names (paper unit, candy barunit, stick unit, etc.) to reinforce the connectionbetween the objects and the diagrams.

Repetition of parts 1-4, using diagrams. Atthis point it is appropriate to repeat the firstfour parts of the fraction-concept sequence,using diagrams in place of concrete objects.

Establish the roles of the unit, the number ofpieces. and the equal size pieces for diagrams.Make the connections between diagrams, oralforms, and written fractions. Use the activitiesin parts 1 through 4 of the fraction-concept se-quence, with diagrams in place of objects.

extending notions of fractions

The topics in this section are included tosuggest directions for extending the fractionconcept. They are not included as masteryitems for young children. They serve as a ; tepbetween initial work with fractions and comj..u-tation with fractions. The topics can easily kietreated in a problem-solving setting to rein-force fraction knowledge as well as to gener-ali:? it.

ell

200 CHAPTER EIGHT

Fractions greater than 1. A common miscon-ception among children is that a fraction mustbe less than a unit. Activities that treat a frac-tion as a certain number of pieces, each ofwhich is a specific part of a given unit. will al,owchildren to generalize about fractions greaterthan 1. Words such as proper and impropershould be avoided, since they suggest treatingfractions greater than 1 as being different fromfractions less than 1.

Give the children paper units. Say. "Showme one-third. Show me two-thirds. Show methree-thirds. Show me four-thirds." Ask, "Whatdoes four-thirds.mean? Is four-thirds greaterthan the unit, the same as the unit, or less thanthe unit?" Repeat with other fractions.

Show the children a sheet of papermarked in six equal size pieces. Say, Thisshows six-fourths; can you show me the unit?"

Mixed forms. Children use mixed forms evenat a young age. A child will say that he is threeand one-half years old when he is over threebut not yet four. These experiences with mixedforms can be made more precise with the fol-lowing types of activities.

Place three paper units and one-half of apaper unit on a desk. Ask. "How many units areon the desk?" Move the complete units into apile and ask. "How many units are in the pile?How much of a unit is not counted? How muchpaper do I have in all?" (Three and one-half)

1Write 32-, and give practice in reading it.

Use mixed forms to find the number ofsheets to cover the desk.

Use stick units and mixed forms to find thelength of a desk, a table, or the chalk tray.

Set model. Children have many opportuni-ties to share sets of objects. Problem solvingusing sets often involves division, but it can be

related to fractions. This means that there is aneed for sume method of determining (1) whatthe unit is. (2) how many pieces (or subsets)there will be, and (3) what a fair share is (equalsize subsets). That is the purpose of these ex-am ples.

Place four red blocks and two pink oneson a table. Say, "Here is a set of blocks. Someare red and some pink. How many are red?How many blocks are there altogether? Whatpart of the whole set is red? What fraction rep-resents the part of the set that is red?" Ask thesame questions for pink blocks and for othersets.U.

Say. "Here is a set of five candies. Threeare caramels and two are creams. Three of thefive candies are caramels. What fraction of the

candies are caramels?" (-3 ) "How many of the5

five car dies are creams?" (2) "What is the frac-

tion?" (-2) "What does the 5 tell you? The 2?5

How much of the set is made up of caramels?"

( ) "How much of the set is made up of5

2creams?" (-5 )

Say, "Here are seven buttons. Some arered; the others are blue. Write the fraction that

tells what part of the set is red." (2) "Blue?"

() "What does the 5 tell you? The 7?"7

Say, "A candy man has eight licoricestickssix black and two red. He ties them inbundles of two as shown. What fraction of the

licorice is black?" (-6) "Red?" () "What2

8fraction of the bundles of licorice is black?"

(-3-) "Red?" ()14 4

1To illustrate the fractions 2 and3. draw

3a set of balls that shows two-thirds of theballs black and one-third of the balls red. Re-peat with other fractions.

Say. "Draw a set of apples for which two-fifths are red and the rest are yellow." Repeatwith similar instructions.

Say, "Here are six candies. If I were to giveyou one-third of them. how many would youget? Let's draw a picture.

Now let's separate the set of six candies intothree sets with the same number in each set.

(00Why did we make three sets?" (Because thefraction has a 3 below the line, meaning threeparts) "How many of these sets should we taketo have one-third?" (1) "How many candies isone-third of six candies?" (2) Do the same typeof activity with other unit fractions, using differ-ent sets of objects.


Say. "Here is a six-foot ribbon. I promisedMary she could have one-third of it. How manyfeet of ribbon should she get?

_40111111111111ftwalsoiwoore- ire

Cut the ribbon into thirdsMary gets two feet.So one-third of six is two."

Say, "A cake is cut into eight pieces. Johnis to get one-fourth of the cake. How manypieces should he get?"

Comparing fractions. Fractions tell "howmuch." It is natural to ask whether one quantityis more than, less than, or the same as another.The purpose here is to use the "how much"idea to introduce the zomparison of fractions.

Say, "Look at these two candy bars. If youget the shaded part of one of them. which onewould you want? Why? In which would you getmore?" (The first one) "What fractions tell how

2 1much you would get in each?' , (-33

, ) "Istwo-thirds of a candy bar more than one-thirdof another candy bar of the same size?" Giveother, similar examples.

41111PAIIIPSay, "Write the fraction for the shaded

part." (-5) For .the unshaded part." (-3) "Which8

is the greater part of the regionthe shadedor the unshaded part? How can you tell? Wesay that five-eighths is greater than three-eights because there are more shaded parts(five) than unshaded parts (three) in the eightequal size parts." Repeat with similar examplesand discuss.

202 CHAPTER EIGHT

Say, "Three-fifths of John's marbles arered, and two-fifths of Peter's are red."

John

Peter

0 0 00 0 0

Ask. "Which has the greater fraction of redmarbles? How can you show that you are rightby looking at the two sets?" Encourage one-to-one correspondence between Peter's red mar-bles and a subset of John's red marbles. Ask.Who has the greater fraction of nonred mar-

bles? Why?"Say, "Here are seven buttons. Some are

pink. What fraction is pink?" (-1) "What frac-

tion is not pink?" (*) Is the fraction of pink

ones greater than, less than, or equal to thefraction of non pink ones?"

Have the children draw three sets of redand blue balls. In the first, see that the fractionof blue balls is greater than the fraction of redballs. In the second, the fractions of blue andred balls should be equal. In the third, the frac-tion of blue balls must be less than the fraction

of red balls. Ask questions so that the childrencan see the relationships between sets.

Equivalent fractions. Much of the later workwith fractions requires the ability to determinewhen two fractions name the same amount orthe ability to produce two fractions that namethe same amount. It may be helpful to showchildren at this age that two different fractionsmay represent the same amount.

Ask, "How much of the region in (a) is2

shaded?" (-8 ) "How much of the region in (b)

is shaded?" (-1) "Is the shaded part in (a) the4

same amount as the shaded part in (b)? Why?2Do you think

4and

8name the same amount?

Why?"

(a)

See . Ask, "What part of the apples is

red?" (4) "Can you group by twos?" (Yes)

"How many groups of two apples are there?"(6) "Show it." See O. Ask, "What part of the

1apples is each set of two?" (-6 ) "How many

sixths are red?" (3) "What part of the apples is

red?" (-36

) "Do you think three-sixths of the

V104060110.0000OP WI* Cn iC)

apples and one-half of the apples are thesame? Why?" Continue similarly to get othernames for one-half.

Transparencies and overlays may be used tohelp children check to see that the sameamount can be named with two different frac-tions.

conclusion

There is much initial work on the concept offractions that can be done with young children.

FRACTIONAL NUMBERS 203X

A good understanding of fractions at this agehelps in later work with computation. In thischapter a sequence was provided that empha-sized the key notions underlying fractions.Concrete objects and their diagrams, oral lan-guage. and written fractions were connected toform the basis for understanding fractions.

Support for many of the ideas suggested in this chapteris found in ongoing research being carried out at The Uni-versity of Michigan The major investigators are JosephPayne. James Greeno. Stuart Choate. Chatri Muangnapoe.Hazel Williams. Patricia Galloway. and Lawrence Eller-bruch.

citi.

edith robinson

206

JUST as the child comes to school with somenumber notions, such as the ability to count byrote or tell how old he is. so does he come withsome geometric, or spatial, notions. As Ilg andAmes point out in their book. School Read-iness, a three-year-old can draw a circle stop-ping after a single revolution (p. 65), in contrastto younger children, who go round and round.Furthermore, a four-year-old can make asquare with acceptable corners, and a five-year-old can copy an equilateral triangle (pp.74, 80).

An examination of the pattern of devel-opment shows why these competencies can beconsidered to be evidence of geometric no-tions. To cite some examples. before age four,the child often reproduces a square by drawingit with "ears" at one or more corners:

Although we do not know for certain why thechild draws the ears, we may conjecture that henotices the corners sticking out but is unawareof how they got there, His means of showingwhat he sees is to draw them in as separateentities. By about age four, he is able to drawthe square somewh at like this one:

AA 0

nr

His drawing may be imprecise, but it does con-sist of four more or less straight strokestwohorizontal and two verticalwith no outwardthrust to the corners. Moreover, a typicalmethod of drawing the square develops. so thatby age five. 60 percent of the girls and 52 per-cent of the boys begin at one corner (usuallythe upper left) and move the pencil in one con-tinuous sweep, usually counterclockwise (II ).

There are variationsbeginning at some othercorner, going clockwtse, or making two distinctmarks as in abovebut by age six, 80 per-cent of the girls and 72 percent of the boys useone continuous motion, with 58 percent of allusing the method exactly as shown in U (1Igand Ames, p. 76).

As the normative ages mentioned earliersuggest, the equilateral triangle is more difficultto execute than either the circle or the square;the problem appears to be that of reproducingthe diagonal stroke. Although very young chil-dren (eighteen months or less) can imitate ver-tical and horizontal strokes with a crayon, a fewchildren still have trouble with the obliquestroke at age six or seven (lIg and Ames. p. 65).Thus, although only 15 percent of the five-year-olds fail at the task of copying the triangle, only17 percent of them produce a well-propor-tioned figure (1Ig and Ames, p. 80). Unlike thesquare. no typical method of drawing the tri-angle is favored by as many as half the chil-dren; it may be made with one, two, or threeseparate strokes (1Ig and Ames, pp. 78-86).

GEOMETRY 207

Hg and Ames give a more detailed dis-cussion of children's performance on thesetasks than is possible here. The few instancescited above were intended to illustrate theavailable evidence concerning what can be ex-pected of preschool children with regard to ge-ometry. In general, the Ilg and Ames studiessupport the following conclusions:

1. By kle five, the majority of children candistinguish circles, squares, and trianglesfrom one another without necessarilyknowing the names; that is, there is visualform discrimination.

2. By age five. most children are aware, atleast to the extent that they can "correctly"reproduce them, that the square and tri-angle have certain characteristicsnamely straight sides. a fixed numberof sides, and vertices (corners).

3. By age five, most children are awarethat all these geometric figures are sim-ple closed curves, for the drawingschange with age. At first they show not-simple or not-closed curves and changeto simple and closed as the child getsolder. At two and one-half, a child's"circle" may look like one of these:

At three and one-half, his circle is a sim-ple closed curve, but his square may looklike one of the following:

At age five, the child draws both a circleand a square with one continuous motion,closing it up and not retracing; that is, hedraws each as a simple closed curve.

4. By age five, most children appear to havesome sense of the direction of a line, sincethe sides of the squares they draw are

elSe7 A

F. 4.

208 CHAPTER NINE

reasonably horizontal and vertical and theappropriate sides of their triangles rea-sonably oblique.

The preschool child also appears to beaware of intersecting lines, and again it is thework of Ilg and Ames that provides the evi-dence. When asked to copy a cross,

a typical effort by a 3-to-3-1 year-old showed2

one of the lines being split (1Ig and Ames, pp.70-73):

By age five, most children could execute thedrawing with two.intersecting strokes, althoughthe resulting segments might not be bisected.

Not until age nine are the drawings made withIDL,th segments bisected by at least half the chil-dren, aad both the change in the drawing fromsplit (P ree segments) to intersecting (two seg-ments) and the method of accomplishing thisresult argue that it is the intersection that thechild is trying to reproduce. Apparently this isthe feature of the model to which he is now at-tentive.

To say that a typical five-year-old has thesegeometric notions is not, of course, to claimthat he can name them or give any sort of ver-bal description of what he sees. It is merely tosay that he has some sort of perceptuffl aware-ness that he can reproduce in a drawing. Fur-thermore, by studying changes in the drawings,in which the type of error made by youngersubjects disappears with age, one is led to theconclusion that the improvement is neither ac-cidental nor solely attributable to better muscu-lar coordination. As a matter of fact, drawing asquare with ears is not easier than drawing atrue square, and drawing the cross "split" takesmore care than making two intersecting seg-

ments. It would seem, then, that the betterdrawings represent some new perception atwork.

On tilt.: other hand, the studies of Jean Piagetand his followers provide strong indication ofwhat not to expect of the preschool child. Mostof these studies relate to the measurement as-pects of geometry. For example, young chil-dren do not seem to recognize that the foursticks in make a path of the same lengthwhen arranged as ins.

Nor do they seem to recognize that the fourpieces of paper in 0 cover the same amount ofspace when arranged as inQ.

0

Apparently the children think that the pieceschange size as they are rearranged. Piaget de-scribes this as a failure to conserve length andarea respectively. It seems unlikely that chil-dren who do not conserve would have muchunderstanding of perimeter and area. Piaget'sstudies also indicate that the concept of"straightness" is slow to develop in children.For his subjects, ideas of straightness came atabout age eight.

Usually the word geometry is used to catalogsuch topics as perimeter, area, congruence.perpendicularity. right angles. and the classifi-cation of polygons. That is, geometry has beenthought of in terms of topics in high school ge-ometryso-called Euclidean geometry. Mostof these ideas are based on measurement orstraightness or both, and as indicated above,Piagetian research suggests that both of theseconcepts are slow to develop. Certainly they donot seem to be possessed by the child enteringschool Thus, one curricular issue raised by theresearch is the following: Should we begin earlywith a systematic introduction to Euclidean ge-ometry. directing our efforts to speeding up theacquisition of the Euclidean concepts ofstraightness and measurement, or should wefocus on geometric notions for which youngchildren can be expected to have some read-iness?

Experiments aimed at lowering the age atwhich children can conserve have so far haddisappointing results. Several explanations forthis are possible. Some of the results are prob-ably due to a lack of knowledge about the exactmechanism by which conservation is attained.Knowledge is also lacking about the role thatlanguage plays in testing for conservation. Fi-nally. the task itself may be impossiblethat is,it may not be possible to accelerate the proc-ess. At this time, we simply do not know.

Fortunately these difficulties can be circum-vented. for mathematics presents alternativesto Euclidean geometry. There are, in fact, manygeometries, and in some of these length andstraightness are not important. Many childrenwill have had experiences with one such geom-etry; it is called topology. Consider a balloonwith facial features printed on it, such as theone in . If not inflated properly, the balloonmay at first look like the one ins.

GEOMETRY 209

The child can notice that although one eye isbigger than the other, the lashes on both eyesare on the lower lid, the right eye is separatefrom the left, and so forth. In other words, cer-tain aspects of the face cannot be changed byany amount of bad luck in blowing up the bal-loon short of bursting it.

Children can be led to discriminate betweenthings that change and things that do notchange without using words to describe whatthey see. For example, they could be askedwhich of these faces they might get by blowingup the balloon:

The teacher can tell from a child's choice whichthings he has noticed and which he has not no-ticed. An advantage in asking questions thisway is that the child need not take anyone'sword for anything; he can find the answers forhimself, and differences of opinion can be set-tled by trying things out to see what happens.

Several important points about the study ofgeometry are implicit in the preceding illustra-tion:

1. Certain ideas of geometry can be learned.For example, children can learn to antici-pate that certain things will always happen

210 CHAPTER NINE

in certain ways and that some things willnever happen in certain ways.

2. It is not necessary to use words to de-scribe what always happens and whatnever happens. Pictures showing bothpossible and impossible events can beused, and the child can be asked to tellwhich is which. To sort such pictures suc-cessfully, the child must be attentive tocertain visual cues. These cues will beright whether or not he knows the precisewords for describing them. For example,if in the preceding display a child says thatthe first result cannot be obtained byblowing up the balloon because there are"no marks on one ear," the teacher knowshe has noticed the right thing. Neither amore precise explanation nor a furthergeneralization is necessary. Thus picturesorting can be used as an instrument forevaluating learning.

3. Learning by doing and settling differencesof opinion by demonstration build thechild's confidence and self-reliance. Hecan thus come to expect that geometryand, it is to be hoped, all mathematicsdescribes reality; that he can master itsuccessfully by 19mself; and that he doesnot always need someone else to showhim what is correct or to legislate what isright and wrong.

The discussion that follows includes a de-scription of a more structured learning situ-ation than the balloon example. "Structured"here means that the ideas that can be learnedin the various geometries have been identifiedand listed and that the representations sug-gested are those in which these particular ideascan be exhibited. Suggestions are also in-cluded for sequencing lessons and for eval-uating the learning. The discussion is orga-nized under three headings:

1. Geometry without straightness or length2. Geometry with straightness but not length3. Geometry with straightness and length

It is hoped that the suggestions given will serveas an impetus, prompting teachers to think ofadditional ideas for presenting these geome-tries to their own classes.

geometry withoutstraightness or length

As indicated in the balloon. example, pre-school children probably have some experi-ence with this kind of geometry, although cus-tomarily no attention is paid to that fact.Nevertheless, as we have seen, there are somegeometric ideas to be learned: namely, thatcertain properties and relations will not changeeven though others, notably size and shape,will.

The basis for this kind of geometry is the no-tion of "elastic m Ins." If we momentarily re-strict our attention to the plane, we can con-sider such things as stretching the plane,shrinking it, pushing humps in it, or makingwaves in it. In fact, we will think about doing justabout anything to it except tearing it or foldingit over on top of itself. Then we will study whathappens to some subsets of the plane (a circle,for example) when any of these motions is go-ing on. Clearly, if the plane is not torn, the circlewill not be split, but it may end up being consid-erably larger or smaller and may not be a circleat all. If you think about the circle as beingdrawn on a broken piece of balloon, you caneasily see that pulling, pushing, bending, andthe like could produce some strange lookingresults. Yet with all this infinite variety, there aresome predictable results. Before consideringlessons for children, we will discuss each ofthese in detail.

some geometric properties thatdo not change

1. Inside and outside. If a simple closedcurve is drawn on a piece of balloon, withan X inside (outside), there is no elasticmotionthat is, no amount of pushing,puffing, and the likethat will result in theX being on the outside (inside).

2. Closed and not closed. If a closed curve isdrawn on a piece of balloon,

it cannot be made to be not closed ex-cept by tearing, and this is not allowedunder the rules of the game.

Likewise, a not-closed curve could notbecome closed without folding the bal-loon onto itself;

again, this is not allowed by the rules ofthe game.

3. Order relations along a path. If on a pieceof elastic thread we paint three spots withdifferent colors (R = red, G = green, B =blue), we might pull, push, stretch, and soforth, so that looks like , yet thegreen patch will still be between the redone and the blue one in the followingsense: in traversing the path, in order toget from the red spot to the blue one, wewill always have to pass the green one.

R

GEOMETRY 211

4. Certain set relations, such as set member-ship, subset relations, and set inter-sections. A piece of clay shaped like adoughnut (as in 0) cannot be made into apiece of clay shaped like a disk (as in 0)without folding it onto itself, which isagainst the rules. The doughnut can, how-ever, be pulled around and pushed in soas to take on the shape of a cup withouttearing or folding onto itself.

0 0If the following figure is drawn on a pieceof balloon, no matter how it is pushed orpulled, there will always be exactly onepoint of intersection.

Some possibilities for deforming thefigure are shown here:

Regardless of the differences in their ap-pearance, they all have three more orless distinguishable parts with exactly

(..----.'-'5"--B one point in common.

4.-41 rz4.10 I

212 CHAPTER NINE

A'

A 8 C D E

5. Connectedness. A geometric figure maybe "all in one piece." ;n which case we sayit is "connected." Some examples of con-nected sets (these are all plane figures)are shown in 111

Some examples of figures that are notall in one piece, hence not connected.are these:

(.08' D'

(Here A'. B', D'. and G' are in two pieces,F' in three. E' in ten). Clearly, to changefrom connected to not connected re-quires tearing (against the rules): tochange from not connected to connectedrequires folding onto itself (also againstthe rules).

If the reader will refer back to the balloonface with which we began this discussion, hewill notice that many of the properties just listedwere exemplified in the face: there are con-nected and not-connected sets. intersections.and order in the eyelashes. Furthermore. itwould be a simple matter to introduce a few ex-amples of simple closed curves into the face:as a matter of fact. if we were to make the eyeslook like this.

we would have two simple closed curves withsomething "inside" each. Although the fiveproperties just listed may seem entirely tooprofound for small children when written intechnical language. they are actually quite easyto exemplify. and studying them is quite withinthe capability of young learners. The technicalvocabulary is for the teacher. not the pupil.

216

/\ CF G H

sequencing the lessons

Assemble the following teaching materials:Things that can be pushed and pulled withoutbreaking are needed; these might be thinsheets of rubbersuch as a piece of brokenballoon. rubber bands. or elastic threadandmodeling clay.

1. The teacher might begin by blowing up aballoon and having the children watch for thedifterent ways it can look. A long thin balloonworks nicely for this because it can assume anyor all of the following appearances, in additionto some others:

If a figure is drawn on the balloon, say a wigglyline (a felt pen can be used), the children canwatch the change in the wiggly line as the bat -loon is blown up. If it is drawn like this.

blowing up the balloon could make the wigglyline look like any of these:

It might be instructive for the children to try todraw pictures on the chalkboard of some funnyways the balloon might look. The teacher might.then ask, "Could it look like this balloon?" (No)"What's wrong with it?"

\ /V2. Draw a picture of a face on a balloon.

perhaps similar to the one here:

Although the features need not exactly repli-cate the ones shown here, note that this ex-ample does incorporate the following from thelist of properties that do not change:

(a) Something inside something; the eyes inthe face illustrate this:

(b) Something not closed, such as the nose:

(c) Something involving orderthe nosebetween the two eyes.

(d) Something intc-secting, such as the eye-lashes with the eyes. or the mouth:

(e) Something connected and somethingnot connected, such as any of the fea-tures drawn on the face.

Then show the children some sketches offaces that are not possible. such as the fol-lowing. and ask. "Could the balloon look likethis? What's wrong?"

GEOMETRY 213

You will notice that a store-bought balloon witha face printed on it would do. provided that thefeatures incorporate the five geometric proper-ties that do not change.

3. Here is Sammy Snake:

'Imagine that Sammy is drawn on a balloon andthat we blew up the balloon funny. "Who candraw a picture on the chalkboard to show howSammy might look? Be sure to put in all hisspots. Did George draw a good picture? Yes?Good. Who can draw a funnier picture? CouldSammy look like that? No? What's wrong? Whocan make it right?" Continue with other volun-teers. "Now I'll show you some I drew. Here aremine":

4^S,T. -4)

214 CHAPTER NINE

"Which of mine are right? What's wrong withthose that aren't Tight ?"

The preceding activities are not suitable foran V11114. VT tcy ,re rather juvenile for, say.third gradersbut they illustrate that this Kindvi geometry is suitable even for kindergartnersif the right sort of activity is chosen.

4. For older children draw figures that aremore abstract on a piece of sheet rubber. Forexample:

ry

Have the children pull the rubber sheeting allsorts of ways to demonstrate how much theycan change the figure. Let the children draw thevarious results on the chalkboard. Offer direc-tions if necessary: for instance, ask if the firstfigure can be made to look like this:

(With enough helping hands, it can!) Ask if thesecond figure can be made to look like this:

5. Here are some fufther suggestions for il-lustrating the properties of geometry withoutstraightness or length:

Can (1)

Can ,ACan ACan FCan FCan FCan -j

be made to look like ? (No)

be made to look like 0 ? (Yes)be made to look like A ? (No)be made to look like F ? (Yes)

be made to look like ? (Yes)

be made to look like H ? (No)be made to look like ? (Yes)

Can be made to look like ?. ? (No)Can be made to look like 0 ? (Yes)Can j be made to look like ? (Yes)

f o iiiusifaZa the three-dimensional case,modeiing clay is very usetut; witn it. tor ex-ample, it can be shown that a ball, a sausage, aplate, a cup without handles, and a block (to listonly a few) can be changed from one into theother in any order. To change a ball into adoughnut would not be possible under therules: however, the doughnut can be changedinto a cup with a handle, a basket with a handle,a sort ofthree-dimensional letter- P, and soforth.

These suggestions should give the teacheran idea of the kind of focus that is required. Ifthe teacher will experiment with balloons, rub-ber bands, elastic thread, and modeling clay.other possibilities will suggest themselves. Therules are simpleno tearing and no foldingonto itselfand correct answers can easily bedetermined by direct experimentation.

evaluating the learning

Evaluation should focus on the five proper-ties listed at the beginning of this section. thatis, inside and outside. closed and not closed.order relations along a path, set relations, andconnectedness. However, the teacher shouldrealize that children must first of all be able torecognize the pairs of figures that can be de-formed into each other according to the rules.Such recognition can be helpful in other partsof mathematics. For example, the fig-ures and Z:\ can be transformed into eachother under the rules of this geometry: whenthe children later study measurement. they willmeasure both in the same wayby perimeter.The figures and $ , however, cannot bechanged into one another under the rules ofthis geometry, and they are not measured in

the same way. The will be measured by pe-rimeter; the II by area.

Sketching the results of experiments can beinstructive for the children, and since straight-ness and length are not important properties inthis geometry. the ability to replicate exactly isnot required. Thus, a.0 would be an accept-able form of deforming Es- . whereasa j would not. And for figures with identifyinglabels,

8would be an acceptable

R G

form of G 8whereas R G

would not. The recognition that relative order-ing always stays the same can be helpful whenstudying the number line; five, for instance, isalways located between four and six regardlessof whether a big number line or a small numberline is used.

As suggested earlier, learning can be eval-uated by means of pictures. The children canbe asked, for example, to look at this collectionof figures and pick out those figures that couldbe o tained from the first figure in the collec-tion. keeping the rules for this geometry inmind.

The five that cannot be so obtained illustratethe properties of "not closed," "new inter-sections," and "not connected." Teachers willneed to make collections of such pictures toaccompany the figures they use in the activi-ties. They will also need to collect some to gowith figures not specifically used in classroomactivities, since ability to predict in new situ-ations is the ultimate test that learning hastaken place.

GEOMETRY 215

geometry withstraightness butnot length

Shadow geometry provides a good repre-sentation of geometry in which straightness isimportant but length is not. At different times ofthe day a person's shadow may be shorterthan, longer than, or the same length as, theperson; so length is not preserved. Straight-ness is preserved. however, as can be seenfrom games of shadows on the wall.


The children should learn through appropri-ate activities that this kind of geometry also hascertain properties that do not change.

1 Straightness. A straight object will cast astraight shadow. A curved object, how-ever, does not necessarily cast a curvedshadow; a curved wire can be'heldso as to cast a straight shadow

2. Inside and outside. A point inside a closedfigure cannot be projected outside thatfigure, and vice versa. For example, if allpoints of the set 0 lie in the same plane,it cannot have a shadow that lookslike ( , although it can have a shadowthat Idoks like CD or .

3. Certain intersections. If a flat wire loop likeQ is used as a model, the shadow can-

not look Ike although it may looklike , 0 , or . Similarly, a flatfigure like c cannot have a shadowthat looks like Q or . However, ifthe loop 0 is not flat (that is. not all inthe same plane), it could have a shadowthat looks like cx-) .

216 CHAPTER NINE

4. Openness. in a restricted sense. A wireshaped like LI cannot have a shadowshaped like ( although it can have ashadow shaped like (. or

5. Connectedness. A shape that is in onepiece cannot have a shadow that is in twoor more pieces. For exam ple. X can-not have a shadow that looks like .

However, an object in two pieces can havea shadow that is all in one piece. It is easyto see, for example. that two sticks can beheld so as not to touch and yet cast ashadow like


Assemble the following teaching materials: asource of incandescent (not fluorescent) lightsuch as an old-fashioned goosenecked desklamp, objects to cast shadows, such as wire fig-ures. cardboard figures. and solids, white card-board or paper on which the shadows can beseen clearly (newsprint works well), crayons fortracing shadows. and pictures for the eval-uation of the learning.

1 Have the children hold objects betweenthe light and the paper and try to make theshadows look different by turning the object.For this purpose, objects found in the class-room will do, such as large-sized paper clips, apencil ("see how little you can make theshadow"), a piece of notebook paper ("see ifyou can make the holes disappear in theshadow"), and so forth.

2. Next, have the children trace the variousshadows. For this purpose, specifically pre-pared objects would be better, such as per-fectly flat shapes fashioned from wire. Attachhandles to these to prevent the children'shands from obscuring the shadow. Some goodshapes to begin with are a 0 , a , and a/\ Let the children make different shadows

with each object by turning the wire and havethem trace each different shadow. The impor-tant thing here is that the children must be

careful to trace only the shadow of the shape it-selffor example, a could "never be ashadow of any of these three shapes. To makesure that the children are learning to associatethe shadow with the shape, bring out their trac-ings the next day and ask them to pick theshape from which each tracing was made.

3. In subsequent lessons, use other spe-cially prepared objectsfor example, pieces ofcardboard, some solid and some with holes.OP or CI or /di "Can you make the dentdisappear? The hole? The tail?" Blocks alsomake good models. They can be in the shapeof cubes, pyramids. cylinders, or cones.

It is important to note that the children neednot know the names for any shape used as amodel in order to learn from these activities.The ultimate test is whether they can success-fully predict the shadows that could or couldnot be cast by a given object. The choices giventhem will, of course. have to be carefully se-lected because even adults might be hardpressed to say for sure which of the followingcould be the shadow of a wire like r :

However, kindergarten children are able to rec-ognize that they cannot get a shadow likeC.,-.) from a flat wire like . A fair ques-tion for older children might be to decidewhether / , can be the shadow of

and whether can be the shadowof v . Also, older children should be ableto choosefrom a cube, a pyramid, a cylinder.a cone, and a rectangular solidall thoseshapes that could produce a shadow like A .

Teachers will want to experiment a bit forthemselves before beginning activities of thesort described above. However, it should beobvious that a teacher need not know all thepossible shadows for a given object before em-barking on such a sequence. Disputes can besettled easily by picking up a model and tryingit outthe child who thinks can be ashadow of C.., can determine for himself thatthis is not possible.

trtiC) i()

The suggestions above do not exhaust thepossibilities for learning geometry from shad-ows. Teachers will think of other ideas for mod-els as well as other ideas for presenting the ge-ometry of shadows. For example. if the objectis a tower of blocks. moving the light instead ofthe object will provide many of the same oppor-tunities for observation.


As with any learning activity. the teacher willwant to make sure the children are learningand not just playing. The age and experience ofthe children. as well as their spontaneous com-ments. will dictate how often and at what pointssome sort of formal evaluation is desirable. Asalready indicated. this evaluation can take theform of presenting pictures and asking childrento pick out from a given set of choices all thosethat could cast a shadow like the picture. Thepictures presented should include some thatcan and some that cannot be shadows of theobjects and should incorporate propertiessuch as straightness, intersections. openness.inside and outside, and connectedness. Thesewords. of course. need not be used. or evenknown, by the children. Successful learningshould be measured in terms of the ability topredict. illustrated with a question such as this:Which of these could be the shadow of

Nor is it necessary to delay working with shad-ows until the children are mature enough tolearn everything (whatever that may be!). Kin-dergarten children can work successfully withtwo-dimensional wire forms, the use of three-dimensional solids to cast shadows can bepostponed until the second or third grade,when the "flattening out" that occurs in theshadow will make sense to them. Conceivably,experience with shadows of three-dimensional

GEOMETRY 217

objects could help with perspective in draw-ings, thus geometry and art may be mutuallyreinforcing.

If the children have done both geometry onthe balloon and shadow geometry. the follow-ing chart might be used to contrast the two.Think about a circular shape. This shape couldbe drawn on a balloon, or it could be a piece ofwire made for casting shadows. Decidewhether each shape shown down the left sideof the chart could result, and write your an-swers in the chart.

Could

made

-this?

X + Or) -if 0 IS ci Wirt

drawn.un balloGii }o malt ci slincloW

j ever bp,

o look like,

G

CD.

(Answers: yes, no; no, no, no, yes; yes, yes; no,no.)

geometry withstraightness and length

Geometry with straightness and length is themost familiar type of geometry to most adults.For example, one of the assumptions of highschool geometry has typically been that "a geo-metric figure can be moved without changingits size or shape." In this section appear activi-ties that illustrate the conditions under whichthis assumption is valid. This is the geometry ofnonelastic, or rigid, motions.

221

218 CHAPTER NINE


1. Length2. Angle measure3. Straightness4. Properties that did not change in other

geometries, that is. connectedness. setrelations. and the like


The first kind of geometry discussed in thischapter was based on elastic motions; thepresent geometry deals with three kinds ofnonelastic motions. These are-

1. turns (sometimes called rotations);2. slides (sometimes called translations),

3. flips (sometimes called reflections).

The children will need to learn about each ofthese motions.

Assemble the following teaching materials:pieces of cardboard on which to draw pictures;an overhead projector; thin paper or trans-parent overlay paper for the overhead projec-tor; small mirrors; Instant Insanity blocks; andwooden models of pyramids, cylinders. and soon.

Draw a picture on a piece of paper; place apin in the paper as shown and turn (rotate) thepaper about the pin.

piri

The picture on the paper can appear in any ofthese ways:

Experiment with the pin in different places toshow that the paper can always be turned sothat the picture looks like any of those abovebut that no amount of turning will make it looklike the one in .

4Since the paper must be flipped, or turned

over, for the picture to look like that in , drawthe picture on transparent paper so the chil-dren can see that it will look like the one inwhen flipped, or like any of the views ins.

To obtain the different appearances shownin and , flip the picture along differentlines, as shown by the dotted lines in thesesketches. (Each of the figures can be made byflipping along some line other than the onesshown in this drawing.)

P.

V

r\

LI

Demonstrate to the children that to slidethe picture does not change its appearance. Aslide means that the paper is placed on someflat surface and pushed along that surface in

any direction, which will never give any of theresults shown: the picture will always appear asit does in the first drawing, just in a different lo-cation in space.

The children should become acquainted witheach of these three motions separately. It maybe instructive for each child to have his ownpiece of paper with a picture on it so that he canturn it and thus be able to convince himself howthat picture can appear.

Have pictures to show on the overheadprojector or sketch them on the chalkboard.Have the children decide in each case whetherthey can turn the paper so as to make theirslook like the one displayed. Include a few ex-amples that cannot result from turns, so thatthe children begin to notice the cues that areassociated with turns. For some children, thisactivity may need to be repeated, using otherpictures. The choice of picture to use as themodel will depend on the maturity of the chil-dren For a simpler one than the choice shownin earlier drawing, block letters of the alphabetcan be used, such as P. 0, or J. Of course. cir-cles, the letter T. and so forth, will not be goodchoices. since differences will not be obviouswhen the paper is turned.

Illustrate flips with pieces of paper, usingthe same type of sequence described for turns.

Illustrate flips with mirrors. The mirrorshows the image that would be obtained if thepaper were flipped along the line on which themirror rests.

There are two reasons for doing the slideslast: (1) since the picture does not seem tochange, the children might not understandwhat they are supposed to be noticing; (2) ifslides are done last, many, if not all, of the chil-dren will realize that in contrast to the ones theyhave been studying, this motion produces nonoticeable change except in location. Thus itmay not be necessary to spend more than a fewminutes on this motion.

GEOMETRY 219


To determine whether the children can rec-ognize the results of the three motions picturescan be used. as before. The children could beasked, for example, which of these picturescould be obtained from the original picture andwhich motion they would use to get it.

Asking the children why could not be ob-tained will emphasize that no amount of sliding,turning, or flipping will make the picture bigger(or smaller) than it was before; likewise, non e ofthese motions will add any features to the pic-ture, and so )11 cannot be a result.

The real test of understanding these motionsis whether the children have the ability to selectcorrectly when the given picture is not one withwhich they have had practice. For older chil-dren, show a picture of an angle, such as thisone:

Ask which of these angles could be obtained byturning, flipping, or sliding.

)

This test would not be appropriate for youngerchildren, since the use of the arrows can beconfusing. Experienced teachers know that thechildren's idea of congruence for angles iscomplicated by their attention to the lengths of

clori4.4u

220 CHAPTER NINE

the sides If they have learned to recognize theresults of turns, flips, and slides, they should beable to understand that the arrows mean onlythat the sides "go on and on." They should thenbe attentive to the cues that have served themin the previous exercises and be able to makethe correct selections in this instance.

If the children have experimented with ge-ometry on a balloon, shadow geometry, andslides, flips, and turns, you may want to con-trast the results. For the chart below, the blockletter K can be thought of as being drawn on aballoon, of being a shape for a wire; or beingdrawn on a card to be flipped, turned, or slid.Then decide whether each picture shown at theleft of the chart could be the result, and fill inthe chart (Note how carefully the picture mustbe drawn!) Answers: no, no, no; yes. yes. yes,no, yes. no: yes, no. no: no, no, no; yes, no,no; yes. yes, yes.

The exercise with angles illustrates the kindof long-range goal that can be achieved from astudy of slides, flips, and turns. The idea of

Calf\ --he

made -I-o

[Ike. -th"

congruence for geometnc figures can be de-scribed in words, but children adept at recog-nizing slides, flips, and turns will, in fact, havebeen selecting congruent figures. For thesechildren, only the word congruent need be sup-plied, they probably already have the concept.Such children should very easily identify thepairs of figures shown here as congruent oncethey have been told the word.

// ./ --h--_-_-..._..---____i-

A nonmathematical example that involvesturns and flips is to be found in crossword puz-zles. Examining the crossword patterns shownhere can show how the pattern can be made togo onto itself by turning or flipping. If a singleflip will do this, the figure is said to have a lineof symmetry; if a half-turn 0801 will do it, thefigure is said to have a point of symmetry.

Slides, flips, and turns are not motions re-stricted to figures in the plane. They are also

< be on olook

<balloon 05 a

<wire on a

(slidesiflips,an

<card,

c-) ?

-i-urns)

applied to three-dimensional figures, but aremore difficult to visualize for most people. In-tent Insanity blocks can help the children be-come acquainted with these motions becausethe faces of the blocks are painted differentcolors However, not all teachers will want topursue the matter to this extent.

shapes and namesfor shapes

The emphasis in the three preceding sec-tions has been on relationships among geo-metric figures' as a child experiments with dif-ferent figures, he learns to examine themselectively for cues that will tell him whetherthey are alike in some sense. He sees, for ex-ample, that [I is just Liturned (rotated) andthat i 1 is just[ istretched out. butthat-4- -is not like= under any of the rules.Furthermore, he learns to ignore or attend tosome cue according to its relevance for a par-ticular geometrysize is not a relevant cue inelastic motions, but it is relevant in flips andturns Moreover, he has derived his own cuesthrough direct experience; no one had to tellhim what to be attentive to and what to ignore.This practice in self-directed. selective scan-ning can stand him in good stead when certaingeometric relationships are studied more me-thodically.

Nevertheless, it is true that in our environ-ment certain configurations occur with suf-ficient frequency to require that children beable to recognize such similarity of shape,

GEOMETRY 221

Some of these instances occur naturallyifchildren view grains of salt under a micro-scope, they can observe that they all have thesame shape (cubical). Likewise. snowflakes areseen under the microscope to have hexagonalboundaries, although the interior patterns vary.If the children experiment with growing crystals(copper sulfate, magnesium sulfate, and citricacid are easy to grow). they can observe theregularity with which the crystalline structure ofpure compounds is built up. Collectors of rocksand minerals can see identifiable crystallinestructure in such minerals as galena. feldspar,and quartzall cleave along well-definedplanes when struck by a hammer so as to con-form to that structure. Mica and shale exhibitstrata, that is, parallel layers that are retainedwhen a chunk of the mineral is broken. Plantsalso exhibit regularity of form. Trees. for in-stance, have shapes that fall into certain gen-eral categoriessome are nearly spherical,others conical, cylindrical, or shaped like fatfootballs. Their shadows are seen respectivelyas round, triangular, rectangular, or elliptical inoutline.

Some degree of regularity of shape is alsoseen in that portion of the environment that isman-made. This is due to in part to certainpractical considerations. A manufacturer of ce-real, for example, must package his product,place the packages in cartons, and ship thesecartons by truck or rail to various locations. Theeconomics of the situation demand that heminimize the cost of packaging and also thewaste space in transportation. Certain shapesmeet the needs of his particular situation betterthan others. As a result, most of the cerealboxes that we see in the store are rectangularprisms, as well as the containers of many other

rri r) 1.7'tve.4.

0...,

222 CHAPTER NINE

dry products. In architecture, aesthetic consid-erations play some role. yet here, too, prac-ticality accounts for some degree of regularity.Local climatic conditions govern such mattersas the pitch of a roof or the location of windows,for example. it has been found efficient in areasof heavy snowfall to have steeply pitched roofs.The mass production of millwork results in uni-formity in the size and shape of window anddoor frames and the like. Bricks and buildingblocks in the shape of rectangular prisms fit to-gether with no gaps. Floor tiles that are rec-tangular. hexagonal, or octagonal can also befitted together with no gaps. Thus certainshapes recur over and over in both the naturaland the man-made environment.

Children's general literacy should includethe names for those objects that they encoun-ter in their daily lives, so along with the namesof animals, nearby towns, and colors. theyshould learn the names for the geometricshapes that they see. However. just as there ismore to art than the ability to distinguish bluefrom green. so is there more to geometry thanthe ability to distinguish a square from a tri-angle. The practical problems that led man tochoose regularity of form for the objects hebuilds or manufactures he in the realms of sci-ence and measurement. Suggestions for in-troducing children to the measurement aspectsof geometry are given in the following sectionand in the next chapter (see especially pp. 245-47).

premeasurementactivities

At the beginning of this chapter it waspointed out that the concept of measurement.as applied to geometric figures. posed somedifficulties for young children. Although meas-urement will be discussed in detail in the nextchapter. this section will present some activitiesthat can help children begin to understandsome of the underlying ideas. These activitiescan be used independently of those suggestedin the preceding sections. However, childrenwho have studied the motions outlined earlier

are likely to find these premeasurement activi-ties much easier. For example, as mentionedearlier, one of the difficulties children have withmeasurement is in recognizing that the samecollection of parts (four sticks, for example) isthe same size- when rearranged, provided the

parts do not overlap. This state of affairs en-sues, of course, because we rearrange by turn-ing, flipping, or sliding, and not by any of theelastic motions. In the example of the foursticks shown at the beginning of the chapter. toget from

to

one stick is turned (rotated), and two othersare rotated and slid.

It will also be noted that in some of the activi-ties already described, there has been oppor-tunity to use such phrases as "larger than" or"smaller than" in an informal way. A given pic-ture, for example, could not come from anotherby a rotation because it was "too big" or "big-ger than" the original. Some of the activities be-low will provide opportunities to recognize twcfigures as being "the same size" without neces-sarily being the same shape.

assembling pictures from parts

Begin with two-dimensional models. Pro-vide two pieces of cardboard exactly the samesize and shape.

Then ask, "Which of these shapes can be madefrom the parts you have?"

Pf16

The correct answer, of course, is that all ofthem can be made. The follow-up questionmight be. "Which of the four shapes is thelargest?" The recognition that all four must bethe same size ,regardless of their appearancecan go a long way toward helping the child ac-cept the results of the measuring process.

The preceding activity could be a class dis-cussion or an individual activity; the teachermust decide which way to use it on the basis ofthe age and experience of the pupils. It will benoted that all the figures are derived by sliding.

or turning the parts. Children familiarwall these motions may be able to recognizeimmediately that all four pictures can be madefrom the parts. Other children may need the ex-perience of actually fitting the pieces togetheron specially prepared pattern. This wouldneed to be an individual activity. It is importantthnit pieces and patterns be made with suf-ficient precision for them to fit together withoutoverlap or gaps.

Other ideas for both pieces and designs willsuggest themselves to the teacher. The piecesmight be square-shaped, for example, andthree or more used instead of two.

Next, have the children work with one-di-mensional models, with pipe cleaners as theparts. Two pipe cleaners shapedlike __J can be put together to form any ofthe following shapes:

I d' L

Many other figures and arrangements am pos-sible.

For work with three-dimensional models.have the children use wooden blocks, the partsof these models. unlike the previous ones. can-not actually be superimposed on a pattern.Whether a picture of the pattern or a modelmade by the teacher is used, a bit of imagina-tion is required on the part of the children. Hereit might be a better idea to ask how many differ-ent patterns the children can make with, say,

GEOMETRY 223

six blocks and then to discuss which arrange-ment is the "biggest" With different arrange-ments made by the children on display for all tosee, the same ideanamely. that regardless ofappearance. all must be the same sizecan bedemonstrated.

Activities of this type can focus attention on(1) what the figure is made of. which is impor-tant later in distinguishing perimeter, area, andvolume; (2) size as an attribute distinct fromshape; and (3) the process of "covering pat-terns with no gaps and no overlap.

The process of covering would seem to be avery simple concept: yet in studying the meas-urement of perimeter and area, it is frequentlyimagined rather than actually carried out. Ex-perience with the one- and two-dimensionalexamples in the preceding activities give thechild practice without specifically calling his at-tention to it. If the patterns and pieces are suffi-ciently well made. the pieces have to be fittedtogether with no gaps and no overlap.

separating the figure to identifyparts

Children are sometimes given patterns to cutout and paste together to make polyhedra. Asan alternative. have them invent their own pat-terns.

Permit the children to examine carefully acube (which is hollow, not a solid block). andthen ask them if they can make one of paper."How many pieces will it take? What shapeshould they be?" Suggest that perhaps insteadof six pieces, one piece might do if it were cutout the right way. Displaying a sheet of squaredpaper should make it obvious that the sixpieces can be cut in one piece instead of sep-arately. The children may need some assis-tance at the beginning, if so, display a piece likethe one shown and ask if it can be folded tomake a cube. Any child who thinks this will

224 CHAPTER NINE

work should be encouraged to try it for himselfto see why it won't work. By so doing, he canlikely figure out how to make a pattern that willavert the difficulty.

Give all the children a piece of squaredpaper and a pair of scissors and allow themto try to make a pattern that will work. Manyideas will have to be tested to determinewhich will and which will not work as a pattern.Since there is more than one pattern for thecube, some children will enjoy finding as manydifferent ones as possible; the question of whatshall constitute "different" patterns can chal-lenge the best students. Refrain from telling achild whether his pattern will work, since tryinga pattern that does not work can suggest to himwhat to try next in order to make one that will.

The cube, of course. is only one possiblespace figure for which children can devise pat-terns. Other possibilities are pyramids, rec-tangular solids, and even cones and cylinders.

Make the following figures from pipecleaners and ask the children which of the fourshown can be made from just two pipe cleanersand what shapes the parts must have.

Since we are not requiring the two parts tobe exactly alike, several answers are possiblefor each figure. Plenty of pipe cleaners shouldbe available for such an activity so the childrencan test their ideas. Some children will selecttwo pieces exactly alike, which gives the oppor-tunity to view the first figure, for example, as the

union of L and . or the unionof . and . They could view it astwo unlike pieces, such as I and

$

Two-dimensional figures may also beused. For example, prepare pictures such asthe following:

Have the children decide on two pieces of pa-per or cardboard to put together to make thesefigures and compare them with the pieces theyput together with pipe cleaners to make the fig-ures in the preceding activity. This activity canemphasize the differences in the figuresdif-ferences that are relevant to the study of perim-eter and area.

Although space permits only a few examplesof possible exercises, many variations of theseactivities can be designed to fit a particularclass.

summary

The study of geometry has both mathemati-cal and pedagogical goals. Pedagogically, it isan ideal subject for-

1. developing self-reliance, since the childcan determine for himself whether his an-swers are right or wrong;

2. acquiring the ability to predict what willhappen, since the visual cues that arepresent, and to which he learns to be at-tentive, allow the child eventually to per-form "experiments" mentally and thuspredict outcomes;

3. developing an expectation that mathe-matics makes sense, since by manipulat-ing pictures and other materials, the childcan see what always happens and whatnever happens. The follow-up discussionthen agrees with his own observationsthe discussion "makes sense" because itdescribes what actually happens.

Mathematically, the geometry presented in thischapter provides an opportunity for the child tobecome acquainted with certain geometricproperties and relations without being con-cerned with vocabulary. The activities haveprovided many examples of properties such asconnectedness, straightness, and closedness.They have also provided examples of two typesof relations between figures: nonmeasurementrelations, such as inside and outside, part-whole, set membership, subset relations, be-stweenness. parallelism, same shape as; andmeasurement relations, such as bigger than,smaller than, same size as, congruent to.

It cannot be overemphasized that thesegoals are long-range goals; the more immedi-ate goals are specified for each activity sug-gested in this chapter. Since the goals arewithin the capabilities of young children, the ac-tivities are interesting as well as easy for them.When more formal instruction in geometry be-gins and a vocabulary needs to be devOoped,the children will already have a rich back-ground of examples to associate with thewords. Likewise, when formal measurement isbegun, the children will have had first-hand ex-perience with the attributes they are attemptingto measure.

GEOMETRY 225Amo

The following sources all contain activitiesthat will supplement those given in this chapter.

Abbott, Janet S. Mirror Magic. Pasadena, Calif.:Franklin Publishing Co.. 1968.

Dienes, Zoltan P.. and E. W. Golding. Geometrythrough Transformations: Geometry of Distortion.New York: Herder & Herder, 1967.

Geometry through Transformations: Geome-try of Congruence. New York. Herder & Herder,1967,

Phillips, Jo M., and Russell E. Zwoyer. Slides, Flips,and Turns. Motion Geometry, vol. 1. New York:Harper & Row, 1969.

Robinson, G. Edith. Shadow Geometry. Practical pa-per, no. 18. Athens, Ga.: Research and Develop-ment Center in Educational Stimulation. Universityof Georgia, 1969.

Shah, Sair Ali, Banding and Stretching. Athens, Ga..Research and Development Center in EducationalStimulation, University of Georgia, 1969.

. Topological Equivalence of Objects.Teacher's guide for use with Bending and Stretch-ing. Working paper 18a. Athens. Ga.: Researchand Development Center in Educational Stimu-lation, University of Georgia, 1969.

references

11g, Frances L., and Louise Bates Ames. School Readiness:Behavior Tests Used at the Gesell Institute. New York:Harper & Row. 1965.

Piaget. Jean. and Barbel Inhelder. The Child's Conceptionof Geometry. New York: Harper Torchbooks. Harper &Row, 1964.

The Child's Conception of Space. London: Rout-ledge & Kegan Paul, 1963.

edith robinson

michael mahaffey

doyal nelson

MEASUREMENT provides one of the mostfrequent needs for number in everyday life: wespeak of the number of miles on a trip; thenumber of days of vacation; the number ofcakes needed for some expected number ofguests; and numbers inform us as to the tem-perature, air-pollution index, and wind velocityon any given day. Yet the teaching of measure-ment in the elementary school presents someof the most frustrating experiences that studentand teacher encounter. There are difficultieswith telling time, difficulties with convertingfrom one unit to another, and difficulties withformulas.

In this chapter the nature of measurement isexplored in some detail; described also aresome studies concerning children's beliefs withregard to measurement. It is hoped that by sodoing, readers are helped to diagnose diffi-culties and plan appropriate learning se-quences for their own classes.

Children today enter school with vastly dif-ferent backgrounds, not only among them-selves, but different from the background ofchildren a generation or so ago. Furthermore,the mobility of our population often enhancesthese differences. Thus one of the major tasksfacing today's teacher may well be the estab-lishment of a common pool of experience fromwhich more formal instruction may sensiblyproceed. It is with this in mind that we examineavailable evidence about what young childrenmay or may not know about measurement andoffer suggestions to help clarify possible mis-conceptions. Teachers will, of course, need toselect and adapt these suggestions to their ownparticular classroom situation.

228

0,0-1

the nature ofmeasurement

what are children's ideas ofmeasure?

Many of the studies of Piaget relate to chil-dren's ideas of measurement, and some ofthese ideas may seem curious to an adult. Forexample, first-grade teachers may wish to trythis simple experiment: Give every child apiece of modeling clay and have them roll it firstinto a ball and then into a worm. Then ask thechildren whether there is more clay in the ballor in the worm. It is surprising to some adultsthat the children often express a preferencesome say there is more clay when it is in a ball,and some say that there is more when it is in aworm. If we stop and think about this for a bit, itseems clear that whatever the phrase "moreclay" means to the children, it does not meanthe same to them as it does to adults. Whetherthe children really believe that the amount ofclay mysteriously changes with the change inform or whether they do not understand thequestion is not known. What is clear is thatsome misunderstanding exists, and that thismisunderstanding could cause difficulty in thestudy of measurement. Piaget, incidentally, de-scribes this misunderstanding by saying thatthe children do not "conserve" quantity.

There are other experiments with similar cu-rious results. One requires two sticks of thesame length. If the sticks are placed like this,

Cftothe children agree that they are the same" (thatis, neither is longer than the other). If, however,one of the sticks is moved slightly so as to looklike this.

MEASUREMENT 229

the children may now respond that the bottomone is longer (or shorter) than the top one. Hereagain, a moment's reflection leads us to ques-tion the child's understanding of what is beingcompared.

In another experiment, equal quantities ofwater are poured into two containers of differ-ent shapes:

When questioned, the children may say thatone of the containers now has "more water"than the other.

For a fourth example, suppose two towers ofblocks are built, one built on a bench:

These two towers are often reported to be "thesame" (neither taller than the other).

In each of these instances, the difficultyseems clear: the children are not comparingthe "right" things. What is not at all clear is howone gets around the difficulty. Certainly insist-ng that a child agree to the adults' judgment inthese instances is no guarantee that his beliefwill change, and belief is the basis for under-standing.

In the experiment with the blocks, it is tempt-ing to try to convince the child by asking him tocount the blocks in each tower. But this may notbe as simple as would appear at first glance.For one thing, we would be asking him to ac-cept numbers as measures of height iii a waythat is in direct opposition to his intuitive feel-ings in the matter. That is. we would be assum-ing that numbers are so well understood by thechild that he would readily discard his per-ceptual judgment in the face of the over-whelmingly persuasive argument that "four isgreater than three." Furthermore, this argu-ment makes use of the fact that the blocks are

(y)9

230 CHAPTER TEN

all the same sizea tower of four blocks couldbe shorter than a tower of three blocks! Clearly,some tacit assumptions are being made, per-haps at this point we should look more closelyat the fundamentals of measurement.

what are the most basic ideas ofmeasure?

Our discussion will focus on the following:

In essence, measurement is a process bywhich numbers are assigned to objects in sucha way that, for some given common attribute, anobject having more of that attribute is assigneda larger number.

To illustrate, part of a school record might looklike this:

George 48 53 80 88 110.

Toni 47 57 78 Pi 113

An adult would readily accept the fact that fivenumbers are assigned to George (and five toTom), along with the understanding that eachof these numbers measures a different attri-bute But is it naturally clear to a child that asingle object can have more than one attribute?

Furthermore, an adult would infer that Tomhas less than George of whatever attribute isrepresented in the first column, but would notinfer that George has less of whatever is repre-sented in the first column than he has of what-ever is measured in the second co'umn. Thefirst inference is based on the obs vation that47 is less than 48; however, the similar obser-vation that 48 is less than 53 does not warrantan valid inference. That is, the appropriate in-ference is not derived solely from the numbers,but must be combined with an understandingof their role in the comparison process.

The process of measurement, then, givesquantitative information, and in our society weaccept this use of number, but only in a veryspecial way. For one thing, not all attributes ofan object are quantified. Color, for instance, isnot usually assigned a measure except in ahighly technical sense. As a matter of fact, fewof the attributes by which objects are per-

ceptually identified, and hence distinguishedfrom one another, are quantified. Yet these at-tributes are frequently the most familiar toyoung children. Moreover, many of the attri-butes that are quantified are difficult to de-scribe in words, and to say that length is "howlong something is" does not shed much light onthe subject. Thus for the child. the first majortask may be to learn to recognize the attributesthat our society elects to measure. A partial listof these attributes, in alphabetical order, isgiven below; many, if not al;, are studied in theelementary school.

area speedcapacity temperaturedensity viscositydistance volumelength weightpressure

If we contrast the items in this list with the worldas the young child sees it, we see an evengreater gap between the child and the adult. Atypical preschool child, for example, identifiesobjects in terms of their use. "A hole is to dig"(from a title of a book by Art Linkletter), a ball Is"to play with," an orange is "to eat." In the earlystages, he may not recognize any way in whicha ball and an orange are alike, although some-what later he may say they are "both round."That is, young children tend to think of like-nesses (common attributes) on the basis of-

1. use ("all toys," "you eat them," or "tools");2. genera ("children," "grandmas," "pets,"

"flowers");3. color ("all green");4. form ("they're round," or "they're flat").

Now none of these is an attribute to which theaverage adult assigns a measure (except in themost technic it sense, which does not concernus here), and it would certainly be a remarkablechild who would claim that a ball and an orangewere alike because both "had weight" or "hadvolume" or any other attribute on the earlierlist. These attributes must then be learned and,compared to such characteristics as "use,""genera," "color," and "form," are not easy toobserve.

In addition to the necessity for recognizingcertain attributes, the assignment of numbers

An 0

as measures requires that objects having"more" of a given attribute be ass,gned largernumbers However, as indicated by the studiesof Piaget cited earlier, children may not recog-nize "more" or "less" or even "the same.amount" of some measurable attribute.

Surely good pedagogy requires that from thevery beginning, all phases of measurement beaccepted by the children as reasonable enter-prises This would seem to imply that the mostbasic learnings would be-

1. the identification of those attributes thatour society considers important enoughto quantify;

2. some intuitive understanding of what"more" or "less" of those attributes shouldbe.

In earlier years, children may have enteredschool with a greater understanding of both,because housewives baked from scratch, toyswere improvised, and grocers measured out toindividual order. Thus children had the oppor-tunity to observe cups, scoops, scales, spoons,and the like used to measure out "just the rightamount of" a wide variety of raw materials.Today's children may hear on television about"lower in phosphates" or "more concentrated,"but there is no visible referent for the words.And the child who reports that his mother "lostten pounds" may know less about measure-ment than the one who reports that he "outgrewhis shoes" (the latter statement definitely has.areferent).

In this first section we shall suggest some ac-tivities designed to supplement out-of-schoolexperiences with measurement, and for thispurpose the activities will focus primarily on theidentification and comparison (in a relativelygross way) of some of the attributes in our firstlist.

Some readers may have questioned the sep-aration of distance and length and the separa-tion of capacity and volume in our list. Both dis-tinctions were made for a reason. Althoughadults tend to equate distance and length, adistinction is nevertheless sometimes made,as, for example, in the phrase, "20 miles as thecrow flies." In the case of children the situationis somewhat different, because it appears thatdistance is a harder notion for them to grasp

MEASUREMENT 231

than length. Teachers will need to take this intoaccount when talking to pupils, as well as whenplanning learning experiences for them.

With respect to capacity and volume, the ini-tial distinction will be as follows: Capacity willbe considered to be an attribute of containers.The rationale for this is that it will be easier tocompare two containers on the basis of "whichholds more" than to compare two blocks ofwood on the basis of the "amount of wood" they"contain."

Some readers may also have wondered whywe omitted time from our It;-..1 Time is not an at-tribute of objects and it will ne discussed in aseparate section.

As mentioned previously, the activities out-lined in this section are to be thought of as sug-gestions for learning about measurable attri-butes and comparisons based on thoseattributes. In particular, exploratory activities ofthis sort should not be thought of as havingprecisely one objective per activity. On the con-trary, rather than "closing in" on some specificitem to be "mastered," their purpose is to openup new aspects of familiar situations for com-parison and contrast. Nor will the teacher beable to assess the learning in the conventionalmanner, such assessment will have to comefrom listening carefully to children's conversa-tions, observing any independent actions takenby them as they work, and noting the amount oftransfer to new settings. We do not, then, offera prescribed sequence that, when followed, willguarantee certain resUlts. Instead, we describea variety of contexts from which the teacher canselect and adapt according to need. Some ofthese are extensions of regular classroom rou-tine, and others are more contrived, but all of-fer children the opportunity to encounter mea-surable attributes and to make comparisons ofthe sort that motivates adults to engage in theenterprise known as measurement. Sincethese activities are foundational, the followingguidelines seem appropriate:

1. Appeal to children's senses, specificallysight and touch, to learn to recognizemeasurable attributes.

2. Select settings with which the children canpersonally identify and to which they canbring knowledge of their own.

232 CHAPTER TEN

3. Use problem situations, that is, situationsin which there is a need to acquire"enough" of something.

4. Provide experiences from which the chil-dren themselves can draw the conclusionthat one thing has "more" (or "less") ofsome attribute than another; that is, pro-vide an experiential basis for the use ofappropriate measurement language.

Although these activities were written withthe young child in mind, teachers of older chil-dren may find in them ideas that could profit-ably be taken up with their pupils. A final re-mark: Quotation marks have frequently beenused to point up .a naturally occurring word orphrase that is so familiar to an adult that heMay not realize that it may be meaningless to achild. In some instances, the activity providesan opportunity for the child to learn what thisword or phrase means to the adult.

activities to developbasic concepts ofmeasurement

using commercially availablematerials

attribute blocks'

These materials are used for a variety oflearning experiences, one of which is recogni-tion that a single object may have more thanone attribute. Once the blocks have become fa-miliar to the children and can be easily sortedaccording to any of their several attributes,their use can be extended to the introduction ofcomparison words.

For example, select a large, blue, thin,round block and a small, blue, thick, roundblock and ask the children how they are differ-ent. The children should be able to name twodifferences. if they do not know the correctwords, provide them. The terms will havemeaning because the children have a per-

Or)

ceptual basis for reference. With children whoalready know the words, show them an emptyspool or piece of dowel rod painted blue andask them how it is different from the other twoblocks. Noticing that it is the "thickest" of thethree but the "smallest around" helps directtheir attention to the different kinds of com-parisons necessary for the understanding ofmeasurement. The reader will note that in thisexample the question of which block is biggestis relative to the attribute in question. Since fail-ure to realize that one .object may be "biggerthan" another in one respect but "smaller than"in another may be the basis for some children'sdifficulties with measurement, these blocksprovide an opportunity for learning such dis-criminations. Scraps of wood or parts of dis-carded toys" can be used along with the regularblocks for activities of this type. Those chosenshould, of course, be the same shape and col-ored the same as in the commercial set so thatneither color nor shape becomes the attributebeing compared. If the children are old enoughto be instructed to "pay no attention" to color,then of course many ordinary objects in theclassroom (a piece of chalk, for example) maybe used in comparison.

nests of boxes

As with attribute blocks, the children need tobe completely familiar with the materialsthrough free play. Once they have becomeadept at assembling the boxes correctly, theirattention can be drawn to comparisons by size.

For example, hold up two of the boxes (itmight be wise to choose two "nonsequential"ones) and ask which is bigger. If the children donot know .the word, ask them which would fitinto which. Then show them a third box and askthe children which is the biggest of all andwhich is the smallest of all..Having the childrenorder up various subccllections of the wholeset may be instructive. Asking "Which boxholds all the rest?" or "Which box will fit into all

the rest?" may help. Care should be taken withthis activity, however, lest the children con-clude that the tallest container is always thebiggest (Ms here, of course): For this reason,the emphasis should be on comparing by "fit-ting in," and teachers may wish to follow this upimmediately with some of the activities under"Capacity" on the following pages.

class projects

finding a rope for spot

For this project, a cutout or toy dog isneeded, along with a doghouse, a dish of food,and a rope (piece of string) with which Spot istied to the doghouse.

//,

Arrangethe.pleceS",ao that Spot can't gethis food. For one reason or another (the in-genious teacher can invent a reason), the prob-lem is not to be solved by moving the dish offood but by finding the "right" rope for Spot.Some teachers might wish to provide severalpieces of twine already cut and have the chil-dren choose a piece that would do, then try outtheir various choices to see which would workand why some will not. Other teachers mightwant to have the children cut off the "right"amount from a ball of twine. In any event, theparts of the display should be movable so thatsome experimentation can be done. Readerswill note that there plenty of opportunity here

MEASUREMENT 233

for comparisons and that the comparisonwords will arise naturally in the discussion:"What's wrong with the rope Spot alreadyhas?" and."What-do we need then?" are exam-ples of questions that will direathe discussionalong measurement lines.

As an alternative, the display could involve,not a dish of food, but a flower bed and a ropethat permits Spot to bury his bones'in.the flow-er bed (Mrs. Jones objects to this):

For older or otherwise more mature_children,a combination of the two suggestions. coul d b,einstruCtive.,Once the "right" rope isiound (longenough' for' him to reach his dinner but shortenoughto 'prevent him orp burying leftovers inthe flower bed), ask the Children which'is closerto Spot, his dog dish or the flower bed. If bothof these objects are movable and the scenerycan be rearranged, using the rope to comparedistances becbmes a reasonable solution. Itshould be noted that this solution is more rea-sonable as an outgrowth of a "simpler" prob-lem or sequence of problems; that is, unlessthe teacher pursues the matter, some childrenmight not realize that the rope does become ameans of comparison. Naturally, teachers willnot want to pursue the activity this far if the chil-dren have difficulty finding the right rope; forsuch children, selecting the "right" rope is suf-ficient when accompanied, of course, by a dis-cussion of what is wrong witiropes that aren't"right."

making a model village

There are ,many .pOssible variations to thisproject, butibasiCally it requires a schoolhouse,some houses. -and streets.

Represent these on a large map (sheet ofpaper) or on a sandtable and use toy buildingsor diagrams. The village could be a representa-tion of the actual neighborhood or some imagi-nary one. Finally, the layout could be in cityblocks or not, as the experience of the childrendictates. The problem to be explored is "Whogoes farther to school?" Thus several in-habitants of the village must be identified andtheir homes located on the map. The story forthe village could involve riding the school bus,

234 CHAPTER TEN

Georgelives Fere Tommy

4. lives here

afi 61:1 e

tztl.Een

4.,,,I, 4 4

in which case it would be wise to have all thecharacters ride the same bus. On the map,then, would be located the bus stop at whicheach of the characters gets on. In simplestform, the map might look like the one in U.For younger children the story could be simpleand the bus route shown as above, with no at-tention given to the fact that some childrenhave farther to walk to the bus stop than others.Depending on the maturity of the children,more complicated arrangements are possible.

Tommylives here

fp tillti13.3 FiffElp

George liveshere

In these cases, the first question might be,"Who rides farther on the bus?" followed by"Tommy has to walk from his house to thebus stop, and so does George; if we figure allthe way from home to school, does Tommy stillhave farther to go than George?" and finally,"How can we find out?" For still older children,two buses might be involved in the storyonecoming to school from each direction, or onewhose route is farther "south" on the map thanthe other. The emphasis should be on devel-oping a concept of the length of a path and onthe comparison of the lengths of two paths togive meaning to "longer" and "shorter." Thus

t3Ifs! r")

L7 L1IJCiii

the questions could be asked in terms of whohas the longest way to go. The means of com-parison here involves a part-whole relationshipfor the simple case. To ensure that the childrencan use this method of comparison, introducea third character. "Pete rides the same bus,and he rides farther than either Tommy orGeorge. Where do you think he gets on thebus?" Then select some possible locations,some right and some wrong, and ask the chil-dren if each of these could be the place wherePete gets on, and if not, why it is wrong. Afourth character might ride "farther thanGeorge but not as far as Pete." To make surethe children understand the attribute in ques-tion (length of a path), it could be imagined thatGeorge misses the bus one day and has to walkall the way to school; does Tommy still have far-ther to go than George? Or, if both boys missthe bus but Tommy runs while George walks,who goes farther to school (assuming, ofcourse, that both follow the route of the bus)?

For the more complicated arrangernent twg-gested above, the comparison of the lengths ofthe paths should, include the walks to the busstop. Tracing each path by "walking" one's fin-gers or a doll may help. The teacher may evenhave to say, "See, the walk is about the samefor each boy." If this is necessary, the arrange-ment is too difficult for the children and shouldbe abandoned.

cages for the animals

Lay out a model zoo with several toy ani-mals. Explain that each of the animals needshis own play yard fenced off, or each of theanimals needs a cage for shelter. Big animalsneed bigger yards or cages. If a supply ofboxes is made available, the chidren can select

what they think is the "right" box for a cage foreach of the animals Cutting holes in the cagesfor doors can also be instructive, since the holeneeds to be "high enough" and "wide enough"for the animal to walk through. If string or toyfencing is made available, the children can setthis up to make the "right" sized pen. The latterproblem may involve making the pen largeenough for the cage selected with some spareroom for the animal to exercise. Young childrengain some good practical experience in "mak-ing things so they will fit" as well as in being in-troduced to some words such as "tall," "high,"'wide," "wider," and the like, since these cancome up naturally in conversation.

learning about specific attributes

capacity

Since this is an attribute shared by con-tainers and it appears easy for children to thinkof objects in terms of their use, the discussionmight begin by asking the children to tell what abox is. If they say it's "to put things in," askthem to name some other objects used to putthings in. Or begin the discussion by displayingseveral pieces of chalk, for example, and ask-ing the children to name some things that couldbe used to put the chalk in. When it is clear thatthey are classifying objects as either being ornot being containers, introduce the notion ofrelative size (capacity).

To do this, display several containers andask the children which could hold some partic-ular object (say, a baseball). The containersbeing displayed should have been chosen sothat the object will not fit in all of them. Thus thechildren can be asked why the object will not fitin the containers not selected (too small). Fromthis it is established that some containers arebigger than others. To reinforce the notion, askthe children to name things in the room "big

MEASUREMENT 235

enough to hold a football," or "too small to holda football," and so forth, choosing objects ofother sizes to ask about.

The activity above introduces the notion ofrelative capacity in rather a gross way, and theobjective is to make children aware that con-tainers do vary in size. The method used is nottoo reliable, since an orange, for example, willnot "fit in" a soft-drink bottle, not because of itscapacity, but because the neck of the bottle istoo small.

To advance the idea of relative capacity ina more productive way, select two bottles of dif-ferent capacity, one short and fat,-the other talland thin (olive jars are a,possibility).-Fill one ofthe bottles with AO (orl, for less mess, usesand, salt, or Cornmeal). Then ask the-childrento guess whether the other jar will hold all thewater. Having 'them guess will give them avested interest in the outcome, and a votemight-be taken before proceeding. Then pourto see wh6 is right. The bottles-should be cho-sen so that the difference is "obviolit" when us-ing water for comparison, but not all that obvi-ous just by sight. The ,outcome of the pouringwill depend on whichla:cittle has been filled first,but when the outcome has been observed, thechildren should be asked if they can now tellwhich bottle holds more, and why. If everyonedoes not agree and they cannot convince eachother, repeat the experiment, being careful tobegin with the same bottle. When everyone hasbeen convinced, fill the other bottle and ask fora prediction on what will happen when you pourinto the first bottle. The correctness of the pre-diction will be an indication of the children's un-derstanding. Even if the prediction is correct,the water should be poured so that the childrensee that their conclusion is correct. To empha-size the method of comparison being usedhere, repeat the experiment with other pairs ofbottles.

temperature

Differences in temperature can be detectedby touch if the difference is large enough. On

236 CHAPTER TEN

certain days, the difference in temperature be-tween the classroom and the outdoors may bevery noticeable.

If the classroom is equipped with runningwater, differences in water temperature can befelt. Thus the environment provides opportuni-ties for comparing the attribute called temper-ature. The way in which the behavior of thethermometer corresponds to the way the airfeels can be very interesting to younger chil-dren. Since the column of liquid changes be-fore their very eyes, they may want to try it invarious parts of the roomover a radiator,near an outside door, and so forthto watch itgo up and down. Experience of this,sort en-ables them-later to accept instruments as ex-tensions of the senses' comparison situ-ations It should be noted that children do notneed to be able to read the number's on the in-strument before using,lt;_it is sufficient if theirexperience gives them the understanding that"the higher,, the warmer" and "the lower, thecolder." To check this understanding', ask themto predict what the thermometer would do ifplaced in some ice water or in some waterheated on the stove. They should, 'of ,course,then be allowed to see for themselves that theywere right, possibly with an assist from theschool cafeteria. Predicting how high-or howlow the liquid will go is good preparation forlater use of numbers, and the use of a china-marking pencil to compare day-to-day temper-ature changes (thus avoiding the reading of thescale) can anticipate the necessity for a refer-ence point in other kinds of measurement.

weightAs with temperature, difference in weight

can sometimes be detected by touch.

Begin the discussion by asking the chil-dren to name some things in the room that theycan lift, then some things that they cannot lift.Make a distinction, of course, between thosethings that can't be lifted because they arenailed down and those things that are "too big"to wt. Since small children use the term "big" ina nonspecific way (a big brother is probably

older, a big house has more capacity, and soforth), this is the opportunity to teach the word"heavy" to describe bigness mhen the attributeof weight is intended. The Experience of heftingenables children to become acquainted withthe weight of objects; so a collection of objectsshould beprovided for the children to heft andnote the difference in the "feel." At first, theseobjects should be chosen so that they varyquite a bit, and teachers may need to practice abit themselves in order to select a good repre-sentation. A stapler and- an sempty cardboardbox, for example, might bea better Choice thantwo books, because itis weight rather than vol-ume that is under :consideration. Having thechildren place several objects in order byweight can be helpful, especiallyif some of thelighter ones "look larger." Repeating the ex-periment with smaller and lighter:objects (say,a bottle cap, half-dollar,;etC.),refineS,the notionof weight. If a ,tvio-;pritbaiaricells available,these smaller ObjectCa-n'f'5e plaCed.onthe bal-ance and its 4phavior observed:. Since theswing of .the;:iiiiile-nce agrees with the "feel" oftwo objects being hefted, the use of thebalanceas an extension of the sense of totrchecomesvery reasonable. Thus paper clip's, thumbthcks,and otherobjectSwhose difterenceinveight istoo small to becieteCtedii*hefting,cen now becompared, using-eninStrument for:thai pur-pose.

height

The difference between height and length ischiefly one of orientation: If the alignment ishorizontal, we frequently call the attribute"length"; if it is vertical, we call it "height." Sinceheight is a very obvious way in which grownupsare "bigger than" children, the introduction of

2:3'

the attribute through the notion of height seemsnatural. The chief difficulty with the conceptseems to be that children judge by looking atthe tops of things without paying attention tothe-bottoms. They may think, for example, thata child becomes taller by standing on a chair.

To isolate the attribute, begin by havingtwo children stand back-to-back to see who istaller. Then have the shorter one stand on achair and ask who is taller. If the children thinkthe one on the chair is now taller, it may be suf-ficient to point out that we are not interested inthe chair, just the children. If this is not Con-vincing, it may be necessary to discus's theprocess of growth as the means by which chil-dren become taller. Comparing someone'spresent height with that when he was a baby orreferring to outgrown clothes are ways of di-recting attention to the attribute called height.The problem may be linguistic, but the childrenneed to know that their height does not changewhen they sit down, stand on tiptoe, or liedown. (We discount here, of course, in-finitesimal differences due to gravity.) To checkon understanding. ask the children which istaller, the wastebasket or some object on theteacher's deska chalk box, for ample. Hav-ing several children line up by height or askingthem to name some objects in the room thatare taller than the teacher or shorter than thewastebasket are some other ways of checkingon thd children's understanding of height.(They may, for example, think of it only as anattribute of people.)

An out-of-school project might be to haveeach child find out who is tallest in his family,then next tallest, and so on. (Should we includethe dog?)

Another project might be to make stickdrawings of a mythical family: father, mother,school-aged child, and baby. Which should betallest? Next? How high should we make theschool-aged child? If the baby is drawn crawl-ing, should the picture be as high as the otherchild? If we held the baby up on his feet, howhigh would he come in our picture?

If the children have already had some workwith capacity, it might be instructive to display a

MEASUREMENT 237

collection of odd-shaped bottles and have thechildren first place them in order by height andthen reorder them by capacity.

incidental problem solving

There are many opportunities for becomingfamiliar with the need to measure as well aswith some of the techniques of measurementapart from activities designed especially forthat purpose. The following list suggests someof these possibilities:

Cutting things to fit. When decorating theroom or making posters, let the children"measure off" the amount of crepe paper,twine, or newsprint needed for some purpose.(Some discreet guidance from the teacher canavoid excessive waste.) By not having every-thing precut or premeasured, children gainpractical experience in determining how muchis needed. For example, cutting a strip of paperto go around a fat Santa for his belt can make itreasonable that we sometimes need to know"how big around" something is; in addiOn, thechildren may be very'surprised at how long astrip this takes! Problems of this sort alsopresent natural opportunities for using phrasessuch as "too long" or "too short" in meaningfulways.

238 CHAPTER TEN

Putting things away. Cleaning up the class-room provides opportunities to discover thatsome books are "too tall" to stand upright onthe shelf, to choose from a collection of emptyboxes one that is "big enough" to hold every-one's potato head, to pick out pieces of chalkor crayons that are "too short to save." Appro-priate questions ("Why will this book stand upand this one won't?'') and appropriate decisionmaking ("Pick out to throw away all the piecesshorter than this") can help the children learnabout comparing length, capacity, and the like.

Counting. Rote counting need not alwaysbe practiced with tips, of a pencil or specialcounters. Counting the cupfuls of sand neededto fill a pail or a dump truck enables the chil-dren to see one means for measuring capacity.The matter may simply be treated as an exer-cise in rote counting. However,.for some chil-dren it might be instructive to fill two containersin this manner, count the cupfuls, and then askthe children which they think "holds moresand." The latter procedure, of course, antici-pates the use of numbers for purposes ofmeasurement and at the same time giVes theteacher some evidence of the children's ac-ceptance of this criterion.

some summary remarks

There are certain problems that arise in anyactivities approach to learning. For one thing,once children's curiosity has been aroused,they can pose unexpected (and sometimes em-barrassing) questions. Careful preparation onthe part of the teacher, including some roleplaying on his part, can often preclude this.Also, a bit more direction can sometimes begiven to an activity if the teacher does part of itand has the children help. But as with anyteaching strategy, one must weigh the dis-advantages against the advantages. Activitiesof the sort described here do provide a com-mon background from which the more preciseaspects of measurement can be developed.The child, for example, who has tried to put twobooks on the shelf and found that one would fitand the other would not has an experience with

"taller than" that makes the results of measure-ment with a ruler sensible"Oh, I see, whenyou measure the taller one you get a bigger an-swer."

As mentioned earlier, it is not intended thatchildren master any of the techniques involvedin these activities. These techniques are in-troduced to make comparisons meaningful, notbecause they are always reliable. A Ping-Pongball, for example, is smaller than a brick in vol-ume, yet it could not literally be "fit into" thebrick.

It was intended that the introductory activi-ties with measurable attributes make use of thechild's senses. Some attributes whose measureis studied much laterlight-years, kilowatthours, caloriesrequire something of an act offaith. Perhaps these abstractions will be moreacceptable to the learner if his first experienceswith measure are seen as perfectly :eason able.

Almost no use is made of number :n theseactivities. The assignment of numbers asmeasures derives from a felt need for moreprecision in comparison. This matter will bediscussed in the next section.

The problem-solving approach requires thatthe child be permitted to make conjectures andthen test them. Not only does this permitgrowth in self-reliance on the part of the child, italso lets him see that his mathematical experi-ences make sense when tested in the realworld. For the teacher, children's commentsand questions as they work will on the onehand provide valuable insight into sources ofmisconception and on the other give assurancethat the children fully understand what they aredoing.

the processof measurement

The process by which a number is selectedto be called the measure of some attribute of anobject is specific to the attribute in question.That is, we use one method for finding the num-ber to be labeled "weight" and quite another toselect the number to be labeled "height." Be-cause of this specificity, it is easy in the class-

((:),".. .1

room to become so embroiled in the details ofthe process that some very basic notions areneglected. For this reason, it is important to ex-plore some of those basic notions before turn-ing to the processes themselves. This we die inthe preceding section, and we shall now en-large on those notions.

Although the process of measurement isspecific to the attribute. there are some sim-ilarities (the use of a standard unit, for ex-ample) that enable us to make some sort ofclassification. The following outline may not becompletely valid. Nevertheless, it will be usefulfor the purpose of instructing young children toconsider that the determination of a number tobe assigned as a measure may be carried outin one of the following ways:

1. Counting2. Reading a scale on a calibrated in-

strument3. Computation4. A combination of the above

countingTo find the number this way, some object is

taken as a unit and physically iterated againstthe object being measured. We use thismethod whenever we pace off a plot of ground.An individual's pace is taken as a unit, is ap-plied successively to the plot, and the numberof paces counted. The count is then taken asthe measure of the length of the plot. Usuallywe think of this as giving a "rough estimate,"possibly because we recognize that the pacemay not be uniform, or possibly because weare not using a "standard" unit of measure. Insome instances, therefore, we may elect to iter-ate a yardstick rather than the pace.

The simplicity of the method of countingmakes it a natural one with which to acquaintchildren. Furthermore, the other methods listedare refinements of this in the sense that theywere invented to introduce greater precision.As a consequence. we can often find ways ofpresenting the other processes first as count-ing. For example, determining the area of arectangular region can be presented first as aprocess of iterating a unit. Now as adults, wetend to mistrust this method of approximating

MEASUREMENT 239

measurewe feel the pace may not be uniformor, if a yardstick is being used, that we lose afew inches as we pick it up and lay it down. Thatis, we sense a lack of "exactness" and hasten toshow children how to read a fifty-foot tape sothat they can measure more 'exactly." How-ever, the studies of Piaget cited earlier showthat children may believe that the length of themeasuring stick changes as it is shifted. If thisis so, then we adults are worried about some-thing that does not concern children in theleast. Perhaps the pedagogical implicationhere is that we delay teaching more sophis-ticated methods until the children sense forthemselves the "inexactness" that the methodof counting entails.

reading a scale on a calibratedinstrument

This is perhaps the most common methodused for selecting numbers as measures. We"read off" the temperature on our thermome-ter, the air pressure on our barometer, ourweight on a dial on our bathroom scales, andthe length of some object on our ruler. Further-more, we tend to place complete faith in thenumbers we read off (except, of course, forthose on the bathroom scales!). That is, we ac-cept the instrument as accurate, we accept thatit does measure what it is intended to measure,and we accept our ability to estimate sub-intervals on the dial. At first glance, it would ap-pear that this process of estimation mightpresent the greatest difficulty for children,since the user of the instrument must note theunits i::) calibrationtenths, fifths, eighths, andso forthand be able to interpret what "half-way between" two marks would mean. Subsidi-ary problems of establishing the correct line ofsight and coping with the meniscus of lir,uidsare also recognized. But for son3 childre.. sheinstrument itself may be baffling. He is handeda ruler, for example, and told that it is for mea-suring length. For some children this may be asinformative as telling them a plark is used towhiffle a thintch." Thus both the behavior of theinstrument as a means. io ao. end as well as themysteries of its calibration may present diffi-culties for children.

-:.% A e)t"..'- b-

240 CHAPTER TEN

computationWe compute to find a number to assign as a

measure whenever we determine someone'sage (subtract the year of birth from the presentone), when we use any of the mensurationformulas (perimeter, area, volume, surfacearea), and when we calculate such measures asIQ. To many adults, this represents the mostexact form of measurement. We do have con-fidence in our ability to compute, and since wecan handle large numbers of decimal places,somehow we are convinced we are being moreaccurate. A moment's thought, however,should remind us that the measure of a side ofa square must be determined before eitherarea or perimeter can be computed and thatthere is no formula for determining that num-ber. We must rely on our second method, theuse of a calibrated instrument, with all its po-tential for human error to obtain the data tothrow into our formula! Children's difficultieswith mensuration formulas are well known.Perhaps these are derived from our zeal toeliminate human error in measurement, result-ing in a rush to establish computational short-cuts. If we give the younger children time tolearn what is meant by these things called pe-rimeter and area, and give them time to appre-ciate the need for accuracy, many of these diffi-culties could be avoided.

combination of methodsThe use of more than one method of compu-

tation was mentioned in the preceding para-graph. Another common example is determin-ing ones weight on the scales in a doctor'soffice. Two separate readings must be madeand the numbers summed. The computationhere is mental, of course, but is computationnevertheless. We include this as a separatemethod merely for the sake of completeness.

What the studies of Piaget and his followerssuggest is that the young child has almost noequipment for the appreciation of the degree ofexactness in measurement that adults hold insuch high esteem. Pedagogically, this impliesthat young children need to spend more timelearning about those attributes that grown-upswish to measure, more time comparing them in

relatively gross ways, and more time iteratingand counting units. When it begins to makesense, for example, that if one container holdsthree scoops of sand and another two, then theformer is larger than the latter according to ca-pacity (never mind the heights), then the chil-dren are distinguishing between two attributesand accepting number as a means of denotingrelative size. But such understandings comeabout through experience, not through legisla-tion.

The activities that follow are intended tocreate a smooth transition from the foundationsof measurement to the more refined tech-niques of instrumentation and computation.Since the aim is to make those processes ap-peal to children as both reasonable and natu-ral, the transitional activities will emphasize theprocess of counting and will incorporate a cer-tain amount of variety.

Instructions cuch as "Measure the line seg-ment below" assume that children accept num-bers as measures, an assumption that may notbe warranted in fact. In order to promote thisacceptance, in some of the activities the chil-dren assemble configurations according toreal-life data and then observe which is largest.The relative size is then associated with thenumber of pieces used by the children (not"contained in"an important distinction).

In other activities, the usual instructions formeasurement exercises are essentially re-versed. Instead of being asked to find themeasure of a given object, the measure will begiven and the children asked to assemble asmany configurations as possible having thatmeasure.

The calibration of instruments is related tothe way in which they operate. Larger numbersare to be associated with "more" of the attri-bute in question. Thus on a thermometer thelarger numbers are higher up because thewarmer the temperature, the higher the columnof liquid. In the foundational activities somesuggestions were given for observing the be-havior of some measuring instruments, in thissection we offer some suggestions for childrenplacing numbers on instruments.

It is also part of this transitional phase thatchildren learn that the number assigned as ameasure is not only related to the amount of the

attribute but alto to the size of the unit. Someactivities are suggested for this purpose.

Finally, some activities are suggested forlearning about the attributes called "perime-ter," "area," "volume," and "surface area."These were placed last because they are moreabstract than, say, icngth or weight. The aim isto distinguish between perimeter and area onthe one hand and between volume and surfacearea on the other. Although it is not necessary,and may not be desirable, to use the words pe-rimeter. area, and so forth, it is impbrtant forthese activities that children believe numberscan convey information about relative amountsof some attribute.

As before. the list of activities is provided asa source of ideas. and teachers are encour-aged to devise variations to suit the needs oftheir own classes.

transitional activities

accepting counting as a measure

making designs on the geoboard

Since a rubber band stretches and canchange length, a piece of string should be usedfor this activity. The string should be preparedso as to be some convenient number of spaceslong (let us say 8), with loops on either end tobe fastened to pegs. The string can then beused to experiment with making different "de-signs" or "paths" or whatever label the teacherwishes to use. Some possible arrangements ofan eight-space string are shown here:

e0---0

Ask the children which path is longest andwhy they think so. Their answers will give valu-able information about how well they "con-serve." If some think one path is longer than

MEASUREMENT 241

another, it probably will not help to tell themthat they are wrong. Instead, try the following.

Draw each of these designs on thechalkboard or overhead projector and askwhich the children could make using the samestring-

--4

Note that the first figure can't be made becausethe string isn't long enough, and the third figurecannot be made without doubling;back, whichshould be ruled out by the teacher. The chil-dren should be encouraged to_try out their con-jectures with the string and geoboard so as todetermine for themselves who is right. Thetrying out will entail some counting if thesepaths are to be reproduced exactly, and thismay help some of the children who thoughtsome of the earlier designs to be longer thanothers. In any event, the discussion accom-panying this activity should be instructive.

making designs with pieces of paper

Give each child four pieces of paper all thesame size and shape; for example, rectangularpieces each 1 inch by 3 inches. Have the chil-dren arrange these into designs. The only rulesare that each piece of paper must touch at leastone other and there must be no overlapping. Ifsuch instructions are too difficult, printed pat-terns may be given out for the children to cover(graph paper helps in the preparation of suchpatterns). Some possibilities are shown in I.Encourage the children to make as many 0-signs as possible. Have each child choose hisfavorite and paste it on the paper; then displaythe designs. The question of interest here is."Which design has more paper in it?" If the

1

242 CHAPTER TEN

children think this varies with the design, theydo not conserve, and teachers will need to usetheir own judgment about how much urging todo in the matter. Some questions' may help, as,for example, "Did you use all the paper for ev-ery design?" It may also help to take out an-other collection of four pieces of paper and fitthese against each design in turn. The problemmay or may not be linguistic. If it is not, carryingout some variation of this activitydifferentnumbers of differently shaped piecesat alater date may help to develop the notion thatall designs made with the same collection ofpieces of paper must have the same amount ofpaper in them. (It is not, of course, intendedthat the children verbalize this generalization.)

tangrams

The difference between tangrams and thematerials of the preceding activity is that thepieces in a tangram are not all the same sizeand shape. They are commercially available,and of great interest to children. They also lendthemselves to the same sorts of questions asthose cited in the preceding activity.

This activity would be a three-dimensionalanalog of the second activity above.

Have the children make designs withblocks (let us say each child has six blocks) andencourage them to make as many different de-signs as possible. Then ask which design is

bigger. If they think some design is bigger, askwhere they got the extra blocks to put in it.Comparing designs made by individual chil-dren"Could eveyone make a design just likeJane's with his own blocks? Let's all try."mayhelp the children see that the shape of a pile ofblocks does not tell the whole story about itssize. That is, piles of different shapes may havethe same number of blocks in them. It is, ofcourse, neither necessary nor even desirable tohave the children verbalize the generalization.Asking appropriate questions can direct theirattention to the phenomenon.

counting as a measure of height

Building towers. If sets of counting blocksof various colors are available (all the samesize, of course), place different numbers ofthree different colors in a box. Have the chil-dren count the different colors separately, andsuppose there are six white ones, four green,and seven yellow. Ask the children which colorthey think would make the highest tower. Whenthey have .made their conjecture, stack theblocks up to check. Repeat with other collec-tions. For one of these, it might be instructive tohave the same number of two different colors.For reinforcement, prepare ahead of time threetowers of three, different heights, not orderedby height. (Placing a box over the dispaly willkeep it from sight until you are ready for it.) Re-move the box ancrask the children which pilethey think haS the most blocks in it, then thenext most, and the least number. Have themcount to check.

Making a graph. Each child should beasked if he knows when his birthday is. If all do,then a graph can be made as follows. Prepare aposter with the months arranged across thebottom. Give each child a square-shaped pieceof paper and have each in turn go up and pastehis paper in the appropriate place according tohis birthday. Part of the resulting graph mightlook like the one in(Lining things up neatly with no gaps or over-laps is nice, but a little sloppiness could be tol-erated.) Then ask* which month has the most

January Febnuir/ March

birthdays and how one can tell. The answers tothe last question could be "taller" or "moresquares." Sinr, the differences in height maybe quite obvious and are clearly related to thenumber of pieces of paper put up, this be-comes not only an exercise in recording dataon a graph but also a means of acceptingcounting as a measure.

counting as a measure of length

Pacing oft. There are many opportunitiesfor pacing off. "Do you walk farther from thepencil sharpener to the door or from your deskto the door?" Step off both paths. Find out howmany paces it is from the door of the classroomto the door of the library or the cafeteria orsome other reasonable choice. For youngerchildren this may simply be an exercise inlearning a way of comparing two fairly longpaths when one is not a subset of the other. Forolder children, the activity prepares the way forseeing the need for a standard unit of measure:George paces the room and gets one number,and Charles paces and gets another; or Georgepaces and the children object that his stepsweren't all "even." These situations can be thebasis for a new problem. In the first case, theproblem would be, "Why do the two boys getdifferent numbers?" In the second, the probhmis. "What could we do to get a more reliablenumber?" And, of course, pacing off the corri-dor or the playground could show the need forsome larger unit.

Using a stick. The janitor comes in to re-place a light bulb and ducks his ladder to get it

MEASUREMENT 243

in the door. Why? Could we find out whether ornot the ladder will go through upright withoutputting it against the door? We can't pace offthe height of the door; so we need another wayto compare the two. (The classroom flagpole,perhaps, or the window hook if there is one,might do.) Comparisons by use of a stringmight be a first suggestion, but what if we haveno string handy? Problem situations of this sortcan be proposed and the children somehowencouraged to use some object such as a stickto solve the problem. Exactness, of course, isnot the goal here; rather, the goals are (1) ac-cepting the method as a means of obtainingneeded information and (2) learning the tech-nique of successive application of the unit (iter-ation of the unit).

Making a graph. Ask the children to reportthe time at which they went to bed Saturdaynight and the time they got up Sunday morning.Show these times on a large graph, and usethis as an opportunity for children to cut stripsof paper "the right length." (Freezer tape mightbe good for making strips.) This activity pro-vides opportunity to explore several aspects ofmeasurement, according to the age of the chil-dren and the kind of questions posed by theteacher. For younger children, it might be suf-ficient that they get the strips of paper the rightlength and can tell from the graph who got upearliest (latest) and who went to bed earliest(latest) as an adjunct to learning about tellingtime. For older children the question might be,"Can you find out from the picture who sleptlongest?" There are several ways in which thiscan be determined, one of which is to count thenumber of hours each child slept. The graphshown here is good for this purpose because ofthe irregularity. That is, if some try to tell bylooking at the endpoints, they may have to bereminded that all children did not go to bed atthe same time. (If a hint is needed, ask if theycan tell from the picture how long Tom slept,how long Keith slept, and so forth.)

7 8 q 10 it 12 I 2.341S t. 7 S

11111111110=1=111MIEMINI

244 CHAPTER TEN

counting as a measure of weight

Once the children understand how to use atwo-pan balance to compare weight (see theactivities in the first half of this chapter) andhow to assess "the same weight as," a problemof the following type can be presented in a nat-ural way: "The staple remover is heavier than abottle cap; is it heavier than two bottle caps?Three bottle caps ?How many -tiottle capswould it take to weigh the same as the stapleremover?" Other combinations can be useddepending on the availability of objects forcomparison. This particular activity lends itselfto extension to standard units if one-ounce orone-pound weights are available. It ego lendsitself to extension to the single-beam balancewith a counterbalancing arm if that is available.

counting as a measure of capacity

Explain that the water in the aquarium isgetting low and some water-must be added bythe panful. Ask the children to guess how manypanfuls it will take to "fill" the aquarium up tosome designated height (to be marked with achina-marking pencil). This activity is particu-larly instructive because of the difference in thecross-section areas of the pan and the aquar-ium. If the children doubt that anything is beingaccomplished, the "beginning level" may alsoneed to be marked. However the problem ishandled, it will undoubtedly be surprising howmany panfuls of water it takes to raise theheight of the water level even a very smallamount.

If small and large school buses are em-ployed by the school system, the children mayknow how many seats are in their bus or howmany children ride it. This information can berelated to the "size" of the bus.

The relative sizes of two toy dump trucksmay be compared by filling them with sand by

the scoopful, by counting the blocks they willhold, or by counting the number of marblesthat can be transported.

For an activity more directly related.to stand-ard units of measure, use individual half-pintmilk cartons to fill quart containers, half-galloncontainers, and the like. As each of these isfilled, count the number of half-pint containersrequired.

relating the measure to the size ofthe unit

Many of the activities suggested above pro-vide opportunities to observe that differentnumbers are also related to different unitsthesize of a "pace," for example. When the time ar-rives for this understanding to receive moreemphasis, some of the activities can be rede-signed for that purpose.

To cite a simple example, fill a large bottlewith water using first a juice glass and then acup. Explain-the difference in count. If the chil-dren have had experience with comparing ca-pacity using a third container, the explanationshould be easy.

In -Never Never Land there lives a giantwho has a funny way of measuring his land. Hedoesn't pace as we do; he goes heel-and-toealong, like this:

In the same country there also lives an elf, whoalways does just the same thing as the giantWhen he tried to heel-and-toe the giant's gar-den, he got an answer different from the gi-ant's. Why? Having some cutouts of the giant'sshoe and the elf's shoe will help, for some ac-tual numbers will be of use here. Let the chil-dren measure a "giant's garden" using bothsets of shoes and record the numbers. "Sup-pose Charles's shoe (Charles is in the class) isused instead. Would we get either of the num-bers we already have? What number do youthink we might get?" See who can come closest

(11:,"*)

with his guess Other children's shoes (or feet,or cutouts of them) can be used.

Explain that in olden times the humanbody was used as a measuring instrument. His-torians tell us that the rod as a unit of lengthwas the length of the right feet, laid end to end.of sixteen men coming out of church on somepredetermined Sunday. The children could beencouraged to reflect on the reliability of this asa "standard" measure, would this be the sameregardless of the Sunday chosen? Experimentswith various subcollections of the childrencould be informative here. Other "body meas-ures" include the span (width of the expandedhand), the cubit (from the elbow to the tip of themiddle finger, the inch (the measure of the topjoint of the right thumb of the reigning monarchof England), and the yard (the length of the armfrom fingertip to nose). Comparisons of thenumbers obtained for different children'sspans emphasize the relationship of the num-bers to the size of the unit chosen. (Interestedteachers are encouraged to consult books onthe history of mathematics.)

calibrating instruments

Making a measuring jug. Select a glass jar,such as an empty mayonnaise jar, for the mea-suring jug. Have the children pour in one juiceglass of water. With a china-marking pencil,mark the level of the water. Then add anotherjuice glass of water, mark .the height, and soforth, making as many marks as you wish. Theappropriate numbers could also be shown. If

MEASUREMENT 245

this measuring jug could then be used for someother experiments or classroom routine, somuch the better.

Calibrating a two -pan balance. If the two-pan balance has a dial, this can be covered upwith masking tape. Have the children mark theempty position with an arrow or a "0." Then addone capful of sand and mark the spot where thepointer stops; add another capful and markthis, and continue as practical. A cap from acatsup bottle, or a cap from a fabric-softenerbottle might be an appropriate size for a capfulof sand. A cup might be too big, and, of course,a flat bottle cap could not as easily be filled"level" as a deeper one. It would probably bedesirable to calibrate for both pans of the bal-ance. The-calibrated balance can then be usedwithout piling on unit weights each time.

Making a ruler. Give every child a strip ofcardboard and have them mark off the lengthof one paper clip, two paper clips, and so forth.The use of paper clips is not mandatory, anyunit being used by the class (elf shoes, for ex-ample) is appropriate. By adding the numbers,the children have their own rulers to measureobjects with. The original cardboard need notbe any integral num berof units long; as a mat-ter of fact, it might be instructive if the stripswere not an integral number of units long.

3

learning the attributes calledperimeter, area, surface area,and volume

Some suggestions for discriminating amongarea and perimeter are to be found in the chap-ter on "Geometry" (chapter 9), and the follow-ingactivities assume some rudimentary knowl-edge of geometry.

246 CHAPTER TEN

Provide each child with a piece of paperruled in one-inch squares and a collection oftwelve or so 1-inch pieces of 1/8-inch dowelstick or pipe cleaner. Have the children useall their sticks to build a simple closed curve(or "fence" if they do not know the term) onthe paper; the fence must be built along thelines on the paper. Have them trace the fenceand then use their sticks to make another fencethat looks different from the first,- putting this inanother place on the paper. Since there aremany possibilities, asking Who can make themost?" can encourage a bit of competition. Themeasurement questions are, "Which fence isthe longest?" and "How many spaces are in-side the fence?" Some of the children mayJump to the conclusion that since the fences areall the same length, there must be the samenumber of "spaces" inside each fence. Havingthe fences outlined allows the children to checkthis idea by counting the spaces enclosed.

I J

On a sheet of paper ruled in one-inchsquares, have each child shade in a "garden"that covers exactly six spaces. Again, there aremany possibilities, and after several gardenshave been laid out, some measurement ques-tions should be posed. "Which garden coversthe most space?" "If each were fenced usingthe sticks of the previous activity, which wouldrequire the most sticks?" As with the previousactivity. the children may think that because allthe gardens are the same size, they require thesame amount of fence. Therefore. some sticksshould be available for checking answers.

Making boxes. Have a display of severaldifferent sorts of boxes. Some possibilities arean oatmeal or salt box, a rectangular cerealbox, a shoe box or other box with a thin lid, abox with a thick lid such as pieces of hardwaresometimes come in, a suit or dress box thatopens up perfectly flat. "These boxes are all

made of cardboard. We have some cardboard.Could we make some boxes like these? But weneed a pattern." The children may be,able to in-vent their own patterns, or it may be necessaryto cut up the boxes to see how they are made.Picking out a big enough piece of tagboard.figuring out how to fasten pieces together. andlearning the advantages of folding to avoidtaping corners are all things that can belearned from this -activity: -Before-tackling thecylindrical box, it might help to have a piece ofmailing tube available. Make the first problemthat of making ends for it in order to convert itinto a box. With a little more experience the en-tire box can then be made. If the children seeminterested, give them each a piece of construc-tion paper (all the same size) and ask them tomake the biggest box they can from the sheet.Then compare different children's products forsize (have them guess first), using sand or saltto compare capacity. If they "weren't thinkingabout it that way," this is a natural opportunityto discuss such things as the amount of card-board related to the cost of makirig boxes com-pared with the amount of "goods" variousboxes would hold. Thus both measures (sur-face area and capacity) are needed in busi-ness; neither is "the right" one.

Making cubes. Have the children make apattern for a cube as described in chapter 9,using squared paper. (There is more than onepossible pattern; so it is not necessary that allthe children's be alike.) When.they have founda pattern that works and become reasonablyhandy at assembling the cube, have themmake a pattern on squared paper for a "biggercube." This is definitely a nontrivial problem. Itrequires time on the part of the childtime tofigure out how to make his pattern "bigger" andtime to try out his patterns and revise themwhen they don't work. It also requires patienceand help on the part of the teacherpatienceso as not to give too much direction, and help inasking such questions as "What seems to bewrong?" and "What do you think you could doto make it right?" But the exercise is instructivebecause the children may be very surprised todiscover how much bigger the pattern must bein order to make even the "next sized" cube.When all the children have finished, display asheet of paper with outlines of the-patterns of

the two different sizes on it and ask the childrento count the number of spaces occupied byeach pattern. This will reinforce numericallywhat they have discovered from inventing thepatterns. The teacher might also wish to pro-pose that the larger pattern be assembled intoa box, with one side loose for a lid. Ask the chil-dren how many of the smaller cubes they thinkwould fit into the box. The children might loansome of their little cubes to try it out and to seewho is right.

More on making boxes. One-inch squaredpaper and one-inch blocks, such as countingblocks, will be needed. The pieces of papershould be 7" by T each. Propose that an openbox be made by first cutting one space fromeach corner of the paper and then folding, asshown in . Ask the children how many spacesare on their sheet (eight rows, each containingseven spaces), and then ask how many wouldbe left for the box after removing one from eachcorner. The children should cut out the cor-ners, tape up the sides, and then the teachershould ask how many blocks they think wouldgo into their box. If they do not have any idea,borrow a box and have someone fill it withblocks while the other children watch. Next,propose that four spaces be cut from each cor-ner (0). "How Many would be left for the box?"With another sheet of paper, have the childrenmake this box. "How many blocks will it hold?Should it hold more or fewer than the first box?Why?" Again, this may need to be demon-strated, but some of the children may have dis-covered a way to figure out the answer. Theraask about removing nine spaces from eachcorner and repeat the experiment. If some ofthe children wonder why you cut squares fromeach corner, ask what they think would work,try out their suggestions (why not remove twospaces from each corner, for instance?), anddiscuss the results. It is not intended that anyformulas be given as the end result of this activ-ity; it is intended that this activity be good prep-aration for later work with formulas., r

MEASUREMENT 247

time duration

It would be artificial if we were to attempt todevelop concepts of time in early childhood in-dependent of the instruments used to measuretime. The frequent use of time and clocksmakes it reasonable to introduce clocks andcalendars. At the same time we can help thechild to develop a sense of time duration and tocompare periods of time.

Children can engage in many activities thathelp them attain some idea of time duration.Some of these activities can be related to theclock and the calendar. Others are dependenton such devices as egg timers, simple waterclocks, or candle timers. Still others involvesimple comparisons independent of any kind ofmeasuring instrument.

Following are a few examples of activities de-signed to help the child gain some idea of timeduration. The teacher will want to develop oth-ers as well.

Races. No special equipment is needed.This simple activity involves the running ofraces of various types. Make two lines abouttwenty yards apart on the playground or in thegymnasium. Teach the children to start "at thesame time," using any method you wish. Alsoteach them to start in the same place" by fixingtheir toes on the starting line. The child whogets to the finish line first is the winner. A dis-cussion of what winning means in terms of timewill establish that the first person to cross theline takes the least time.

Variations in the type of race should bemade. Include changes in the start and finishline as well as in the kind of race. Insect racesor turtle races are common variations.

The stopwatch can be introduced in racingactivities fairly early. Children should see thatthere is a relationship between the distance thehand moves and the length of the race.

Beat the clock. Provide an egg timer or asimple water clock. (A water clock can be madeby punching a small hole in the bottom of asoup can. When it is filled with water, the waterwill take about the same time to run out each

248 CHAPTER TEN

time it is filled. If different periods of time are tobe measured, the can can be graduated at thehalfway or quarterway points. Shortening thelength of time can be accomplished by enlarg-ing the hole.) These devices can be used to an-swer such questions as "how many times canyou walk from the front of the room to the backand return before all the sand runs out?" or"How many times can you bounce a ball beforeall the water is out of the can?"-Have the childestimate when he thinks all the sand or all thewater has run out. Turn the timer over and askthe members of the class facing away.from thetimer to put up their hands when they thinkthere has been enough time for the sand or wa-ter to run out. They will have had to have hadconsiderable experience with the time periodmeasured by these devices before such esti-mating is attempted.

As a follow-up activity, punch holes of differ-ent size in three soup cans of the same size.Paint each can a different color. Call one can"short time," another "medium time," and thethird "long time." This will help build the idea ofa need for variation in lengths of time units,such as minutes, hours, days, weeks, and soon.

Time's up. A large clock with a secondhand and a flow pen with washable ink will beneeded. The object of this activity is to have thechild make estimates of short periods of time.For example, make a black mark on the clock-face at twelve- and another at six. Have themembers of the class turn their backs to theclock. Give them a signal when the secondhand is at twelve. When they think th, hisreached six, ask tht m t 4u. n ur .

who estimate too little will be able to wah_'.rest of the time run out. Those who go over thetime will be able to get a better notion of howquickly the hand moves.

How long? A large clock, masking tape,and masks for the clockface with pielike openspaces are needed. To help the children get anidea of the relative motion of the minute andhour hands, affix the masks to the face of theclock. Set the children to some task and tellthem to begin some other task when the minutehand disappears behind the mask. The handand the edge of the mask should just coincideat the beginning of the activity.

Later on, use the hour hand instead of theminute hand. Mark with a strip of tape justwhere the hour hand is at the beginning of theday, for example. Then, just before the dayends, make another mark so that the child cansee how far the hour hand has moved.

Provide a calendar of the current monthfor each child and have it pasted in a book. Atsome time in a day, let the child make a mark inthe spaces representing the day. This can be abasic activity for the development of a sense ofthe duration of a week, a month, or a schoolyear.

Variations would involve the use of symbolsto indicate the type of weather, the temper-ature, or some other aspect of the days of themonth or the week.

Vocabulary such as "Wednesday, the sixth ofNovember" should be used verbally wheneverconvenient. To help children get familiar withboth the written and oral names, write the dayof the week and the day of the month on thechalkboard each d ,,c(rd, orally

of measurethemetric system

Eventually, the children will need to learnthose units of measure that are standard, orconventional, in contemporary society. As theUnited States moves toward the adoption of themetric system, it will become necessary forchildren to learn to use metric units in theirmeasuring activities.

(-a(

The metric system is a decimal system de-veloped as an outgrowth of the French Revolu-tion; the units are derived from nature ratherthan from the dimensions of some mortal mon-arch The following brief description is summa-rized under four headings.

length

The basic unit of length is the meter. Origi-nally the meter was defined as one ten-mil-lionth of the earth's quadrant (one-fourth of thelength of a meridian) and was determined bysurvey. Howeyer, with advancing technology ithas become possible to redefine the meter interms of the wavelength of light. By so doing, ithas become possible to reproduce the meterwithout the fuss of sending out an expedition.One meter is slightly longer than one yard. Ameterstick is subdivided into 100 equal parts;each of these is called one centimeter. A cen-timeter is slightly less than half an inch. Thesetwo units are adequate for measuring lengthsaround the classroom or school building. Formeasuring such things as automobile trips, thekilometer is used. One kilometer is 1.000 me-ters, a length slightly more than half a mile.

capacity

The basic unit of capacity is the liter. It is thecapacity of a cube measuring ten centimeterson an edge and is slightly more than the Eng-lish quart. Liquids such as gasoline or milk aremeasured in liters. When a smaller unit is desir-able (liquid medicines, for example), the millili-ter is used. One milliliter is one one-thousandthof a liter, that is, the liter is subdivided into1,000 parts, each of which is called a milliliter.Since a cube ten centimeters on an edge has avolume of 1,000 cubic centimeters, one millili-ter is equivalent to one cubic centimeter.

weightThe basic unit of weight is the gram. It is de-

fined to be the weight of one cubic centimeterof distilled water under certain specific condi-tions of temperature and atmospheric pressure

MEASUREMENT 249

(necessary because liquids expand and con-tract with changes in temperature). Since thegram is a very small unit (about the weight of apaper clip), the weight of a person would typi-cally be given in kilograms. A kilogram is 1,000grams, and hefting a two-pound bag of sugargives a fair idea of the weight of one kilogram.

temperature

The basic unit of temperature is the degreeCelsius. It is one one-hundredth of the differ-ence between the freezing and boiling points ofdistilled water when the atmospheric pressureis that of sea level. Unlike units of length, ca-pacity, and weight, no prototype of the unit canbe housed in some bureau of standards. In-stead, a thermometer would be calibrated bymarking the level of mercury when the in-strument is placed in freezing water and when itis placed in boiling water. The first mark is thenlabeled "0" and the second "100," and the in-terval between is subdivided into 100 equalparts Each of these parts represents "1 degreeCelsius" on that particular thermometer; thescale may then be extended by laying off thedesired number of units above and below theoriginal marks and labeling them appropri-ately. It will be noted that if the thermometerwere being calibrated as a Fahrenheit scale,the two original marks would be assigned thenumbers "32" and "212" respectively, and theinterval between would be subdivided into 180parts.

The summary above is not intended to be acomplete description of metnc units, rather, itis intended to call attention to those units mostuseful to a beginner. The beginner needs only afew units, and those few should be ones thatare appropriate for measuring things in his im-mediate environment. Relationships among theunits in a single system or among different sys-tems are not needed for the beginner: Thus,references to the customary (formerly English)system given in the summary above are in-tended to direct the reader's attention to theappropriateness of the unit for small, medium,or large quantities, they are in no way intendedto suggest that young children be taught toconvert from one system to another. Years of

tr...., ',3/ .'.r r-vi

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250 CHAPTER TEN

experience give the typical adult the feeling thatthe customary units are somehow naturalwhereas the metric units are artificial and hardto learn without a prolonged introduction to themeanings of the prefixes "centi-," "milli-,""kilo-," and the like. What we must recognize isthat for beginners, such as young children, anysystem of standard units of measure is arti-ficial No system is more natural than any other,and the simultaneous introduction of the inchand the centimeter will cause no more con-fusion than the simultaneous introduction ofthe inch and the foot. For the beginner, havingan appropriate unit and knowing its name willenable him to record the results of his meas-urementafter all, that's what the units are for.

The April 1973 and the May 1973 issues ofthe Arithmetic Teacher contain a number of ar-ticles on the metric system that many readersmay find helpful.

summary

The overall aim of this chapter has been todevelop in children an inner conviction thatthose activities to which we refer as measure-ment are reasonable means to practical ends.To accomplish this goal, we have describedways in which children can be made aware ofcertain attributes of objects that are useful tomeasure, and we have suggested ways inwhich children can be made aware of the use-fulness of measuring these objects. We havesuggested ways in which children can be madeaware of what "more" or "less" of some attri-bute means, and we have shown how evenyoung children can see that instruments can beextensions of their senses when making com-parisons. In the matter of assigning numbers toquantify "more" and "less," we have empha-sized the value of counting. We have stressedthis because it seems futile to insist on a pre-cision for which children have no appreciation.Furthermore, ifas the studies of Piaget in-dicatechildren believe that a unit changessize as it is moved from place to place, neithercomplicated tools nor computational shortcutswill alter this belief. Finally, we have includedactivities in which two or more attributes of a

set of objects were explored simultaneously. Inso doing, preparation is made for recognizingthe difference between, for example, the pe-rimeter and the area of some region.

A problem-solving approach was used in allthe activities because it provides the child withthe opportunity to see the need for knowing"how much" of something and because it giveshim the opportunity to expenm.ent with variousideas that he may have. Presumably, he willcome to accept as reasonable the conventionalsolutions to measurement problems. Althougha problem-solving or activities approach maybe considered time consuming, account mustbe taken of the time spent in review and re-teaching under the usual plan of instruction. Ifseveral weeks are spent in each year for fouryears on the subject of, say, converting fromone unit of measure to another, perhaps a fewdays spent pouring sand from one box to an-other or "walking" elf and giant shoes could re-duce this time considerably. When viewed inthis way, activities may not seem so time con-suming after all. The goals of an activities ap-proach are thus long-term goals, and "mas-tery" in the usual sense is not one of the goals.Consequentlythe activities are often present-ed as a request for a prediction about what willhappen or as encouragement to find more thanone way to carry out a set of instructions. Achild can hardly fail at either of these tasks,and so there is no stigma about not knowingthe "right" answer. When the results are ex-amined, there still need be no stigma. Initial"wrong" predictions were only guesses any-way, and no prize is offered for the greatestnumber of solutions. The examination of re-sults is for the purpose of seeing what reallyhappens, and this understanding can be as-sessed by asking for more predictions andchecking experimentally. Thus understandingdevelops within the strictures of the real world,and it is in the real world that we have need ofmeasurement.

references

Piaget. Jean The Child's Conception of Number NewYork: W. W. Norton & Co.. 1965.

Piaget. Jean. and Barbel Inhelder. The Child'I, Conceptionof Geometry. New York. Harper & Row. 1964.

r'l,-' ).I .'.... 0

relations,number sent nces,other topics

henry van engen

douglas grouws

THIS chapter contains suggestions and activi-ties from other mathematical areas that haverelevance for mathematics programs at theearly childhood level. Content areas includedare mathematical relations; number sentences;and other topics, such as logic, probability, sta-tistics, number theory, and the integers.

mathematical relations

There are many mathematical relations thatare appropriately studied in the early elemen-tary grades. In fact, many of the topics pre-viously discussed in this book are in essencemathematical relations, although they may nothave been so characterarl Topics such ascongruence and similarity and ideas such asmore than, as many as, less than, one morethan, is a subset of, and longer than are all ex-amples. This section describes the nature ofmathematical relations and shows their funda-mental role with topics in the elementaryschool.

one-to-one correspondenceThe notion that counting is the basic idea of

arithmetic has been accepted and promotedfor a long time by many people interested inelementary school mathematics. Counting isnot the most basic idea of arithmetic! Ideassuch as one-to-one correspondence and more

252

RELATIONS, NUMBER SENTENCES, AND OTHER TOPICS 253

than are much more fundamental and are, infact, prerequisite to a meaningful developmentof counting.

"One-to-one correspondence" and "morethan" are basically mathematical relations andare not dependent on the concept of numberbut are the foundation for number. Determiningthat there are as many chairs in,a classroom asthere are children, for example, requires onlythat each child sit in a chair, that all the childrenhave a chair to sit in, and that there are nounused chairs, that is, making a one-to-onecorrespondence between chairs and children.Similarly, if after this pairing there are somechildren who have no place to sit, it is knownthat there are more children than chairs. In nei-ther case must the chairs or the children becounted.

If the objects of two sets match one to one,there is more than one way to demonstrate it(unless the sets have only one object each).Two such pairings for sets of geometric figuresand fruit are shown.

It is necessary that children come to realize thatif two sets can be put in ore-to-one correspon-dence, then the sets will always be in one-to-one correspondence regardless of the way theobjects are matched.

The failure of children to recognize that twosets are in one-to-one correspondence regard-less of how they are rearranged may result in

inadequate number concepts and may alsoplay a vital part in Piaget's conservation tasks.Consider the classic situation where squaresand circles are arranged so that a one-to-onecorrespondence between them is visually obvi-ous and then the squares are brought closer to-gether. If the child says initially that there are asmany squares as circles but changes his mindafter the squares are brought together, then hecannot conserve one-to-one correspondence.

There are three important properties of theone-to-one correspondence relation, the re-flexive property, the symmetric property, andthe transitive property. Although the names ofthe properties are not needed by children in theelementary school, questions about relationsthat illustrate the properties are needed.

Reflexive property. It is quite obvious (atleast to adults) that any set can be put into one-to-one correspondence with itself. The simplestpairing is to match each member of the set withitself. This property of one-to-one correspon-dence is referred to as the reflexive property.Its importance is best understood and appreci-ated when relations that are not reflexive are

254 CHAPTER ELEVEN

considered. This property will be discussedfurther in such a context when the more thanrelation is examined.

Symmetric property. The symmetric prop-erty of one-to-one correspondence states thatif set A can be put into one-to-one correspon-dence with set B. then set B can be put intoone-to-one correspondence with set A. Put insimpler terms, if set A has as many membersas set B. then set B has as many members asset A. For example, if there are as many hats asmen, then there are as many men as hats.

Transitive property. The transitive propertyof the "as many as" relation states that if set Ahas as many members as set B and set B hasas many members as set C, then set A has asmany members as set C. For example, if there

are as many balls as chickens and as manychickens as blocks, then there arc: as manyballs as blocks. This result is immediate without

resorting to physical matchings of any kind,once it is known that the transitive propertyholds. Of course, many physical matchings inan activity format are necessary before chil-dren grasp that one-to-one correspondence istransitive.

In summary, the as many as" relation satis-fies three important properties, namely, the re-flexive property, the symmetric property, andthe transitive property.

congruence relation

The idea of congruence is introduced in theschools after basic notions about size andshape have been established. This leads to the"same size and shape" relation, called the con-gruence relation. The congruence relation isused to compare and classify plane figures,such as triangles, squares, circles, rectangles,and pentagons. Through an examination of theproperties of congruence, similarities betweenit and the one-to-one correspondence will be-come apparent.

Just as with one-to-one correspondence,there is a need to find an "action method" ofdetermining if two figures are congruent. Con-gruence can be determined by attempting tosuperimpose one figure on the other. If figure Acoincides with figure B. then figure A has thesame size and shape as figure B; that is, A iscongruent to B.

Notice that the congruence relation is reflex-ive, since any figure has the same size and

RELATIONS. NUMBER SENTENCES. AND OTHER TOPICS 255

shape as itself. Further. note that if A has thesame size and shape as B, then B has the samesize and shape as A. Hence, the congruencerelation is symmetric. The symmetric property

.can easily be developed-in the classroom withactivities similar to this:

Use cutouts and make .physical com-parisons. Ask questions such as these: "Doesmask A have the same size and shape as maskB? (Yes) "Jim. can you show that mask A hasthe same size and shape as mask B?" (PlacesA on B) "Do they fit exactly?" (Yes) "Does mask8 have the same size and shape as mask A?"(Yes) "How do your know? Would someone liketo put B on A and show that they fit exactly?"

So far it has been established that the reflex-ive and symmetric properties hold for bothone-to-one correspondence and congruence.The as many as relation is also transitive.Does this property hold for congruence too?The answer, of course. is yes, and this can bedemonstrated in the obvious way by using cut-outs on a flannel or magnetic board.

equivalence relations

Many other relations, such as has the sameteacher as" and has the same number of sidesas." satisfy each of the three important proper-ties: reflexive, symmetric, and transitive. Askthe children if they can think of other relationsthat satisfy all three properties. What about 'Isthe same color as" or -lives on the same streetas"?

Relations such as the ones just mentioned.which are reflexive. symmetric. and transitive.are given the special name equivalence rela-tions Are all relations equivalence relations?Although many relations are equivalence rela-tions. we shall later see that many importantones. such as "longer than" and more than."are not equivalence relations.

One final characteristic of all equivalence re-lations cannot be overlooked. Any equivalence

relation provides a means of forming distinct,nonoverlapping classes of objects. For ex-ample, if an envelope full of polygon-shapedcutouts is emptied onto the floor, we can formpiles using the equivalence relation has thesame number of sides as." All the cutouts inany pile will then be related (i.e., have the samenumber of sides). Further, any two cutouts fromdifferent piles will not hove the same number ofsides. This formation- of distinct classes is animportant characteristic of equivalence rela-tions. Other examples that can be used to getacross this compartmentalization idea in-tuitively are included in the illustrative devel-opmental activities that follow.

1. Make two cutouts each of various geo-metric shapes. Attach one of each to a bulletinboard within reach of the children. Fasten apiece of string or yarn to each of the remainingcutouts. Have the children attach the other endof the string to the cutout on the board that hasthe same size and shape.

Variation. Make a number of different cut-outs of each shape. Fasten one of each to a bul-letin board and place strings on the remainingcutouts. Proceed as before. When finished, askquestions about the cutouts attached to anygiven one on the bulletin board. "Do they allhave the same size and shape?"

Activities of this type develop the con-gruence relation in a physical setting and alsobring out intuitively the idea of equivalenceclasses.

2. Place on a flannel board in separaterows, or in scattered arrays. a set of flowers.vases, and watering cans. Ask, "Are there asmany flowers as there are vases? As manyvases as flowers? As many vases as wateringcans?" After each question, have the childrenexplain their answer and verify it by matchingappropriate objects on the flannel board onefor one by using yarn or string.

Variation. Place a number of flowers on adisplay board. Have the children place vaseson the board such that there are as many vasesas flowers. Repeat, but have them place morevases than flowers on the board.

256 CHAPTER ELEVEN

These activities provide practice in actuallyconstructing one-to-one correspondences andIn looking at the properties of the relation.

3. Read stories to the children, such as"Goldilocks and the Three Bears" or TheThree Little Pigs." Use cutouts of the bears.pigs, houses, and.so forth on a display board toillustrate-parts-of the story. Have-the childrenname the sets of things that match the set ofbears (pigs). Have them describe and verifytheir answers by matching cutouts one for one.

Variation. Place several different sets of ob-jects on the display board. Have the childrencreate stories about them. Follow up with ap-propriate questions like those suggested pre-viously.

These activities develop the idea of one-toone correspondence and its properties. In or-der for the properties to be developed, how-ever, questioning must be appropriately struc-tured and at some time summarized.

4. Make about thirty-cards with one, two.three, or four squares drawn on each card.Shuffle the cards and give one card to eachchild. Tell the children to find other childrenwhose cards have as many squares as theircard. Eventually (perhaps with some additionaldirection from the teacher) four distinct groupsof children will be formed. Questions con-cerning cards from the same group and fromdifferent groups should be asked and then veri-fied. "Will David's card have as many squaresas Michael's card? Are you sure? How can wecheck?"

Variation. Place one geometric shape oneach card and use the relation "same size andshape."

7.19) C.11)

These activities develop the particular rela-tion used and its properties. They also empha-size the partitioning into disjoint subsets in-duced by an equivalence relation. Theseactivities will not work well for relations that arenot equivalence relations.

order relations

In activities for the "as many as" relation, sit-uations must be included sometime where thepairing of objects does not result in a one-to-one correspondence. In such situations, thereare opportunities for the introduction of impor-tant order relations. Suggestions for teachingthe "more than" order relation follow.

Consider a group of children investigatingthe "as limy as" relation between sets, usingthe matching process previously mentioned.The teacher enters the scene and asks howthings are going. The children quickly respondthat "this is easy" or "this. is cinchy." Theteacher, rising to the occasion. asks them if setA (consisting of eight cups) has as many mem-bers as set B (consisting of six saucers). Almostimmediately the air is full of responses: "yes,""as many as," "A has more than B," "A has twomore than B," and perhaps others. The stage isnow set for a discussion of the 'more than" re-lation.

After some discussion, one of the childrenpairs the elements of set A with set B. indicatesthat two of the objects from set A do not havepartners, and concludes, It is not right to say Ahas as many as B." More discussion. Finallyagreement is reached that in this situation it isappropriate to say that "set A has more than setB." Can we then also say that B has more thanA, as we could with the "as many as relation?The symmetric and reflexive properties arequickly ruled out. They are not satisfied by thisnew relation. But what about the transitive

4-1%rci3


property? Is it true that if A has more than Band B has more than C, then A has more thanC? This calls for experiments!

David and Michael construct sets A and Bsuch that A has more than B. Judy and Janetfind a set C such that B has more than C. (Judyand Janet will have to make set C after Davidand Michael finish constructing set B.) Then ev-eryone agrees that when A has more than Band B has more than C, it is certainly true that Ahas more than C, and so the transitive propertyworks for the "more than relation.

The more than" relation is an order relationand not an equivalence relation. The transitiveproperty is satisfied. but the reflexive and sym-metric properties are not. Many seriation (or-dering) types of activities can be used to de-velop the "more than" relation and itsproperties. These activities: will also bring outthe fact that the idea of the equivalence class isa unique feature of equivalence relations.

A special form of the "more than" relation isthe cornerstone for the meaningful ordering ofthe cardinal numbers. This relation is the "onemore than" relation. It can be developed in theclassroom in much the same way that the"more than" relation is developed.

Place cutouts of four children and threetoys on a flannel board. "Are there as manychildren as toys?" (No) "Is there a child foreach toy?" (Yes) "Can you show me?" (Yes. likethis) Is there a toy for each child?" (No) "Arethere more children than toys?" (Yes) "Howmany more children than toys?" (One) "We saythat the set of children has one more than theset of toys."

Place cutouts of three men' on a displayboard "Margaret. put umbrellas on the boardso that there are as many umbrellas as men.Now who will place a cutout on the board sothat there is one more man than umbrellas?What can you do so that there is one more um-brella than men?

Repeat with different materials and variedquestions. For example, "How many menwould get wet when there is one more man thanumbrellas?"

The one more than relation is clearly nei-ther reflexive nor symmetric, and this can bebrought out in the same way as in the "morethan" relation. But what about the transitiveproperty? Is the "one more than" relation tran-sitive? That is, if set A has one more memberthan set B and set B one more than set C, thenwill set A have one more member than set C?Showing this with cutouts allows the children toconclude quickly that the "one more than rela-tion is not transitive.

A A A

"longer than" and "as long as"relations

The relations "longer than" and "as long as"are important and have implicitly been a stand-ard part of the elementary mathematics cur-riculum for years. The terms as long as andlonger than can be thought of in terms of num-bers, and this is true in most current programs.However, teaching these relations to youngchildren in this way assumes that they have astable concept of number and measure andcertainly does little to connect the relations tothe children's physical surroundings. Such anintroduction, based on abstraction, is particu-larly disturbing, since these relations can soeasily he defined operationally. For instance.rod A and rod B are laid side by side so thatone end of A coincides with an end of B. If thesecond end of A extends beyond that of B, wesay that rod A is "longer than rod B. What arethe properties of this relation? When rods ofvarious lengths are used and appropriatequestions are asked, children soon find

258 CHAPTER ELEVEN

through actual experimentation that the aslong as" relation is reflexive, symmetric, andtransitive. Further experimentation yields theconclusion that the relation "longer than" is nei-ther reflexive nor symmetric but is transitive.

1. Cut strips of construction paper in dif-ferent lengths so that there is one strip for eachchild and one strip to be used as a standard.Place the standard strip on a display board andhave each child come forward and compare hisstrip to the strip on the board. The child shouldthen describe what he finds, for example. "Mystrip is longer than this strip."

Variation. Use string or cord that will notstretch easily. Repeat the previous activity andfollow by having children compare lengths witheach other two at a time in front of the class.Have thekn, describe what they find. This couldbe followed by ordering the children accordingto the length of their string.

These activities develop the "longer than"and "as long as" relation in a concrete setting.Depending on the follow-up questions used,they can also develop the symmetric and tran-sitive properties of this relation.

2. Place on a flannel board a number ofpieces of yarn varying considerably in length.Have the children order them by length.

Variation. Place several pieces of yarn on thedisplay board in serial order, shortest to long-est. Then conceal a piece of yarn behind yourback and have the children ask questions ("Is itlonger than the green yarn on the board?'') todetermine where the hidden piece of yarnshould be placed to maintain the shortest-to-longest ordering.

If the children,have had little experience withsedation activities, it would be best to use onlythree pieces of yarn initially. Further, in the verybeginning it may be best to use strips of paperrather than yarn to avoid the necessity of hold-ing the yarn taut.

general instructional strategy forrelations

By now it should be clear that relations pro-vide a means of examining objects with respectto a particular rule, attribute, or dimension.Real-world situations abound and should beused. A list of real-world situations that canproperly be interpreted as a relation would in-clude such relations as "is the brother of," "istaller than," "is the same color as," "belongsto," "was, born in the month of," "is colderthan," "is sitting beside," "has the same shapeas," "lives on the same street as," "weighsmore than," and so forth. Clearly, these areonly a few of the everyday situations that can beconceptualized as a mathematical relation.

An overall instructional plan involves the fol-lowing stages: introducing relations initiallythrough familiar real-world situations, ex-panding the idea of a relation by studying im-portant mathematical relations in situationswhere physical actions are employed by thechildren, and then refining the latter relationsby examining them later in a more abstractframework. As the child's study of relationsprogresses through these stages, a meaningfulsymbol system for representing relations iscarefully developed.

The opportunities for learning about rela-tions should not cease at the end of the mathe-matics class period, never to be mentionedagain; rather, as other topics (e.g., linear meas-urement) are studied, the idea of a relationshould again be employed (e.g., "is longerthan," "is shorter than," and so forth). Thestudy of relations is also an ideal time forstressing -the interrelatedness of various dis-ciplines. For example, as part of a science-lab-oratory activity in which the relative hardness ofvarious materials is tested by attempting toscratch one with the other, the idea of a relationentitled "is harder than" or "can be scratchedby" could be brought out by asking appropriatequestions of the students. Another possibility inscience is the relation "will float in" when study-ing the idea of density. Still other possibilitieswould include the relations "has a higher melt-ing point then," "has a lower freezing pointthan," "will dissolve in," "has a longer life span


than," "grows taller than," "migrate earlierthan." and "is sweeter than." In social studies,relations such as "is the capital of," "receivesmore rainfall than," and "is closer to the equa-tor than" could be considered. Interesting situ-ations in other disciplines can be found if timeand attention is given to the task. After manyexperiences in different areas of study, chil-dren will begin to recognize situations that canbe conceptualized as a relation. As they matureintellectually, they will gradually capture theidea of a relation in a more general sense, thatis. as a classifying and unifying concept.

number sentences

There is a strong need to represent andcommunicate ideas in mathematics, and be-cause of the nature of mathematics these ideascan often be stated very precisely. Mathemati-cal sentences provide a means of representingideas precisely and compactly. This precisionis achieved without lengthy verbal explanationsand descriptions. This feature enhances thevalue of mathematical sentences as a means ofrepresenting mathematical conditions and stat-ing properties and principles.

The ability to represent the mathematicalconditions in a physical situation or verbalproblem is well accepted as a desirable tech-nique in any person's problem-solving reper-toire. Symbolic representation should naturallyfollow the need to record physical phenomenaor the results of experimentation and manipu-lation.

As the ability to represent physical situationswith mathematical sentences develops, the roleof a given sentence as a model for many situ-ations should be emphasized. Following areexamples of some fundamental open sen-tences involving the basic operations. Thesesentences are models for different physical sit-uations.

1. Three children are at the front of theroom. Two other children join them.How many are in the new set?

2. Three birds are on a fence. Two birdsfly down and join them. How manybirds are now on the fence?

3. Three apples are in a basket. Put 2more apples in. How many applesare now in the basket?

3 i 2 0 is a model for all the situations inproblems 1, 2, and 3.

Joining may also be thought of in the follow-ing situations, although the number of objectsinone of the sets is not known.

4. David had 3 pencils, and then he gotmore pencils for his birthday. He has5 pencils now. How many did he getfor his birthday?

5. There are 3 marbles on the floor.John drops some more marbles onthe floor. There are 5 marbles now.How many marbles did John drop?

6. Scott has 3 jacks. How many moredoes he need if he must have 5jacks?

The sentence 3 N 5 is a model for the sit-uations in problems 4, 5, and 6. In fact, 3 N5 is a model for all situations where there is aset of 3 members and a set of unknown size isjoined to it to form a set containing 5 members.

There is another situation involving the join-ing of disjoint sets where one set is of unknownsize and a set of known size is pined to it.

7. There are some fish in the aquariumand 2 new fish are placed in theaquarium. There are now 5 fish in theaquarium. How many fish were in theaquarium at first?

8. Kelly has some apples. She needs 2more apples so that she will have 5apples for her party. How many ap-ples did Kelly have to begin with?

1 Li

260 CHAPTER ELEVEN

N f 2 5 is a model for any situation wherewe have a set of unknown size joined by a setcontaining 2 members, whereby the new setformed has 5 members.

There are many physical situations where aseparating or removing action is involved.

6-2=n

/ 9. If there are 6 glasses on the shelf and2 fall off the shelf and break, thenhow many glasses remain on theshelf?

10. John has 6 candies and gives 2 ofthem away. How many candies doeshe have left?

The open sentence 6 2 - N is a model forboth of the situations in problems 9 and 10 aswell as for every other situation where there is aset of 6 objects and a subset of 2 objects is re-moved.

There are many physical situations that in-volve several sets all of the same size (i.e.,equivalent sets) where the question of interestis how many in the new set when these sets arecombined or considered in totality.

11. John has 3 boxes of toys. There are 4toys in each box. How many toys inall the boxes?

12. An egg carton has 3 rows with 4spaces in each row..How many eggsdoes the carton hold?

13. Jim buys 3 packages of baseballcards. When he counts the totalnumber of cards that he has pur-chased, he finds he has 12. Howmany cards.were in each package?

14. Mary has a flower garden with 6 flow-ers in each row. She has 30 flowersin her garden. How many rows offlowers does Mary have?

The models for problems 13 and 14 show sit-uations where the total number of objects isknown but one of the other numbers is un-known.

The relationship just illustrated betweenmathematics and the physical world is an im-portant one. As part of their study of mathemat-ical sentences, children should learn to repre-sent mathematical conditions in physical-worldsituations by mathematical sentences. As thisskill is developed, an appreciation for the inter-relationship of mathematics and the physicalworld is promoted. Making up instructional ac-tivities that focus on the use of a mathematicalsentence as a means of recording activitieswith objects or diagrams is one way to developthis ability and to establish a meaningful rela-tionship between actions on physical referentsand number sentences. The development ofthis ability is part of a larger ability of represent-ing the physical world by mathematical models.

After children become adept at writing anappropriate open sentence as a model of a-physical situation or problem, they should haveexperience in "going the other way." For ex-ample, when given the open sentence 4 x 12N as a model, children should be able to trans-late the sentence into a real-life situation. Thistranslation may take many forms. The childmay be encouraged to express the idea withconcrete objects by forming 4 sets of 12 ob-jects each and combining them. Another possi-bility is to write a verbal problem to "go with"this open sentence:

)John has 4 bags of marbles, and each bagcontains 12 marbles; how many does he 1 x12 =11have in all?

A creative teacher can devise other ways totranslate from ,a number sentence to the physi-cal situation.

Some further activities to help with thesetranslations follow.

1. Put a number of verbal problems on in-dex cards, one problem to a card. Have thechildren sort the cards into piles on the basis ofsentence types. For instance, the following twoproblems would be placed in the same pile,


since 6 0 = 10 is a model for both of them.John has 6 marbles. Bill gives him some marbles.John now has 10 marbles. How many marbles didBill give to John?

Marys bracelet has 6 charms on it. She receivessome new charms for her birthday. She now has10 charms. How many charms were given to herfor her birthday?

The following problem would not be placed inthe previous -pile, since it represents a separat-ing type, of action; which is modeled by a sub-traction sentence.

There are 10 boys playing football. Six boys leaveand go home. How many boys are now playingfootball?

2. Put basic addition and subtraction factson 3" ic 5" index cards, one fact to:a card. Placethe cards in a box. Make available a quantity ofmanipulative- materials. Have the childrencome forward, draw a card, read the numbersentence on their card to the class, and thendemonstrate a situation that represents thegiven sentence, using the available materials.For example, if the number sentence was 5 4

9, the child could form a set of five blue disksand a set of four red disks and 'then combinethe two sets into a single set. Perhaps at somepoint and for some children, a verbal ex-planation such as "I put a set of five with a set offour and there are nine in the new set" shouldbe encouraged.

3. Have one child write a number sentenceand another child translate the sentence into aphysical situation, using the concrete materialspresent. For instance, if one child wrote thesentence 10 8 - 2, then the other child couldtranslate this into a physical-world situation byforming a set of ten and then removing a set ofeight. Of course, forming a set of ten disks anda set of eight buttons and then matching eachbutton with one disk, 'leaving two disks un-matched, would be a perfectly acceptable al-ternative to the former situation. The possibilityof having more than one interpretation pro-vides opportunities that should be capitalizedon whenever possible. At the same time, it em-phasizes the need for teacher supervision ofactivities in the classroom. Later, have the twochildren switch roles: the one who was writingbegins to translate, and vice versa.

solving sentences

Number sentences without variables can beclassified as true or false, as shown in these ex-amples:

2 x 3= 6 (true) 5 1 = 5 (false)1 + 3 < 6 (true) 9 + 8 < 10 (false)

Solving open sentences such as 3 x N = 15,2 + 0 < 6, or N X 1 = N means finding thenumbers that make true sentences.

As part of a child's introduction to mathemat-ical sentences, practice should be provided indetermining the truth or falsity of mathematicalsentences. These experiences can and shouldbe constructed in a way that maximizes the useof concrete materials for decision making andfor verification. Further, this experience shouldprovide a base for a sequence of carefully se-lected activities designed to introduce the no-tions of open sentence and solution set.

Some of the activities outlined here are builton the "longer than" relation. The means of in-troducing this relation and presenting its prop-erties have been discussed previously. It is as-sumed that these activities will not be thechildren's first acquaintance with this relation:::or with the materials described.

1. Divide the children into small groupsand provide each group with a set of rods ofvarious sizes and colors. The lengths of therods should differ only slightly, thus promotingphysical comparisons; rods of'the same lengthshould be of the same color. Give each group alist of comparisons between rods, stated in theform of mathematical statements. The notationused in forming the sentences on the list shouldagree with the notation previously developed inthe classroom. For example, sentences such as"Red longer we could be used if such nota-tion was familiar to the children. The task setfor the children is to determine the truth or fal-sity of each of the sentences on the list by mak-ing comparisons between the two rods in-volved.

Variation: This activity can be expanded inmany ways. For instance, ask the children to

4-Ap tt..4

262 CHAPTER ELEVEN

expand the list of sentences by recording aspecified number of true sentences and asimilar number of false sentences about therods and the "longer than" relation. Alterna-tively, one child could write additional sen-tences and the other children could determinetheir truth values Of course, the entire activitycould be repeated with a different relation ordifferent materials.

Possible objectives of the preceding lessonsare manifold, and a long list of such objectivescould be generated. One of the more importantobjectives, however, is the attainment of theability to determine the truth or falsity of a givenmathematical statement. This is an obviousprerequisite skill to the development of theconcepts and skills associated with solutionsets of open sentences.

2. The essence of this activity is to deter-mine whether a-given mathematical sentence istrue, false, or open. It should again allow thechildren to use concrete materials in drawingtheir conclusions and should, perhaps, be re-stricted initially to the use of. one relation. Letthe relation be "as many as." Put materials likeplastic counters into bags and identify eachbag with a letter Print sentences such as

as any sA m asp on index cards. Have each

child in the group working on this activity drawa card from the deck and try to determine if thesentence on his card is true, false, or open (i.e..cannot tell if it is true or false). His conclusionwill be based initially on pairing the objectsfrom the two sets identified in the given sen-tence or on realizing that only one of the twonecessary sets has been specified. This type ofactivity can be extended to other relations andother materials. If the children have studied thebasic addition facts, sentences such as 2 3 =5, 4 r 3= 6, and 4 4 1= 0 could be used.The concrete materials available for use mightbe a set of counters, Cuisenaire rods, or othersimilar materials.

Activities of this type provide a backgroundthat is helpful for students in the next activityand in their gradual development of the con-cept of a solution of a sentence.

3. Provide a set of index cards with anopen sentence written on each. The open sen-

tences should involve only one variable andshould involve only relations and operationswith which the children have had considerableexperience. Give each child one card and askeach in .turn to read the sentence on his cardaloud. Assign each child an element (or havehim moose one) from the domain of the van-able in his sentence (e,g., a number). Have thechildren one at a time read their sentences. re-placing the placeholder with the given number,and tell whether the resulting statement is trueor false.

This activity, like the preceding ones. servesonly as an illustration of an appropriate typeof activity to fulfill the purpose stated. Otheractivities written in the same spirit are certainlyneeded to help children grasp the idea of agiven element satisfying an open sentence.

4. This card game is to be played by agroup of three to five children. On index cards.place open sentences with which the childrenhave had experience. Use sentences that haveonly one number in their solution setfor ex-ample, 7 4 0 = 15. Distribute the cards to thechildren so that each child receives the samenumber of cards. To begin, instruct each childto turn up a card. The child whose card has thelargest number for a solution wins that trick andplaces these cards under his stack. Play con-tinues until one child has acquired all the cards.Rules will have to be developed as needed; forexample, a rule concerning ties will have to bemade.

Children's first experience with specifyingsolution sets should involve open sentencescontaining only one variable. Further, it is de-sirable that the domain of the variable containonly a small number of elements and that theseelements be explicitly stated. The task can thenbe stated as, "Which of these elements makethe sentence a true statement? Write themdown." Most children will complete this activityby checking each of the specified replace-ments. Children need lots of experience in thistype of activity, and the teacher should be sure

265


that the children become acquainted with themany different situations that exist.

The solution of sentences such as 7 r 5 0 ,

15 6 0 7 , 9 0 , and 60 : 10 0 can befound directly by application of the operationshown in the number sentence. Sentencessuch as 8 N 3. 4 N 11, and 13 6 t N.however, are not solved directly and are moredifficult for children. A perusal of contemporarymathematics textbooks at the primary levelclearly indicates that much attention is given todeveloping the ability to solve the direct typeof open sentence. This is in sharp contrast tothe haphazard approach often taken to devel-oping children's skill in solving open sentencesthat are not in direct solution form. Developingthe ability to solve open sentences of the lattertype may be the more difficult task. Some ex-plicit attention can and should be given to solv-ing such open sentences in the primary grades.

Children can often solve open sentencessuch as 8 N 3 and 4 - N 11 without for-mal instruction on methods of solving. A greatvariety of methods is used. Most of these meth-ods are informal and intuitive. For instance, achild might solve a sentence such as 8 N 3by using a tallying procedure or by countingbackwards; he might solve 4 N 11 by"counting on." The use of such methods shouldnot be discouraged In fact, a child using thesemethods is often displaying ingenuity and ini-tiative. At some point, however. when much in-formal experimenting has been allowed, it maybe desirable to suggest a more systematic wayof solving these open sentences. The proce-dures suggested below are more systematicbut are certainly not an abstract, formal ap-proach to solving equations.

When addition and subtraction are mean-ingfully taught and are interrelated, childrenbecome ery adept at identifying the knownaddend, sum, missing addend, missing sum.and so iarth. in various open sentences Theyreadily recognize that sentences such as N 4

9, 5 N 9. 9 N 5, and 9- 4 N involvea known addend, a known sum, and a missingaddend. Similarly. they become aware thatopen sentences such as 5 4 N and N 4

5 involve two known addends and that the sumis to be found. Carefully and regularly relatingthese sentences to appropriate physical repre-

sentations can usually lead to the meaningfuldevelopment of the two generalizations neededto solve all open sentences of these types. Inthe missing addend situation, it is necessary tosubtract the known addend from the sum tofind the missing addend, to find the sum whenboth addends are given, the given addends areadded. Obviously, these generalizations couldbe taught by rote, but this is not the authors' in-tent. Rather, these principles should be mean-ingfully taughtand developed.

A substantial number of multiplication anddivision open sentences not in direct form canalso be systematically solved if several prin-ciples are known. The basis for these two prin-ciples is again the relationship between the twooperations. If these operations are mean-ingfully related as suggested and detailed in aprevious chapter, then children can grasp theidea that in open sentences such as 18 - N 3.18 -: 6 N, 3 > N = 18, and N > 6 18, aproduct and a factor are known and the re-maining factor is to be found. The principleneeded to solve any sentence of this type, re-gardless of the size or type of numbers in-volved. is Given the product and one factor, tofind the missing factor, divide tne given productby the known factor. Sentences such as N . 8

24 and 3 8 N are recognized by childrenas situations where the product is to be foundand two factors are known. The principle ap-propriate for these conditions is, If given twofactors and the product is to be found, thensimply multiply the one known factor by theother.

In summary, children's ability to solve opensentences should be gradually developed. Itshould proceed from the informal to the moresystematic, but not necessarily to the abstractin the primary grades. Many important opensentences involving the four basic. operationscan be systematically solved if several gener-alizations are meaningfully developed. The so-lution to other vaieties of open sentencesshould be handled on an intuitive basis withplenty of opportunity allowed for solving themin a variety of informal ways. The systematic al-gebraic method of solving most open sen-tences should be deferred until the childrencan meaningfully operate on a more formallevel.

rrn -14

F e,"?b

264 CHAPTER ELEVEN

other mathematical topics

Other mathematical topics belong in a com-prehensive program. There are productive ac-tivities relating to logic, probability, statistics,and number theory that are quite appropriatefor the primary grades. Many of the activitiesand problem situations in chapters 4 and 5 re-late to some of these mathematical topics. Aswas shown in those chapters, approaches tothese topics must be informal for young chil-dren.

logic

The most basic part of logic is recognizingwhether a sentence that is a statement is true orfalse. (Sentences can be open and neither truenor false, e.g., n 4 3 = 7. Sentences that can beclassified as true or false are called state-ments.) Such statements often arise inclassifying objects, in studying relations, and inworking with number sentences.

The following activities focus on true andfalse statements.

1. Select a block from a set of attributeblocks, keeping it hidden from the children.Ask some child to make a statement about thehidden block. Prior to showing the block,record each child's averpt on the chalkboardand analyze it to if 4 can be determined tobe either true or false when the block is shown.That is, was a statement made or not? The sen-tences that are not statements should beerased or marked out. After this is completed,show the block to the children 'and help themjudge each of the statements true or false.

2. Print simple statements on strips of tag-board, one statement on a strip. For example:Place two or three strips at a time on a displayboard. Have children say "true" or "false" assomeone points to each strip.

Tolyri is Dater 11-)o-n h1s father.

Sue has black hoar.

43 . 5.

David i5 511 years

30.ck has a ssier.

12..- '3.10 .

A blue, block the \Eat-

FA squo.v. block %s on 16_ table.

Variations. Point to a false statement. Ask,"How can you change this to make it true?" En-courage replies such as The block is not blue,""The block is red," "The block is not square,"and "The block is round."

Use simple statements where some of thestatements involve the word not. Occasionally,to emphasize the function of this word, state-ments such as "Mike is absent today" and"Mike is not absent today" should both be puton the display board at the same time.

Make simple statements using the words all,there is, and none. (All means every single one;there is means at least one; none means not asingle one.)

3. Join two statements from the precedingactivity with and. See if the resulting statementis true. (A statement with and is true only if bothparts are true.)


A red block Is on the

True

iA red block is on the andFalse

Variation. Make a large clockface on thefloor and place one statement in each hour'sposition. (The statements from the precedingactivity could be used for this purpose.) Havethe class_ decide orally the truth or falsity ofeach statement. On completion of this task,place a tag board strip labeled "and" at the cen-ter of the clock. Move the hands of the clock sothat each hand points to a statement. Havesomeone read the compound statement in-volving the connective'word and. Then ask onechild if the statement is true or false. If he an-swers correctly, he gets to reset the hands andchoose from the volunteers another child, whomust then judge the truth value of the new com-pound statement. Continue the game in thismanner.

te4

F-,

Li" nc .1 soh.

Fro, tar cJrt beer's]

A variation with the clock is to label the cen-ter with the word not and use only one hand.

11.. 4. Begin with a false compound sentencecontaining and. Make one part of the com-pound sentence true. Ask, "Is the other parttrue or false?" (False)

The block is blue. andFalse

The block is round.

A square block i5 or the 1a131e

False

A scluuxe l:Aock is on table.

Variations could include making the originalcompound sentence true and making one parttrue. Another variation would begin with a falsecompound statement with one part false, in thiscase you cannot tell for sure if the other part istrue or false.

Other activities with logic involve the use ofor and if-then. Such ideas are more difficult.and although they may be begun informally inthe early grades, their full understanding is de-veloped later.

At the primary level, the main objective withrespect to the logical ideas must be intuitive ex-perience. This experience should be based asfar as possible on real-world situations withwhich the children can identify.

probability

Although there is substantial agreement thatprobability concepts should be included in theschool curriculum, there is also some dis-agreement on when these ideas should betaught. These differences concern, in part, theage at which a child is sufficiently developed in-tellectually to comprehend concepts asso-ciated with chance and randomness. Con-comitant with the consideration of this issue,attention must be given to the fact that thereare large differences between the children in agiven ci _ ss as well as between classes. Teacherjudgmect, and flexibility, therefore, are requiredif any portion of the sequence of probability ac-tivities suggested here is to be incorporatedintc. classroom instruction.

An objective of an early lesson on probabilityshould concern the ideas of certainty and un-certainty. Many everyday events are predict-able, whereas others are uncertain. That thesun will rise tomorrow, that there will be no

rf,r.The block is round.] ti

True

266 CHAPTER ELEVEN

school on Saturday, or that there will be cars onthe streets tomorrow are events about whichwe can be quite certain. That it will be cloudytomorrow or that there will be no absencesfrom class tomorrow, however, are not as pre-dictable; they are more uncertain. Children canbe helped to gain an intuitive feel for theseideas by discussing such statements and ques-tions.

1. On a display board, label one column"Certain" and another column 'Uncertain."Present cards one at a time to the class withsuch statements on them as these: "Tomorrowwe will have two recesses." "It will be a clearday tomorrow." "Fifty airplanes will fly over theschool today." "If a carton of eggs is dropped,all the eggs will break." "Two children will beabsent tomorrow." As each card is presented, itshould be discussed and placed in the appro-priate column on the display board. Use ofwords such as for sure, always, probably not,not very likely, impossible, and never, to namea few, will certainly be useful in discussing thecertainty or uncertainty of the event listed oneach card. Note that under the "Certain" col-umn, events that are both very likely to happenand very likely not to happen wille included.

Variation. Divide the display board into threecolumns representing "certain to happen,""uncertain," and "certain not to happen" cate-gories. Repeat the previous activity, placingeach card in the appropriate column after classdiscussion.

These activities develop the ideas of cer-tainty and uncertainty and a working vocabu-lary for discussing situations that involve un-certainty in various degrees.

2. Use guessing games involving situationslike the following:

I am thinking of a whole number between 4 and8.What number am I thinking of?I have a square block, a round block, and a tri-angular block in this bag. I am going to take oneblock out of the bag. What shape will it be?

In each of these situations possible outcomesmay be listed. Questions concerning the cer-tainty of a guess may. be discussed. After one

incorrect guess has been made, the previousquestions can fruitfully be discussed again.

Variation. Use three blocks of the sameshape, but vary the color of the blocks. Tomake a much more complex activity, put twosquare blocks of different colors and one blockof each of the other two shapes into a sack andrepeat the activity.

These activities introduce the idea of out-comes and provide additional practice with theideas of certainty and uncertainty.

3. Use something such as a coin tosswhere there are just two outcomes. Ask ques-tions such,as these:

When the coin is tossed, will the side that landsup always be either a head or a tail? (Yes)Will a head come up on the next toss? (Can't tellfor sure)

Can you be certain which side will come up thenext time the coin is tossed? (No)

4. Make a spinner having more than twooutcomes. "Suppose I spin this spinner; to whatnumber could the spinner point? Is there an-other number to which the spinner could point?Any other? Could someone write down each ofthe numbers to which the spinner could point?How many different numbers would be on thelist? Will the spinner land on a 3 next time? ..."

Variations. To introduce the likelihood ofvarious events, which is a cornerstone of allprobability theory, make two spinners, one ofthem half red and half blue and the other three-fourths red and one-fourth blue. Explain that inhis turn at some game a child moves forwardone space if his spinner stops on the blue andmoves zero spaces forward if his spinner stopson the red. "Which spinner would you choose?"Children very quickly realize that if both spin-ners are used, the game is not fair to whoeverhas the second spinner and that certain out-comes are more likely to occur on certain spin-ners.


Have two children use a spinner divided intofive congruent sections, numbered onethrough five. Let one child receive a counterwhen he spins and lands on an odd numberand the other child receive a counter when hespins and lands on an even number. Discusspossible outcomes for each child and favorableoutcomes for each child. Discuss the fairnessof the game in terms of who is most likely to winand why. Vary the activity with spinners that arenot divided into congruent.. sections, such asthose illustrated.

These kinds of games foster the devel-opment of most of the probabilistic notionspreviously mentioned if they are accompaniedby probing questions and meaningful dis-cussions.

4 Put one red block and one blue blockinto a cloth bag. Have teams take turns drawinga block from the bag. A point is scored eachtime a red block is drawn. Allow twenty drawsby each team for a game. Follow up this gamewith a related, but more complex, game. Thistime use two bags, each contdining one redand one blue block. On a given turn a blockmust be drawn from each bag. A team scores apoint on that turn only if both blocks are red.Comparing the size of the scores between thetwo games leads to interesting discussionsabout the likelihood of two outcomes both hap-pening.

Variation. Play the two-bag game, but let apaint be scored whenever a red block is drawnfrom either bag. (A point is scored when a redblock is drawn from bag 1 or from bag 2 orfrom both bags in a given turn.)

These activities bring out the fact that it isless likely that two events will both happen thanthat one of the two will happen.

The extent to which very many young chil-dren can go beyond the activities suggestedhere is questionable. Therefore, no further con-cepts or activities are discussed. For additions!suggestions or ways to extend the ideas listedhere, see the References at the end of thechapter.

statistics

Contrary to the reservations expressed con-cerning the teaching of probability, there is agreat deal of evidence and support for the fea-sibility of introducing basic notions of statisticsearly in a child's school experience. These no-tions center on descriptive statistics and in-volve the processes of counting, collecting,measuring, classifying, comparing, recording,displaying, ordering, interpreting, and repre-senting. Each of these processes is important,and they are all highly interrelated. These factsshould be reflected in instructionthat is. eachof these processes needs to be given explicitattention, and activities should be formulated insuch a way that many of the processes areused in the activity. Clearly there are many ap-propriate projects and activities in which chil-dren can take an active part and in so doing de-velop the skills and concepts associated withthe preceding list of processes. Rather than listsuggestions concerning each process. someaspects of pictorial and graphical representa-tion that incorporate many of these processeswill be explored.

Graphical and pictorial representation areimportant means of communication and pro-vide interesting ways of developing an abun-dance of mathematical concepts and skills. Inorder for a development of this topic to be mostproductive (in the sense of maximizing learningand interest). care should be taken to insurethat the lessons proceed from the simple to themore complex and from the concrete to theless concrete. Giving children the opportunityto make choices wherever possible serves as agood motivational influence. Some guidelinesfor producing activities that develop represen-tational skills follow. These guidelines are notto be interpreted as rigid rules.

13,-' 1g

268 CHAPTER ELEVEN

1. Focus initial work with pictorial represen-tation on discrete variables (variables that in-volve only whole numbers). For example, havethe children work with such interesting exam-ples as the number of siblings, the number ofchildren having lunch at school each day, thenumber of cars passing b; the school duringrecess, the number of children absent eachday, the number of birds seen on the, way toschool, the number of days in each month, andthe number of pets each child owns. Later, con-tinuous data should be examined. This couldinclude such things as measuring (1) the heightof a bean plant daily, (2) each child's height andweight, (3) inches of rainfall a week, (4) the pe-rimeter of leaves, or (5) the area of classrooms.

2. Keep the number of categories small atfirst; in fact, the first representation of datashould involve only two categories: present orabsent, boy or girl, and similar dichotomies.Later, situations where a larger number of cate-gories is required would be examined. Exam-ples could include each child's favorite day ofthe week or the month of each child's birthday.

3. Avoid assigning more than one value to acategory in early work with discrete variables.For example, recording the results of activitieslike rolling a die ten times should be repre-sented initially by six categories, one for eachpossible outcome. Later, when it is clear thatchildren have a good grasp of this one-value-to-a-category situation, record the results ofthrowing a die again, but this time record oddoutcomes (1, 3, and 5) in one category (or col-umn) and even outcomes (2, 4, and 6) in an-other category. Here a three-values-to-a-cate-gory situation is represented. Notice thatassigning values to a category is generally re-quired when representing data involving con-tinuous variables.

4. Focus introductory work with representa-tion on V 'ne -to -one correspondence be-tween the .igs being represented and themeans of representation in the pictorial dia-gram. For instance, if the composition of aclassroom is being represented according tosex, then each child must be represented in thediagram. Give each child a concrete object(e.g., block, matchbox, etc.) and use these ob-jects in making the representation. This em-

phasizes the one-to-one correspondence be-tween the objects in the representation and thechildren in the classroom. The specific repre-sentation might involve two rows of blocks ortwo sets of washers placed on two pegs. Labelone row (or peg) '.boys" and the other "girls."

5. Proceed in the representation from theconcrete to the more abstract. Some definitestages in this progression are shown.

(a)

3

(6)

Girls i3okiS biris Days

(c) (d)

In (a) the composition of the classroom IS com-municated by a systematic lining up of the chil-dren in the class with boys in one row and girlsin the other. In (b) each child places a match-box with his name on it in the appropriate cate-gory. In (c) each child shades in a square in theappropriate column to represent himself on thegrid type of diagram. In (d) the number of boysand the number of girls are first counted, andthen a bar is used to represent the number ineach category. Note that in this type of repre-sentation the scaling on the vertical axis plays avery important role.


6 Keep in mind the following important ideasassociated with making grans (pictorial repre-sentations):

a) Labeling and interpreting graphs is im-portant.

b) Using a common baseline and equalunits is necessary if graphs are to coin-municate information

c) The order in which the categories ap-pear on the baseline of the graph issometimes important (i.e., when thebaseline is scaled).

The number of opportunities for doing ex-citing, interesting and meaningful things withpictorial representation is practically unlimited.It is hoped that this topic will become a verypopular one in the elementary school mathe-matics program.

number theory

Few areas of mathematics offer more oppor-tunities for children to make interesting discov-eries than number theory. Fortunately, the ba-sic tools needed to make these discoveries areneither elaborate nor highly abstract, and inter-esting activities involving concrete materialscan ea-Hy he venerated for teaching the basicconcepts.

1. Have the children construct as many ar-rays as they can for a given number of objects.For instance, the following arrays could beformed for a set containing twelve objects:

1:<3

.

3v-1 dr

I X12- (2.,x

Variations. Ask questions like these; "Find anumber for which you can build exactly four ar-rays." (6 is one; 8 is another) "For what num-ber(s) can you build only two arrays?" (Anyprime number: 2, 3, 5, 7, 11, ...) "Are therecertain kinds of arrays that can be constructedfor every number?" (One-row or one-columnarray) "Wi'lich numbers have more than twoways to make an array?" (Composite numbers:4, 6, 8, 9, 10, . ..) "Can an array having six rowsbe formed from a set of twenty=four objects?"(Yes) "Which numbers can be arranged to forma two-row array?" (Even numbers: 2, 4, 6,8, .. .) "Which can't be put in a two-row array?"(Odd numbers: 1, 3, 5, 7, ...)

List the odd and even numbers. Choose twoeven numbers and, ask, "Is the "sum even orodd?" (Even) Repeat for "odd plus odd" (even)and "odd plus even" (odd). Verify by using ar-rays:

even ntimber.

odd number

even +.odd = odd

2. Use arrays to show square numbers."For what numbers can a square array beformed, that is, an array having the same num-ber of rows and columns?" (1, 4, 9, 16, 25, 36,49, ...)

4,

Ito

Variations. Ask, "How many do you add toeach square to get the next one?" This showsthe relation between square numbers and thesum of odd numbers.

272

270 CHAPTER ELEVEN

143 '436 143i 5t7

3. Make dots to show triangular numbers.Variations. Relate triangular numbers to

sums of whole numbers. Illustrate some ofthese numbers on the chalkboard and ask."What will the next one be? Can you relate tri-angular numbers to sums of consecutive wholenum bers?

10

14 1 4 2H-6 IA

What must be added to 1 to get the next tri-angular number? To 3 to get the next triangularnumber? To 6 to get the next triangular num-ber?' Thinking of the first four triangular num-bers in the following way makes seeing therelationship easy: 1 1, 3 1 + 2. 6 1 2 t3. 10 = 1 + 2 t 3 4. The nth triangular num-ber is clearly the sum of the whole numbers 1,2, 3, .... n.

Another variation miht be to relate tri-angular numbers to square numbers. "How aretriangular' numbers related to square num-bers? Are there some triangular numbers thatcan be added together to form a square num-ber?" An examination of arrays divided like this

\V

should verify their guess that any square num-ber is the sum of two consecutive triangularnumbers. Children can often be guided tomake this discovery and then intuitively verify itby demonstrations with counters similar tothose just illustrated.

integers

The idea of an integer is considerably morecomplex than that of a cardinal number. Theset of integers consists of all the whole num-bers and the negatives of the whole numbers.The integers can be denoted using set notationin this way: 3, -2, 1, 0, 1, 2, 3.. Longbefore a child formally studies this set of num-bers, he has been exposed to many situationsthat can be considered representations or usesof integers. Focusing attention on such situ-ations in the primary grades builds a founda-tion on which a more formal and detailed studycan later be based.

Many physical situations familiar to childrencan be described using the integers:

5 in the hole -5

24 degrees below zero 24

gain of 6 yards 6

a hundred dollars in the red 100

Each of the following activities, designed forthe primary grades. is structured in a way thatallows for "hands on" experience by the chil-dren and should be used in the classroom inthis way.

1. When the child understands and canuse the number line as shown, ask, "What kind

o I z 3 '1 S to 7 S c1

of numbers go to the left of zero?" (Negative.or "in the hole" numbers) "How could you showa score of five in the hole?" (5 units to the left)"A temperature of three below zero?" (3 to theleft) Build the expanded number line.

<-2. -I 0 l 2. 3

RELATIONS. NUMBER SENTENCES. AND OTHER TOPICS 271

Emphasize the unit distance between points.2. When playing games where both nega-

tive and positive scores are possible, such asthe beanbag game, use the expanded numberline in keeping score. If the number line is onthe chalkboard, represent each child's score bya different mark.

3. Given a partially labeled number linesuch as the one shown, mark a specific point

on it. and ask for volunteers to write the correct-number" by the point.

4. Begin with an unlabeled number line likethis:

Have a student come forward and la bel'one ofthe marked points zero. Then have him call outan integer and select a volunteer to label thepoint associated with this integer. Continue thelabeling of the number line in the same man-ner

5. Stage a weather show where a list ofcities and their low temperatures are reported.Have some child display each city and its tern-

references

Dienes. Zoltan P and E W Golding. Learning Logic. Logi-cal Games New York Herder & Herder, 1971

Gibb. E Glenadine. and Alberta M. Castaneda TeachingArithmetic Concepts to PrePrimary Children Glenview.IN Scott. Foresman & Co.. 1969.

Grouws, Douglas A. "Differential Performance of ThirdGrade Children in Solving Open Sentences of FourTypes Doctoral dissertation. University of Wisconsin.1971

"Open Sentences. Some Instructional Consid-erations from Research Arithmetic Teacher 19 (No-vember 19721:595-99.

'Solution Methods Used in Solving Addition andSubtraction Open Sentences Arithmetic Teacher 21(March 1974)255 -61

Heddens. James W Today s Mathematics 2d ed Palo Alto.Calif Science Research Associates. 1971

Lankford. Francis G.. Jr, "What Can a Teacher Learn abouta Pupils Thinking through Oral Interviews/' ArithmeticTeacher 21 (January 1974):26 32.

Nuffield Mathematics Proiect Pictorial RepresentationNew York Johti Wiley & Sons, 1567.

perature as they are read. Ask questions aboutwhich city had the coldest temperature. Thenask comparison questions. Was Madisoncolder than Minneapolis" How much colder?

6. Ask questions about a rabbit who hopsforward and backward on a number line and al-ways starts hopping from zero.Emphasize thaton the one hand if the rabbit jumps three hopsforward, he will land on 3, and that on the otherhand if he jumps two hops backward, he willland on 2. Then begin asking questi,ms aboutwhere the rabbit will land if he jumps so manyhops forward and then so many backward.Have the children move a rabbit on a numberline to help them.

7. Use games where a postman deliversbills and checks. Represent, for example, thedelivery of three bills by the integer -3. A deliv-ery of four checks would then be associatedwith 4. Go a step further to represent the de-livery of two bills and one check by t

Teachers should feel free to experiment withany of these activities for the integers as long asintuitive experience, not mastery or abstrac-tion. is the goal. The emphasis on integersthe grades should be kept under control, how-ever, since the operations with these numberscan cause real difficulty, as any teacher of first-year algebra can testify.

O'Brien. Thomas C. and June V Richard Interviews toAssess Number Knowledge. Arithmetic Teacher 18(May 1971):322 26.

School Mathematics Study Group Probability for PrimaryGrades. Rev. ed, Pasadena, Calif A. C Vroman, 1966

Van Engen. Henry 'Epistemology. Research, and Instruc-tion In Piagetian Cognitive-Development Research andMathematical Educalloo, edited by Myron F Rosskopf.Leslie P. Steffe. and Stanley Taback pp 34 52 Wash-ington. D.C., National Council of Teachers of Mathemat-ics. 1971,

Weaver. J. Fred The Ability of First-. Second-. and Third-Grade Pupils to Identify Open Addition and SubtractionSentences for Which No Solution Exists within the Set ofWhole Numbers School Science and Mathematics 72( November 19721:679 91

. Some Factors Associated with Pupils Perform-ance Levels on Simple Open Addition and SubtractionSentences. Arithmetic Teacher 18 (November1971) :513 19...

17

directionsof

I

alan osborne

nibbelink

THE glimpse of a different concept of pre-school and kinci.garten education for all hasgiven impetus to change 7n the mathematicseducation of young children. During the last tenyears, the sense of urgency that accompaniedthe belief in, and desire for, compensatory edu-cation as a treatment for societal ills has led tothe rise of unprecedented resources for the de-velopment and implementation of curricularmaterials for the preschool and the early ele-mentary school child.

The features of the mathematical materialsdeveloped within this context will be examinedin this chapter. The quantity of materials aloneargues against exhaustive cataloging of alldevelopmental efforts or giving a complete andthorough description of more than a few pro-grams. Consequently, the discussion of materi-als is focused on those program features thatprovide examples of some of the distinctive di-rections that have emerged in the developmentprocess.

This development has been multidirectional.At the preschool level the new emphasis oncognitive objectives has encompassed mathe-mat'cai concepts and skills (Evans 1971; Kamii1971). But few traditions existed for the designof materials for, orthe-teadhing of mathematicsto, the young child. Demand for programs andmaterials rapidly exceeded the capabilities ofthe available specialists in mathematics educa-tion for the very young. This has resulted in ma-terials and methods being created by a largenumber of people with widely varying back-grounds. The programs developed represent

4 ;

the differing philosophical, psychological, andpedagogical briefs held by this divergentgroup.

The range of differences in the materials de-veloped for grades 1 through 3 is almost asbroad as that of the programs. Stemming froma variety of causes, the differences representreponses to dissatisfaction with materialscreated in the post-Sputnik reform period andreflect the influence of the stress on com-pensatory education. Clearly, the potential ofthe precursive experience in preschool hascaused some authorities to reconsider tne tra-ditional content dimensions of the curriculum.

Four categories of curricular developmentprovide a classification scheme for the dis-cussion that follows: activity learning, languagedevelopment. mathematics, and compensatoryeducation. In addition to these major directionsof curricular development, a fifth section de-scribes directions not readily s.itject to classifi-cation but of potential influence.

Programs are not necessarily classified ac-cordirg to their primary characteristics. Rather,the classification is made in the programs ex-emplifying particular directions of curricularchange It should also be noted that many pro-grams share characteristics with programs thatexemplify other directions of change.

activity learning

Activity learnmy has a long, rich tradition inthe teaching of arithmetic to young children.The trend toward activity-based instructionrepresents a return to a familiar tradition of us-ing motor activities as a base for developingcognitive skills and abilities.

Three examples of activity-based instruc-tional programs are provided in this section.The first, that of Montessori, is directed at chil-dren of ages three through eight and is the onlyexample of a preschool mathematics programwith a tradition dating from the turn of this cen-tury The second, the Developing MathematicalProcesses program, attempts to incorporate

DIRECTIONS OF CURRICULAR CHANGE 275

recent psychological findings into the design ofan activity-based program. The third, the Brit-ish Infant School, is described more accuratelyas a movement than as a program. It illustrateshow activity-based instruction can provide adifferent emphasis on the content of mathe-matics.

montessori

No effort to describe preschool practices inmathematics education would be completewithout paying tribute to the influences of MariaMontessori (1870-1952). Not only do Montes-sori schools provide a backlog of experience,but a number of increasingly popularsetS ofmanipulative materials (including Cuisenairerods and Dienes blocks) have equivalent ornearly equivalent forerunners in Montessorimaterials. With the recent attention given to thework of Piaget, it is also in order to point outthat a number of Maria Montessori's assertionsabout the nature of the child parallel those ofPiaget (Sheenan 1969). For example. bothMontessori and Piaget "emphasize the norma-tive aspects of child behavior and developmentas opposed to the aspects of individual differ-ence" (Elkind 1967, p. 536).

Several difficulties are involved when speak-ing in generalities about the Montessori ap-proach to mathematics. In fact, some may con-sider it unfair, since the Montessori approachintends to deal with the whole child, not with acompartmentalized curriculum. Further diffi-culties in defining the mathematics program re-sult from -

1. variance from school to school in theamount of play allowed;

2. variance from school to school in theamount of usage of classroom materialsthat would not be classified as "Montes-sori materials";

3. variance from school to school resultingfrom differences in Montessori teacher-training institutions.

However, the Montessori contribution to earlychildhood education is worthy enough of con-sideration to run the risk of some misrepresen-tation relative to any given Montessori school.

r:"I. /.r ;

276 CHAPTER TWELVE

Materials common to most Montessori pro-grams, listed in sequence according to their us-age by pupils. include the following:

stairs (or number rods)sandpaper numeralsspindle boxcounterscolored bead materialbead framesnumber framesaddition board for rods*golden bead materials (base ten)number (place value) cardsblock material (base ten)multibase bead materials'multiplication boardabacus equivalent'division board'Place in sequence vanes from school to school

Pictures or diagrams of most of the materialslisted above may be found in books written byMaria Montessori (1917; 1948; 1965); anyonemore than casually interested is advised to visita Montessori school. The materials are con-structed and designed with children in mind.The "staircase" materials, for example, areavailable in different sizes for children at differ-ent levels of physical development.

The Montessori approach tends to use eachmodel (set of materials) for a single or limitednumber of mathematical activities. even wherethe same model could serve for more activities_Of particular interest relative to the devel-opment of number concepts is the use of the"bead" materials prior to using solid rectangu-lar blocks. Also of interest is that the Montes-sori blocks appear with "beads** painted on thesurface. whereas most commercially availableblock materials either refrain from marking"units" or mark them inconspicuously:

OneTen

'Ve seissa*-)

ONINIEEM

One

Ten

One

Ten

Montessori heads

Montessori blocks

tit:nu:Irked blocks

The bead materials afford the child the op-tion of managing the learning tasks success-fully without an appreciation for representingnumbers by lengths; at the same time, the beadmaterials clearly allow for representations bylengths. The colored bead materials, as well asmany commercially available block materials,allow for whatever payoff may come of asso-ciating numbers with colors.

The ease of transition to rational- and real-number concepts may be lost by the more ex-plicitly discrete representations of number bybeads. It is, perhaps, easier to perceive anddeal with a 2 1/3-unit stick than with 2 1/3beads. An attempt to determine an advantageor a handicap from using beads would suggestan investigation of questions like the following:

1. Do children perceive unmarked block ma-terials as continuous, as discrete, or as ei-ther, depending on the task at hand?

2. If there are initial differences in per-ceptions, do the differences persistthrough the use of Montessori block ma-terials?

3. If differences persist, do they appear to bea factor in success with subsequent learn-ing tasks?

Perhaps more important to the program thanthe materials is the teaching methodology.Each set at materials is intended to be used in atightly prescribed way. A child who uses mate-rials according to the prescription is doing his"Montessori work"; a child who chooses to usethe materials for purposes other than those in-tended is at "play." In some schools "piay" Isnot 'permitted, and failure to use materials asintended results in their removal. In otherschools a child at "play" may be asked to takethe materials to a designated area in the roomreserved for such activity. "Play" may or maynot be overtly scorned, but in either event it isclearly second place to work.

The premium placed on "work" so definedmay be, on the surface, puzzling in view of theclaim by Montessori teachers that an importantcharacteristic of the approach is the guardingof the child's "freedom." If the child is free, thenclearly the meaning of freedom is somethingother than an absence of externally imposedvalues or a license to follow whims. Either there

F. i

is little freedom or, as is the case, the definitioninsists that feedom is gained by learning self-discipline relative to fixed values and tightly

escribed activities. This latter definition iscertainly not new. Plato would insist that a fail-ure to control lower instincts is a sure path toslavery, Christianity insists that freedom isgained by submission.

A second possible source of confusion is theMontessori use of the word discover. AlthoughMontessori teachers generally agree that thechild "discovers" mathematics, the means ofgetting h:m fr..= an initial demonstration of in-terest in a set of materials (by approach behav-ior) to the intended usage of the materials usu-ally goes something like this:

1. The teacher asks the child if he would liketo be able to work with the materials.

2. Given a yes, the teacher "names" thecomponents or demonstrates the in-tended manipulation of the materials. Ver-balization is kept to a minimum.

3. The child is asked to name indicated com-ponents or is asked to perform indicatedsubtasks to the total task.

4. The child is asked to identify componentswhen given the name or is asked to per-form the task.

There is a heavy dependence on the child'swillingness and ability to imitate. Means like theabove do not, however, evidence a strong com-mitment to the gestalt "aha."

"Individualization" in the mathematics pro-gram may be seen relative to pacing, point ofentry with the materials, and one-to-oneteacher/pupil interaction. Individualizatii asclaimed by Montessori teachers does not Implythe existence of a variety of acceptable ends ora variety of means to ends.

A key notion both to establishing the appro-priate point of entry for a new pupil and to pac-ing is that of "readiness." In practice, readinessseems to mean simply approach behavior fol-lowed by an acceptable rate of advancementtoward accomplishing the task. Beyond the in-tuitively easy decisions, there is no machineryfor concluding that the child is ready for a givenlearning task other than observing his encoun-ter with the task. Neither does there appear tobe a formula for distinguishing between a


child's being unwilling to follow the prescriptionand his being unable to follow it.

The importance of the notion of readiness tothe Montessori approach derives from one ofthe Montessori claims. the best time to teach apupil any given content is at the point of hisbeing ready. Without some means of determin-ing readiness external to encountering thelearning task itself, this axiom reduces tosaying that the best time to teach a pupil anygiven content is when the child is first able andwilling to learn it. Beyond lacking research evi-dence that shows it to be valid. the statement issomewhat weak toward making the decisiun of"when" if any premium is to be placed cn firstexperiences being successful experiences.

Among questions suggested by the aboveparagraphs are the following:

1. Is a heavy dependence on the child's abil-ity and willingness to imitate appropriateto the teaching of mathematics?

2. Is the application of the Montessori defini-tion of freedom a souce of security or athreat to the child? Is it an aid or a hin-drance to sound and/or creative mathe-matical thinking?

developing mathematicalprocesses

The Developing Mathematical Processes(DMP) program for grades 1 through 3, cur-rently being developed, field tested, and eval-uated by the Wisconsin Research and Devel-opment Center, offers an example of a tightlydefined, segue Lai approach to activity-basedlearning (Centel for Cognitive Learning 1971).

Each set of activities in the DMP materials in-cludes a listing of "regular objective-, and'preparatory objectives." The regular objec-tives are those for which mastery is to bechecked. These objectives are stated in behav-ioral terms, but they are not so given to pre-cision in statement that they deny admittingcertain goals as objectives. For example, onesuch objective reads, Given a geometric figureconstructed on a geoboard or from plastic orwooden objects, the child' pictor ially represents

t

278 CHAPTER TWELVE

that figure on specially prepared paper." Obvi-ously a degree of subjectivity will be required indeciding if the child's picture is adequate: thechild's familiarity with the geometric figure maybe an important consideration, the degree ofneatness the child usually brings to his workmay be a factor in judging it, and some figurescould likely be concocted to bring out errors.The preparatory objectives are those that canbe pusued naturally in a given activity, that willbe treated later, and that need not be''checkedfor mastery until the later treatment.

The DMP materials may place an additionalload on the teacher, since the required materi-als may not ce readily available. In attempts toimplement the program on a broader scale,this additional demand on the teacher may be akey factor to its acceptability. The introductionto the DMP materials encourages the teacherto develop a "pack rat" habit, cautions that forany given teacher some activities may "fail mis-erably." acknowledges that implementing theprogram is "a difficult and time-consumingtask." and warns that "it may take severalmonths with this program before (the teacher)and students are comfortable with it." A fewsentences (admittedly selected for dramatic ef-fect) from the direbtions given to teachers forprelesson preparations may serve to make thepoint: "For every cup of flour add one cup ofsalt and one tablespoon of cornstarch. The wa-ter should be added slowly as you stir.... Pre-pare cans by making holes of different cubes,frizzier. etc. ... Using masking tape, make theletters A, M, E, F. H, K, N, W, and Z on thefloor.... Borrow five balance beams from an-other teacher."

The flavor of the materials may be capturedby an outline of the early treatment of linearmeasure At the kindergarten level, the experi-ences (listed in order) include comparinglengths in pairs, "equalizing" lengths, orderingmore than two objects by length, representinglengths by physical means, and representinglengths pictorially. At the primary level (K -2),children are given experiences in measuringlengths with arbitrary units. Suggestions toteachers include using toothpicks, drinkingstraws, Lots-a-Links, and Unifix Cubes. Theworksheet accompanying this activity is the fol-lowing:

11.0123 4 5678910M

0123 4 5678910M

013 4 567289

......""" 0123 4 5678910M

In the space at the top of the worksheet, thechild draw.; a picture of the object being meas-ured. The object pictured in the column on theleft is used to measure the object drawn in thespace at the top; the "M" is to be circled for"more than ten." The middle column is for tallymarks.

The instruction sheet for the teacher sug-gests that the children be encouraged to guessthe number of units to be needed before theymeasure. This may serve two functions, First, ithelps focus the _children's attention on the goalat hand, second, over the long haul, it may de-velop their ability to detect unreasonable "an-swers" to mathematical problems. Standardunits of linear measure ("inch, centimeter,etc.") are introduced after experiences witharibtrary units.

A three-step sequence for concept dev,el-opment is employed by the DMP materials.This sequence begins with physical experi-ence, moves on to pictorial experience, and

culminates with a use of more abstract sym7bolism Insistence on precise vocabulary is de-layed until the probability of the child's usingthe technicEil term before he has learned theconcept has been minimized.

Provisions for individualization allow for al-ternate means to mathematical ends but not foralternate ends. Refraining from a wider allow-ance for individualization is justified on the fol-lowing grounds: First, the program developerstake seriously those implications of Piagetiantenets and research on the nature of the childthat emphasize the normative aspects of childbehavior and development. Second, presentknowledge about individual differences isjudged inadequate to make provisions for a va-riety of cognitive styles.

Viewing the Wider range of activities carriedon by the Wisconsin Research and Devel-opment Center, one could argue that the DMP

dais are largely determined. by prior re-ser.,,ch. Although it is too early to determine ifsuch care nays off, it is comforting to find ex-amples of curricular development that gobeyond the practice of merely appealing tosupportive tidbits from learning theory to justifya product after the fact. Findings on con-servation and numerousness, for example,have been heeded in making curricular deci-sions.

One question suggested by the DMP materi-als is whether they embody an attempt to accel-erate cognitive stage development. The hintthat stage acceleration may be intended by thematerials derives from the inclusion of the"preparatory objectives." If the function of suchpreparatory objectives is to provide what Ausu-bel would call "advance organizers," then itwould seem reasonable to require confirmationof their being established. Whether such accel-eration is possible has been an object for con-troversy and will remain an undeckled issueunless someone proves convincingly that it ispossible. Another question, perhaps more im-portant, is whether or not such acceleration isdesirable. For example, six-month-old kittensare winners over six-month-old children in anumber of stage-related Piagetian tasks. Suchobservations at least suggest that the capabili-ties enjoyed by a later developmental state maybe determined to some degree by the number


of experiences encountered at the earlierstage. If so, stage acceleration certainly holdspotential perils.

The several references to Piaget in describ-ing the DMP materials were not meant to assertthat the program is an example of one totallyprescribed by Piaget's developmental psychol-ogy. Several of the programs that use Cuise-naire rods- as the exclusive model for earlytraining in mathematics also claim Piaget. Incontrast to such, the DMP materials use a vari-ety of models. A case can also be made for par-allels between Montessori and Piaget on theappropriate treatment of the child. Again, theDMP materials and the Montessori approachdiffer regarding the provision for alternatemeans to a given end. In short, if learning the-ory as known today is prescriptive, many of itsprescriptions appear to be somewhat vague.This is not to say that they are unworthy ofbeing heeded; rather, it is to question the na-ture of such prescriptions.

A last, somewhat obvious, question raisedby the materials is whether or not the elemen-tary school teacher can be expected to acceptthe demands of using the materials.

british infant school movement

Informal classroom procedures are used inapproximately one-thiri of the infant schools(ages 5 to 8) in Britain (Featherstone 1967). Nosingle group or agency is responsible for gen-erating and implementing the approach; rather,a variety of people in many settings have devel-oped many techniques over a period of years.The Plowden report (Central Advisory Councilfor Education 1967), which described the stateof British education in 1967 and proposed aguideline for reform, captures the essence ofthe attempts at reform. Because of the widevariation from school to school, the phrase Brit-ish Infant School (BIS) is used as a generic la-bel for the movement.

The BIS movement is more of a philosophythan a specific curricular development withscope and sequence. The BIS movement hasbeen perceived, perhaps inappropriately, as ameans to deal with "dehumanization" and

280 CHAPTER TWELVE

"relevance." As such, it has possessed a fasci-nation for both the lay and the professionalpress on this side of the Atlantic and encour-aged emulation.

The BIS program has three cu, ricular thruststhat contribute to the strength and uniquenessof the approach. First, the activity orientationembeds mathematics in other subject mattercontexts and provides an opportunity to usethe skills and concepts of other subjects. Sci-ence and mathematics are unified as the childstudies his environment. The conditions im-posed by the individual and small-group in-structional format require children to be some-what self-directive in following writtendirections and require the use of reading skills.Children typically keep a diary of their activi-ties. Representative activities can be found inEdith Biggs's Mathematics for Younger Chil-dren- (1971),-the Nuffield Mathematics Projectmaterials (1967, 1969 [a j and (1) j), and HaroldFletcher's teacher resource books (1970, 1971[a J and [b ]). The impact of these activities onchildren's reading and writing skills and on thelearning of science may be of grea.er impor-tance than the nature of the mathematicslearned. Clearly longitudinal research can pro-vide only information , elative to this question ofvalues, the answers are to be determined onother grounds.

The second curricular thrust, the treatmentof geometric concepts, transcends the namingactivities common to many American pro-grams. Attention is given to developing visual-perception skills. The Nuffield EnvironmentalGeometry (1969 [a j) uses the child's environ-ment to help the child look for geometric regu-larity and constancy. Children experiment withthe strength of various shapes not only in theplane by comparing the triangle to other poly-gons but also in three-space by comparing thesupport provided by cylinders, prisms, tetra-hedra, and so forth. This geometrical expen-mentation is supplemented by mea .urementactivities of a more scientific than mathematicalcharacter. These activities provide a setting forusing computational skills.

A third feature of the BIS materials is thestress on collecting and analyzing data. TheBIS materials are reminiscent of the recom-mendations of the Progressive Education As-

ry

sociation's chapters in Mathematics in GeneralEducation (1940) concerning data and thefunction concept. BIS materials involve chil-dren with coordinate systems that are used tograph "nice" linear relationships stemmingfrom problems concerning milk money, meas-urernent, and a variety of other down-to-earthcontexts (Nuffield Mathematics Project 1969(b)). The materials do focus on the eventual ex-perience of children with algebra, but the im-mediate payoff of the stress on the functionconcept is the graphical interpretation of data.

The preceding description of BIS material iscontent oriented. It is based. on materials thatdo not describe a complete mathematics pro-gram but rather provide examples of the typeand spirit of activities. The nature of the ap-proach entails more freedom for both teacherand child. It seldom features activities by awhole, intact class. Children work in smallgroups and by themselves at different learningstations for several content areas. The teachermust keep careful records of an individual'sprogress to assure that mathematics is not fa-vored to the exclusion of reading or geometricconcepts to the exclusion of arithmetic. The ap-plication of the Montessorian idea of having"family- groupings of children of at least threeage levels in one class and the fact that schoolentry is not just in the fall further confound therecord keeping. Techniques for coping withthese problems are discussed in Palmer'sSpace, Time and Grouping (1971).

Corresponding to the complexity of adminis-!ration is the payoff of not needing a set of ma-terials for each laboratory activity for eachchild. Only a limited number of children use aset of materials at a given time. Consequently,the BIS movement uses activity learning to amuch greater extent than that allowed by typi-cal American classroom practices.

A current, common assumption in the UnitedStates is that features of the BIS program canbe lifted intact from one cultural setting andone school context and applied in another.Through premature and thoughtless misappli-cation, however, several characteristics of theBIS mathematics program may be sacrificedand lost. British schools, unlike those of theUnited States, have a tradition of early child-hood education. Government inspector-adv:-

sors and headmasters have a different style ofoperation. British school personnel use psy-chologyparticularly developmental psychol-ogydifferently from their counterparts in theUnited States (Fogelman 1970). British childrentraditionally begin cognitive learning tasks ear-lier than American children. These facts pro-vide a different context for implementing theBIS program (Elking 1967). More research incomparative education is needed beforewholesale application of the BIS program is be-gun in another setting.

The preceding discussion portrays the spiritof the BIS approach. The philosophic base ofthe program is radically different from some ofthe rigidly structured materials found in theUnited States. Do children participating in suchprograms differ significantly in problem-solv-ing skills and approaches? Do they differ intheir application of mathematics?

Clearly the BIS approach places distinctivelydifferent responsibilities on the shoulders ofteachers and demands considerable skill fromthem in designing a comprehensive, thoroughmathematical experience for children. If theteacher is skilled and knowledgeable, the childcould have a rich and rewarding experience, ifnot, not.

language development

Talk is a critical variable in teaching mathe-matics. Limitations on communication are onesort of pedagogical problem. An example ofthese limitations is found in the second-lan-guage problems of Mexican-American chil-dren. Their problems led the Southwes.t Educa-tional Development Laboratory (r. d.) todevelop bilingual tapes and materials or theimplementation in grade 1 of the IndividuallyPrescribed Instruction mathematics program,whose content and mode of instruction are in-dependent of the language variable. Within thissection are described two programs that weredeveloped around different assumptions con-cerning the role of language. Rather than treat-ing the limitations imposed by language or the


lack of it, these programs were designed to usethe characteristics of verbal interaction as asupporting, integral component of teachingmethodology.

The DISTAR program uses language as a ve-hicle to help students habituate to mathemati-cal responses. rhe Interdependent LearningModel program uses games extensively as avehicle of instruction. The gameS are designedto take pedagogical advantage of the verbal in-teraction implicit in the gaming.

distar

The DISTAR arithmetic program, created bySiegfried Engelmann and Doug Carnine, is oneof the more unique approaches to mathematicsfor the young child (Engelmann and Carnine1970). The uniqueness stems partly from themathematical content but most particularlyfrom the instructional methodology. The in-structional procedures detailed in theTeacher's Guide are built on philosophical andpsychological foundations markedly differentfrom those of the currently popular laboratoryand British Infant School approaches. How-ever, several learning difficulties of young chil-dren in arithmetic are identified, and strategiesfor coping with these difficulties are described.The teacher's precise control of her language isan important component of the instructionalstrategy.

The DISTAR arithmetic program evolvedfrom the compensatory preschool educationprogram developed by Carl Bereiter and Sieg-fried Engelmann at the University of Illinois inthe mid-1960s. Features of this compensatoryeducation program establish a basis for under-standing the DISTAR Arithmetic InstructionalSystem. The first task in planning was the iden-tification of a limited set of objectives that thechild should attain if success were to be ex-pected in grade 1. Most of these objectiveswere oriented toward the acquisition of verbalskills. The second part of instructional planningwas a search for ''a practical way of getting allchildren to learning the specified content" (Ber-eiter 1967). The methods that evolved for the

282 CHAPTER TWELVE

single-minded pursuit of the limited set of ob-jectives have created controversy in the com-munity of preschool educators, particularlythose subscribing to the whole child and thechild development points of view. Beyond thescope of this book, the reader may turn to Ber-eiter and Engelmann (1967), Hodges (1966),and Hymes (1966) to examine the issues in-volved.

The instructional approach of the DISTARarithmetic exhibits this same single-mind-edness. Based on an extensive task analysis,such as that described in Engelrnann's Con-ceptual Learning (1969), an efficient chain ofskills and concepts leading to the content of thetypical grade 2 arithmetic program was con-structed. It consists of approximately four hun-dred lesson formats progressing slowly frommatching tasks through addition and sub-traction of numbers named by two digits andmultiplication. Negative numbers are consid-ered, but few geometrical concepts.are devel-oped. The rate of the presentation of the sym-bols of arithmetic indicates the pace ofDISTAR. 150 formats must be accomplishedbefore the symbols for the numerals throughten, addition, equality, and subtraction havebeen introduced.

DISTAR shares many of the features used incurrent approaches to the teaching of lan-guage. Pattern drill ai;d practice procedurestypical of the teaching of foreign language(based on behavior modification) are amongthose methods used in teaching mathematics.The hierarchy of the skills and concepts ofarithmetic provides an efficient structure for thelanguage-oriented teaching methodology.Arithmetic is a special form of language. Thech.'d develops this language by small, in-cremental steps.

Counting provides the basis for developingmost concepts. Developed from matching ex-periences, the counting activities progressthrough two types of activities, labeled with thewords rote and rational. The rote activities donot relate number to objects or events but arelanguage drills. Two types of rote counting arecritical to the later developments of additionand subtraction. The first type emphasizescounting to a ;Umber. The format of the in-troductory lesson of the rote counting se-

quence taken from the Teacher's Guide bothexhibits the emphasis of the counting activityand characterizes the pattern drill and practiceapproach of the entire program. The teacher isprovided a script (in regular type) and direc-tions (in italics). For example (p. 29, fig. 14):

Task Counting to a NumberListen to me count. Tell me what I count to.a. 1,2,3,4.5,6-6. I counted to six.b. 1.2,3,4,5,6,7,8 -8. I counted to ... Wait. Eight.c. 1,2,3,4,5.6,7 -7. I counted to ... Wait. Seven.d.1.2,3,4,5-5. I counted to ... Wait. Five.e. 1,2,3,4.5,6. 1 counted to ... Wait. Yes, what did I do?

Wait.f. c( rated to ... Wait. Yes, what did I

do? Wait.g. 1,2,3,4,5,6,7. 1 counted to . .. Wait. Yes, what did I

do? Wait.h. 1,2,3,4,5. I counted to ... Wait. Yes, what did I do?

Wait.To Correct. Repeat the counting sequence, saying

the last number with emphasis. Forexample: 1,2,3,4,5,6-6. I counted to... Wait.

The second type of rote counting emphasizescounting from a number to a number. Eachof the remaining lesson -formats on countingoffers a variation of the stimuli associatedwith the counting response. The variationscorrespond to small increments in the skillsof counting. Suggestions for eliciting groupand individual responses are provided. Theserange from the control of pacing and ca-dence in oral counting to the consideration ofappropriate reinforcement techniques. Thesuggestions are precise and often prescribetreatment for difficulties exhibited by learners.

Some teachers might find the tightly struc-tured nature of the lesson formats uncomfort-able and unrealistic. However, the careful taskanalysis of the increments in the hierarchy pro-vide useful ideas and insights for those who donot accept the highly structured classroom ap-proach. For example, many children do finddifficulty in addition. Diagnosis may indicate aninability to conserve numerousness or a lack ofunderstanding of the part-part-whole relation-ship. Or it may show difficulty with the skills ofcounting implicit in addition. For example, insolving the problem 9 t 4 0, the child mayfirst count the nine objects in the first set andthen exhibit an almost allegiancelike propen-sity to onehe starts the counting process over

on nibeing to the set of four objects. The habit-uated process of counting from a number canprovide linguistic support for the child havingtrouble associating the addition operation withthe union of sets. The linguistic pattern pro-vides but one entry into the child's establishingthe desired conceptual connection.

Counting skills are the basic components inthe chain of steps leading to the concept ofequality and to the operations of arithmetic.Equality is identified as the single most impor-tant concept of the program. The counting hereis rational instead of rote because objects arecounted in the paradigms used in teaching theoperations. The sets of objects to be countedare limited to "lines." In the paradigms used toteach the operations, the lines are in closeproximity to the mathematical sentences usedfor the operations. Equality is considered as acharge to the child to produce the same num-ber of lines on one side of the equal symbol asthe other. For the sentence

11111111_

the child counts and draws the eight lines asso-ciated with the 8 and the eleven lines asso-ciated with the 11. Using the skills previouslyestablished, he should (a) draw below theframe the lines that (b) correspond to the num-ber necessary for counting from 8 to 11 and (c)place the correct numeral in the box. Early inthe track of lessons for addition, the word plusis used both as a name for the operation and asa language cue for working the problem. In theproblem above the child would "plus threelines." This distinctive verbal usage for opera-tiChs pervades the program.

The paradigm for teaching subtraction issimilar to that used for addition. Countingbackward and crossing out vertical lines areassociated with "minusing" as shown by the fol-lowing:

4 =Hxyx,r


Steps in teaching are broken down into the fol-lowing subactivities (Engelmann and Carnine1970, p. 99):

1. Find the equal.2. State the rule for equality.3. Find the empty box.4. Point to the side of the equal on which

they start counting (the side without theframe).

5. Make the lines for the(first group.6. Minus the lines (crosscut for the second

group).7. Indicate how many remain.8. Indicate what numeral goes in the empty

box.

Each of these steps corresponds to a pre-viously acquired skill. The conjoining of the op-eration of subtraction and the act of crossingout provides the rationale on which the conceptof negative number is developed.

Multiplication is also presented as a countingoperation. The skills of group counting arecarefully established before multiplication isencountered. The problem "5 X 4 = is de-coded according to the following steps (p. 144):

1. The numeral 4 tells you to count fourtimes. That means you must make fourboxes.

5 x 4Hifi

2. The 5 x tells you that you are to count bygroups of five. That means you are to putfive lines in each box.. ..

3. You count the lines by fives.4. Then you apply the equality rule....5. So the completed statement is

5 X 4 = 20.

Children are to use a multiplication chart as aguide to counting groups. They are to touch thenumerals above the appropriate groups as theycount. If they are to count by groups of seven,then they are not to recite "seven, fourteen,twenty-one, ..." but rather to touch those nu-merals as they count "one time, two times,three times, . . ." The Teacher's Guide states

284 CHAPTER TWELVE

that confusion arises for many children if theyrecite the numbers rather than the number oftimes they are supposed to count a group.

One of the distinctive features of the DISTARarithmetic is the delay of emphasis on the con-cept of place value. A special symbol is usedfor the decades. Children do not encounterhigher-decade addition and subtraction untillate in the program. When children first en-counter ten, the symbol 10 is used. Rather thanfuss with the different role of the symbol 2 in thenumerals 32 and 27, the children are to write302 and 207.

In summary, Engelmann and Carnine havedeveloped a rigid approach for teachingarithmetic in the early elementary grades. Thelessons are based on a thorough analysis of theconcepts and skills necessary for the next stepin learning R would be easy to create a flowchart of ideas from the lessons. At many pointsin the program, shrewd insight into a difficultyof learning is exhibited by the nature of thesmall incremental step designed as treatment,for the difficulty Linguistic and perceptual cuesare carefully programmed into the instructionalactivities Some of these are extinguished orwithdrawn later The Teacher's Guide providesseveral suggestions that teachers might findhelpful for the handling of group activities. Atesting program subsuming both diagnosticand achievement functions is a part of the DIS-TAR arithmetic. Used and evaluated in HeadStart and Follow Through programs exten-sively. the program has been found successfulin helping children acquire skills in arithmetic(Steely 1971).

The DISTAR program may have some severelimitations Several strong assumptions implicitto the DISTAR approach need to be tested. Forexample, DISTAR may be too efficient. Chil-dren have little opportunity to "monkeyaround" in a mathematical context. The role ofmathematical play in developing intuition is notknown precisely. Second, the perceptual mod-els used for'developing concepts are limited intype and in use. Sets of lines in close physicalproximity to equations are used almost exclu-sively to establish operations. Do children needseveral models to increase the probability oftransfer and application? Is proximity of themodel and the symbols a critical variable it

learning? Is the counting model used too exclu-sively? Finally, mathematics appears to be alanguage. The doing of mathematics becomesa problem of translation, of decoding and en-coding. This is but one view of the nature ofmathematics. Is this too limited a perspective toestablish early in a child's experience?

interdependent learning model

The Interdependent Learning Model (ILM)uses a games approach to mathematical in-struction. ILM is a Follow Through program de-signed by the Institute for Developmental Stud-ies, directed by Martin Deutsch of New YorkUniversity. Socialization is considered a pri-mary variable in the learning of young children.Language and games are identified as two im-portant components of the socialization of theyoung in any culture. The mathematical cur-riculum and methodology is designed to takeadvantage of the interactive nature of languagein gaming. The word transactional is used todescribe the value-laden, verbal negotiationbetween players that provides motivation, im-mediate feedback of the consequences ofone's learning, and the socialization necessaryto learning. The mathematical games are de-scribed as transactional (Gotkin 1971).

The purpose of the ILM program is to pre-pare the inner-city, disadvantaged child tocope with the demands of schooling in the laterelementary and junior high school years.Learning and schooling are recognized as ver-bal an.1 social activities. Thus, the instructionalgames are language games that provide chil-dren with the opportunity for social, verbaltransactions as they apply the rules of thegames. The game is a socializing activity thatenjoys the advantage of not separating socialand emotional development from cognitivelearning and development.

The stress on language permeates the directinstruction portions of the program as well asthe games. Accounting for less than 40 percentof the total program, the direct instruction ma-terials are adapted from the work of Engel-man n and Carnine. The scope and sequence ofthe curriculum' is similar to the DISTAR pro-

gram but encompasses more traditional con-tent areas of the early elementary school cur-riculum, such as geometry, telling time, Romannumerals, estimation, and measurement. Multi-plication receives greater emphasis than inmost typical pre-third-grade mathematics pro-grams Negative integers are introduced. Al-though place value is introduced, no attempt toteach to completion is made. Counting pro-vides entry into most subsequent arithmetic.Appropriate game formats were sought only af-ter objectives were identified. Games of thelotto style are found throughout the programbut receive particularly heavy use as a strategyfor teaching correspondence. The rules andscore-keeping procedures provide further op-portunity for children to talk about the match-ing process and extend the use of the game toteaching addition and subtraction concepts.The use of the rules, and indeed the in-troduction of children to the rules, establishesthe transactional mode of verbal interactionidentified as tine critical feature of the gamingapproach (Winters 1971).

It is beyond the scope of this chapter to dis-cuss each ILM game There are several usefulgames in t` e ILM materials that teachers canconstruct cheaply. A careful, summative eval-uation of the transactional variable in the gam-ing approach has not been conducted to date.The expected outcomes relative to the verbal-ization and socialization objectives are impor-tant and extend beyond the achievement ofmathematical concepts and skills. Using thetransactional characteristics of games for theseobjectives and for those more specific to math-ematics suggests many researchable hypothe-ses.

mathematics

comprehensive schoolmathematics program

The nature and uses of mathematics in ourculture have determined the design of theComprehensive School Mathematics Program(CSMP) for early elementary school chadren.


The CSMP project collected groups of mathe-maticians and mathematic:: educators for thepurpose of reflecting on the ideal content ofschool mathematics and attempted to translatethese ideas into realistic materials for youngchildren. Few development projects havestressed the study of the question of whatmathematics to teach to the extent that CSMPhas.

The CSMP effort, at the elementary schoollevel is relatively recent; in the mid-1960sCSMP concentrated its resources on the imple-mentation of the Cambridge conference cur-riculum at the secondary school level (CSMP1971). The first year of CSMP pilot materials inthe elementary school was 1968. Since the,tmoderate beginning, a complete K-12 curricu-lum has been outlined, and materials havebeen developed and field tested for grades K-3. The CSMP project was supported by theCentral Midwestern Regional Educational Lab-oratory.

The selection of content was, and is, criticalto the disciplinary orientation of CSMP. In or-der to free individuals for dreaming, the initialselection of content was to be unfettered by tra-ditional notions of "what children can do" or"what teachers can teach" (CSMP 1971). Givena determination of what is important mathe-matically, the curriculum designers of CSMPthen assumed the responsibility of creating orfinding appropriate, effective materials andmodes of instruction.

Three criteria were provided for judgingwhat is important mathematically. These cri-teriaadequate coverage, the viewing of math-ematics as both art and science, and utilityprovided direction for content selection andwere carefully detailed to stress the unity ofmathematics in terms of both concepts (suchas relations, sets, and mappings) and mathe-matical methods and tools (such as algorithms,mathematical logic, and heuristics).

These criteria led to content that is atypicalof the majority of elementary school programs.The CSMP advisors and staff attempted tocreate materials that help children begin to de-velop concepts of probability and corn-binatorics. The CSMP program stresses thedevelopment of preliminary concepts of logicbut does not reserve them to the linguistic- or

286 CHAPTER TWELVE

communication-skills component of the cur-riculum, which is typical of many Head Startand Follow Through pi ograms. The instruc-tional materials for logic at the first-grade levelare independent of the instructional sequencein mathematics. The concept of function servesto unify the entire curriculum. Many of the de-vices and materials for teaching arithmetic con-cepts and skills help children develop an in-tuitive basis for the concept of mapping ingeometry, another conceptual thread spanningthe CSMP curriculum. Mathematical structuresand the relationships between them and theirreal-world models were identified as conceptsfitting the criteria.

Possessing content ideals is a long step fromhaving a curriculum ready to implement. TheCSMP staff, under the guidance of Burt Kauf-man, elected to build instruction around theconcept of individualization by means of activ-ity packages, although the experience arid in-fluence of Mme Frederique Papy tempered atotal commitment to individualization (CSMP1971). The activity packages use ideas frommany different sources, such as Cuisenaire,Minnemast, Dienes, Nuffield, and Papy.

The CSMP program begins in the kindergar-ten and progresses through , bird grade. Thepackages are designed in sequence, with a spi-raling of content. Special packages for enrich-ment are part of the program. Many activitypackages have a corresponding remedialpackage, although the teacher is warned thatspiraling provides a means of treating childrenwho are unsuccessful in acquiring concepts onthe first introduction to the concept. Manypackages are cheap newsprint materials of asemiprogrammed nature. Some are used todiagnose whether a child should progress tothe next activity or be channeled to a remedialpackage. Many of the packages are based onmanipulative, laboratory materials. Color cuesare used extensively. A garr.ing format is usedwith many of the packages.

One device, the Papy minicomputer, is usedextensively. Recently introduced from Belgium,this device, through the CSMP use, is receivingone of the first extensive tests of its effec-tiveness in the United States. The device and itsappropriate use are complex and exceed thescope of this chapter, the article "A Two-di-

mensional Abacusthe Papy Minicompu:er" inthe October 1972 issue of the ArithmeticTeacher (Van Arsuel and Lasky 1972) providesa description of the device.

Activities on the minicomputer involve chil-dren in placing counters on boards corre-sponding to place vai.les in a base-ten numer-ation. system. Embedded within each board is alimited binary system determining the valuecorresponding to counters placed on theboard. The use of the minicomputer Is con-stantly oriented toward minimizing the numberof counters on the boards. The use of a particu-lar problem orientation, such as this minimization task, as an integral part of the instruc-tional strate7ins used to eAablish a number ofdifferent concepts and skills has not been re-searched thoroughly. Does this use of an over-riding minimization problem establish a con-structive orientation similar to that espoused byZoltan Dienes (Dienes 1967)? Is tnis a criticalfactor contribi. ':ng to the effectiveness of labo-ratory learning uids?

The CSMP approach to developing compu-tational skill provides a primitive introduction tofunctions as mappings Addition and sub-traction are associated with arrows or map-pings as shown 1:y the addition exercise in(an understanding of the concepts of multipli-cation and division is developed around similarmappings).

The "arrow" diagrams are used to develop adynamic "feel" for operations as functions ormappings. The concept of the composition offunctions develops nicely in this context, as wellas the concept of the inverse function. This pro-vides a treatment of fractions not so thoroughlygrounded on the physical representation offractional parts. Porhapc the greatest potentialof the extensive use of mapping in establishingarithmetic concepts will be found in geometryrather than arithmetic. The behaviors childrendevelop in the context of arithmetic a, e used byCSMF' as an intuitive entry to performing sideand reflection transformation activities.

The CSMP materials are being field testedand evaluated. To date, the evaluative efforthas been Yormative in nature, leading to the re-vision of materials and administrative proce-dures. Comparative, hard data of the plIot1970/71 school year have not been published.

The year 1970/71 was the third year of imple-mentation of the program, which was used inthe classrooms of four schools for evaluativepurposes. The variation of facilities and per-sonnel make interpretation of data specious atbest. but the need for careful evaluation is rec-ognized by the CSMP project. Informal eval-uation based on data collected up to midyear1971 reports that children are enthusiasticabout CSMP materials and that students whowere able to work indepen iently tended towork completely through great amounts of ma-terials Pilot StOdy teachers. after training, wereable to cope with the new tasks of the program,including a complex system of individualization(CSMP 1971). The cost of CSMP is. however,quite high.

The CSMP materials suggest a variety of re-search and evaluation problems. If the prog-


nostic judgment of eminent mathematicians isaccepted as justification for the introduction ofnontypical content into the early elementarycurriculum, the question of determining suit-able means of instruction still remains. Inteaching probability without a background ofrational-number concepts, one is forced toconcentrate on building a primitive intuitionalbase for future learning, an area of researchnut well explored. The Papy minicomputershares many of the characteristics of the Cuise-naire rods and suggests comparable researchand pedagogical problems. The task of seekingminimal displays of counters does impose aproblem-solving orientation that pervades sev-eral activities. Few laboratory teaching devicesestablish a task-determinate mind set extend-ing over a protracted time period. The impactof an extended problem task on learning needs

Red is for +3

Write in Vie missing mimber names.

288 CHAPTER TWELVE

s'udy The possibilities of exploiting the map-ping experiences in the early grades warrantexamination.

CSMP is the only development project thathas given first priority to the question of themost appropriate content for the future (Len-nart 1970) Answers to this presumptive ques-tion are of the utmost importance to society.CSMP has concentrated on determiningwhether it is possible to teach the identifiedconcepts, but the project has not yet dissemi-nated widely the results of its deliberations.Other mathematicians and educators involvedin the task of curriculum design would benefitfrom the collective wisdom focused by CSMPon this question.

compensatory education

The DISTAR. and ILM programs had theirroots in the federally sponsored efforts to edu-cate the disadvantaged. The characteristics ofthese materials are testimony to the potentialimpact of the Head Start and Follow Throughprograms on the education of the young. Interms of the number of students alone, theseprograms would be judged significant to thefield; in 1967, only three years after inception,there were 12.927 Head Start centers (Cicirelliet al 1969) But more significant for the futureof early childhood education is the testing ofdistinctive approaches in the teaching of theyoung Many of the ideas being found produc-tive in the education of the disadvantaged willbe used in the classrooms of the more fortu-nate.

The implications of the Head Start and Fol-low Through programs are not clear for mathe-matics The growth of the number of programsfor the preschool child from 1964 to the presenthas exceeded the capabilities of the mathemat-ics education community to participate in thedesign of the programS. The evolution of pro-grams has taken place with no genetic blue-print comparable to the 1959 Report of the Col-lege Entrance Examination Board (CEEB),which was so fundamental in shaping the sec-ondary school curriculum (CEEB 1959). The

impact of no curricular guidelines was corn:pounded by the noncontent mind set of manyspecialists in the field of early childhood educa-tion. Ellis Evans, an authority in the field of earlychildhood education. states, -Childhoodeducators who are content-oriented seem tohave been viewed with suspicion by those whofeel such an emphasis implies a priority higherthan children themselves" (Evans 1971). Theimportant objectives of school readiness andsocialization have appeared exclusive of con-tent-oriented objectives.

The planning guidelines for Head Start pro-grams reflect this bias. They do not emphasizeconceptual objectives nor refer specifically tomathematics (Evans 1971). Indeed. six of theseven aims stated in the federal guidelineswere noncognitive. These guidelines set thepattern for the majority of funded curriculumprojects outside as well as inside governmentsponsorship. This is not to say that the aims ofmost Head Start programs are not important.Building responsible family attitudes. imdrov-ing physical health, increasing self-confidence,and the like are important.

Head Start programs typically provide pre-school experience for at least a summer butpossibly up to a year's duration prior to schoolentrance. The broad. general goal is to providelearning experiences to supplement and com-pensate for the disadvantaged background at-tributable to the social context of the first yearsof life. Follow Through programs differ in that aconcerted effort is made to modify the total en-vironment of the child in addition to providingthe preschool experience. It is assumed thatfurther environmental planning will increasethe probability that the gains made in pre-school through tile' first years of elementaryschool will be sustained. Follow Through wasauthorized in 1968. and in 1969 nineteen FollowThrough projects enrolled over 16,000 children(Evans 1971).

The implications of compensatory educationfor mathematics may well be realized in termsof the applications of more general character-istics of the programs. For example, parentalinvolvement in the preschool program is an es-sential characteristic of many Head Start andFollow Through projects. The Florida ParentEducation Program at the University of Florida

(Gordon 1971), the DARCEE Training Programfor Mothers at George Peabody College (Bar-brack. Gilmer. and Goodroe 1970; Cupp 1970:Gray 1971; Stevens 1967), and the Home-School Partnership Program at. Southern Uni-versity in Baton Rouge (Johnson and Price1972) have each used and studied carefully theintervention of parents as a mechanism forteaching the disadvantaged. Typically the par-ent is incorporated into the instructional proc-ess and. for these three programs. may be in-volved in mathematical activities.

Behavior analysis is another means of in-struction that pervades al! aspects of manycompensatory education programs. Skills andconcepts are carefully analyzed to determinethe order of teaching. Careful scheduling of re-inforcement and rewards as well as the order ofpresentation of the concepts typii!es most suchprograms. The DISTAR and ILM programs ex-emplify the extension of this instructional ap-proach from behavior and language to mathe-matics Other projects making extensive use ofthis technique are the Follow Through projectat the University of Kansas (Bushell 1970.Bushell 1971) and the Tucson Early EducModel program at the University of Arizona(1968).

Concern for helping children develop a moreadequate self-concept is the primary aim ofmany programs. Behaviors labeled with wordssuch as initiative. self-direction, self-respect.curiosity, and commitment are important de-terminants of instruction. Specific cognitive ob-jectives are not as important as these behav-iors. and mathematics may serve simply as thevehicle for acquiring such behaviors. Childrentypically participate in determining their owneducational goals The Bank Street College ofEducation Follow Through program (n.d.) istypical of such programs. The attention tomathematics in such settings is incidental andoften accidental. The present knowledge ofreadiness. efficiency. and the critical period forlearning mathematics may be inadequate to re-quire creating and implementing mathematicsprograms at the preschool level at the expenseof realizing other goals. The absence of a defin-able program in mathematics need not implythat mathematics is either poorly treated or ig-nored. Mathematical instruction in such pre-


school settings may remain incidental, but itmay also be high-quality instruction. even care-fully planned instruction.

Two types of evaluation have been applied toFollow Through and Head Start programssummative and comparative. Summative eval-uation of Head Start indicates that Head Startgenerally has not been successful in generatinggrowth in cognitive performance by children.The joint Westinghouse -Ohio University eval-uation team for Head Start lists more than thirtyevaluative studies agreeing with their findingthat Head Start has had -only limited impact oncognitive performance of children.' (Cicirelli.Evans. and Schiller 1970). This lack of payofffor the compensatory education dollar was per-haps predictable, the guidelines for Head Startproposals emphasized other than cognitivegrowth. The programs stressing cognitive ob-jectives have tended to be more successful inpromoting growth that was maintained two andthree years later. Weber's Early ChildhoodEducation. Perspectives on Change (1970) andEvans's Contemporary Influences in EarlyChildhood Education (1971) describe in detailsome of these more successful Head Start pro-grams.

Comparative evaluation efforts have resultedfrom the variation from program to program ininstructional design and priorities of educa-tional objectives. The integrity of the approachof various programs has been tested by com-parative analysis. Soar and Soar (1972) in theNSSE yearbook devoted to early childhoodeducation report systematic observations ofteacher and pupil behaviors that indicate thepurity and consistency of approach for five Fol-low Through projects. Kamii's analysis of theobjectives of programs offers another exampleof the comparative evaluation efforts (1971). Abasis for studying the long-term impact of dis-tinctive preschool programs on subsequentschooling is being built by such studies.

Compensatory education has had a stimu-lating. salutary effect on preschool education.Based on a social pathology model of pre-school education (Baratz and Baratz 1970) ithas raised a variety of issues and problems(Rosenthal et al. 1968). The store of techniquesand materials applicable to mathematics hasbeen made richer by compensatory education.

290 CHAPTER TWELVE

other directions ofpotential influence

AS

Two "programs** are discussed in this sec-tion, the television program Sesame Street'.and the phenomenon of local curriculumprojects. Judgments of the effect of each aredifficult. "Sesame Street" is not school based.The ineffable character of the viewing habits ofchildren means that the relation of "SesameStreet" to mathematics teaching in the schoolsis difficult to ascertain. Curriculum projects thatare locally sponsored are quite common. Mostare bootstrap operations depending on teachercommitment. They are often directed towardlaboratory learning or behavioral objectives,and the range of the quality of the materials isgreat.

"sesame street"

Preschool and kindergarten teachers shouldbe acquainted with TV offerings like "SesameStreet" Although subtraction may not be ade-quately or purely modeled by a monster eat-ing cookies, the monster's habits do provide areferent for the process of establishing an ap-preciation for subtraction. The reason for dis-cussing "Sesame Street" in this context is notto pass judgment on the quality of instructionbut to suggest its potential for providing pointsof entry and familiar examples. We refrain frommaking judgmental statements because we donot at present know what impact such TV pro-grams may have on education in mathematics.The assumptions about learning on which thedesign of "Sesame Street" is based are dis-cussed in the May 1972 issue of the HarvardEducational Review (Lesser 1972). Severalquestions, however. of special concern to pre-school and early elementary education areraised:

1. Is a high degree of stimulation necessaryor appropriate to attempt to educatesmaller children? Do children develop adependence on such stimulation that can-not be honored by the public schools?

)(-- 0 9fli..r..0

i

2. Do the stimuli that are intended to hold in-terest but that are superficial to the in-tended learning interfere with the learningor change the nature of the learning? If so,do they leave a detrimental effect? (For awidely known example to help make thepoint, some kindergarten children whoinitially learned the alphabet by song re-ceive a mild jolt on finding out that "L -M-N-O" is not a single letter.)

3. Is a child's attention span as fragile a thingas the "Sesame Street" programming as-sumes it to be? If not, does refraining topush for longer attention to a single stim-uli set have an effect on the potential for alonger attention span at a later time?

4. Although both culturally deprived and ad-vantaged children appear to profit fromviewing "Sesame Street," do indicationsthat the program does nothing towardclosing the achievement gap betweenthem suggest that its potential is re-stricted to complementing or clarifyingexperiences that involve the child moredirectly?

5. More specifically to the concern of thischapter, is the fact that "Sesame Street'viewers often learn to "say" things mathe-matical long before developing corre-sponding concepts an advantage or a dis-advantage in learning mathematics? (Forexample, children learn to "say" the num-ber names and repeat basic facts of addi-tion before being able to "count" a smallset of objects.)

The questions raised above are not solelythe concern of "Sesame Street." They areamong those questions that should be consid-ered in any attempt to educate young children.

p.

locally developed curriculum

A phenomenon spawned in part by the avail-ability of government funds at the local levelduring the 1960s is that of the "locally devel-oped curriculum." Other factors contributing tothe development of this type of curriculum area weariness over the number of new topics in-cluded in textbooks and the corresponding ne-

cessity to be selective; commitments to behav-ioral objectives that do not fit standardizedachievement tests or the unit examinationsprovided by textbook publishers, and com-mitments to the proposition that relevance isgained by fitting the curriculum to the commu-nity setting.

By "locally developed" is meant "developedby a school system's employees under contractobligation." Patterns range from forming teamsof curriculum developers with federal funds ad-ministered by the school system to appointir.gcommittees of teachers who assume responsi-bilities without additional reimbursement.

The search for "model" locally developedmaterials that did not erl'e3y federal funds wasthwarted by the fact that such materials are of-ten being considered for publication by com-mercial firms The better of these materiais arelikely to gain visibility without the aid of thischapter and are likely to undergo enough revi-sion to advise against evaluating what is cur-rently available.

A perusal of locally developed materials sug-gests that few can be named that will have awide impact on early elementary school prac-tices. The primary impact may remain at the lo-cal level, since development efforts character-istically demonstrate teacher commitment andenthusiasm on the part of those directly in-volved but not by others. In fact, the com-mitment and enthusiasm exhibited in creatingeven poor and mediocre materials suggeststhat the degree of the teacher's involvementmay be the key to his success with them.Among the questions raised by this observa-tion is the following: Given a choice between in-ferior materials with strong teacher com-mitment and superior materials with weakteacher commitment, which set most facilitatespupil achievement and the development ofpositive attitudes?

A number of local attempts to revise themathematics curriculum have begun with list-ings of behavioral objectives as the first step.The most common scheme observed in arriv-ing at the listing of objectives is that of outliningthe content of several commercially availabletextbooks, then screening and ranking objec-tives according to the degree of their impor-tance, and finally deciding on the flow diagrams


for treating objectives. Many such attempts dogo through the second step of creating a reser-voir of test items for the objectives. However,rarely does the development of these curricularmaterials go beyond identify;:by specific pagenumbers in textbooks fir the objectives andperhaps supplying short, teacher-made work-sheets.

It seerr, safe to suggest that the primary ad-vantages, if any, derived by the local schoolsystem from such attempts are a higher degreeof teacher commitment to the mathematicsprogram and the degree of teacher educationresulting from the effort. The listings and rank-ings of objectives are usually no better thanthose suggested by the table of contents of anysingle commercial series; often they are infe-rior. For example, one such set of objectivesfails to include establishing the area of a rec-tangle as length times width, thereby denyingthe option of treating the multiplication of ra-tional numbers in terms of area.

One effort to develop a program locallywhich has taken care in creating instructionalmaterials is that by the Winnetka, Illinois,schools (n.d.). The packaging format is that ofshort, consumable student workbooks withcorresponding teacher manuals and exam-ination items. Verbalization in the student ma-terials is minimal. Many of the booklets treat asingle concept or topic, with a number of keyconcepts (e.g., place value) covered by severalbooklets.

These elementary materials may be viewedas a compromise between extreme positionson teaching for understanding and teaching byrepetitive drill, between practices that dependheavily on manipulative materials and practicesthat deal exclusively in symbolism, and be-tween insisting that each child must be a dis-coverer and believing that each child is a ma-chine to be programmed. The sequence andpacing of content does not require an early us-age of precise mathematical symbolism or ter-minology in dealing with partially developedconcepts. Nor, in attempts to foster conceptformation, do the materials dart from one stim-uli set to the next at a rate that might frustrateall but the brightest pupils.

The early treatment of "less than" and"greater than" may serve to demor.strate the

,'(13l',.-.)

292 CHAPTER TWELVE

(1)

style of the Winnetka materials. The first sets ofexercises ask children to cross out the"greater" set:

(3)

Items in set 2 provide a transition from set 1 toset 3. Although the word greater is used with in-struction for the first set. examples given witheach worksheet may allow the assignment tobe completed without recognizing the word.

Later in the unit the sequence from pictorialto numeral is repeated with the seesaw:

(2) (3)

The unit on "greater than" and "less than"also devotes some exercises directly to the or-dinality of counting numbers.

in summary, most attempts at curriculum de-velopment at the local school level tend to beoriented toward behavioral objectives or labo-ratory materials The strength of such attemptsappears to derive from teacher involvement.The main concerns are the amount of moneyand time spent and the quality of the resultinginstructional materials.

retrospect

The curricular development of the past dec-ade has been extensive. Burgeoning enroll-

ments, the change of perspective on objec-t;:es, new knowledge of children's learning,and plentiful resources have all contributed to.encouraging creativity in designing mathemati-cal experiences for young children.

Most program creators at least honor psy-chology by stating that their instruction is psy-chologically based; sometimes this is true. Theinfluence of Gagnean organization of objec-tives is evident in many programs. The increas-ing usage of active, manipulative instructionalmaterials has contributed to the diversity ofmaterials and to the significantly broadenedrange of quality of instructional approaches.The commitment to "psychologizing" the in-struction of young children has led to the im-plementation of several instructional tech-niques that have not withstood the test of time.In some programs, such as With the transac-tional variable in the ILM games approach, theprecisb nature of the variable providinguniqueness has not been clearly identified. Inbrief, an evaluation of programs is often notenough; research to isolate and refine the op-erant factors of the approach is needed.

The preschool and kindergarten mathemat-ics program deserves careful attention by themathematics education community. Preschoolprograms have been established in many com-munities in the last five years. Most programshave no carefully designed mathematical com-ponent, since no set of traditions existed in thisarea and the quantity of appropriate materialsis small. The mathematics education commu-nity can ill afford to ignore the stake it has in thenew emphasis on preschool education. It is notknown whether an informal, precursive explo-ration of mathematical topics ol a formalized,hierarchical treatment directed toward specificgoals has the greater potential for subsequentschooling. Clearly preschool and kindergarteneducation is more cognitively oriented than inthe past. The directions for the future are beingdetermined now. The compelling need forcareful, comparative, longitudinal evaluation isevident.

The historic issues identified in chapter 2 areevident in the recent curricular thrusts. Thepremium to be placed on skill learning, the useof activity-based teaching strategies, and the

relation of mathematics to the rest of the cur-riculum remain significant problems.

The final generalization concerning devel-opments in early childhood education is that

references

Bank Street College of Education Bank Street Apprbachto Follow Through Working paper. Mimeographed.New York: The College. n.d.

Baratz. Stephen S.. and Joan C. Baratz. "Early ChildhoodIntervention' The Social Science Base of InstitutionalRacism." Harvard Educational Review 40 (1970): 29-50.

Barbrack, C R.B R Gilmer. and P. C. Goodroe Informa-tion on Intervention Programs 0/ the Demonstration andResearch Center for Early Education. DARCEE Papersand Reports. vol. 4. no. 6. Nashville, Tenn.: George Pea-body College for Teachers. 1970.

Bereiter, Carl. Instructional Planning in Early Com-pensatory Education. Phi Delta Kappan, March 1967.pp. 355-58.

Bereiter. Carl. and Siegfried Engelmann. The Educatorand the Child Develop mentalist: Reply to a Review." Edu-cational Leadership 25 (October 1967). 12-13.

Biggs. Edith. Mathematics for Younger Children. New York.Citation Press. 1971.

Bushell. D. The Behavior Analysis Classroom. Lawrence.Kans.. Development Center for Follow Through, 1970.

. Behavior Analysis Teaching, Lesson 3: MeasuringObjectives. Lawrence. Kans.: Development Center forFollow Through. 1971.

Center for Cognitive Learning. 'Developing MathematicalProcesses." Experimental copy. Mimeographed. Madi-son. Wis.. Wisconsin Research and Development Center.1971.

Central Advisory Council for Education (England I. Childrenand Their Primary Schools, A Report 0/ the Central Advi-sory Council for Education. 2 vols. Vol. 1. Report. vol. 2.Research and Surveys. London: Her Majesty's StationeryOffice. 1967.

Cicirelli. Victor G.. et al. The Impact 0/ Head Start. An Eval-uation 0/ the Effects 0/ Head Start on Children's Cogni-tive and Affective Development. Report of a study under-taken by Westinghouse Learning Corporation and OhioUniversity under 0E0 contract B 89-4536. Preliminaryreport. April 1969.

Cicirelli. Victor G.. John W. Evans. and Jeffry S. Schiller."The Impact of Head Start' A Reply to the Report Analy-sis." Harvard Educational Review 40 (1970): 105-29.

College Entrance Examination Board (CEEB). Commissionon Mathematics. Programs for College Prepatory Mathe-matics, New York: CEEB. 1959,

Comprehensive School Mathematics Program (CSMP).Basic Program Plan St Ann, Mo Central MidwesternRegional Educational Laboratory. 1971.


federal funds have had a strong influence. Theseminal ideas leading to unique program de-signs were developed, refined, and imple-mented only because resources were available.

Cupp. Janet (Camp). A Skill Development Curriculum forThree. Four, and Five-year-old Disadva,itaged Children.DARCEE Papers and Reports. Nashville. Tenn.: GeorgePeabody College for Teachers. 1970.

Dienes. Zoltan P. Building Up Mathematics. 3d ed. London.Hutchinson Publishing Group, 1967.

Elkind. D. "Piaget and Montessori." Harvard EducationalReview 37 (Fall 1967): 535-45.

Engelmann, Siegfried. Conceptual Learning. Belmont,Fearon Publishers. 1969.

Engelmann, Siegfried, and Doug Carnine. Teacher's Guide.DISTAR Arithmetic I and II. Chicago. Science ResearchAssociates. 1970.

Evans. E. D. Contemporary Influences in Early ChildhoodEducation. New York. Holt. Rinehart & Winston. 1971.

Featherstone. J. The Primary School Revolution in Britain.Reprint of four articles published in the New Republic,1967. Washington. D.C. New York: Pitman PublishingCorp.. 1967.

Fletcher. Harold. et al. Mathematics for Schools, Teacher'sResource Book, Level 1. London: Addison-Wesley Pub-lishers. 1970.

(a). Mathematics for Schools, Teacher's ResourceBook, Level II Books 1 and 2. London. Addison-WesleyPublishers. 1971.

_ (b) Mathematics for Schools, Teacher's ResourceBook, Level II. Books 3 and 4. London: Addison-WesleyPublishers. 1971.

Fogelman. K. R Piagetian Tests for the Primary School.London. National Foundation for Educational Researchin England and Wales. 1970.

Gordon. I. J. The Florida Parent Education Program."Mimeographed. Gainesville. Fla.. Institute for the Devel-opment of Human Resources. 1971.

An Instructional Theory Approach to the Analysisof Selected Early Childhood Programs." In Early Child-hood Education, edited by I. J. Gordon. Seventy-firstYearbook of the National Society for the Study of Educa-tion (NSSE). Chicago: NSSE, 1972.

Gotkin, Lassar G. "The Interdependent Learning Model. AFollow Through Program. Mimeographed. New York:Institute for Developmental Studies, New York University,1971.

Gray, Susan W. Home Visiting Programs for Parents ofYoung Children. DARCEE Papers and Reports. vol. 5, no.4. Nashville. Tenn.. George Peabody College for Teach-ers, 1971.

C111.7.-..

294 CHAPTER TWELVE

Hodges, Walter L "Review of Carl Bereiter and SiegfriedEngelmann in Teaching Disadvantaged Children. En-glewood Cliffs. N.J. : Prentice-Hall, 1966. Also in Ameri-can Education& Research Journal 3 (November 1966):313-14.

Hymes, J L "Review of Carl Bereiter and Siegfried Engel-mann " In Teaching Disadvantaged Children. EnglewoodCliffs, N.J.' Prentice-Hall, 1966. Also in EducationalLeadership 24 (February 1967): 463-67.

Johnson. E. E., and A. E. Price. "Home-School Partnership:A Motivational Approach." Mimeographed. BatonRouge. L3 . 1972.

Karim. Constance "Preschool Education. Socio-emotional.Perceptual-motor, and Cognitive Development." InHandbook on Formative and Summative Evaluation ofStudent Learning, edited by Benjamin S. Bloom, J.Thomas Hastings. and George F. Madans. pp. 281-344.New York: McGraw-Hill Book Co., 1971.

Lennart, R.. ed. The Teaching of Probability and Statistics.Report of the international conference sponsored by theComprehensive School Mathematics Program. St. Ann,Mo Central Midwestern Regional Educational Labora-tory. 1970.

Lesser. Gerald S. "Learning. Teaching, and Television Pro-duction for Children. The Experience of Sesame Street."Harvard Educational Review 42 (May 1972). 232-72.

Montessori. Maria. Advanced Montessori Method. 1917.Reprint. Cambridge, Mass.: Robert Bentley, 1964.____. The Discovery of the Child. Madras. India: Kalak-

shetra Publications, 1948,..... _ _ . The Montessori Method. Reprint. Cambridge.

Mass.: Robert Bentley, 1965.

Nuffield Mathematics Project. Computation and Structure.New York: John Wiley & Sons. 1967.

_ (a) Environmental Geometry. New York. John Wiley& Sons, 1969.

(b). Graphs Leading to Algebra. New York: JohnWiley & Sons, 1969.

Palmer, R Space. Time and Grouping. New York. CitationPress. 1971.

Progressive Education Association, Committee on theFunction of Mathematics in General Education of theCommission on the Secondary School Curriculum.Mathematics in General Education. New York D. Apple-ton-Century Co.. 1940.

Rosenthal, Robert, and Lenore Jacobson. "Teacher Ex-pectations for the Disadvantaged." Scientific American,April 1968. pp. 19-23.

Sheenan, M. C. "A Comparison of the Theories of MariaMontessori and Jean Piaget in Relation to the Bases ofCurriculum, Methodology, and the Role of the Teacher."Doctoral dissertation, St. Johns University, 1969.

Smith, Marshall, and Joan S. Bissell. "Report Analysis: TheImpact of Head Start." HarvardEducational Review 40(1970): 51-104.

Soar, R. S., and R. M. Soar. "An Empirical Analysis of Se-lected Follow Through Programs: An Example of a Proc-ess Approach to Evaluation" In Early Childhood Educa-tion, edited by I. J. Gordon. Seventy-first Yearbook of theNational Society for the Study of Education (NSSE). Chi-cago: NSSE, 1972.

Southwest Educational Development Laboratory. "FirstGrade Mathematics Component Conceptual Design."Mimeographed. Austin, Tex.. The Laboratory, n.d.

Steely, D "The Effectiveness of Direct Verbal Instructionand Behavior Modification Techniques with ExtremelyLow-Performing Children." Master's thesis, University ofOregon, 1971.

Stevens. Joseph, Jr. Instructional Materials and Their Usein an Early Training Center, Paper delivered at the FourthInternational Childhood Education Conference. St. Louis.26 March-1 April 1967. DARCEE Papers and Reports,vol. 2, no. 1. Nashville. Tenn.: George Peabody Collegefor Teachers. n.d.

"The Tucson Early Education Model." Mimeographed. Tuc-son: Arizona Center for Early Childhood Education. 1968(revised 1971).

Van Arsdel, Jean, and Joanne Lasky. "A Two-dimensionalAbacus-the Papy Minicomputer." Arithmetic Teacher19 (October 1972): 445-51.

Weber, Evelyn. Early Childhood Education: Perspectiveson Change. Worthington, Ohio: Charles A. Jones Pub-lishing Co., 1970.

Weber, Lillian. The English Infant School and Informal Edu-cation. Englewood Cliffs, N.J.. Prentice-Hall. 1971.

Winnetka Materials. Mimeographed. Winnetka, Ill.: Win-netka Public Schools. n.d.

Winters. Andrea. ed. "A Math Manual for the Inter-dependent Learning Model." Mimeographed. New York:Institute for Developmental Studies, New York University,1971.

index

Abacus. 53. 177-78Achievement, relation of to other

traits, 45-52Activity learning, active participation

in, 9-10. 275-81. 292Activities

addition and subtraction, 166-81classification. 82, 98-102, 120comparing and ordering. 102-4,

120estimating. 84-85, 151-52, 189,

239, 243fractions. 195-202geometry, 79-81, 84, 107-10, 122-

23, 212-14. 217-19. 222-24graphing and organizing data, 78,

106-7. 122. 267-69integers. 270-71logic, 264-65measurement. 78, 81, 104-6, 120-

22, 232-38, 241-48multiplication and division, 181-89number. whole. 83-84. 112-23.

131-36number theory. 269-70number sentences. 259-62patterns. 79, 111-12, 122-23premeasurement. 222-24. See also

Activities. comparing and order-ing

prenumber, 98-112probability, 86-87, 266, 267relations, 255-58role of, 39. 238

Addition. 24. 161 -62. 165-68. 173.178-81

two-digit numbers. 175-78Addition and subtraction, 162-63,

165-81as related or inverse operations.

163. 169-74. 263number sentences for problems.

259-61preparation for, 137research on learning. 47 -48, 52-53.

55-58, 60. 63Affective factors in learning. 45 -46Algorithm, research on, 52, 56-61Applications. role of, 27, 32-37, 280.

See also Real-world situationsApproximation. See Estimation and

approximationArea, 85. See also GeometryAssociative property. See Properties

of operations

Attention to relevant atributes. 4Attitudes, 45-46Attribute blocks, 53, 63, 77, 232Attributes, attention to in learning,

4-5

Base, in numeration, 137. See alsoNumeration

Behavior modification, 10Bereiter-Engelmann program, 50British Infant School. 54. 279-81Brownell, William, 19-21, 48-49. 52-

54. 57-59Bruner, Jerome. 96

Cambridge Conference. 285Capacity and volume. 106, 122, 231,

244. 246. 249. See also Measure.ment

Cardinal number. 47. 55, 113, 130.137

Checking. in computation, 61Class inclusion, 12Classification activities, 17, 47-48, 77.

82. 98-102. 120. 258-59Classroom organization. See Instruc-

tionCognition. cognitive development, 1-

14. 50-52, 116. 274Commutative property. See Proper-

ties of operationsComparison. See also Order, Se-

quence of numbersactivities, 102-3. 120prenumber. 126-27whole numbers, 130. 134-35fractions, 201-2

Compensatory education. 284. 288-89

Composite number, 269Comprehensive School Mathematics

Program, 285-88Computation

in curriculum, 16-20, 25, 27, 32-37in measurement. 240research, 52, 55-61

Computer-assisted instruction. 52Concepts, 20. 32-37Concrete materials

in curriculum, 25, 37-39, 292with fractions. 192 96, 198 -99, 203with graphing. 268

295

with measurement, 231-32, 250with number and numeration, 131.

137, 159with number sentences, 261with operations, 162, 189-90research on, 50-54, 60-61. 63

Congruence, 220, 194. 254-56Conservation, of number or quantity,

4-5, 47-48, 53, 128-30, 171. 208-9, 229, 253. 279. 282

Counting, 17. 113. 117-19, 127. 137,252

in DISTAR program, 282-84partial, 166related to operations, 165, 182related to measurement, 238-39,

241 - 44,250related to numeration, 149tens, 143

Cuisenaire rods, 53. 104, 262, 275.277. 286

Curriculum, 15-41. 273-94balance in. 17-18. 22-23, 28-40early age (preschool), 17, 273-94issues, 17-28spiral, 23

Data collection. See GraphingDecimal numeration system. See Nu-

merationDeveloping Mathematical Processes

Program. 275. 277-79Developmental aspects. See also In-

dividual differencesof learning. vii, 22. 46-51in geometry, 206-8

Diagnosis for instruction, 52. 158-59.See also specific content areas

Dienes materials, 54, 275, 286Disadvantaged and special pro-

grams. 54. See also Com-pensatory education

Discovery in learning, 25. 29, 277Discrimination in learning, 5. 8-9DISTAR Arithmetic program, 281-284Division. See also Multiplication and

divisionalgorithm, 60-61, 187-89concepts, 34, 56, 164, 186facts, 186-87in problem solving, 72-76

Drill-and-practice programs, 52, 58Drill theory, 19. 21, 57

296 INDEX

Early age. See also Kindergartenactivities. 78-82. 95-124. 131-36.

212-14, 222-24. 232-38curriculum. vii. 17, 95-24. 275-77.

288-89. 292Follow Through program, 288Head Start program, 54, 288knowledge possessed by. 48-50.

55. 130 -31, 206-8. 228-30programs. 288-89

Equilibration. 46Equivalence relations. 18, 255-56Error analysis. 52Estimation and approximation. 84-85.

151-52, 189. 239, 243Evaluation of learning. See also spe-

cific content areasin geometry, 214-15. 219-21in number and numeration. 158-59

Even number. 25. 269

Fact learning. 24. 57-59. 174additior. 165-67.169-70. 173. 178-

80division. 186-87multiplication. 184-86subtraction, 168 71. 178-80

Feedback, importance of. 130Follow Through program. 288Fractional numbers, fractions. 29. 49.

61, 191-203

Geoboard, 80. 241Geometry

activities, 79-81, 84, 107-10. 122-23. 212-24

concept formation. 8.11. 55in curriculum, 17, 24. 26, 28-29, 32.

61, 206-11, 224-25. 280topology. 209-15transformational, 215-21

Goals, 16, 21Grade placement of content. 25, 29Graphing. organizing and presenting

data, 27. 29. 78, 80. 106-7. 122.242-43. 267-69, 280

Head Start programs, 54. 288Hypotheses making, 9

lig and Ames. 46.48, 52.206. 207Impulsive-reflective style. 48. 51-52Incidental learning. vii, 19. 21Individual differences. 3-4

organization for instruction. 27. 39-40

Individualization, individualized in-struction. 39-40. 277. 286. Seealso Instruction

In-service, effects of, 53Instruction. See also Activities; Learn-

ing: Meaningclassroom organization, 39 -40. 87

92. 174

planning. 3-4, 97-98, 123-24, 174-75, 189-90

research on, 50-63Integers, 270-71Intellectual development, 46Intelligence and learning, 3, 46Interdependent Learning Model, 284Interpersonal relations. 13Inverse operations, 56. See also Ad-

dition and subtraction: Multi-plication and division

Interview technique, 52

Kindergarten. See also Early ageknowledge of children on entering.

48-50, 55, 130-31, 207-8, 216effect of experience, age. or

socioeconomic level in, 50-51

Languagebilingual. 54, 281development, 7-8, 55, 97. 279,

281-85for numbers, 135, 141-44, 193-95,

197-98Learning. See also Instruction, Mean-

ingfactors for success in, 16research in, 1-14research in, mathematics, 43-67

Linear measurement. See Measure-ment

Logic and logical thinkinghypothesis making. 9-10in the curriculum. 62, 261-62, 264-

65in problem solving. 78transitive inferences, 11-12

Manipulative materials. See Concretematerials

Mathematical model, 30, 70-71, 97,259-61

Mathematical sentences. See Num-ber sentences

Maturation, 46Meaning. meaningful learning. 17-24,

36. 238. 263. See also Learning;Instruction; Activities; specificcontent areas

research on, 51-60Measurement. See also Activities

area, 84-85. 208, 241-42, 245-46basic concept and process, 208.

230-35. 238-40capacity. 84, 229. 231-33. 235. 244.

246-47. 249in the curriculum. 29. 61. 280linear, 81. 121, 229. 233-34. 236,

241, 242. 243. 244-45, 249metric system. 248-50perimeter. 85-86. 208premeasure, 102-6, 222-24scale reading and making. 239-40,

245

standard units in, 245. 248-50temperature, 235-36. 249time duration. 247-48units, 244-45, 248-50weight, 78, 105-6, 121-22. 236.

244, 249Memory in problem solving, 11-13Metric system, 248-50Mexican-American programs, 54, 281"Modern" mathematics, 19, 22-27Montessori, 275-77, 279Motivation, 45, 54Multiplication, 163-64. 181-85

algorithm. 187-89concept, 30-31, 163, 181-82tact learning, 184-86

Multiplication and division, 163-65.181-89

number sentences. 260-63as related or inverse operations,

164-65, 186-87, 263research on. 54-58

Nonmetric geometry. See GeometryNuffield Mathematics Project, 280Humber. See also Numeration. Fact

learningfrom 0 to 10, 112-17. 126-36from 10 to 100, 137-53from 100 to 1.000. 154-57cardinal, 47, 55, 113, 130, 137even and odd, 25. 269foundational ideas, 127, 253fractional, 191-203integers. 270-71and measurement, 230. 238ordinal, 47, 55, 130, 134-35. 194prenumber. 98-112

Number line use, 152, 168-69. 270-71Number sentences, 31, 59. 71. 171,

259-63Number theory. 269-70Numeration. 20. 24, 38. 116. 137-59

as used in algorithms, 174-78, 187-89

Observation and learning, 9Odd number. 25, 269One-to-one correspondence. 102-3.

127, 131-34. 252-56. 268Operations on numbers. 20. 34. 55-

61, 161-90. See also Addition:Subraction; Multiplication: Divi-sion

Order. ordering. See also Com-parison; Sequence of numbers

activities, 103-4. 120numbers 0 to 10. 118. 134-35numbers 10 to 100, 148-49relations. 256-57relations along a path. 211tens and ones. 150-51

Ordinal number. 47. 55. 130, 134-35.194

Patterns. 79, 111-12, 122-23Perception, perceptual. 110. 127. 129Perimeter, 85-86. 208Personality factors, related to

achievement. 51Physical materials See Concrete ma-

terials; Real-world situationsPiaget. Jean. 4. 11, 46-50, 56. 62. 97.

110, 128. 208-9, 229. 231. 240.250. 253. 275, 279

Place value. See NumerationP-actice, ways to organize. 51Preschool. See Early agePrime number. 269Probability. 27, 62, 86-87. 265-67Problem solving. 3, 10-13, 17. 27, 29.

51, 63, 69-93. 172. 181, 189, 199-202, 232, 237-38, 250

Progressive Education Association.280

Programmed instruction. 10Properties of operations (associative.

commutative. etc.). 24.35, 57-58,162-65. 173, 178, 183, 185

Physical world. See Real-world situ-ations

Quantitative thinking and information.21-22. 57, 126. 130. 230

Readability of mathematics materials.55

Readiness. 22. 50. 277Real-number system. 24Real-world situations, vu. 16. 21-22.

27. 29, 70-71. 162. 189-90, 250,258-60. 278-79. See also s aecihccontent areas

Reflective-impulsive style, 48, 51Relations. 225, 252-59Remediation 52Research

on cognition and learning. 1-14on mathematics learning, 43-67

Retention, 51

School Mathematics Study Group. 54Sedation. 47-48 See also Sequence

of numbersSequence of numbers. 130, 148-49,

157. See also'Order; ComparisonSequencing content. 17. 30.36Sesame Street. 54, 290Set. Sets

Cartesian product. 56, 163concepts. 24, 99empty. 115. 136equivalent, 113, 116. 171. 260and fractions. 195, 200inclusion, 137intersection. 211partitioning. se pa1a at or re-

moval. 136-37. 163. 169, 17374relations, 211union or joining, 119. 162. 165, 167.

259subset. 136

Sex differencesin attitudes. 46in achievement. 52-53

Shapes. See GeometrySimilarity, 252Skill learning. 19. 32Socioeconomic level and achieve-

ment. 53Spatial. See Geometry

cbcifih.-JO

INDEX 297

Subtraction. 31, 36-37, 55-56, 137,163. 168-69. 172, 178-81. See

« also Addition and subtractionalgorithms. 59-60two-digit numbers, 175-78

Statistics. 267-69. See also GraphingStory problems. 63. 172, 181. 183.

189. 259-61Strategies for thinking. 31-32. 35. 70Symmetry. 220-21. See also Geome-

try

Tangrams. 242Teacher/child relationship. 13 14Teacher attitudes and achievement.

46Teaching. See InstructionTemperature. See MeasurementTesting with young children, 45.51Time duration. 247-48Topology. 209-15Transfer in learning. 1 5-7, 51.57Transformation geometry, 215-21Transitive inference. 11-12

Understanding See Meaning

Verbal problems. See Story problemsVolume. See Capacity and Volume:

Measurement

Weignt, See MeasurementWord problems. See Story problems

Young children. See Early age: Kin-dergarten

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