This article was downloaded by: [Andrews University]On: 07 August 2013, At: 16:38Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Journal of Research in ChildhoodEducationPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/ujrc20

Place Value: An Explanation of ItsDifficulty and Educational Implicationsfor the Primary GradesConstance Kamii aa Curriculum and Instruction, The University of Alabama atBirminghamPublished online: 28 Feb 2011.

To cite this article: Constance Kamii (1986) Place Value: An Explanation of Its Difficulty andEducational Implications for the Primary Grades, Journal of Research in Childhood Education, 1:2,75-86, DOI: 10.1080/02568548609594909

To link to this article: http://dx.doi.org/10.1080/02568548609594909

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Journal of Research in Childhood Educat ion19 6. Vol. 1. No. 2

Place Value: An Explanation of ItsDifficulty and Educational Implicationsfor the Primary Grades

Constance KamiiCurr iculum and InstructionThe Universit y of Alabama at Birmingham

Copyr}ght 1986 by the Association forChildhood Education Intern ational

0256-8543/86

Abstract. R ecen t research has shown that place valueremains difficul! in third an dfo urth grad e, in spite of the fa ct that it is taugh t repeatedly in eoeru prim ary grade.This study was conduc ted to understand the cognitive processes underlying th isdifficulu]. A counting task was devised, based on Piaget's theon; of number , to findout if children in grades 1- 5 are constructing a sys tem of tens on a sys tem of ones.Only som e children in grad es 2- 5 evidenced this construction . The implications ofthefindinqsfor place value in struction are discussed. with obseroationsfroni secondgrad e classrooms in which children are encouraged to in vent their own ways ofdoin g doubl e-colum n addit ion.

Place value refers to the fact that in 333, forexample , the first 3 means 300, the secondone means 30, and the last one means 3. Thissystem of notation is very difficult for youngchildren to learn , and it is ta ught in everysingle grade of elementary school. Teachers inthe primary grades have long known that car­r ying , borrowing, and /or regrouping causetrouble for many children, and re searcherssuch as Ginsburg (1977) and Ashlock (1982)have documented thi s difficulty in detail. Inrecent years, some re searchers have focusedmore specifically on children's understandingof place value. M. Kamii (1980, 1982) nearBoston and C. Kamii (1982a, 1985) in the Chi­cago area found that only half of the childrenin fourth grade understand the 1 in 16to mean10. (Most of the others say that it means one.)In Canada, Bednarz and Janvier (1982, 1984)have shown similar re sults with a variety oftechniques, and concluded that place value re­mains very difficult in third and fourth grade.In Switzer land, Brun, Giossi, and Henriques(1984) likewise found that most of the childrenin the first two grades do not understand the1 in 12 to mean 10.

I would like to expre ss appreciation to Mieko Kamii,Wheelock College, Her mina Sinclair, niversity of Ge­neva. Ann Dominick and Linda Joseph. Homewood, Ala­bama public schools . and Suzanne Str inger, Selma. Ala­bama public schools for crit ically reading a draft of thi sarticle and making helpful suggest ions.

75

This article describes re search conductedto understand why place value is so difficultand t o determine the grade level where itmight be appropriate to introduce this deepconcept of notation.

Before proceeding, however, I would like tomake three points about ta sks that appear todeal with place value but in reality do not.F irst graders generally do not have troubleputting out the correct number of chips whenthey are shown the numeral 16. Likewise ,they have little difficulty writing thi s numeralwhen they are shown sixteen chips and areasked to write how many there are . Thesetasks are easy because first graders can learnthe cyclical order of digits from °to 9, and thewriting of the same series in the tens place.As long as these numerals designate wholenumerical quantities, first graders have no dif­ficulty. It is when part-whole relationships areinvolved in the notation (6, 10, and 16, for ex­ample) that trouble begins to appear.

Second, first graders can easily tell whether61 is gre ater or less than 16. They know that61 is more because it is written after 16, andthe counting word sixty -one also comes aftersixteen. In other words, it is not necessary toknow that the 6 in 61 means 60 to know that61 is gre ater than 16.

Third , many first graders easily rememberthat ten plus ten equals t went y and thattwenty and twenty equal s forty. Tens are easyto operate on as long as they are homogeneous

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

CONSTANCE KAMII

because one ten can be assimilated to onebunch without numerical precision. Whenten s and ones are mixed together, however,this heterogeneity causes trouble.

Wh y Is P lace Valu e So Ha rd ?Authors of all math series in existence today,as well as other empiricist math educators,believe that number concepts are abstractedfrom groups of object s , and that numerals canbe taught by association with these groupsand/or pictures of groups of objects . Forthem, place value is simply a problem of rep­resentation , i.e., a problem of teaching sur­face rules about which numeral in which col­umn represents the hundreds, ten s, and/orones that are observable in reality.

Piaget's theory and re search about the na­ture of number concepts and representationproved these empiricist as sumptions to bewrong for three reasons.

1. Number concept s belong to logico-math­ematical knowledge, the source of whichlies in the child's mental action , and notin sets of object in external reality (C.Kamii, 1982b, 1985).

2. Number takes many years to construct,and children cannot construct the sys­tem of ten s on the system of ones untilthey can conserve the system of ones intheir heads as they cut the system ofones into parts of ten. Figure 3 showsthe construction of the system of tens onthe system of ones. The conservation ofthe sy stem of one s will be clarifiedshor tly. Thi s article will pr esent evi­dence suppor ting thi s point.

3. Place value involves multiplication. Forexample , sixty-one means six times tenand one more. Multiplication has to beconstructed on addition , and is not asimple extension of addition. No re­search has been conducted to suppor tthi s point.

Th e Nature of Logico-MathernaticalK nowledge and the Construction of theS ystem of OnesPiaget distinguished among three kind s ofknowledge according to their ultimatesource -physical knowledge , logico-mathe­matical knowledge, and social knowledge.Physical knowledge is knowledge of objects

76

that exist in external reality. The color andweight of a chip are examples of physical prop­er ties that are in objects in external reality,and can be known by observation. Physicalknowledge is thus knowledge of what isobservable in external reality.

Logico-mathematical knowledge , on theother hand, consists of relationships con­st ructed by each individual. For instance,when we see a red chip and a blue one, andthink that they are different , thi s differenceis an example of logico-mathematical knowl­edge. The chips are observable, but the dif­ference between them is not. The difference isa relationship created mentally by the indi­vidual who puts the two objects into thi s re­lationship. The difference is neither in the redchip nor in the blue one, and if a person didnot put the objects into thi s relationship , thedifference would not exist for the observer.

Other examples of rela tionships an individ­ual can create between the chips are similar,the same in weight, and two. It is just ascorrect to say that the red and blue chips aresimilar as it is to say that they are different.If the individual wants to compare the weightof the chips, on the other hand , the objectsare the same in weight. If one thinks aboutth e objects numerically, however, t here aretwo. The two chips are observable, but thetwoness is not. Number is a relationship cre­ated mentally by each individual. Pia get 'stheory is different from t he empir icis t as­sumpt ion on which math education has tradi­tionally been based. He located the ultimatesource of logico-mathematical knowledge ineach child's mental action , rather than in setof objects.

Children go on to construct logico-mathe­matical knowledge by coordinating the rela­tion ships they created earlier between ob­jects. For example, by coordinating therelationshi ps of "same ," " differ ent ," and"more;' children become able to deduce thatthere are more animals than cows in theworld. It is likewise by coordinating the rela­tionship between two and two that they cometo deduce that two and two makes four andlater, that two times two is four. '

The ultimate sources of social knowledgeare conventions worked out by people. Ex­amples of ocial knowledge are that Christ­mas comes on December 25, that a chip is

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

PLACE VALUE

Figure 1

Counting withou t Ordering the Objects (a)and by Ment ally Orderinq the Same Objects

( b)

called chip , and that tables are not to standon. The main characteristic of social knowl­edge is that it is largely arbitrary in nature.The fact that a chip is called chip is arbitrary.In another language, the same object is calledby another name. The words one, two, threeand the wri t ten numerals 1, 2, 3 thus belongto social knowledge, but the numerical con­cepts underlying them belong to logico-math­ematical knowledge.

Piaget went on to elaborate a t heory ofnumber by demonstrating that number is asynthesis of two kinds of relationships-orderand hierar chical inclusion. I will begin by clar­ifying what Piaget meant by order and thengo on to discuss hierar chical inclusion.

All teachers ofyoung children have seen thecommon tendency among four- year-olds tocount objects by skipping some and countingsome more than once. When given eight ob­jects, for example, a child who can recite "one,two, three . . :' correc tly up to ten may endup claiming t hat th ere are t en things bycounting as shown in Figu re l( a). This ten­dency shows that the child does not feel thelogical necessity of putting the objects in anordered relationship to make sure not to skipany or count the same one more than once.The only way we can be sure of not overlook­ing any or counting the same object more thanonce is by putting all of them into a relat ion­ship of order; The child, however, does not haveto put the objects literally in a spatial orderto put t hem int o an ordere d relat ionship.What is important is to order them mentallyas shown in Figure l(b).

If ordering were the only mental action onobjects, the collection would not be quantified,since the child could consider one object at atime rather than a group of many at the sametime . For example, after counting eight ob­jects arranged in an ordered relationship asshown in Figure 2(a), four-year-olds usuallystate that there are eight. Ifwe then ask themto show us eight , some of them point to thelast one, the eighth object . This behavior in­dicat es that , for thes e children , the wordsone, two and three are names for individualelements in a ser ies, like Monday, Tuesday andWednesday. To quantify the collection of ob­jects, the child must put them mentally into arelationship of hierarchical inclusion. This re­lationship, shown in Figure 2(b), means thatthe child mentally includes one in two, two inthree, three in four, and so on. When pre­sented with eight objects, children can quan­tify the collection numerically only if they canput all the objects into a single relationshipthus synthesizing order and hierarchical inclu­sion.

The child at 7 or 8 years of age is st ill con­structing small numbers through the opera­tion of + 1. When we read about the long timeit takes for the child to become able to makethe quant itat ive part-whole relationship ofclass inclusion, 7 or 8 years according to In­helder and Piaget (1964), we can appreciatethe difficulty of the idea of twenty. This ideaconsists of twenty hierarchical levels.

The trouble with traditional place value in­struction is, in part , that it is based on thewrong ass umption that numerals simply rep­resent numbers of objects and/or groups ofobjects. The difficulty of place value is not aproblem of conventional representation butone of conceptual abstraction . Piaget made adistinction between abstraction and represen­tation, and also between two kinds of abstrac­tion-rejiective abstraction and empirical ab­st raction.

In empir ical abstraction, all the child doesis focus on a cer ta in property of an object andignore the others. For example, when ab­stracting the color of an object , the child sim­ply ignores th e ot her properties such asweight and the material with which the objectis made.

Reflective abstraction, by contrast, involvesthe const ruction of relationships between/(b)(a)

77

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

CONSTANCE KAMII

Figure 2

The Term Eight Used to Refer Only to the Last Element (a)and the Same Word Used with the Mental Structure of Hierarch ical Inclusion (b)

0 0 0 0 0 0 0 A<:»

/cisht

(a)

among objects. Relationships, as stated ear­lier, do not originate in external reality. Thesimilarity or difference between two chipsdoes not exist in one chip or the other, noranywhere else in external reality. The rela­tionship exist s only in the minds of those whocan create it between the objects. The termconstructive abstraction might be easier tounderstand than reflective abstraction to in­dicate that this is a veritable construction bythe mind rather than a focus on somethingthat already exists in objects. Number andarithmetic ar e constructed by reflective ab­straction and not by empirical abstractionfrom the extern al world. In other words, theheart of mathematics is reflective abstraction,but this is precisely what math educatorsoverlook.

If children have not constructed number byreflective abstraction (i.e., through the syn­thesis of order and hierarchical inclusion),they cannot give to words like one , two ,three .. . the meanings adults can. This iswhy four-year-olds often count the same ob­jects more than once and/or skip some. Thisexample illustrates the difference between ab­straction and representation. Words do notrepresent numbers of objects in external real­ity. It is the child, and not the words, whodoes the representing. People give meaningsto words, and the meaning of eight has to beconstructed by each person through reflectiveabstraction bearing on the whole series. Thisis why expressions such as one ten or onehundred often mean no more than one observ-

78

(b)

able bundle to first graders. Although spokennumber words and written numerals belongto social knowledge, their meanings belong tologico-mathematical knowledge, which cannotbe taught by direct transmission from the en­vironment.

The preceding discussion focused on thefirst system of numbers constructed throughthe operation of + 1. Let us now turn to theconstruction of the second system through theoperation of +10.

The Construction of the System of TensOne ten, two tens, three tens , etc., involvesthe construction of a second hierarchical sys­tem on the system ofones already constructedby the child. As can be seen in Figure 3, theconstruction of the second system requiresthe mental cutting of the first system intoequal parts of tens while keeping the firstsystem intact. After cutting the whole intoequal parts, the child has to go through thesame process as before of ordering theseparts and hierarchically inc luding them(i.e., including ten in twenty, twenty in thirty,thirty in forty, etc.). Since children have tobuild the second system (of tens) on the firstsystem (ofones), they cannot build tens solidlyuntil ones are structured at least to some ex­tent. When children are prematurely givenplace value instruction , they sometimes buildtwo systems separately, and juxtapose a sys­tem of ones and an indep endent syst em oftens.

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

Figure 3

The Construction of the System of Tenson the System ofOnes

According to Piaget (1960), quantities arearithmetized very slowlyby the child, and thefollowing four levels can be distinguished atsix to seven years of age. It will be noted thatthese levels become less and less well struc­tured as the number increases, and that eventhe system of ones is shaky beyond fifteen orthereabout.

Quantities up to seven or eight: Numbersare quasi-structured.Quantities from eight to fourt een or fif­teen: Units are both ordered and equidis­tant, but ordinality and cardinality are notwell coordinated (see Kamii, 1985).Quant ities f rom fifteen to thirty or forty:Units are ordered but not equidistant.Even for adults, an increase of $1 from $10to $11 seems greater than the same in­crease from $100 to $10l!Quantities above thirty or forty: Numbersare conceived as a lot without precision.Even for adults, the federal budget is justa lot!Shortcomings of traditional place value in­

struction are that it focuses only on the car­dinal aspect of one ten, two tens, etc., andcompletely overlooks the relationship betweenthe system of tens and that of ones, especiallyits ordinal aspect. For example, it overlooksthe fact illustrated in Figure 3 that the firstone in the second ten is the eleventh object inthe system of ones, that the second one in thesecond ten is the twelfth object in the systemof ones.

79

PLACE VALUE

It also overlooksthe child's inability to con­fidently use numbers larger than fifteen. Thefollowing study was conducted to investigatechildren's process of constructing the secondsystem on the first one.

MethodSubjectsOne hundred children were individually inter­viewed in a public school in Geneva, Switzer­land, in a lower middle-class neighborhood.The children belonged to one class each ofgrades 1 through 5. The interviews took placein May and June, at the end of the school year.

Materials and ProcedureThe material used was 200 identical plasticchips. The interviewer asked children to bothestimate and count the chips.

Estimating. I told the children that I hadhidden chips under a folder, and that I wouldshow them for a very brief moment that wouldbe too short to count them. I demonstrated 3seconds of exposure with my hand. I toldthem that I wanted them to write an esti­mated number before going ahead and count­ing the chips. First graders were shown 73­78 chips, and the older children were shown98-120.

The decision to show only 73-78 chips tofirst graders was made because this numberwas difficult for many of them to count. I didnot want to frustrate them unduly and de­cided to give priority to this consideration overthe scientific rigor of presenting the samenumber to all the children. Besides, for firstgraders, the difference between 75 and 110seemed negligible as will be seen shortly.These large quantities were chosen to makecounting by ones seem tedious and conduciveto inaccuracy. The exact number was changedfrom child to child, and the class was told thatthe number would be different for each child.

The rationale for asking children to guessthe number of chips was twofold. First, Iwanted to give them a reason to want to counta large number of chips later. I felt that theywould be intrinsicially motivated to count thechips after guessing the number. Second, Iwanted to know about their arithmetizationof large quantities. I wanted to know howclose they would come to the number actually

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

CONSTANCE KAMII

presented, and whether their estimation im­proved with age.

Spontaneous counting. When the childrenwrote down the number guessed, I askedthem to count the chips. This request wasmade to find out how they spontaneouslycounted the large number of chips (by ones,twos , fives or tens). Adults usually makegroups of ten when they quantify large collec­tions. I wanted to see how children comparedwith this adult procedure.

Counting by tens . I then imposed the con­straint of counting by tens. I did this to seethe proces s of constructing a system of tenson the system of ones (see Figure 3). Thushaving to quantify a large heap numericallynecessitated the child's use of both systems.I particularly wanted to see how children cutup the series of ones (from one to about ahundred) into segments of ten while conser v­ing the ordered and hierarchical relationshipof the system of ones.

To require the children to count a quantitythat was not exactly the same as the one theyhad just counted , I hid two or three chips inmy hand and asked them to count those re­maining on the table by tens, to figure outhow many were in my hand.

Only the first graders were then asked whatthey thought of what another child in anotherschool said last week. I made heaps of ten ,and told each first grader that in anotherschool last week I had shown another childhow I counted , demonstrating counting bytens. I went on to inform the first grader thatthe other child had said, "It's hard to countthe chips when they are in heaps. It 's betterto mix them up because we can count thembetter when they are mixed together." Manyfirst graders agreed with this statement madeby another child, and when this happened , Iasked one final question: "That child said it'sbet ter to mix the chips up real well, like this. . . (demonstrating mixing the objects thor­oughly). What do you think? Is it better tomix them very well, or is it enough just topush them together?"

This question was put to first graders tofind out whether or not they were bothered bythe separation of chips into heaps. Severalyears ago, I asked first graders in a classroomwhether they thought my way of countingchips by making groups of ten was a good idea

80

(Kamii, 1985). Most responded with disdainand a shrug. When I asked them to show mea better way of counting the chips, all exceptthe most advanced children mixed them upthoroughly, as if they had felt the need to ho­mogenize the chips (i.e. , to change them allback into ones). Since I could not expect chil­dren to be as spontaneous in interviews asthey were with a frequent visitor, I decided toask these questions about a fictitious child Isupposedly saw last week.

ResultsE stimating the Number of ChipsTable 1 shows the number s guessed by all thechildren in grades 1-5. Although the numberpresented to first graders was smaller (73-78instead of 98-120), this difference, for reasonsthat will be given shor tly, is not serious.

While it is not easy to speculate why chil­dren guessed certain numbers positively, it iseasier to interpret the small numbers theynever wrote. First graders never guessed anumber smaller than 12. Second gradersnever guessed a number smaller than 20, andthird graders never guessed one smaller than30. In fourth and fifth grade s, the number sincr eased to 40 and 50 re spectively, withamazing regularity.

These number s confirm the progressive ar­ithmetization of numerical quantities found byPiaget (1960). Large numbers are first con­sidered qualitatively as a lot, a few, or a wholelot, and the child progressively st ruct uresthem more precisely by ordering and hierar­chically including the unit s by reflective ab­st raction. The data presented in Table 1 sug­gest that this numerical structure was solidat least up to 11for all the children by the endof first grade. None of the first graders wrotethat a collectionof 73-78 chips contained fewerthan 12. This numerical structuring was doneat least up to 20 by the end of second grade ,at least up to 30 by the end of third grade,and at least up to 40 and 50 respectively bythe end of fourth and fifth grades.

This es t imat ion ta sk revealed that whenchildren are presented with a large group ofobjects, they see different numerical quan­titie s at different age levels. They interpretcollections by attributing to them the numer­ical ideas that they have built. It is significant

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

PLACE VALUE

TABLE 1The Number E stimated by Children in Each Grade

73-78 98-120 chipschips

Grade 1st 2nd 3rd 4th 5thNumberestimated N =21 N=18 N=21 N =22 N =18

10-19 420-29 7 330-39 4 6 340-49 2 4 3 650-59 2 2 3 2 460-69 2 5 1 370-79 2 1 180-89 1 190-99 2 3 2100-109 1 1 5 4110-119 1 3120-129 1 2200 1210 11010 1

Grade Level

TABLE 2Percents of Children at E ach Grade Level Who

CO'l~nted by Ones, Twos and Tens

to note , for example, that some second grad­ers looked at about a hundred chips and wrotethat there were twenty, but that this neverhappened at a later age.

Spontaneous CountingAs can be seen in Table 2, all the first gradersand most of the others counted the largequantity of chips by ones. Counting by tens,by separat ing the whole into heaps of ten, ap­peared in the fourth grade for the first time,among a few advanced children. A frequ entlyused technique was counting by twos, but thismay be considered as only a faster way ofcounting by ones.

Counting by TensThe results of asking children to count bytens, presented in Table 3, were the most un­expected findings. They revealed the con­struction of the system of tens to be a problemof part-whole relationships. The lowest levelshown at the bottom ofTable 3, called No ideahow , includes a large variety of behaviorsranging from saying I don 't know how tocounting each chip by saying, ten, twenty,thirty, etc.

The next category, called Making heaps often without conservation of the whole, is re­lated to modern math, which is still beingtaught in Geneva. Part of this instruction in­volves different bases, and children are in­st ructed to make groups of two, three, four,etc., and to code and decode numbers ofgroups and of uni t s in a variety of base s.When I asked first graders to count the chipsby tens, those in this category immediatelyasked, Do you want bunches of ten? I repliedthat I wanted them to count the chips by tensin any way they liked, and they made heapsof ten. I then had to ask them how many chipsthere were altogether, and they replied Seven,referring to the seven heaps. When I asked,"Seven chips altogether?" implying that I wasnot satisfied with this answer, these first

52272

14453210

197110

694100

1st 2nd 3rd 4t h 5t hN =21 N= 18 N = 21 N= 22 N =18

By tensBy twosBy onesOthers

81

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

CONSTANCE KAMII

TABLE 3Percent s of Children in Each Grade

Reported by Four Different Levels of Counting by Tens

Grade Level

1st 2nd 3rd 4th 5thN=21 N=18 N=21 a N=22 N=18

IV, Making heaps of ten 39 71 36 78with conservation ofthe whole

III. Not separating the 38 56 19 64 22whole into parts

II. Making heaps of ten 29 0without conservationof the whole

I. No idea how 33 6

"The behavior of 2 children (10%) did not fall into any of the 4 categories.

graders changed their response to Ten , mean­ing that there were ten chips in a heap. Theseanswers are not surprising in view of the factthat quantitative part-whole relationships arevery difficult until seven to eight years of age(Inhelder and Piaget, 1964). Most first grad­ers can think about the whole and the partssuccessively but not simultaneously. Whenthey made heaps of ten, for them, the wholedisappeared, and the only things left to countwere seven heaps or ten chips in each heap.This is why first graders continued to sayseven or ten when I repeatedly asked themhow many chips there were on the table alto­gether. Some of them resorted to counting byones to answer this question.

The third category, called Not separatingthe whole in to parts , is different in the follow­ing way from the fourth category, Makingheaps of ten with conserooiion of the whole.The children in the latter group made sepa­rate heaps of ten , and counted the heaps af­tenoards to determine the total number ofchips. This is what adults usually do.

The children in the third category, by con­trast, in reality counted by ones . Theycounted out ten chips first and left them in agroup. They then counted out ten other chips,making a separate heap , but then said,twenty, as they joined the second heap to the

82

first one making a heap of twenty. They thencounted out a third heap of ten, spatially sep­arating them from the heap of twenty. As theythen pushed this heap of ten to combine itwith the first twenty chips, these childrensaid, thirty. They continued this process untilthe end as shown in Figure 4.

When children count chips by ones, theybegin by taking one at a time and movingeachone to make a new heap. When they havemoved each chip one by one in an orderedrelationship from the original heap to the newone, and there are no more left in the original

Figure .4-

Counting by Tens withoutSeparating the Whole in to Parts

"forl(

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

heap , they know that they have counted allthe chips. In their heads, they have thus in­cluded one in two, two in three, three in four,etc. When asked to count the chips by tens,the children in the third category count outten, but mentally count all of them by ones.They temporarily make a separate heap of tento comply with my reque st but feel the needto keep together the heap of those alreadycounted to keep the whole intact. In otherwords, these children cannot mentally cut theseries of 1+ 1+ 1+ 1 . . . into segments of tenwhile conserving the whole, the sys tem of1+ 1+ 1+ 1 . . ., in their minds.

Some children in this category did not evenmake a spatially separate heap of ten. Theysaid, twen ty , for example, and then silentlyadded one chip at a time to the heap of twentyas shown in Figure 5. As they added the tenthone to the heap, they said, thirty, etc. Onechild, whom I did not put into this third cat­egory, moved the chips in this way but said,twenty-one, twenty-two, twenty-three, etc . asshe moved each chip. When I pointed out toher that she had not counted by tens as I hadrequested , she prote sted that she had said,thiris], f011,y, fifty, etc. more loudly than thenumbers in between!- When I later realized that fourth and fifthgraders had a tendency to count large quan­tities by ones (category III , Table 3), I addedabout a hundred more chips and asked thechildren how they would count them (about

Figure 5

Counting Aloud by Tens without E venTemporarily Making a Separate Pile of Ten

+1

"th i rt ,

83

PLACE VAL UE

two hundred ) by tens. Only a few childrenshifted from category III to category IV atthis point (three children out of the seven incategory III , or 17% of all fifth graders).

Category IV, Making heaps of ten withconservation of the whole, appeared for thefirst time in second grade . In these children'sheads, the system of ones was solid enough toremain intact even when it was cut up intosegments of ten. In other words, these chil­dren could hold the order and hierarchical in­clusion of ones from one to a hundred, ormore, while simuluineouslu cutting up thissystem into parts of ten, and including ten intwenty, twenty in thirty, thirty in forty, etc .Because the two systems were available tothese children simultaneously, they couldmake heaps of ten , intending to go back andcount them later.

Mixing HeapsIt will be recalled that only the first graderswere asked what they thought about a ficti­tious child who said, "It' s hard to count thechips when they are separated into heaps. It 'sbetter to mix them up because we can countthem better when they are mixed together."

As can be seen in Table 4, 12out of 18 (66%)of the first graders agreed that it was betterto mix up the heaps. I forgot to ask this ques­tion with three children. First graders aredisturbed by the separation of the whole intoparts. Eight of the 12 (or 44% of all the 18questioned) went even further and said it wasbetter to mix the chips up thoroughly thanmerely pushing them together. This thoroughmixing seems to them to homogenize the chipsand to make them become equal units againin a whole that has been restored!

The other children, the 33% who repliedthat it is not good to mix up the piles, allbelonged to the highest category found in firstgrade, category III in Table 3. While thesechildren did not separate the whole into partswhen asked to count the chips by tens, theycould see when this procedure was suggestedthat it was a good method.

Discussion and Educational Implications

Although the number of children representingeach grade level is small in this study, it islarge enough, in light of the 60 years of re­search by Piaget and others, to answer in part

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

CO STANCE KAMII

TABLE 4First Graders Who Were or Were Not in Favor

of Mixing Up the Heaps

Not in FavorIn Favor of of MixingMixing Up Up thethe Heaps Heaps

I I I. Not separating the 2 6whole into parts

I I. Making heaps of 5 0ten without conser-vation of the whole

I. No idea how 5 0

Total- 12 6

the questions posed at the beginning of thisarticle: (a) Why is place value so difficult? and(b) At what grade would it be desirable tointroduce it?

The part-whole relationship involved in cut­ting up the system of ones is a prerequisitefor understanding place value. We saw in Ta­ble 1 that the children were st ill in the processof constructing the system of ones. When theywere free to count the chips in any way theywished, they demonstrated a strong prefer­ence for using this system (Table 2). Evenfourth and fifth graders counted a hundredchips by ones or by twos.

The ability to count by tens appeared forthe first time in second grade (Table 3). Firstgraders often knew the words ten-twenty­thirty ... but used them , for example, tocount the first chip, the second one, and thethird one. As can be seen in Table 3, the otherfirst graders either did not separate the wholeinto parts (category III) or made heaps of tenand became unable to think about the whole(category II ). These behaviors revealed thatfirst graders have only the system of ones.When they think about seventy-five, they canthink about it as seventy-five ones, but not asseven groups of ten, and some left over. Fur­thermore, when the chips were physically sep­arated into heaps of ten , most of the firstgraders thought it was desirable to mix themup before counting them.

Category IV, Making heaps of ten wi thconservation of the whole, appeared for the

84

first time in second grade and continued toappear in every subsequent grade. Unlike thechildren in category II , those in category IVmade heaps of ten and counted them after­wards. Because these children had the twosystems shown in Figure 3, they could cut upthe system of ones into parts of ten withoutlosing the whole (the first system of ones).This finding is supported by Inhelder and Pi­aget (1964), who state that quantitative part­whole relationships become possible generallyaround seven or eight years of age, when chil­dren's thought becomes mobile enough to bereversible.

The appearance of category IV in secondgrade suggested the possibility of trying tointroduce place value instruction for somechildren at this grade level. The question thatimmediately came to mind was what kind ofinstruction might be desirable. Traditionalplace value instruction begins with the mak­ing of bundles of ten straws, toothpicks, etc. ,on the assumption that tens and ones arelearned by empirical abstraction from exter­nal reality. Piaget's theory indicates that tensand ones cannot be learned by empirical ab­st raction. Children have to create a system oftens, by reflective abstraction, on the systemof ones they have already constructed in theirheads.

As can be seen in Figure 3, children seemto think about thirty-two as thirty and two,and not as two and thirty. This idea suggestedthe undesirabili ty of teaching the procedureof double-columnaddition that makes childrenproceed from the ones to the tens. When pre­sented with a problem such as

32+ 12

children are now taught to begin with 2 + 2.This instruction goes counter to the way chil­dren think. If they think about 32 as thirtyplus two, and about 12 as ten plus two, itwould be more natural for them to add thirtyand ten first .

Madell's (1985) observation supports thepreceding argument. He reported that whenchildren are encouraged to think in their ownnatural way, "they universally proceed fromleft to right (p. 21):' If seven- and eight-year­olds are given a problem such as

36+ 46

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

with~ut being told to work from right to left,they Invariably compute the tens first as fol­lows:a) Some will actually record a 7 in the tens column

before looking at the ones. These children thencome back and erase.

b) Others, having arrived at 7 as the sum of 3 and4, do not record that 7 before checking the onescolumn to see if it contains another ten.

c) A few of the most sophisticated students checkthe ones first. Noting (often by estimation) thatthere are more than 10ones in the ones columnthey come back to sum the tens and record 8before returning to the ones and the last detailof the computation.

This last process is the closest that the chil­dren get on their own to the standard right­to-left procedure. Even for the addition of 3­and 4-digit numbers where a right-to-left pro­ce~s would seem more efficient, the childrenuniformly prefer the other direction (p. 21).

Based on Madell's observation and the re­search presented in this article, I have beenworking with three second grade teachers ina public school near Birmingham , Alabama.The teachers have been encouraging childrento invent their own procedures to add 2- 3­and 4-digit numbers. All the children kne~ b;the end of first grade that 10+ 10 = 20, andthey all knew how to read 2-digit numbers(Karnii, 1985). In second grade , therefore, theteachers began by giving problems such as

20 20 20+ 10 + 20 , and + 12

without going through any bundgles of tenstraws or pictures of objects to circle to showgroups of ten. The great majority of secondgraders became able to add 2- and 3-digitnumbers without writing anything. Whenpresented with a problem such as

266+ 146

for example, they said:200 + 100 = 30060 + 40 = 100

300 + 100 = 4006 + 6 = 12

So the answer is 412.

The detail s of this way of teaching is beyondthe scope of this paper. The point I want to

85

PLACE VALUE

make here is that the widespread difficulty ofplace value may be at least in part due to theteaching of standard procedures. The proce­dure of starting with the ones column encour­age s children to crank out answers mind­lessly This is why second graders often write

36+ 46

712

without any signs of embarrassment. In theprocedure of starting with the tens, by con­trast, children think and therefore never get700 by adding 30 and 40.

Standard procedures seem efficient toadults. But young children do not think likeadults, and we need to focus on how childrenthink if we are to improve our teaching. Placevalue and standard procedures are productsof centuries of construction by adults. Chil­dren cannot swallow these products in ready­made form. They have to go through a processof reconstruction with their own way ofthink­ing if they are to master place value and go onto other things that depend on this foundation.

ReferencesAshlock , R. B. (1982). E rror patt ern s in com putation.

Columbus, Ohio: Charle s E. Merrill.Bednarz, . & Janvier, B. (1982). The underst anding of

numer ati on in primary school. Educational S tud iesin Math ematics , 13, 33-57.

Bednarz, N. & Jan vier, B. (1984). La numeration [Nu­meration]. Grand No. 33, 5-31 & No. 34, 5-17.Grenoble, France: Centre Nat ional de Documenta­tion Pedagogique,

Brun, J. , Giossi, J. M. , & Henriques , A. (1984). A propo sde l'ecri ture decimale [Concerni ng decimal wri ting].MA TH-ECOLE, 23(112) , 2- 11, (P ublished in Ge­neva, Switzer land) .

Ginsburg, H. (1977). Children's arithmetic: The learn­ing process. New York: Van Nostrand.

Inhelder, B. & Piaget , J . (1964). The early growth oflogicin the chi ld . New York: Harper & Row.

Kamii, C. (1982a). First graders invent arithmetic: UsingPiaget's theory in the cla ssroom. In S. Wagner(Ed.) , P roceedings of the Fou rth Annual Meeting ,

orih A merica 11 Chapter, In ternational Group forthe Psychology of Mathematics E ducation (pp. 93­99). Athens , GA: University of Georgia.

Kam ii, C. (1982b). Num ber in preschool and kind ergar­ten. Washingt on, D.C .: National Association for theEducation of Young Children.

Kamii , C. (1985). Young chi ldren rein vent ari thmetic .New York: Teachers College Press.

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3

CONSTANCE KAMII

Kamii, M. (1980, May). Place value: Children's eff ortsto find a correspondence between digits and num­bers of objects. Paper presented at the Tenth AnnualSymposium of the Jean Piaget Society, Philadelphia.

Kamii, M. (1982). Children's graphic representation ofnumerical concepts: A developmental study. Un­published doctoral dissertation, Harvard University.

Madell, R. (1985). Children's natural processes. Arith ­met ic Teacher , 32, 20-22.

86

Piaget , J. (1960). Problemas de la construction du nombre[Problems in the construction of number] . In P.Greco, J. B. Grize, S. Papert & J. Piaget , (Eds.),Problemes de la construction du nombre (pp. 1-68).Paris: Presses Universitaires de France.

Received April 21, 1986 •

Dow

nloa

ded

by [

And

rew

s U

nive

rsity

] at

16:

38 0

7 A

ugus

t 201

3