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Using a Base-Ten Blocks Learning/Teaching Approach for First- and Second-Grade Place-Valueand Multidigit Addition and SubtractionAuthor(s): Karen C. Fuson and Diane J. BriarsSource: Journal for Research in Mathematics Education, Vol. 21, No. 3 (May, 1990), pp. 180-206Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749373.

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Journal or Research n MathematicsEducation1990,Vol. 21, No. 3, 180-206

USING A BASE-TENBLOCKSLEARNING/TEACHINGAPPROACHFORFIRST- ANDSECOND-GRADEPLACE-VALUEANDMULTIDIGITADDITION AND SUBTRACTIONKARENC. FUSON,NorthwesternUniversityDIANE J. BRIARS,PittsburghPublic Schools

A learning/teaching pproachused base-tenblocks toembodytheEnglishnamed-value ystemof numberwordsanddigitcardstoembodythepositionalbase-tensystemof numeration. tepsin additionandsubtraction f four-digitnumberswere motivatedby the size of the blocks andthen were carried out with the blocks; each step was immediatelyrecordedwith base-tennumerals.Childrenpracticedmultidigitproblemsof from five to eightplaces aftertheycouldsuccessfullyadd or subtract mallerproblemswithoutusing the blocks. In Study 1 six of theeightclasses of firstand secondgraders N= 169)demonstratedmeaningfulmultidigitadditionand place-valueconcepts up to at least four-digitnumbers;average-achieving first gradersshowed more limitedunderstanding.Threeclasses of secondgraders N = 75) completedtheinitialsubtractionearninganddemonstratedmeaningful ubtractiononcepts.InStudy2 mostsecond graders n 42 participatinglasses (N = 783) in a large urban chool district earnedatleast four-digitaddition,andmanychildren n the35 classes (N = 707) completingsubtractionworklearnedat least four-digitsubtraction.

The English spokensystem of numberwords is a named-valuesystem for thevalues of hundred, housand,andhigher;a numberword s said andthenthe valueof thatnumberwordis named. For example, with five thousand seven hundredtwelve, the "thousand" ames the value of the "five" to clarifythat it is not fiveones (= five) but is five thousands.In contrast,the system of writtenmultidigitnumbermarks s a positionalbase-tensystemin which the values areimplicitandareindicatedonly by the relativepositionsof thenumbermarks.Inorder o under-stand these systems of English wordsand written numbermarks for largemul-tidigit numbers,childrenmust constructnamed-valueand positional base-tenconceptual structures or the words and the marks and relate these conceptualstructureso each otherandto the wordsand the marks.

Englishwords for two-digitnumbersare irregularn severalways and are notnamed-value, n contrast o Chinese (andBurmese,Japanese,Korean,Thai, andVietnamese)wordsin which twelve is said"ten two" andfifty seven is said"fiveten seven."These irregularitiesmake it much moredifficult forEnglish-speakingStudyI was fundedby a grant o theUniversityof ChicagoSchool MathematicsProjectfromthe Amoco Foundation.Thanksgo to MaureenHanrahan or handlingall of the fielddetailsfor Study 1;to GordonWillis for carryingout the dataanalysesfor bothstudies;toFredCarr,TracyKlein,and ThucHuongfor carefulgrading,dataentry,and erroranalysesforbothstudies;andespeciallyto the teachersof both studieswhowerewillingto trysome-thingnew becausetheythought t might help theirchildren earnbetter.Thanksalso to ArtBaroody,PaulTrafton,and severalanonymousreviewers who madehelpfulcomments onearlierdrafts.


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181

children hanfor Chinese,Japanese,orKoreanchildren o constructnamed-valuemeanings for multidigitnumbers(Fuson, in press a; Fuson & Kwon, in press;Miura,1987; Miura,Kim, Chang,& Okamoto, 1988;Miura& Okamoto,1989).English-speakingchildren use for a long time unitaryconceptualstructures ortwo-digit numbers as counted collections of single objects or as collections ofspokenwords(Fuson,Richards,& Briars, 1982; Fuson, 1988a; Steffe, von Glas-ersfeld, Richards,& Cobb, 1983; Steffe & Cobb, 1988); these early conceptualstructures an interferewith children's later constructionof named-valuemean-ings. The lack of verbalsupport n the Englishlanguagefornamed-valueor base-ten concepts of ten makes it particularly mportant hatsupport or constructingsuch ten-structured onceptionsbe providedin otherways to English-speakingchildren.Inthe United Statessuchsupports rarelygivenor is insufficient.Childrenmorecommonlyaretaughtmultidigitadditionand subtraction s sequentialproceduresof addingandsubtracting ingle-digitnumbersandwritingdigits in certain oca-tions(Fuson, npressc). Theseexperiencesresult nmanyU.S. childrenconstruct-ing conceptual structuresfor multidigit numbers as concatenated single-digitnumbers,a view that is inadequate n many ways and results in many errors nplace-value tasks and in multidigitadditionand subtraction Fuson, in press a;Kouba et al., 1988). Even manychildrenwho carryout the algorithmscorrectlydo so procedurallyanddo notunderstand easonsfor crucialaspectsof theproce-dureor cannotgive the values of the tradesthey arewritingdown (Cauley,1988;Cobb & Wheatley,1988;Davis & McKnight,1980; Labinowicz, 1985;Resnick& Omanson,1987).U.S. childrenalso showquitedelayedunderstandingf place-value concepts(Kamii,1986;Koubaet al., 1988;Labinowicz, 1985;Miuraet al.,1988;Ross, 1989;Song & Ginsburg,1987).Furthermore,n theUnitedStates,instructionn the additionand subtraction fwhole numbers ypically s bothdelayedand extended acrossgradesmore than ncountries ike China,Japan,Taiwan,and the Soviet Unionthat have been charac-terized as fostering high mathematics achievement (Fuson, Stigler, & Bartsch,1988). In the UnitedStates the single-digitsums anddifferences to 18 consumemuch of the first two grades,andworkon the multidigitalgorithmswith trading(carryingandborrowing) s distributed ver 4 or 5 yearsbeginningwithtwo-digitproblemsin second gradefollowed by the introductionof problemsone or twodigits largereachyear.Incontrast,othercountriesstressmasteryof sumsand dif-ferences to 18 in the first grade,andthey complete multidigitinstructionby thethirdgrade.In orderto use andunderstandEnglishwords and base-tenwrittenmarks andadd and subtractmultidigitnumbers,childrenneedto linkthe words and the writ-tenmarks o eachotherand need to give meaningto both the words and themarks.Thelearning/teaching pproachusedin thepresentstudies was developedto meetthese goals. It is an adaptationof anapproachusedby the first authorwith teach-ers and children or 20 years(theteacherversionis inBell, Fuson,& Lesh, 1976).It provideschildrenan opportunity o construct he necessary meanings by using


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182 Base-TenBlocksLearning/Teaching pproach

for each system a physical embodimentthatcan direct theirattentionto crucialmeaningsandhelp to constrain heiractions with the embodiments o those con-sistentwith the mathematical eaturesof the systems. The Englishnamed-valuesystemof words is embodiedby a set of base-tenblocks (Dienes, 1960), and thepositionalbase-ten writtenmarks are embodiedby digit cards(numeralswrittenon small individualcards).Englishwords,words for the block embodiment,andwordsfor the digit cards(see Figure1) were used to help direct children'satten-tion to criticalfeaturesof the mathematical ystems andembodiments, acilitatecommunicationamong the participants n the learning/teachingapproach,andsupport he constructionof linksamongthe differentsystemsandembodiments.

f o u r t h o u s a n d t w o h u n r e i t y s e v e n

fourbigcubes twoflats fivelongs seven 4== =four two fivesevenlittle ubes

Figure1. The learning/teaching pproach.Featuresof the approachn actionareas follows:

"*Whenaddingandsubtractingwith theblocks,theblocks-to-written-marksinksare made stronglyand tightly:Each step with the blocks is immediatelyre-cordedwiththe writtenmarks."*inksamongtheEnglishwords,base-tenblocks,digitcards,andbase-tenwrit-ten marksarestrengthenedby the constantuse of the threesets of words."*hildrenworkwith the learning/teaching pproach ormany days; they areal-lowed to leave the embodimentsanddo problems ust in written form when-everthey feel comfortabledoing so."*When childrenbeginto do writtenproblemswithoutblocks, theirperformanceis monitored o ensure thatthey arenotpracticingerrors."*dditionand subtraction othbeginwithfour-digitproblems orin somecases,these problems mmediately ollow initialworkwith two-digitproblems).


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Karen C. Fusonand Diane J. Briars 183

"*hildren pendonly 1 to 4 days on place-value conceptsinitially;muchplace-value learning is combined with the work on multidigit addition andsubtraction.

"* modificationof the usualalgorithm s used for subtractionsee the methodssectionfor Study 1).Thesefeatures,and thereasoningbehindthem,are discussed in Fuson(inpressa),wheredistinctionsbetween named-valueandpositionalbase-tensystemsare dis-cussed morefully andliteraturepertaining o bothadequateandinadequatecon-ceptualstructures hildrenconstruct or multidigitnumbersarereviewed.

Results of an earlierstudywith this learning/teaching pproachwere reportedin Fuson (1986a). In thatstudysecond gradersand some first graders earnedtoadd and subtractmultidigitnumbersmuch moreaccurately hanreported orusualschoolinstruction.Most of these childrensuccessfullyand ndependently xtendedthe procedures earnedwith the blocks to five- through en-digitsymbolicprob-lems done withoutthe embodiment.Childrenwho madeerrorswereinterviewed,and those still makingerrorswere told to thinkaboutthe blocks as they solvedproblems.Most of these children were able to use a mentalrepresentation f theblocks to self-correct heirwrittenerrors,andthis use of theblocks showed under-standingof place-valueconcepts.This studyleft unanswered everalimportantquestionsthatwereaddressedbythe two studiesreportedhere.First,the grade evel, achievement evel, andsocio-economic level of the studentswho could benefit from the learning/teaching p-proachwas not clear from the limited sampleused in that initial study. Study 1reportedhere extendedthesampleto secondgradersof all achievement evels andto first gradersof above-averageandaveragemathematicsachievement.Study2extended the sampleto second graders n a largeurbanschool district.The goalforboththe age/achievementand theresidentialextensionswas not to manipulatethese variousbackgroundvariables n order o determine heirdifferentialeffectson performance.It was simply to examine whether the effects of the learning/teaching approachcould be considered to generalize across a heterogeneouspopulation.Second, there were the practicalquestions of whetherthe learning/teachingapproachcould be distanced from its designer,communicated n a fairly smallamountof in-servicetime, andimplementedby teacherswith little field support.Theseseemto be crucial ssuesdetermininghefeasibilityof wide-scale use of thelearning/teachingapproach.Distancing focused on three majoraspects of thislearning/teachingntervention:the classroomteaching,the in-serviceteachingofthe involvedteachers,andteachingandsupervisionof field supportpersonnel.InFuson (1986a), projectstaff members did some of the teaching, the projectde-signerconducted the teacher n-service, andthe field supportpersonwas taughtandsupervisedclosely by theprojectdesigner.Inbothof thestudiesreportedhere,all of the teachingwas doneby classroomteachersusing lesson plansandstudentworksheetsdeveloped by the projectdesigner.In the second study, the project


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184 Base-Base-TenBlocksLearning/Teaching

designerdid not conduct the in-service sessions nor supervisethe field supportpersons.Theamountof in-service time wasfairlysmall for both studies: a 1-houroverviewof the learning/teaching pproachn the firststudyandone or two 2V2-hour n-servicesessionsin the secondstudy.Fieldsupportwasprovided nthe firststudy by two teachersin each school who had taughtthe learning/teachingap-proach n the firstyear.In the secondstudy,threeelementary K-8) mathematicssupervisorswere available oprovide ieldsupport or the 132second-gradeeach-ers targetedfor the learning/teachingapproach,but these supervisorsalso hadmanyotherduties.The results of the two studiesreportedhere areanalyzedwith respectto threegoals of the learning/teaching pproach:1. understandingmultidigitadditionand subtraction ndjustifying procedureswithnamed-value/base-tenoncepts;

2. understanding lace-valueconcepts;3. being able to add and subtractmultidigitnumbersof severalplaces, includ-ing subtraction roblemswith zeros in the topnumber.The literature oncerningperformancen these areasby childrenreceivingusualinstruction s brieflysummarizedn the discussion of the resultsof each studyinorderto providea contextwithinwhich to interpretheresults.

STUDY1Method

SubjectsChildrenfrom two schools in a small city on the northernborderof Chicagoserved as subjects.Teachersgroupedchildrenby mathematics achievement inthese schools dependinguponrecommendations f thepreviousteacher;childrenweremovedto a differentroom at anytimea teacherthought hat a move shouldbe made. In each school there were sufficientfirstgraders or three mathclasses,one each of low, average,andhigh math achievement.The high-achievingfirst-gradeclasses from both schools wereasked to participaten thestudy.The teach-ersof theaverage-achievingirstgraders n both schools asked ater n theyeartoparticipate nd were allowedto do so. Inone school therewerethreesecond-grademathclasses, one each of low, average,andhigh math achievement.Manyof thechildren n thehigh-achievingclass had received additionmultidigit nstruction sfirstgraders n thestudyreportedn Fuson(1986a), so only the low- andaverage-achievingclasses participated.nthe otherschool there wereonly enoughsecondgradersto form two classes. The five lowest achieving second gradersweregroupedwith a low-achievingfirst-grade lass, and the remainingchildren weregrouped ntoa high/averageand anaverage/lowclass. Manyof the children n thehigh/average lasshad received additionmultidigit nstruction s firstgraders,butthis class was retained n the present studyin orderto studysubtractionearningfor all childrenandaddition earningfor the new children.All eight classes (N =


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KarenC. Fuson andDiane J. Briars 185

169) participatedn the addition nstruction,and threesecond-gradeclasses (N =75) receivedthe subtraction nstruction.Teachers

Four teachers(two from each school) hadparticipatedn the multidigit nstruc-tion in the Fuson(1986a) study.The otherteachersweregivena brief overview ofthe instruction,esson plans,studentworksheets,and tests. Forquestionsand fur-therhelp, they wereto relyon the two teachers n theirschool who hadtaughtthematerials before.A researchprojectassistantalso visited the schools weekly tocheck on teaching progress.Instruction

All children irst learned o find sums anddifferencesto 18 by countingon andcounting up with one-handedfinger patterns(see Fuson, 1986b, 1987, 1988b;Fuson & Secada, 1986;Fuson & Willis, 1988).These countingprocedurescouldbe used for any additionand subtraction acts children did not know.They havebeen found to be efficientand accurate noughforuse in themultidigitalgorithms(Fuson, 1986a).Each class hadat least one set of base-ten blocks.The firstphaseof instructionfocusedon explorationof therelationshipsbetween the differentblocks andon useof theblocks words(littlecubes, longs, flats,big cubes,ornames chosenby chil-dren) andEnglish words (ones, tens, hundreds, housands).Both the consistentone-for-ten and ten-for-one tradesbetween adjacentplaces and the nonadjacenttrades(one-for-hundred ndone-for-thousand)were discussed anddemonstrated.Then the blocks were used to make different hree-andfour-digitnumbers(e.g.,3725), and index cards each containingone numeralwereused to make thebase-ten version of the numberbesidetheblocks(e.g., fourcardscontaining he numer-als 3, 7, 2, and 5 were selected and were put down in order to the right of theblocks).Thesecards,and numeralswrittenon children'sworksheets,werereadbybase-ten words(e.g., "threeseven two five").These activities were accompaniedby much verbalizationof the block words, the English words, and the base-tenwords.Additionandsubtractionwith theblocks were doneon a largecardboard alcu-lating sheet (see Figure2). Addition was considered irst. A writtenproblemwasgiven. Blocks for the top number were placed in the top row of the calculatingsheet, andthenblocks for the bottom numberwereplacedin the second row (seeFigure 2). Addition was done column by column, beginning on the right. Theblocks in a given column were addedtogether(pulleddown) intothe bottomrow.If the sum was nine or less, it was recordedwith the digit cards.Each child alsorecordedeach step on his or herown worksheet.If the sum was overnine, ten ofthe smallerpieces were tradedfor one of the next largerpieces, and the resultrecordedwithdigitcardsand on individualworksheets.Muchverbalizationof allthreesets of wordsaccompaniedall additionandsubtraction, ndrecordingwithwrittenmarkswas done aftereach action with the blocks. Thenecessityof trading


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Thousands Hundreds Tens Ones

T 3725

D D i 1647000

Figure2. Calculatingboardwith an additionproblem.was raisedby showingwhathappenswiththe digitcardsif a two-digitnumber swritten n any column(the otherdigit cardsget moved over to the left, makingabiggernumber).The fairnessof the ten/onetrades,and the idea of trading o getmore(in subtraction) r tradingwhenyou had too many(in addition),arosefromthe size of theblocks: ten of theblocksinanycolumn wereequivalent o one blockin the columnto the left.

Multidigitsubtraction an be shown in variousways with the blocks, and thesubtractionwithineach value can be phrased n differentwaysin words. Thechil-drenin this studyhadmultiple interpretations f subtractionavailable(as take-away,comparison,andequalize,see Fuson,1986b, 1988b;Fuson&Willis, 1988).We suggestedthatteachersverbalize he subtractionwithinvalues as "Sevenplushow many to make twelve?"or "Twelveminus seven is how many?"(becausethese fit children'suse of countingup to find these differencesbetterthanusingthe words"take-away") ndthatthey separate he blocks for the topnumber ntothose thatmatchthe bottomnumber the subtrahend) ndthe leftoverblocks (thedifference)andthenmove the differencenonmatchingblocks to the bottomrowas the answer.A simplificationof the usualalgorithmwas also used. Children irst checkedeachcolumn of the topnumber o be surethat t was larger hanthebottomnum-ber in that column. If a top digit was not as large, a one-for-ten trade(borrow,regrouping)was madefromthecolumnon the left.Afterall thenecessarytradinghad beendone to the topnumberso thateachtopnumberwas as largeas or largerthaneach bottomnumber, ubtractionwas done columnby column.Boththe trad-ing and the subtractingcan be done from either direction,but teachersusuallymodeledthetypicalU.S. right-to-leftapproach.This trade-first lgorithmreduces


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Karen C. Fuson and Diane J. Briars 187

the difficultalternationof tradingandsubtractingused in the commonalgorithmand thus eliminatesthe need for children o switchrepeatedly rom a named-valuerepresentation or tradingto a unitaryrepresentation or subtracting(Fuson &Kwon, in press).The initial sustained ocus on makingall the top columnslargeralso helps to avoid the common errorof subtractinghe topnumber romthe bot-tom numberwhenthe topnumber s smaller.Teachersorganized heirclassrooms n differentways for thisinstruction.Someworkedwiththe wholeclass, havingchildrenparticipaten solvingproblemswiththe blocks and the index cards. Others divided their class into small groupsandeither worked with groups simultaneously or serially while the other groupsworked on othertopics. In the formercase, childrenwho had learned the blocksprocedure he yearbeforeor older childrenshown how to use the blocks workedwith each group initially to ensure that the blocks andwritten-marks rocedureswere correct andthat children n the groupwere understandinghe relationshipsinvolved. In all cases all childrenhadworksheets,and all recordedeachproblemas it was worked with the blocks.Children n the averageandhigh-achievingsecond-gradeclasses were able todo three-andfour-digitadditionandsubtraction roblemswith theblocksinitially.In the low second-grade lass andfirst-grade lasses, childrenhaddifficultyrelat-ing the four columns of blocks to the fourcolumnsof writtenmarks.Therefore, nthese classes two-digit problemswere done first, and then three- and four-digitproblemswere done with the blocks and writtenmarks.Whenever children saidthey understoodthe written-marksprocedureand did not need the blocks anymore, they were allowed to go to their seats to work on worksheetscontainingthree- andfour-digitproblems.Theirprocedurewas checkedby someonebeforethey were allowed to leave the blocks. Worksheetswith larger problems(up toeight digits) were availableforchildrenwho wished to trythem.Work on subtractionwas followed by very shortunits focusing on aspects ofmeaningful addition (alignment of problems with different numbers of digits,adding 3 two-digit numbersrequiringa tradeof 2) andplace value (translatingfrommixed orderwordsto numeralsand vice versawith no trades,doingthesamewithtradesrequired,andchoosingthe largerof two multidigitnumbers).The les-son plansdescribedhow attentioncould be directedwithin the learning/teachingapproach o facilitatethe learningof these concepts.The time necessary to complete each unit variedconsiderablyfrom class toclass. The initial introduction/additionnit took from 3 to 6 weeks, andthe sub-tractionunit took from 2 to 4 weeks. Eachmeaningfuladdition andplace-valueconcepttook abouta day.

All of these classes were also participatingn an instructional esearchprojectfocusedon teachingadditionand subtractionwordproblems.Thesetopicsandthemultidigit topics went farbeyondthe districtgoals. Teachershadto meet districtgoals as well as teachingthese extratopics. In some classes teachers also had tocover considerablegroundbefore the multidigitworkcould begin (e.g., learningaboutsingle-digit sums and differencesto 18). Thereforedifferent classes com-


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pleted differenttopics. The low-achieving second-gradeclass and one averagefirst-gradeclass only completedaddition of two- andthree-placenumbers.Theteacherof theotheraverage first-grade lass only taughtmultidigitaddition o 10of the24 children n herclass, butshedidcompletethegeneralization f the algo-rithmpastfourplaceswiththeparticipating hildren.All otherclasses completedthe generalizationof the additionalgorithmto problemswith as manyas sevendigits.Thehigh andaveragesecond-gradeclasses completedsubtraction, nd theaverage/low second-gradeclass completed ordinary ubtractionandbegan workon problemswith zeros in the minuend. The workon meaningfuladditionandplace-value concepts was completed only by the high- and average-achievingsecond-gradeclasses.Measuresof Skill and Understanding

Additionand subtractioncalculation tests. All children were given two addi-tionpretests.TheTimedAdditionTest contained12problems,with 2 two-digit,2three-digit,3 four-digit,1 five-digit, 3 six-digit,and 1 seven-digit problem; chil-drenworkedontheseproblems or 2 minutes.All problemsrequiredrading n oneor moreplaces (thenumberof tradesranged rom one to five). TheTen-DigitAd-dition Test was a single ten-digitproblem(6385740918+ 8557586736).All prob-lems were writtenalignedin verticalform.These same two tests were also givenasposttests.The lowerachievingandyoungerclasseswere also given anUntimedAdditionMinitest of fourproblems(2 two-digitand2 three-digitproblems,eachrequiringone trade).Parallelsubtraction ests (TimedSubtractionTest,Ten-DigitSubtractionTest, Untimed SubtractionMinitest) were made by using inverseproblemsfrom the additiontests; children were given 3 minutes for the TimedSubtractionTest because subtractionhad been slower thanaddition n the earlierstudy.A fourth ubtractionest(ZerosSubtractionTest)consistedof fourproblemswith zeros in thetopnumber: 1 two-digit,2 three-digit,and 1 four-digitproblemwithone, one, two, andthreezeros,respectively.The tests for each child were first evaluated to determine whether the childshowed any evidence of correcttrading; wo correctlytradedcolumns were re-quiredfor the child to bejudgedas showingsome indicationof trading.Eachtest was then scoredto permita finer evaluationof performance.Scoringwas basedon eachdigit in the answer: one pointwas given foreachcorrectdigit.Thisprocedurewas adoptedbecausescoringeach problemonly as corrector in-correctdoes not differentiatea solutionin which all columns butone are correctfrom a solution in which a child demonstrated o notion of multidigitadditionorsubtraction.An analysisof thekinds of errorswas made on theten-digitproblem.The errorsidentified n Fuson(1986a) wereclassifiedinto fourcategoriesreflecting ncreas-ing amountsof knowledgeaboutmultidigitadditionor subtraction s follows:

1. Preaddition/presubtractionrror:Columnswere left blankor filled in withseeminglyrandomnumbers;presubtractionrrorsalso includedadding.


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KarenC. Fuson and Diane J. Briars 189

2. Column addition/subtractionerror: Addition/subtractionproblems wereapproached olumnby column: In additionthe sum of each columnwas writtenbelow thatcolumn even when the sum was a two-digitnumber(e.g., 28 + 36 =514); in subtraction he smallernumber n eachcolumnwas subtractedrom thelargernumber e.g., 36 - 28 = 12).3. Tradingerror: Tradingerrors nvolved some partiallysuccessful attempt otrade carry,borrow);n additionproblems heseerrorsncluded hefollowing: thetradewas not writtenor addedin anywhere,a tradewas madewhen the sum wasnot over 9, the tens digit rather hanthe ones digit was traded,a trade was madebutignoredwhenthatcolumn was added(thiserrormighthave been a fact error-such errorswere countedas both tradeandfact errors), he trade was subtractedfrom rather han addedto the top number; n subtractionproblemsthese errorsincluded the following: the left column was not reducedby one even thoughatradewas recordedin the right column, a trade was made even thoughthe topnumberwas already arger,more thanone tradewasmade from a given column, 1was subtractedromthe traded-to olumn,therightcolumnreceived11rather han10, 1 was subtractedroma left column even thoughno tradewas recorded o theright.4. Fact error: Facterrors nvolvedcorrecttradingbut incorrectaddingor sub-tracting n a column.Two coderscoded all errors.Coderagreementwas 97%.Because not every columnin every problemrequireda trade,childrenmakingconsistent column addition/subtractionrrorscould get 20% correctdigit scoreson both UntimedMinitests and 9% on the additionTen-DigitTest,andchildrenmakingtrading rrors hatwereincorrectnonlyone columncouldget digitscoresrangingbetween 36% (on the Ten-DigitTests) and 60% (on the UntimedMin-itests).

Place-value and meaningfulmultidigitaddition written tests. Threeaspectsofplace-valueunderstanding ndtwo aspectsof meaningfulmultidigitadditionwereassessedthroughwritten ests.The MixedWords oNumeralsTestrequireda childto write a three- or four-digitnumeralfor numeral/wordnamed-valuecombina-tionsgiven in mixed order e.g., 6 hundreds, tens,5 thousands,and7 ones). TheTradedWord/Numeral estrequireda childto writea three-orfour-digitnumeralfornumeral/word amed-value ombinationsgivenin standard rder e.g., 2 thou-sands, 16hundreds,1ten,and4 ones) or to fill in a numeralblank when the three-orfour-digitnumeralwas given with thenumeral/word amed-valuecombination(e.g., 2643 is 2 thousands, hundreds,14 tens, and 3 ones). All of these itemshad one numeral/word airthatexceeded 10and thus had to be traded o the left inthe former temsor to theright n the latter tems to make the correctanswer; heseitems were modeled after those in Underhill (1984). The Choose the LargerNumberTestrequireda child to choose thelargerof apairof three-through even-digitnumbersby circlingthe largernumberandby insertinga < or> betweenthepairof numbers.The five pairsof numberswere all misleading n that all digits in


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the smaller numberexcept one were equal to or greaterthan the correspondingdigits in the largernumber.The AlignmentTest presented horizontally-writtenproblemswhose addends had differentnumbersof digits; children were told towrite the problemso thatit could be addedeasily. This tested a combinationofplacevalue and additionunderstanding-understandinghatone addedand there-fore alignedlike places;the differentnumbersof digits werechosen to maximizethefrequent rrorof aligningsuchproblemson theleft rather hanon theright.TheTrading2 Insteadof 1Test consistedof problemswiththreeaddends hatrequireda tradeof 2 tens rather han 1 ten becausethe sum in the ones columnexceeded20; the first item had a sum of 21 to maximize thepossibilitythatchildrenwouldrotelytrade he 1astheyhad beendoingforproblemswithtwo addendsratherhantrading henumberof tens (2, in theseproblems).These testshadbetween two andsix items.Each test item was markedas corrector incorrect,and test means wereconverted o percentages or ease of comprehensionof the test results.Understanding fAddition,Subtraction,andPlace Value

Individualinterviews were carried out to assess children'sunderstandingofaddition, subtraction,and place value. Eight childrenfrom one class at eachachievement evel wererandomly elected to be interviewed theaverage-achiev-ing firstgraderswere fromthe class in whichall childrenparticipated).Therefore,the addition nterviewsamplecontained40 children,andthe subtractionnterviewsampleconsisted of the24 secondgraders n the addition nterviewsample.Inter-views wereconducted ndividuallyn a roomoutsidethe classroom.Childrenwereshown solved multidigit problems,each written on a separate ndex card. Eachproblemsolution was written in a color different from the color of the originalproblem.Two additionproblemsweresolvedcorrectly:a two-digit problemwithatrade romtheones to thetens and a four-digitproblemwitha trade romthe hun-dredsto the thousands.Two additionproblemswere solved incorrectly.The twomost common additionerrorsbeforeinstructionwereused: (1) columnaddition,forexample,for 8 + 6 writing14in the ones columnand(2) ignoring hetensdigitof a two-digitsumand ust writing he ones digit. Five subtraction roblemsweregiven. Two were solved correctly and paralleledthe correctlysolved additionproblems, except that differentnumberswere used.A thirdshowed the commonerrorof columnsubtraction-subtracting hesmaller romthelargernumberevenwhen thesmallernumber s on thetop.Twothree-digitproblemswithtwo zerosinthe top number were given. One was solved correctly,and the other showed 1hundred raded or 10 ones.Childrenwere told thatthey would be shownproblems hatsomebodyelse hadsolved andthatsomeproblemswere correctandsome werewrong.Theywere thenshown an index card with a problemwrittenon it andaskedif thatproblemwasrightor wrong.After a judgmentwas made,they were askedwhy it was rightorwrong.The interviewerwrotedown verbatim he child'sresponsesandanyinter-viewerprompts.Childrenwererandomlyassignedto one of two differentordersof problems.Onesequencebeganwith a correctproblem,andtheotherbeganwith


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an incorrectproblem.The more difficultproblems(the four-digitadditionprob-lem and the subtractionproblemswith zeros) were given in the last half of theinterview.The interviewrecordswere classifiedby the interviewerand one of the authors.Theclassificationof aproblemas correctorincorrectwasevaluated irst.If a childchangedhis or heranswer,the last assignmentwas coded. The ratersagreedon100% of these classifications. The interviews were coded forplace-valueunder-standingof the tens or hundredsvalues of writtennumeralswithin anexplanationof additionor subtraction;o receive credit,a child had to use the word "ten"or"hundred" o identify a numeralcorrectlysometimeduringan explanation.Theaddition nterviewswerecodedfortwo aspectsof additionandplace-valueunder-standing: (a) explainingthe writtenprocedureas trading10 ones for 1 ten or 10tensfor 1hundred,and(b) identifying hetraded1as a ten or as a hundred.For(a)a child had to explainexplicitly the tradingor say that the ten came from the 13ones or the hundred ame fromthe 16 tens.The subtractionnterviewswere codedfor threeaspectsof subtraction ndplace valueunderstanding: a) explainingthewrittenprocedureas trading1 tenfor 10 ones or I hundred or 10 tens;(b) identi-fying the traded1 as a ten or as a hundred;and(c) explainingthe doubletradingover two topzeros, i.e., thetradeof 1hundred or 10tens and thetradeof 1tenfor10 ones.All of these aspectswereevaluated or tens andfor hundreds.Coderagreementwas 95%. Children'sexplanationsdid not always spontaneouslycover all of thecodedaspectsof theinterview.A series of promptswas used to tryto ascertain uchknowledge. These includedquestions about the traded 1 ("What's he one?"or"One what?")anda questionabout the 8 tens in the four-digitadditionproblem("Eightwhat?").The most explicit promptwas to ask a child to think about theblocks; this was used when a child failed to give any answerto otherprompts.However,due to thecomplexityof the interview and thefact thatthe attributes fthe responses to be coded were finalized after the interviews were completed,needed prompts were not always given. Thus, the data may underestimatechildren'sknowledge.

ResultsAdditionMultidigitComputation

On thepretestsonly 9 of the 169childrenshowedanyindicationof correct rad-ing, whereasontheposttests160 of the 169 childrenshowed suchevidence,averylargeand statisticallysignificant change (McNemar'stest chi-square= 151,p


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192 Base-Ten Blocks Learning/Teaching Approach

Posttestdigitscores are shown in Table 1. Theseindicateexcellentperformancefor all classes except the low-achievingsecond-gradeclass and the averagefirst-gradeclass in which all childrenparticipatedn the learning/teachingapproach.Even the lattertwo classes demonstrated ome learning,because theirUntimedMinitestscores were well above those obtainableby carryingout column errors(75%and 69%compared o 20% for columnerrors).Teachersreported hatchil-drenwere enthusiasticaboutthe multidigit nstructionandenjoyed solving largeproblemsandthatmanyof thehigher-achievingecondgradersknew mostof theiraddition actsorusedthinkingstrategies o find sumstheydid not know and mostof the otherchildrencountedon withone-handedingerpatternso solve sumstheydidnotknow.Table 1AdditionComputationPosttestDigit Score Means or Each Class and AchievementLevel in Study1

Grade/achievementevel2 2 2 2 1 1 1 1Tests High/av Av Av/low Low High High Av Av

n 29 23 21 14 26 25 10 21Percentageof correctdigits in answers

Untimed Minitest ng ng ng 75 92 98 92 69Ten-DigitTest 99 93 90 58a 88 91 93 ngTimedTest 98 91 94 74 92 94 91 ngMeannumberof correctdigits completed n 2minuteson Timed Test 28 25 26 15 17 24 12 ngNote. Percentageof correctdigits in theanswer s out of all digits in the UntimedMinitestandTen-DigitTest andout of thecolumnsattemptedby a given child in the Timed Test. ng means the test was not given.aThe low-achievingsecond-gradeclass only completed2- and3-digit addition.

The errorsmade on the Ten-Digitpretestsandposttests are given in Table 2.These analysesshow a largereduction n the numberof errorsmade.Few of theprimitivepreaddition ndcolumnadditionerrorsweremadeon theposttest.Therewas a reduction n the tradingerrorsand no increasein the fact errors n spiteofthefact thatalmostall childrenwereaddingandtradingon almost allproblemsontheposttest.Table2Numberand Kindsof Pretestand Posttest Additionand SubtractionErrors in Study1

Preaddition/ Column Trading Factpresubtraction add/sub error errorTests Pre Post Pre Post Pre Post Pre Post

AdditionTen-DigitTest 527 28 837 18 109 79 57 45SubtractionTen-DigitTest 135 4 650 14 0 96 8 22Note. There were a possible 1859 errors n additionand825 errors n subtraction alculatedby multiplyingthenumberof digits in the answer(11) by the numberof subjects(N = 169 for the AdditionTest, N = 75 for theSubtractionTest).


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KarenC.FusonandDianeJ.Briars 193

SubtractionMultidigitComputationOn the pretests2 of the 75 childrenparticipating n the subtraction earning/teaching approach howedsomeevidenceof trading;on theposttests72 of the75childrenshowed such evidence, a very large and statisticallysignificant change(McNemar'stest chi-square= 70, p < .0001). Five of these 72 childrendemon-stratedsuch trading or two- or three-digitproblemsbut not for largerproblems.Paired -testanalysesof pretest-posttest ifferencesonthedigitscores foreach testfor eachclass separatelyrevealedsignificant mprovement oreverytest foreveryclass,p < .001 in all cases.Meandigit scores for each test for each class aregiven inTable3. Performance

by the high/averageclass was excellent on all tests, and for the other two classesperformancewasgoodon theTimedTestandtheUntimedMinitest.Scores for theaverageand average/lowclasses on the Ten-DigitTestand on the ZerosTestre-vealed weakerperformance hatwas neverthelessabove the level of consistenttradingerrors 36%and33%,respectively).Teachersreported hat some childrenknewsubtractionacts orusedthinkingstrategies o determinedifficultdifferencesbutthatmost countedup with one-handed ingerpatterns o determine acts theydid not know.

Table 3SubtractionComputationPosttest Class Means by AchievementLevel in Study1Achievement evel

Tests High/av Av Av/lown 29 23 23Percentageof correctdigits in answersUntimed Minitest ng 89 87Ten-Digit Test 95 72 75Timed Test 95 84 84ZerosTest 92 78 (49)Meannumberof correctdigits completedin 3 minutes on Timed Test 22 15 16Note. Percentageof correctdigits in the answer s out of all digits in theUntimedMinitest,Ten-DigitTest, andZeros Test and out of the columnsattemptedn the Timed Test. The Zeros Test for the Av/low class is inparenthesesbecause this class only beganworkon zeroproblems. ng means the test was not given.

The erroranalysespresented nTable2 indicateanalmostcompleteeliminationon the posttestof the largenumberof presubtraction nd column subtraction r-rors madeon the pretest.Substantialnumbersof tradingerrorsweremadeon theposttest,butmostposttesttradingwas correct over80%of thetrades nvolvednoerror n eithercolumn).Few fact errorswere made on the posttest, only half asmanyfact errorsas weremadein addition.Place-ValueandMeaningfulMultidigitAdditionWrittenTests

Results of the writtentest measures of place-valueandmeaningfulmultidigitadditionaregiveninTable4. Almost all children akingthesetests weremisledby


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194 Base-TenBlocksLearning/TeachingApproach

these items on thepretest(exceptfor the Circlethe LargerNumberTest).On theposttest, children in both classes showed very considerablegains on the place-value tests, all childrencorrectlyaligned problems,and most childrentraded2whenthey had 20-some ones or tens.Table4PercentageCorrecton Place-Value andMeaningfulAdditionWrittenTest in Study1

Grade/achievementevel2 High/av 2 Av

Tests Pre Post Pre PostPlace-value testsMixed Wordsto NumeralsTest 3 98 8 83TradedWord/NumeralTest 2 90 3 72Choose theLargerNumberTestCirclethelargernumber 50 96 34 84Insert> and< symbolsin the numberpairs 0 96 44 98

Meaningfuladdition estsAlignmentTest 0 100 5 100Trading2 Insteadof 1 Test 0 100 12 73Note. TheClass 2 High/av pretestsweregiven earlyin the year,and the Class 2 Av pretestsweregiven midyear.

Understanding f Place-Value,Addition,and SubtractionEveryinterviewedchild correctlyclassified all four additionproblemsas hav-ing been solved correctlyor incorrectly,94%correctlyclassified the subtractionproblemswith no zeros, and 94%of the childrencompletinginstructionon thesubtraction roblemswith zerosclassified suchproblemscorrectly.Resultsof theinterviewmeasuresaregiven in Table5. Everychild but one identifieda numeralin the tensplace as x tens at least once during heirexplanations.Similar dentifi-cation of a hundredsnumeralwas doneby 92% of the secondgradersbutby only50%of the firstgraders.Almosteverychildexplained heten-for-ones radingandidentified the traded1 as a ten for bothadditionandsubtraction;hree-fourths fthese explanationswere spontaneouswithoutany prompts.The problemswitherrorsweremuchmore effective thanwerecorrectproblems neliciting spontane-ous explanations, ndicatingthat the childrenwere notjust repeatingmemorizedverbalexplanations or correctproblems.For the hundreds onceptspromptswererequired oraboutthree-fourths f theresponses,but this seemedto stem as muchfrom the fact thatonly a correctproblemwas given for the hundred radeas fromhundredsbeingmore difficult. Children n the second-gradeaverage/low-achiev-ing class andespecially in the average-achieving irst-gradeclass showed morelimitedunderstanding f the ten/hundredradethandid the children n the otherthreeclasses. Most children ailingto identifythetraded1 as a hundred dentifiedit as a ten, and most of these identifiedthat 1 as coming from the "8 tens plus 8tens is 16 tens." Thus,they hadlearneda generalaspectof multidigit rading, otradethe tensdigit fromany two-digitsum,buttheycould not simultaneously itthis generalview of tradingwithinthe named-valueplaces to name the new value


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Table 5Percentage of StudentsDemonstratingUnderstandingof Place Value,Addition,and Subtractionin Study

Grade/achievementevel2 Hi/av 2 Av 2 Av/lo 1 Hi 1 Av

Tests Ten Hun Ten Hun Ten Hun Ten Hun Ten HunPlace-valueunderstandingIdentifythe tens and hundreds 100 100 100 100 100 75 100 75 88 25valuesof writtennumerals (13) (25)

Addition andplace-valueunderstandingExplainwrittenprocedureas 100 88 100 100 88 50 100 100 88 13trading10 ones for 1ten (13) (25) (50) (13) (13)or 10 tens for 1 hundredIdentifythe traded1 as a 100 88 100 88 100 63 100 100 88 13tenor a hundred (38) (25) (38) (13) (25)

Subtraction ndplace-value understandingExplainwrittenprocedureas 100 100 100 100 100 75 ng ng ng ngtrading1 ten for 10 ones (13)or 1 hundred or 10 tensIdentifythe traded1 as a ten 100 100 100 100 100 75 ng ng ng ngor a hundredExplainthe doubletradingover 100 100 75 88 38 38 ng ng ng ng

two top zeros:hundredsto tens and tens to onesNote.Percentagesnparenthesesrechildren horespondednlyafterheywerepromptedothink boutheblocks. ng means the test was not given.

of the traded1. Not a single interviewedchild identifiedthe traded1 as a one, insharpcontrast o childrenreceivingtraditionalnstruction.Discussion

The secondgradersandhigh-ability irstgraders howedmultidigitadditionandsubtractionomputationperformancehatwasveryconsiderablyabovethatshownby thirdgraders receiving traditional nstruction(cf. Kouba et al., 1988). Thesubtracting-smaller-from-largerrror hat is so commonin multidigitsubtractionwas almostcompletely eliminated. These childrenalso showed competence farabove thatusually demonstratedby thirdgraders n verbally labelling tens andhundredsplaces, in changingwords to numeralsandvice versaeven when thesewere given in mixed orderor required rading, n choosing the largernumber, naligningunevenproblemson therightrather hanon theleft, in showingthequan-titativemeaningof tens andones, andin identifyingthe traded1 in additionandsubtraction as a ten or as a hundred ratherthan as a one (cf. Cauley, 1988;Ginsburg,1977; Kamii, 1985;Kamii &Joseph,1988;Labinowicz,1985;Resnick,1983;Resnick andOmanson, 1987; Ross, 1986, 1989;Tougher,1981).Kamii (1985; Kamii & Joseph, 1988) andRoss (1986, 1989) reported hat ondigit correspondenceasks most secondgradersandmanythirdand fourthgraders


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receivingtraditional nstructional howno understandinghatthe tensdigitmeansten things (thesechildren show one chip-rather than ten chips-to demonstratewhat the 1 in 16means),orthey aremisledby nontengroupingsandshow only agrouping ace-valuemeaning(for 13 objects arranged s threegroupsof fourob-jects andone left-overobject, they say thatthe 3 means the threegroupsandthe 1means the one left-over object). These tasks were not availableat the time thisstudywascarriedout,butreviewersraisedthequestionof whetherchildren n thestudywouldhavedemonstrated lace-valueunderstanding n these tasks. At thattime two teacherswere still carryingout reasonable acsimiles of the instructionwith theirabove-averageand average-achievingsecond-gradeclasses. In an at-tempt to provide some informationon this issue, half the children from eachachievement-levelgroupingwithineach class were randomlychosen to be indi-viduallyinterviewed(n = 22). They weregiven these two tasks and a subtractionproblemwith zeros in the topnumber.On the Kamii task (showing with chips whatthe 6 and the 1 in 16 mean), 12childrenimmediatelyshowed ten chips as the meaningof the 1, another4 firstshowed one chip but showed ten chips when askedto show withthe chips "whatelse could thispart(the 1)mean?"anotherchild showed tenchipswhengiven thetaskagainafterworkingthe four-digitsubtractionproblem,and 3 childrenfirstshowedone chip but showed ten chips when askedto "look at the places"in 16(tens and ones were not mentioned).Thus, morethanhalf of these childrenhadtens and ones availableas theirfirstmeaningfor a two-digitnumeraland fouroth-ers hadit readilyavailableas a secondchoice, while fourmore first showed theirunitarymeaningbut showed a tens and ones meaningwhen a multidigitcontextwas elicited for them;overall,91% of the interviewedchildrenshowed thatthe 1meanttenobjects.Not a singlechild showedagrouping ace-valuemeaningon theRoss task;performancewas the same as performance n the Kamii task.Thus,onthese tasks also, second gradersusing the base-tenblocks showed performanceconsiderablyabove thatordinarily hownby secondgradersreceivingtraditionalinstruction.

STUDY2Method

Subjectsand TeachersPotentialsubjectswere all secondgraders n the 132 second-gradeclassroomsin the PittsburghPublic School system.A 2V2-hourn-servicetrainingsession onusing base-tenblocks to teachmultidigitadditionand subtractionwas offeredtoall second-grade eachers nAugust.This in-service session was voluntary;each-ers were paid salaryto attend.The workshopwent through he teacherplansforthelearning/teaching pproach,ocusing particularly n usingtheblocksand ink-ing actionson the blocksto stepsin the writtenmultidigitadditionand subtractionprocedures. n November,a follow-up 2V2-houression focusing more intenselyon subtractionincludingthe new trade-firstalgorithm)was given to these teach-ers, anda 2V2-houression on bothadditionand subtractionwas given for teach-


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ers who had not attendedthe August session. These sessions were given by thesecond author,who is experienced n using the base-ten blocks to teach the mul-tidigit algorithms.A mathsupervisorwho had no previous experience with thebase-tenblocksgaveanother2?-hour in-servicesession in December orthose notableto attendearliersessions.Most second-grade eachers 91%)attendedatleastone of these sessions.Teachers were urgedto use the base-ten blocks and lesson plans to teach themultidigit algorithms.Threeelementarymathematicssupervisorswere availableas questionsarose,though theyalso hadmanyotherdutiesconcerning eachersatothergradelevels. The supervisorsencouraged eachers to trythe approach,butbecause the goals wentconsiderablybeyondthe districtsecond-gradegoals, par-ticipationwas voluntary.Manyteachersstarted eaching multidigitaddition andsubtraction omewhat atein theyearandexpresseddoubtsthat heywould be ableto finish all of the units. In orderto increasethe numberof teachersfinishing atleasttheadditionand subtractionwork,thesupervisors uggestednotcoveringthemeaningfuladdition andplace-valueunits but finishing the subtractionwork atleast up to the problemswith zeros. The number of teachers and children whoparticipatedn variousaspectsof the studyarediscussedin the final sectionof themethods section.Instruction

Teacher esson plansanda class set of studentworksheets n individualstudentbooklets(bothas described n Study 1) were sent to each second-grade eacher nthe district. At the in-service sessions some teachersexpresseda preference orusing the blocks to show subtractionas take-away nsteadof as comparisonbe-cause the take-awaymethod fitted better theirconceptionof subtractionas take-away.Teacherswereallowedtouse take-away f theywished: Thetopnumber theminuend) was made with blocks and blocks were taken away for the bottomnumber.One class set of base-ten blocks (theEducationalTeachingAids neutral-coloredblocks, metricversion)was available n each school.Testing

Tests.The additionand subtractioncalculationtests and the place-value andmeaningful-additionwritten ests usedin Study 1 wereused in thisstudy.All testswere given as pretestsat the beginningof the year.The same tests were given asposttestsas each phase of the learning/teachingapproachwas finished (e.g., theadditioncalculationtests were given at the completionof the additionteaching).Teachersgradedall tests accordingto writtendirections. They returned o thecentral district office the pretests accompaniedby a class list containingpretestscores. Posttestsaccompaniedby a class list withposttestscoreswerereturned othe centraloffice as teachersgave them. For both thepretestsand theposttests,thetests of four children n each classroom,two boys andtwo girls, wererandomlyselected andgradedby researchstaff members n order o check the teachergrad-ing. The few teacherswithsystematicgradingerrorshadtheir scorescorrected.


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Criterionscoresand errorclassification.Criterion coreswereadopted or theadditionand subtractionUntimedMinitests, he additionand subtractionTen-DigitTests, and the subtractionZeros Test. These were based on the digit scores de-scribed orStudy 1. Thetradingcriterion corewas 8 or more forthe additionandsubtractionUntimedMinitestsand the additionandsubtractionTen-DigitTestsbe-cause a child making tradingerrors hat were incorrect n only one column couldobtain scores of 6 out of 10 on the Untimed Minitestsand4 out of 11 on theTen-Digit Tests.A score of 8 requireda child to make at least two correcttradeswithno facterrorson the UntimedMinitestsand four correct radeswithno facterrorson theTen-DigitTests. For the subtractionZerosTest,a criterion coreof 9 (of the12digitscorrect)wasselected becausethisscore meant hat he child demonstratedcorrect rading or at least two of the three zeroaspectstested.Erroranalyseswerecarriedout on fourTen-DigitTestsdrawn atrandom romeach of 30 classroomsrandomlyselectedfor eachtestand time (pretest,posttest).Errorswereclassified into the fourcategoriesusedin Study 1. The classificationwas doneby the sametwo coders used in Study 1;coderagreementwas 96%.The PretestandPosttestSamples

Of the 132teachers,125(95%)returned retests or 2723 children.Pretestswerereturned romat least one classroom for everyschool in the district.Across all ofthe tests the numberof completed pretests rangedbetween 2531 and 2378. Toascertainwhetherthepretestsrepresented he whole sampleof childrenwith oneor more returnedpretests,on each test the scores of childrenwho hadcompletedataon all tests werecompared o scores of childrenwho had oneor moremissingscores on othertests.There were no significantdifferencesbetween these groupson anytests.Only partof the potentialsampleof classroomscompletedthe work with thelearning/teaching pproach ndreturned heposttests.The numberof childrenwithreturnedposttests is given for each test in Table 6. The numberof teacherswhoreturned dditioncalculation,subtraction alculation,andplace-value/meaningfuladditionposttestswas 42, 35, and 16,respectively.These teacherscame from 18,18, and9 differentschools, respectively.This partialreturnraised the obviousquestionof whetherthe childrenfor whomposttestswerereturneddiffered fromthechildrenwithoutreturnedposttests.The additioncalculationpretestswere thefocus of the differenceanalysesbecause all otherpretestsshowed floor effects.Severalaspectsof the additionpretestsfor thechildrenwith no returnedposttestswerecompared o pretests orthechildrenwith returnedposttests.Thepercentageof childrenwithpretestscoreson the UntimedMinitest at or abovecriterionwas abit higher or thechildrenwithno postteststhanforthose withposttests,themeandigit scores on the UntimedMinitest and the TimedTestwere about the same forboth groups, and childrenwith no posttests showed somewhat more advancederrors handid the children with posttests(moreof the formermade at least onetradingerrorwhile more of the lattermadepreaddition r column additionerrors).Thus,theposttestsamplechildrenwere,if anything, nitiallya bit worseatmulti-


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KarenC. FusonandDianeJ. Briars 199

digit addition calculation than the children not participating,and all childrenshowed flooreffects on the multidigitsubtraction alculationandplace-valueandmeaningfuladditionpretests; hat s, bothgroupshadthe same low level of initialknowledge.Most (90%)of theposttestteacherscamefrom a school in whichall the second-gradeteachers returnedposttests.Teacherswithin a given school almostalwaysreturnedexactly the same posttests. Thus, the performancedata to be reportedcome fromallachievement evels of secondgraders.Theschoolswith all teachersparticipatingwere distributedacross the whole rangeof schools in the city withrespectto location,ethnicity,andsocioeconomic level. Participatingeachersdidnot seem to differmuchfromnonparticipatingeachers n theirrateof attendanceat the in-service sessions: 76%, 15%,and 10% of the participating eachers at-tendedtwo, one, and zero sessions, respectively,while these percentages or thenonparticipatingeacherswere68%, 23%,and 9%.Two classes from a magnetschool were dropped romthe additionsamplebe-cause more than half the children were above criterionon the pretest,indicatingprevious addition instruction.One of these classes was also droppedfrom thesubtraction amplefor the same reason.

ResultsAdditionMultidigitCalculationPerformanceOn the pretest 10%of the instructedsamplemet the criterionon the UntimedMinitest,and on the posttest96% of the childrenmet this criterion.This shift forthe Ten-DigitTest was from 5% on the pretestto 90%meeting criterion on theposttest. Both of these changes were significant, McNemar'stest of correlatedproportions hi-square= 674 and659,p < .0001. The childrenwerequiteaccurateadders,withdigit scores on the three tests showingthatthey solvedbetween 89%and 96%of the columnscorrectly(Table6), andthey solved a mean of 24.3 col-umnsof multidigitproblemscorrectly n 2 minutes.These children showed the same large reduction in preadditionand columnadditionerrors rom thepretest o theposttestas shownby thechildren n Study 1(see Table7). Tradingerrorswere also reducedconsiderably, ven thoughalmostall childrenweretradingon theposttest.

SubtractionMultidigitCalculationPerformanceHardlyanychildrenmet thetrading riteriaon thesubtraction retests(2%, 1%,and 0.4% on the UntimedMinitest,Ten-DigitTest,and ZerosTest,respectively),but84%, 70%,and81%of the instructed hildrenmet thecriterionon therespec-tive posttests.These changes were all significant,McNemar'schi-square= 580,487, 486, p < .001. Children obtaineddigit scores on the various tests rangingbetween 80%and90% correct seeTable6). Children olved subtraction roblemsmore slowly than additionproblems,solving a mean correct 18.4 columns in 3minutes.


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200 Base-TenBlocksLearning/TeachingApproachTable6PercentageCorrecton theAdditionand SubtractionComputationPosttestsand the Place-ValueandMeaningfulAdditionPosttestsin Study2

n % CorrectAdditioncomputationestsUntimed Minitest 783 96

Ten-DigitTest 776 89Timed Test 780 92Subtraction omputationestsUntimed Minitest 707 90Ten-DigitTest 705 80Timed Test 669 85Zeros Test 602 85

Place-valuetestsMixed Wordsto NumeralsTest 360 88TradedWord/NumeralTest 360 53ChoosetheLargerNumberTestCirclethe largernumber 360 67Insert> and< symbolsin thenumberpairs 363 65Meaningfuladdition estsAlignmentTest 300 85Trading2 Insteadof 1Test 278 80Note. The % correct or the additionand subtraction omputation ests arethe percentageof correctdigits out ofthe totaldigits in the UntimedMinitests,Ten-DigitTests, and Zeros Test andout of the digitsattemptedby agiven child in the Timed Tests.

Table 7NumberandKindsof PretestandPosttestAdditionand SubtractionErrorsin Study2Preaddition/ Column Trading Factpresubtraction add/sub error errorPre Post Pre Post Pre Post Pre Post

AdditionTen-DigitTest 341 7 798 3 79 45 11 83SubtractionTen-DigitTest 282 8 984 26 6 187 1 58Note.Therewereapossible 320errorsoreach estcalculatedymultiplyinghenumber fdigitsntheanswer11)bythenumber f subjectsN= 120).

The subtraction-errornalysesindicateda substantialmovementfromthe pre-subtractionand column subtraction rrors o the more advancedtradingandfacterrors(see Table7). The percentagesof posttesterrorsfalling within each errorcategoryare similarfor Study 1 andStudy2.Place-ValueandMeaningfulAdditionTests

Thepretestscoresonmost of theplace-valueandmeaningfuladdition ests wereverylow, indicating hat childrenwereresponding o the misleadingnatureof theitems.Forexample,on theAlignmentTest,mostchildrenalignedthe numbersonthe left, recopied the problemshorizontally,or treated each digit as a separatenumberand formednew problems(e.g., 67 + 1385 was writtenverticallyas 67 +13+ 85). On the testgivingmixed orderwords,38%ignoredthe words andwrotethe numerals n theirgiven orderand 39%left blanksorwroteseeminglyrandom


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Karen C. Fuson andDiane J. Briars 201

responses; only 23% showed even any partial knowledge. About a sixth of thechildrendid get threeof the five itemscorrecton the Choose the LargerNumberTest and anothersixthgot four or five itemscorrect, ndicatingsome pretestabil-ity to comparemultidigitnumbers.Performance n theposttestMixed Words o SymbolsTest,theAlignmentTest,and theTrading2 Insteadof 1Test was good, ranging rom80%to 88%(see Table6). Individual lass meanson these testsranged rom lows of 59%to 66%to highsof 100%.Performanceon the Choose the LargerNumberTestimproved o mod-erate evels of accuracy,with littledifferencebetweenscores obtainedby circlingthe largernumberor inserting< or> between the numbers 67%and65%).Classmeanson the TradedWord/NumeralTestwere extremelyvariable,rangingfrom3%to 88%,with an overall meanperformanceof 53%of the itemscorrect.

DiscussionInformal teacherreportsvia the supervisorsand directcommunication o thedistrictmathematicsdirector ndicatedconsiderableenthusiasmandenjoymentofthe learning/teaching pproachby both teachersandchildren.Being able to solvelargeproblemsseemed to empowerchildrenand make themfeel goodabout hem-selves and about mathematics.Children learnedmultidigitadditionquite well,

though heystill made some addition act errorsandoccasionaltrading rrors.Thesubtraction est scoresanderroranalysesindicated hatmostchildrencould tradecorrectly and that few continued to make the presubtractionand the subtract-smaller-from-largerrrors o common on thepretests.However,manychildrendidnotcompletelymastersubtraction omputationand continued o make some trad-ing and facterrors,especiallyon theten-digitproblem.Bothadditionand subtrac-tion performancewas considerablyabove thatordinarilyreported or thirdgrad-ers,as wasperformance n theAlignmentTest,theMixedWords o SymbolsTest,and the Choose theLargerNumberTest. Childrenshowedmore limitedabilitytogeneralizetrading o the new TradedWord/NumeralTest.There were obvious limitations to this study. Because systematic classroomobservationswerenot made, it is not clear how closely the work with the blocksfollowed the lesson plans.Thus,no inferences can be made aboutwhichfeaturesof the learning/teachingapproachmight be crucial and whetherany might beexpendable.It is not clearwhy teachers n some schools participatedwhile thosein other schools did not. Informalreportsto field supervisors ndicated thattheschool-baseddecisionsto participatewere sometimes nitiatedby theprincipalandsometimesby the teachers.The field supervisorsreported hatsome teachersex-pressed skepticismthat secondgraderscould learn materialso much abovegradelevel even thoughthe success of the approachwith the childrenin Study 1 wasdiscussedin the in-servicesessions;this skepticismmayhavecontributedo deci-sionsnotto use the approach.Thepartialparticipation y teachersdid not seem tobias the samplewithrespectto initialknowledgeof theparticipatinghildren.Theteacherassignmentandtransferpoliciesof the districtmake tunlikelythatthe bestteachersareheavilyconcentratedn certainschools (i.e., only in the participating


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schools), but there still might have been some bias towardparticipationby thebetter teachers in the district.Finally, althoughthe scores on the addition andsubtraction omputation ests and the shiftsin errors rompretest o posttestweresimilar n Study 1 andStudy2, the lackof interviewdata n Study2 means that tis not clear whether hechildren nStudy2 understood ndcouldexplainmultidigitadditionand subtraction s well as could the children n Study 1.GENERAL ISCUSSION

On all tests and interview measures, performanceby second gradersof allachievement evels considerablyexceededthatreported n the literatureor thirdgraders eceivingusual instruction.Most children earned o trade nfour-digitad-dition and subtraction roblems,columnerrors requently esulting romusualin-structionwerevirtuallyeliminated,and childrenshowed considerablegeneraliza-tionof multidigitadditionand subtractiono multidigitproblems arger hanfourdigits. Mostchildrenalignedunevenadditionproblemson the right,traded2 in-stead of 1whennecessary,and couldtranslate rom mixed words and numerals omultidigitnumerals.Childrenin Study 1 showed in the interview quantitativeunderstanding f writtenmultidigitnumeralsandused this understandingo ex-plainone/tenandten/hundredradingproceduresnboth additionand subtraction.Theseresultsindicatethatsecond-gradeclassroom teacherscan use the learn-ing/teachingapproach ffectivelyto supporthighlevels of meaningful earning nmanyof theirchildren.Children roma smallcity/suburban eterogeneouspopu-lation and childrenfrom a wide rangeof schools in a largeurbanschool districtdemonstrated uch learning,so the learning/teaching pproachcan be used suc-cessfullywith a fairlywiderangeof children.Thesuccessfullearning nboth stud-ies indicated hatthelearning/teaching pproach ouldbe implementedon a broadscale with a moderateamountof in-servicetime, materials,and teachersupport.Manyparticipatingeachers n Study2 did askfor their own set of blocksfor thecoming year,so one set of blocks per buildingis clearlynot ideal. In particular,moresets of blocksmayfacilitatetheuse of place-valueunits in the crowdedend-of-the-yearschedule.The approachdid not result in maximallearningin all areasby all children.Some childrencontinuedto make occasionaltradingandfact errors,particularlyin subtractionwith theten-digitproblem.Some childrenwere not ableconsistentlyto choose the largerof two three-digit hroughseven-digitpairsof numbers,andmanychildren n Study2 didnotgeneralize rading o all of the items ontheTradedWord/Numeral est.Whether hese limitationsareinherent n thisapproach r aredue to inadequatemplementation f certain eaturesof the approachor simplytoinsufficient ime with theapproachor some children s not clear.Inthe firststudyof the approach Fuson, 1986a),telling childrento "thinkabout the blocks"wassufficientfor most of themto self-correcterrors hey were still makingafter theinitiallearningor to self-correcterrors hatbeganto appearon delayed posttestsafter correct initial learning. Thus, the blocks can be a powerful supportforchildren'sthinking,but many childrendo not seem spontaneouslyto use their


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knowledgeof the blocksto monitor heirwrittenmultidigitadditionor subtraction.This suggeststhatfrequentsolving of one multidigitadditionor subtractionprob-lem accompanied by children's thinking about the blocks and evaluatingtheirwritten-marks roceduremightbe a powerfulmeans to reduce heoccasional trad-ing errorsmadeby children.A limitationof both of thesestudies s thattheirdesignsdidnotpermitanevalu-ation of any of the specific featuresof the learning/teachingapproach.The ap-proachhad many features,not all of which may be crucial to its success. Thesefeatures stemmed from the need to providechildren an opportunity o constructconceptualstructures or the mathematicallydifferentEnglish named-valuesys-tem of numberwordsand the positionalbase-tensystem of writtenmarksand tothinkabouthow these systems workin multidigitaddition andsubtraction;howthe featuresrelateto children's earningarediscussedinFuson(inpressa). Thesestudies are also limited because they were not intended to provide a completeadditionand subtractionor place-value experience. Obviously important opicswereomittedthatrelateto the goals of understandingmultidigitadditionandsub-traction(e.g., estimation,alternativemethods of addingandsubtracting).Futureworkmightexplorehow well the learning/teaching pproach ould support hesemore extensivegoals.These two studiesraise several issues for futureresearchconcerning he use ofembodiments n learningmultidigitadditionand subtraction ndplace value (seealso Baroody, n press;Fuson, in pressb). First,we took no positionconcerningwhether the teacheror the children moved the blocks or whetherlearning pro-ceeded withina total class approach,withinsimultaneous mallgroups,or withinserial small groups. In Study 1 differentteachers used all of these, and they allseemedto be effective. Otherpossibleoutcomesof thesedifferentapproaches forexample,beliefs thatsuccess dependson effort,attempts o understand, ndcoop-eration with peers as reported n Nicholls, Cobb, Wood,Yackel, & Patashnick,1990, for a small-group problem-solvingclassroomorganization)might be ex-plored.Second, relative benefits of using the learning/teaching pproach o sup-port prechosenmultidigitadditionand subtractionprocedures,as in the presentstudies, versus using the approach o supportprocedures nventedby children,might be examined. Thus, the focus of the present studies on computationasmeaning(on understandingmultidigitaddition and subtractionandplace value)mightbe contrastedwithcomputation sproblemsolving(Labinowicz,1985).Thelatterdoes not necessarilyresult in morecompetence(for example, only 34% ofthe thirdgraderswho had reinventedarithmeticwithout traditional nstructionsolved 43 - 16 correctly,Kamii, 1989), but the supportof the learning/teachingapproach n Figure 1 mighthelp children nventmultidigitadditionand subtrac-tionprocedures.Third,severalaspectsof a moregradualuse of base-tenblocks asproposedelsewhere do not seem to be necessaryforhigh levels of skill andunder-standing,becausetheywerenot implemented n ourapproach.These includepro-longed work with two-digitnumbers,followed considerablylaterby work withthree-digitand even laterby four-digitnumbers; ather xtensiveexperiencewith


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204 Base-Ten Blocks Learning/Teaching Approach

tradingbeforetrading s set withinadditionproblems;extensivepractice ustwiththe blocks with no recording;pictorialrecordingbeforerecordingwith base-tenwritten marks;and use of the blocks to count on by tens and hundreds(e.g.,Baroody,1987;Davis, 1984;Labinowicz, 1985;Wynroth,1980).Futureresearchmay establish that these aspectsdo bring particularbenefits,but it seems wise toundertake uch researchrather hanmerelyto assertthesebenefits.The resultssuggesta gradeplacement ormultidigitadditionandsubtraction ndplace-value concepts with this approach.Even thoughmany average-achievingfirst graderswere able to learnthe multidigitadditionalgorithm, heirrelativelypoorerperformanceon some aspectsof the interviewsuggests thatthe approachin these studiesriskspushingchildrenbeyondtheircomfortable earningrange.Some of these childrenmay still requireperceptualunit items for thinkingaboutsingle-digit numbersand thus may have troubleusing the blocks to constructconceptualten-unit,hundred-unit, ndthousand-unittemsmade out of collectedones. Therefore, for first gradersof average and below-average mathematicsachievementandperhapseven for manyhigh-achievingfirst graders, t may bebetterto concentrate n the first gradeon helpingchildrento build and use theirunitary equence/counting onceptualstructures oraddingandsubtracting ingle-digit numbers(i.e., sums anddifferencesto 18). Tryingto build simultaneouslythese unitaryconceptualstructuresand the multiunitnamed-value/base-tenon-ceptualstructuresneededformultidigitadditionandsubtraction, speciallygiventhe interference he irregularEnglishnumberwords create forthis task(cf. Fuson&Kwon,inpress), maybe too difficultformanyfirstgraders.Thelearning/teach-ing activitiestestedin thesestudiesdo seem to be developmentallyappropriateorsecond-grade hildrenof all achievement evels exceptperhaps hosewithspecialdifficulties.Teachersreported hatsecond-gradechildren n both studiesenjoyedthe learningactivitiesand felt good aboutthemselvesand theirabilityto do suchproblemswith understanding.Thus, the typicaltextbookextension of multidigitadditionandsubtraction roblemsoverGrades2 through4 or5, addingone or twodigits each year(Fuson, in press c), underestimateswhat our childrencan learn.Theconceptualbases forgeneralmultidigitadditionandsubtraction lgorithmsarewell withinthe capacityof most secondgraders f they arelearnedwith the sup-portof physicalmaterials hatembodytherelative size of the base-tenplaces anddemonstratehe positionalnatureof the multidigitwrittenmarksandif the focusof such learning s understanding nd notjust procedural ompetence.

REFERENCESBaroody,A. J. (1987). Children'smathematicalhinking:A developmentalrameworkor preschool,primary,andspecial educationteachers. New York: TeachersCollege Press.Baroody,A. (in press).How and when shouldplace-valueconceptsand skills be taught?Journal orResearch in MathematicsEducation.Bell, M. S., Fuson,K. C., & Lesh,R. A. (1976). Algebraicandarithmetic tructures:A concreteap-proachfor elementary chool teachers. New York: The FreePress.Cauley,K. M. (1988). Construction f logicalknowledge: Studyof borrowingn subtraction.ournalof EducationalPsychology, 80, 202-205.


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Cobb, P:, & Wheatley,G. (1988). Children's nitialunderstandings f ten. Focus on LearningProb-lems in Mathematics,10, 1-28.Davis,R. B. (1984). Learningmathematics:Thecognitivescienceapproachto mathematics ducation.Norwood,NJ:Ablex.Davis, R. B., & McKnight,C. C. (1980). The influenceof semanticcontent on algorithmicbehavior.Journalof Children'sMathematicalBehavior,3, 39-87.Dienes, Z. P. (1960). Buildingupmathematics.London:HutchinsonEducationalLtd.Fuson,K. C. (1986a). Roles of representation nd verbalizationn the teachingof multi-digitadditionandsubtraction.EuropeanJournalof Psychology of Education,1, 35-56.Fuson,K. C. (1986b). Teachingchildrento subtractby counting up.Journal or Research in Mathe-maticsEducation,17, 172-189.Fuson,K. C. (1987). Research ntopractice: Adding by countingon with one-handedingerpatterns.ArithmeticTeacher, 35(1), 38-41.Fuson,K. C. (1988a). Children'scountingand conceptsof number.New York: Springer-Verlag.Fuson, K. C. (1988b). Researchinto practice: Subtractingby counting up with one-handedfingerpatterns.ArithmeticTeacher,35(5), 29-31.Fuson, K. C. (in press a). Children'sconceptualstructures or multidigitnumbers: Implications oradditionandsubtraction ndplace-value learningandteaching.Cognitionand Instruction.Fuson,K. C. (in pressb). Issuesin place-valueandmultidigitadditionandsubtractionearning.Jour-nalfor Researchin MathematicsEducation.Fuson,K. C. (inpressc). Researchon learningandteachingadditionandsubtraction. nG. Leinhardt& R. T. Putnam Eds.),Cognitiveresearch: Mathematics earningandinstruction.Fuson,K. C., &Kwon,Y. (inpress).Chinese-based egularandEuropean rregular ystemsof numberwords: ThedisadvantagesorEnglish-speaking hildren. nK.Durkin& B. Shire(Eds.),Languageand MathematicalEducation.MiltonKeynes,GreatBritain:Open UniversityPress.Fuson,K. C., Richards,J., & Briars,D. J. (1982). Theacquisitionand elaborationof thenumberwordsequence.In C. Brainerd Ed.),Progress in cognitivedevelopment:Children's ogical andmathe-maticalcognition,Vol. 1 (pp. 33-92). New York:Springer-Verlag.Fuson,K. C., & Secada,W. G. (1986). Teachingchildren o addby countingon withfinger patterns.CognitionandInstruction,3, 229-260.Fuson,K. C., Stigler,J.W., & Bartsch,K. (1988). Gradeplacementof additionand subtractionopicsin China,Japan, he Soviet Union, Taiwan,andtheUnitedStates.Journal or Research in Mathe-maticsEducation,19, 449-458.Fuson,K.C.,&Willis,G. B. (1988). Subtracting y countingup: Moreevidence.JournalforResearchin MathematicsEducation,19, 402-420.Ginsburg,H. (1977). Children'sarithmetic:How theylearnit andhowyou teach it. Austin,TX: Pro-Ed.Kamii,C. (1985). Youngchildren reinventarithmetic.New York: TeachersCollege Press.Kamii,C. (1986). Place value: An explanationof its difficultyandeducational mplicationsfor theprimarygrades.Journalof Research in ChildhoodEducation,1, 75-86.Kamii,C. (1989). Young hildrencontinue o reinventarithmetic-2nd grade: ImplicationsofPiaget'stheory.New York: TeachersCollege Press.Kamii, C., &Joseph,L. (1988). Teachingplacevalue anddouble-columnaddition.ArithmeticTeacher,35(6), 48-52.Kouba,V. L., Brown,C.A., Carpenter,T.P.,Lindquist,M.M., Silver,E.A., & Swafford,J.O. (1988).ArithmeticTeacher,35(8), 14-19.Labinowicz,E. (1985). Learning rom children: New beginnings or teaching numericalthinking.Menlo Park,CA: Addison-WesleyPublishingCompany.Miura,I. (1987). Mathematics chievementas a functionof language.Journalof EducationalPsychol-ogy, 79, 79-82.Miura,I. T., Kim, C. C., Chang,C., & Okamoto,Y. (1988). Effects of language characteristicsonchildren'scognitive representationf number:Cross-nationalomparisons.ChildDevelopment, 9,1445-1450.


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Miura,I. T., & Okamoto,Y. (1989). Comparisonsof AmericanandJapanese irstgraders'cognitiverepresentationf numberandunderstandingf placevalue. Journalof EducationalPsychology,81,109-113.Nicholls, J.G., Cobb,P., Wood,T., Yackel,E., &Patashnick,M. (1990). Assessing students' heoriesof success in mathematics: ndividualandclassroomdifferences. Journal or Research in Mathe-maticsEducation,21, 109-122.Resnick,L. B. (1983). A developmental heoryof numberunderstanding.nH.P.Ginsburg Ed.), Thedevelopmentof mathematicalhinking.New York: Academic.Resnick,L. B., & Omanson,S. F. (1987). Learning o understand rithmetic. n R. Glaser(Ed.),Ad-vances in instructionalpsychology(Vol. 3). Hillsdale,NJ: Erlbaum.Ross, S. H. (1986, April).Thedevelopmentof children'splace-value numeration onceptsin gradestwo through ive. Paper presentedat the annualmeeting of the American EducationalResearchAssociation,San Francisco.Ross, S. H. (1989). Parts,wholes,andplacevalue: A developmentalview. ArithmeticTeacher,36(6),47-51.Song, M., & Ginsburg,H. P. (1987). Thedevelopmentof informalandformalmathematicalhinkingin KoreanandU.S. children.ChildDevelopment,57, 1286-1296.Steffe, L. P., & Cobb,P. (1988). Constructionof arithmeticalmeaningsand strategies.New York:Springer-Verlag.Steffe,L.P., von Glasersfeld,E., Richards,J.,& Cobb,P. (1983). Children'scounting ypes: Philoso-phy, theory,andapplication.New York: PraegerScientific.Tougher,H. E. (1981). Too manyblanks What workbooksdon't teach. ArithmeticTeacher,28(6),67.Underhill,R. (1984). Externalretentionand transfer ffects of special place value curriculum ctivi-

ties. Focus on LearningProblems n Mathematics,1 &2, 108-130.Wynroth,L. (1969/1980). Wynrothmathprogram: The natural numberssequence. Ithaca,NY:WynrothMathProgram.

AUTHORSKAREN C. FUSON, Professor,School of Educationand Social Policy, 2003 SheridanRoad,North-westernUniversity,Evanston,IL60208-2610DIANE J. BRIARS,Directorof the Division of Mathematics,PittsburghPublicSchools, 850 BoggsAvenue,Pittsburgh,PA 15211

base ten blocks fuson 1990

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