Using a Base-Ten Blocks Learning/Teaching Approach for First- and Second-Grade Place-Value and Multidigit Addition and Subtraction Author(s): Karen C. Fuson and Diane J. Briars Source: Journal for Research in Mathematics Education, Vol. 21, No. 3 (May, 1990), pp. 180206 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/749373 . Accessed: 12/03/2011 14:28 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=nctm. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Journal for Research in Mathematics Education. http://www.jstor.org Journalfor Researchin MathematicsEducation 1990, Vol. 21, No. 3, 180-206 USING A BASE-TEN BLOCKS LEARNING/ TEACHINGAPPROACHFOR FIRST- AND SECOND-GRADEPLACE-VALUEAND MULTIDIGITADDITION AND SUBTRACTION KAREN C. FUSON, NorthwesternUniversity DIANE J. BRIARS, PittsburghPublic Schools A learning/teachingapproachused base-tenblocks to embodythe Englishnamed-valuesystem of numberwordsanddigit cardsto embodythe positionalbase-tensystem of numeration.Steps in additionand subtractionof four-digitnumberswere motivatedby the size of the blocks and then were carried out with the blocks; each step was immediately recorded with base-ten numerals.Childrenpracticedmultidigitproblemsof from five to eight places afterthey could successfully add or subtractsmallerproblemswithout using the blocks. In Study 1 six of the eight classes of firstand second graders(N = 169) demonstratedmeaningfulmultidigitaddition and place-value concepts up to at least four-digit numbers;average-achieving first graders showed more limited understanding.Three classes of second graders(N = 75) completed the initial subtractionlearninganddemonstratedmeaningfulsubtractionconcepts.In Study2 most second gradersin 42 participatingclasses (N = 783) in a large urbanschool districtlearnedat least four-digitaddition,and manychildrenin the 35 classes (N = 707) completingsubtraction work learnedat least four-digitsubtraction. The English spoken system of numberwords is a named-valuesystem for the values of hundred,thousand,andhigher; a numberwordis said andthenthe value of that numberword is named. For example, with five thousandseven hundred twelve, the "thousand"names the value of the "five" to clarify that it is not five ones (= five) but is five thousands.In contrast,the system of writtenmultidigit numbermarksis a positionalbase-tensystem in which the values are implicit and are indicatedonly by the relativepositionsof the numbermarks. In orderto understand these systems of English words and writtennumbermarks for large multidigit numbers, children must construct named-value and positional base-ten conceptual structuresfor the words and the marks and relate these conceptual structuresto each otherand to the words and the marks. English words for two-digit numbersare irregularin several ways and are not named-value,in contrastto Chinese (and Burmese, Japanese,Korean,Thai, and Vietnamese)words in which twelve is said "tentwo" and fifty seven is said "five ten seven."These irregularitiesmake it much more difficult for English-speaking Study I was fundedby a grantto the Universityof Chicago School MathematicsProject from the Amoco Foundation.Thanksgo to MaureenHanrahanfor handlingall of the field details for Study 1; to GordonWillis for carryingout the data analyses for both studies;to Fred Carr,Tracy Klein, and Thuc Huong for careful grading,data entry,and erroranalyses for both studies;andespecially to the teachersof both studies who were willing to try something new because they thoughtit might help theirchildrenlearnbetter.Thanksalso to Art Baroody, Paul Trafton,and several anonymousreviewers who made helpful comments on earlierdrafts. 181 childrenthanfor Chinese, Japanese,or Koreanchildrento constructnamed-value meanings for multidigit numbers (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 unitaryconceptual structuresfor two-digit numbers as counted collections of single objects or as collections of spoken words (Fuson, Richards,& Briars, 1982; Fuson, 1988a; Steffe, von Glasersfeld, Richards,& Cobb, 1983; Steffe & Cobb, 1988); these early conceptual structurescan interferewith children's later constructionof named-valuemeanings. The lack of verbal supportin the English languagefor named-valueor baseten concepts of ten makes it particularlyimportantthat supportfor constructing such ten-structuredconceptions be provided in other ways to English-speaking children. In the United Statessuch supportis rarelygiven or is insufficient.Childrenmore commonly are taughtmultidigitadditionand subtractionas sequentialprocedures of adding and subtractingsingle-digit numbersand writingdigits in certainlocations (Fuson,in press c). These experiencesresultin manyU.S. childrenconstructing conceptual structuresfor multidigit numbers as concatenated single-digit numbers, a view that is inadequatein many ways and results in many errorsin place-value tasks and in multidigit addition and subtraction(Fuson, in press a; Kouba et al., 1988). Even many children who carryout the algorithmscorrectly do so procedurallyand do not understandreasons for crucialaspectsof the procedure or cannot give the values of the tradesthey are writingdown (Cauley, 1988; Cobb & Wheatley, 1988; Davis & McKnight, 1980; Labinowicz, 1985; Resnick & Omanson,1987). U.S. childrenalso show quite delayedunderstandingof placevalue concepts (Kamii, 1986; Koubaet al., 1988; Labinowicz, 1985; Miuraet al., 1988; Ross, 1989; Song & Ginsburg,1987). Furthermore,in the United States, instructionin the additionand subtractionof whole numberstypicallyis both delayed and extendedacross gradesmore thanin countrieslike China,Japan,Taiwan,and the Soviet Union thathave been characterized as fostering high mathematics achievement (Fuson, Stigler, & Bartsch, 1988). In the United States the single-digit sums and differencesto 18 consume much of the first two grades,and work on the multidigitalgorithmswith trading (carryingand borrowing)is distributedover 4 or 5 years beginningwith two-digit problems in second grade followed by the introductionof problems one or two digits largereach year.In contrast,othercountriesstress masteryof sums and differences to 18 in the first grade, and they complete multidigit instructionby the thirdgrade. In orderto use and understandEnglish words and base-ten writtenmarksand add and subtractmultidigitnumbers,childrenneed to link the words and the written marksto each otherand need to give meaningto both the wordsandthe marks. The learning/teachingapproachused in the presentstudies was developed to meet these goals. It is an adaptationof an approachused by the first authorwith teachers and childrenfor 20 years (the teacherversion is in Bell, Fuson, & Lesh, 1976). It provides childrenan opportunityto constructthe necessary meaningsby using Base-TenBlocks Learning/TeachingApproach 182 for each system a physical embodimentthat can direct their attentionto crucial meanings and help to constraintheir actions with the embodimentsto those consistent with the mathematicalfeaturesof the systems. The English named-value system of words is embodiedby a set of base-tenblocks (Dienes, 1960), and the positional base-ten writtenmarksare embodied by digit cards (numeralswritten on small individualcards). English words, words for the block embodiment,and words for the digit cards (see Figure 1) were used to help direct children'sattention to critical featuresof the mathematicalsystems and embodiments,facilitate communication among the participantsin the learning/teachingapproach,and supportthe constructionof links among the differentsystems and embodiments. fourthousand fourbig cubes two hundred two flats fi ty seven 4 2 5 7 five longs seven 4== =four two fiveseven littlecubes Figure 1. The learning/teachingapproach. Featuresof the approachin action are as follows: links "* Whenaddingand subtractingwith the blocks, the blocks-to-written-marks are made strongly and tightly: Each step with the blocks is immediately recorded with the writtenmarks. "* Linksamongthe Englishwords,base-tenblocks, digit cards,andbase-tenwritten marksare strengthenedby the constantuse of the three sets of words. "* Childrenwork with the learning/teachingapproachfor many days; they are allowed to leave the embodimentsand do problemsjust in writtenform whenever they feel comfortabledoing so. "* When childrenbegin to do writtenproblemswithoutblocks, theirperformance is monitoredto ensurethatthey are not practicingerrors. "* Additionand subtractionboth begin with four-digitproblems(or in some cases, these problemsimmediatelyfollow initial work with two-digit problems). Karen C. Fuson and Diane J. Briars 183 "* Childrenspend only 1 to 4 days on place-valueconcepts initially;much placevalue learning is combined with the work on multidigit addition and subtraction. "* A modificationof the usual algorithmis used for subtraction(see the methods section for Study 1). These features,andthe reasoningbehindthem, arediscussed in Fuson (in press a), where distinctionsbetween named-valueand positionalbase-ten systems are discussed more fully and literaturepertainingto both adequateand inadequateconceptual structureschildrenconstructfor multidigitnumbersare reviewed. Results of an earlier study with this learning/teachingapproachwere reported in Fuson (1986a). In that study second gradersand some first graderslearnedto add and subtractmultidigitnumbersmuch more accuratelythanreportedfor usual school instruction.Most of these childrensuccessfullyandindependentlyextended the procedureslearned with the blocks to five- throughten-digit symbolic problems done withoutthe embodiment.Childrenwho made errorswere interviewed, and those still making errorswere told to think about the blocks as they solved problems.Most of these childrenwere able to use a mental representationof the blocks to self-correcttheirwrittenerrors,andthis use of the blocks showed understandingof place-valueconcepts. This study left unansweredseveral importantquestions that were addressedby the two studiesreportedhere. First,the gradelevel, achievementlevel, and socioeconomic level of the studentswho could benefit from the learning/teachingapproach was not clear from the limited sample used in that initial study. Study 1 reportedhere extendedthe sampleto second gradersof all achievementlevels and to first gradersof above-averageand averagemathematicsachievement.Study 2 extended the sample to second gradersin a large urbanschool district.The goal for both the age/achievementand the residentialextensions was not to manipulate these variousbackgroundvariablesin orderto determinetheir differentialeffects on performance.It was simply to examine whether the effects of the learning/ teaching approach could be considered to generalize across a heterogeneous population. Second, there were the practical questions of whether the learning/teaching approachcould be distanced from its designer, communicatedin a fairly small amountof in-service time, and implementedby teacherswith little field support. These seem to be crucialissues determiningthe feasibility of wide-scale use of the learning/teachingapproach.Distancing focused on three major aspects of this learning/teachingintervention:the classroomteaching,the in-service teachingof the involved teachers,and teaching and supervisionof field supportpersonnel.In Fuson (1986a), project staff members did some of the teaching, the project designer conductedthe teacherin-service, and the field supportperson was taught and supervisedclosely by the projectdesigner. In bothof the studiesreportedhere, all of the teachingwas done by classroomteachersusing lesson plans and student worksheets developed by the project designer. In the second study, the project 184 Base-Base-TenBlocks Learning/Teaching designer did not conduct the in-service sessions nor supervise the field support persons.The amountof in-servicetime was fairly small for both studies: a 1-hour overview of the learning/teachingapproachin the first study and one or two 2V2hourin-servicesessions in the second study.Field supportwas providedin the first study by two teachers in each school who had taught the learning/teachingapproachin the first year.In the second study,threeelementary(K-8) mathematics supervisorswere availableto providefield supportfor the 132 second-gradeteachers targetedfor the learning/teachingapproach,but these supervisors also had many otherduties. The results of the two studies reportedhere are analyzed with respect to three goals of the learning/teachingapproach: 1. understandingmultidigitadditionand subtractionandjustifying procedures with named-value/base-tenconcepts; 2. understandingplace-valueconcepts; 3. being able to add and subtractmultidigitnumbersof several places, including subtractionproblemswith zeros in the top number. The literatureconcerningperformancein these areasby childrenreceiving usual instructionis briefly summarizedin the discussion of the resultsof each study in orderto providea context within which to interpretthe results. STUDY1 Method Subjects Childrenfrom two schools in a small city on the northernborderof Chicago served as subjects. Teachers grouped children by mathematicsachievement in these schools dependingupon recommendationsof the previousteacher;children were moved to a differentroom at any time a teacherthoughtthat a move should be made. In each school there were sufficientfirst gradersfor threemathclasses, one each of low, average, and high math achievement.The high-achievingfirstgradeclasses from both schools were asked to participatein the study.The teachers of the average-achievingfirst gradersin both schools askedlaterin the year to participateand were allowed to do so. In one school therewere threesecond-grade mathclasses, one each of low, average,and high mathachievement.Many of the childrenin the high-achievingclass hadreceivedadditionmultidigitinstructionas first gradersin the studyreportedin Fuson (1986a), so only the low- and averageachieving classes participated.In the otherschool therewere only enough second graders to form two classes. The five lowest achieving second graders were groupedwith a low-achieving first-gradeclass, and the remainingchildrenwere groupedinto a high/averageand an average/lowclass. Many of the childrenin the high/averageclass hadreceived additionmultidigitinstructionas firstgraders,but this class was retainedin the presentstudy in orderto study subtractionlearning for all childrenand additionlearningfor the new children.All eight classes (N = KarenC. Fuson and Diane J. Briars 185 169) participatedin the additioninstruction,and three second-gradeclasses (N = 75) received the subtractioninstruction. Teachers Fourteachers(two from each school) had participatedin the multidigitinstruction in the Fuson (1986a) study.The otherteacherswere given a brief overview of the instruction,lesson plans, studentworksheets,and tests. For questionsand furtherhelp, they were to rely on the two teachersin their school who had taughtthe materialsbefore. A researchproject assistant also visited the schools weekly to check on teachingprogress. Instruction All childrenfirst learnedto find sums and differencesto 18 by countingon and counting up with one-handed finger patterns (see Fuson, 1986b, 1987, 1988b; Fuson & Secada, 1986; Fuson & Willis, 1988). These countingprocedurescould be used for any additionand subtractionfacts childrendid not know. They have been found to be efficient and accurateenough for use in the multidigitalgorithms (Fuson, 1986a). Each class had at least one set of base-tenblocks. The first phase of instruction focused on explorationof the relationshipsbetweenthe differentblocks andon use of the blocks words (little cubes, longs, flats, big cubes, or names chosen by children) and English words (ones, tens, hundreds,thousands).Both the consistent one-for-ten and ten-for-one trades between adjacentplaces and the nonadjacent trades(one-for-hundredand one-for-thousand)were discussed and demonstrated. Then the blocks were used to make differentthree- and four-digitnumbers(e.g., 3725), and index cards each containingone numeralwere used to make the baseten versionof the numberbeside the blocks (e.g., fourcardscontainingthe numerals 3, 7, 2, and 5 were selected and were put down in order to the right of the blocks). These cards,andnumeralswrittenon children'sworksheets,were readby base-ten words (e.g., "threeseven two five"). These activities were accompanied by much verbalizationof the block words, the English words, and the base-ten words. Additionand subtractionwith the blocks were done on a large cardboardcalculating sheet (see Figure 2). Addition was consideredfirst. A writtenproblemwas given. Blocks for the top number were placed in the top row of the calculating sheet, and then blocks for the bottomnumberwere placed in the second row (see Figure 2). Addition was done column by column, beginning on the right. The blocks in a given column were addedtogether(pulleddown) into the bottomrow. If the sum was nine or less, it was recordedwith the digit cards. Each child also recordedeach step on his or her own worksheet.If the sum was over nine, ten of the smaller pieces were traded for one of the next larger pieces, and the result recordedwith digit cardsandon individualworksheets. Much verbalizationof all three sets of words accompaniedall additionand subtraction,and recordingwith writtenmarkswas done aftereach action with the blocks. The necessity of trading 186 Base-TenBlocks Learning/TeachingApproach Thousands T Tens Hundreds Ones 3725 D D000D H Hi 1647 000 Figure 2. Calculatingboardwith an additionproblem. was raisedby showing what happenswith the digit cards if a two-digit numberis writtenin any column (the other digit cards get moved over to the left, making a bigger number).The fairness of the ten/one trades,and the idea of tradingto get more (in subtraction)or tradingwhen you had too many (in addition),arose from the size of the blocks: ten of the blocks in any columnwereequivalentto one block in the column to the left. Multidigit subtractioncan be shown in various ways with the blocks, and the subtractionwithineach value can be phrasedin differentways in words.The children in this study had multiple interpretationsof subtractionavailable (as takeaway, comparison,andequalize,see Fuson, 1986b, 1988b;Fuson& Willis, 1988). We suggested thatteachersverbalizethe subtractionwithin values as "Seven plus how many to make twelve?" or "Twelve minus seven is how many?"(because these fit children'suse of counting up to find these differencesbetterthan using the words "take-away")and thatthey separatethe blocks for the top numberinto those thatmatch the bottomnumber(the subtrahend)and the leftover blocks (the difference) and then move the differencenonmatchingblocks to the bottom row as the answer. A simplificationof the usual algorithmwas also used. Childrenfirst checked each column of the top numberto be sure thatit was largerthanthe bottomnumber in that column. If a top digit was not as large, a one-for-ten trade (borrow, regrouping)was made from the column on the left. After all the necessarytrading had been done to the top numberso thateach top numberwas as large as or larger thaneach bottomnumber,subtractionwas done columnby column. Both the trading and the subtractingcan be done from either direction, but teachers usually modeled the typical U.S. right-to-leftapproach.This trade-firstalgorithmreduces KarenC. Fuson and Diane J. Briars 187 the difficult alternationof tradingand subtractingused in the common algorithm and thus eliminatesthe need for childrento switch repeatedlyfroma named-value representationfor trading to a unitary representationfor subtracting(Fuson & Kwon, in press). The initial sustainedfocus on making all the top columns larger also helps to avoid the common errorof subtractingthe top numberfrom the bottom numberwhen the top numberis smaller. Teachersorganizedtheirclassroomsin differentways for this instruction.Some workedwith the whole class, having childrenparticipatein solving problemswith the blocks and the index cards. Othersdivided their class into small groups and either worked with groups simultaneously or serially while the other groups worked on other topics. In the formercase, childrenwho had learned the blocks procedurethe year before or older childrenshown how to use the blocks worked with each group initially to ensure that the blocks and written-marksprocedures were correctand that childrenin the group were understandingthe relationships involved. In all cases all childrenhad worksheets,and all recordedeach problem as it was workedwith the blocks. Childrenin the average and high-achieving second-gradeclasses were able to do three-andfour-digitadditionand subtractionproblemswith the blocks initially. In the low second-gradeclass and first-gradeclasses, childrenhad difficultyrelating the four columns of blocks to the fourcolumns of writtenmarks.Therefore,in these classes two-digit problems were done first, and then three- and four-digit problems were done with the blocks and writtenmarks.Wheneverchildrensaid they understoodthe written-marksprocedure and did not need the blocks any more, they were allowed to go to their seats to work on worksheets containing three- and four-digitproblems.Their procedurewas checked by someone before they were allowed to leave the blocks. Worksheetswith largerproblems (up to eight digits) were availablefor childrenwho wished to try them. Work on subtractionwas followed by very short units focusing on aspects of meaningful addition (alignment of problems with different numbers of digits, adding 3 two-digit numbersrequiringa trade of 2) and place value (translating frommixed orderwordsto numeralsandvice versawith no trades,doing the same with tradesrequired,and choosing the largerof two multidigitnumbers).The lesson plans describedhow attentioncould be directedwithin the learning/teaching approachto facilitatethe learningof these concepts. The time necessary to complete each unit varied considerably from class to class. The initial introduction/additionunit took from 3 to 6 weeks, and the subtractionunit took from 2 to 4 weeks. Each meaningful addition and place-value concept took abouta day. All of these classes were also participatingin an instructionalresearchproject focused on teachingadditionand subtractionwordproblems.These topics and the multidigit topics went far beyond the districtgoals. Teachershad to meet district goals as well as teaching these extra topics. In some classes teachers also had to cover considerablegroundbefore the multidigitwork could begin (e.g., learning about single-digit sums and differences to 18). Thereforedifferentclasses com- 188 Base-TenBlocks Learning/TeachingApproach pleted different topics. The low-achieving second-gradeclass and one average first-gradeclass only completed addition of two- and three-placenumbers.The teacherof the otheraveragefirst-gradeclass only taughtmultidigitadditionto 10 of the 24 childrenin her class, but she did complete the generalizationof the algorithmpast four places with the participatingchildren.All otherclasses completed the generalizationof the addition algorithmto problems with as many as seven digits. The high and averagesecond-gradeclasses completedsubtraction,and the average/low second-gradeclass completed ordinarysubtractionand began work on problems with zeros in the minuend. The work on meaningful addition and place-value concepts was completed only by the high- and average-achieving second-gradeclasses. Measuresof Skill and Understanding Additionand subtractioncalculation tests. All childrenwere given two addition pretests.The TimedAdditionTest contained 12 problems,with 2 two-digit, 2 three-digit,3 four-digit, 1 five-digit, 3 six-digit, and 1 seven-digit problem; childrenworkedon these problemsfor 2 minutes.All problemsrequiredtradingin one or more places (the numberof tradesrangedfrom one to five). The Ten-DigitAddition Test was a single ten-digitproblem(6385740918 + 8557586736). All problems were writtenaligned in verticalform. These same two tests were also given as posttests.The lower achievingand youngerclasses were also given an Untimed AdditionMinitest of four problems(2 two-digit and 2 three-digitproblems,each requiringone trade).Parallelsubtractiontests (TimedSubtractionTest,Ten-Digit SubtractionTest, Untimed SubtractionMinitest) were made by using inverse problems from the addition tests; children were given 3 minutes for the Timed SubtractionTest because subtractionhad been slower than additionin the earlier study.A fourthsubtractiontest (ZerosSubtractionTest)consistedof four problems with zeros in the top number: 1 two-digit, 2 three-digit,and 1 four-digitproblem with one, one, two, and threezeros, respectively. The tests for each child were first evaluated to determine whether the child showed any evidence of correcttrading;two correctly tradedcolumns were requiredfor the child to be judged as showing some indicationof trading. Each test was then scoredto permita finer evaluationof performance. Scoring was based on each digit in the answer: one point was given for each correctdigit. This procedurewas adoptedbecause scoring each problemonly as corrector incorrectdoes not differentiatea solution in which all columns but one are correct from a solution in which a child demonstratedno notion of multidigitadditionor subtraction. An analysisof the kinds of errorswas made on the ten-digitproblem.The errors identifiedin Fuson (1986a) were classified into fourcategoriesreflectingincreasing amountsof knowledge aboutmultidigitadditionor subtractionas follows: error:Columns were left blank or filled in with 1. Preaddition/presubtraction random numbers; presubtractionerrorsalso included adding. seemingly Karen C. Fuson and Diane J. Briars 189 2. Column addition/subtractionerror: Addition/subtractionproblems were approachedcolumn by column: In additionthe sum of each column was written below that column even when the sum was a two-digit number(e.g., 28 + 36 = 514); in subtractionthe smallernumberin each column was subtractedfrom the largernumber(e.g., 36 - 28 = 12). 3. Tradingerror: Tradingerrorsinvolved some partiallysuccessful attemptto trade(carry,borrow);in additionproblemsthese errorsincludedthe following: the tradewas not writtenor addedin anywhere,a tradewas made when the sum was not over 9, the tens digit ratherthan the ones digit was traded,a tradewas made but ignoredwhen thatcolumn was added(this errormight have been a fact errorsuch errorswere counted as both tradeand fact errors),the tradewas subtracted from ratherthan added to the top number;in subtractionproblems these errors included the following: the left column was not reduced by one even though a trade was recorded in the right column, a trade was made even though the top numberwas alreadylarger,more thanone tradewas made from a given column, 1 was subtractedfromthe traded-tocolumn,the rightcolumnreceived 11 ratherthan 10, 1 was subtractedfrom a left column even thoughno tradewas recordedto the right. 4. Fact error: Fact errorsinvolved correcttradingbut incorrectadding or subtractingin a column. Two coders coded all errors.Coder agreementwas 97%. Because not every column in every problemrequireda trade,childrenmaking consistent column addition/subtractionerrorscould get 20% correctdigit scores on both Untimed Minitests and 9% on the additionTen-Digit Test, and children makingtradingerrorsthatwere incorrectin only one columncould get digit scores ranging between 36% (on the Ten-Digit Tests) and 60% (on the Untimed Minitests). Place-value and meaningfulmultidigitaddition writtentests. Three aspects of place-valueunderstandingand two aspectsof meaningfulmultidigitadditionwere assessed throughwrittentests. The Mixed Wordsto NumeralsTestrequireda child to write a three- or four-digitnumeralfor numeral/wordnamed-valuecombinations given in mixed order(e.g., 6 hundreds,4 tens, 5 thousands,and 7 ones). The TradedWord/NumeralTest requireda child to write a three-or four-digitnumeral for numeral/wordnamed-valuecombinationsgiven in standardorder(e.g., 2 thousands, 16 hundreds,1 ten, and4 ones) or to fill in a numeralblankwhen the threeor four-digitnumeralwas given with the numeral/wordnamed-valuecombination hundreds,14 tens, and 3 ones). All of these items (e.g., 2643 is 2 thousands, hadone numeral/wordpairthatexceeded 10 andthushadto be tradedto the left in the formeritems or to the rightin the latteritems to make the correctanswer;these items were modeled after those in Underhill (1984). The Choose the Larger NumberTestrequireda child to choose the largerof a pairof three-throughsevendigit numbersby circling the largernumberand by insertinga < or > between the pairof numbers.The five pairsof numberswere all misleadingin thatall digits in 190 Base-TenBlocks Learning/TeachingApproach the smaller numberexcept one were equal to or greaterthan the corresponding digits in the larger number.The Alignment Test presented horizontally-written problems whose addendshad differentnumbersof digits; children were told to write the problem so that it could be added easily. This tested a combinationof place value and additionunderstanding-understandingthatone addedand therefore aligned like places; the differentnumbersof digits were chosen to maximize the frequenterrorof aligningsuchproblemson the left ratherthanon the right.The Trading2 Insteadof 1 Testconsistedof problemswith threeaddendsthatrequired a tradeof 2 tens ratherthan 1 ten because the sum in the ones column exceeded 20; the first item had a sum of 21 to maximize the possibility that childrenwould rotelytradethe 1 as they hadbeen doing for problemswith two addendsratherthan tradingthe numberof tens (2, in these problems).These tests hadbetween two and six items. Each test item was markedas corrector incorrect,and test means were convertedto percentagesfor ease of comprehensionof the test results. UnderstandingofAddition, Subtraction,and Place Value Individual interviews were carried out to assess children's understandingof addition, subtraction,and place value. Eight children from one class at each achievementlevel were randomlyselected to be interviewed(the average-achieving first graderswere fromthe class in which all childrenparticipated).Therefore, the additioninterviewsamplecontained40 children,andthe subtractioninterview sampleconsisted of the 24 second gradersin the additioninterviewsample.Interviews were conductedindividuallyin a roomoutsidethe classroom.Childrenwere shown solved multidigit problems, each written on a separateindex card. Each problem solution was written in a color different from the color of the original problem.Two additionproblemswere solved correctly: a two-digitproblemwith a tradefromthe ones to the tens anda four-digitproblemwith a tradefromthe hundreds to the thousands.Two additionproblemswere solved incorrectly.The two most common additionerrorsbefore instructionwere used: (1) column addition, for example,for 8 + 6 writing 14 in the ones columnand (2) ignoringthe tens digit of a two-digit sum andjust writingthe ones digit. Five subtractionproblemswere given. Two were solved correctly and paralleled the correctly solved addition problems,except that differentnumberswere used. A thirdshowed the common errorof column subtraction-subtractingthe smallerfromthe largernumbereven when the smallernumberis on the top. Two three-digitproblemswith two zeros in the top numberwere given. One was solved correctly,and the other showed 1 hundredtradedfor 10 ones. Childrenwere told thatthey would be shown problemsthat somebody else had solved andthatsome problemswere correctandsome were wrong.They were then shown an index card with a problemwrittenon it and asked if that problemwas right or wrong.After a judgmentwas made, they were asked why it was right or wrong. The interviewerwrote down verbatimthe child's responsesand any interviewer prompts.Childrenwere randomlyassigned to one of two differentorders of problems.One sequencebeganwith a correctproblem,andthe otherbeganwith 191 KarenC. Fuson and Diane J. Briars an incorrectproblem.The more difficult problems (the four-digitadditionproblem and the subtractionproblems with zeros) were given in the last half of the interview. The interviewrecordswere classified by the interviewerandone of the authors. The classificationof a problemas corrector incorrectwas evaluatedfirst. If a child changed his or her answer, the last assignmentwas coded. The ratersagreed on 100%of these classifications. The interviews were coded for place-value understandingof the tens or hundredsvalues of writtennumeralswithin an explanation of additionor subtraction;to receive credit, a child had to use the word "ten"or "hundred"to identify a numeralcorrectly sometime during an explanation.The additioninterviewswere coded for two aspects of additionand place-valueunderstanding: (a) explaining the writtenprocedureas trading 10 ones for 1 ten or 10 tens for 1 hundred,and (b) identifyingthe traded1 as a ten or as a hundred.For (a) a child had to explain explicitly the tradingor say that the ten came from the 13 ones or the hundredcame fromthe 16 tens. The subtractioninterviewswere coded for threeaspects of subtractionand place value understanding:(a) explainingthe writtenprocedureas trading1 ten for 10 ones or I hundredfor 10 tens; (b) identifying the traded 1 as a ten or as a hundred;and (c) explaining the double trading over two top zeros, i.e., the tradeof 1 hundredfor 10 tens andthe tradeof 1 ten for 10 ones. All of these aspects were evaluatedfor tens and for hundreds.Coderagreement was 95%. Children'sexplanationsdid not always spontaneouslycover all of the coded aspectsof the interview.A series of promptswas used to tryto ascertainsuch knowledge. These included questions about the traded 1 ("What'sthe one?" or "One what?")and a question about the 8 tens in the four-digitadditionproblem ("Eight what?").The most explicit promptwas to ask a child to think about the blocks; this was used when a child failed to give any answer to other prompts. However, due to the complexity of the interviewand the fact thatthe attributesof the responses to be coded were finalized after the interviews were completed, needed prompts were not always given. Thus, the data may underestimate children'sknowledge. Results AdditionMultidigitComputation On the pretestsonly 9 of the 169 childrenshowed any indicationof correcttrading, whereason the posttests 160 of the 169 childrenshowed such evidence, a very large and statistically significant change (McNemar'stest chi-square= 151, p < .0001). Of these 160 children, 156 correctlytradedon a four-digitor largerproblem. Of the 13 childrenfailing to demonstratecorrecttradingor doing so only for two- or three-digitproblems,7 were in the average-achievingfirst-gradeclass and 5 were in the low-achieving second-gradeclass. Pairedt-test analyses of pretestposttest differences on the digit scores for each test for each class separatelyrevealed significantimprovementfor every test for every class, p < .001 in all cases. 192 Base-Ten Blocks Learning/Teaching Approach Posttestdigit scores are shown in Table 1. These indicateexcellent performance for all classes except the low-achieving second-gradeclass and the averagefirstgrade class in which all childrenparticipatedin the learning/teachingapproach. Even the latter two classes demonstratedsome learning, because their Untimed Minitest scores were well above those obtainableby carryingout column errors (75% and 69% comparedto 20% for column errors). Teachersreportedthatchildren were enthusiasticabout the multidigitinstructionand enjoyed solving large problemsandthatmanyof the higher-achievingsecond gradersknew most of their additionfacts or used thinkingstrategiesto find sums they did not know andmost of the otherchildrencountedon with one-handedfingerpatternsto solve sums they did not know. Table 1 AdditionComputationPosttest Digit Score Meansfor Each Class and AchievementLevel in Study1 Grade/achievementlevel Tests n Percentageof correct digits in answers UntimedMinitest Ten-Digit Test Timed Test Mean numberof correct digits completedin 2 minuteson Timed Test 2 High/av 2 Av 2 Av/low 2 Low 1 High 1 High 1 Av 1 Av 29 23 21 14 26 25 10 21 ng 99 98 ng 93 91 ng 90 94 75 58 a 74 92 88 92 98 91 94 92 93 91 69 ng ng 28 25 26 15 17 24 12 ng Note. Percentageof correctdigits in the answer is out of all digits in the Untimed Minitestand Ten-Digit Test and out of the columns attemptedby a given child in the Timed Test. ng means the test was not given. a The low-achieving second-gradeclass only completed 2- and 3-digit addition. The errorsmade on the Ten-Digit pretests and posttests are given in Table 2. These analyses show a large reductionin the numberof errorsmade. Few of the primitivepreadditionandcolumnadditionerrorswere madeon the posttest. There was a reductionin the tradingerrorsand no increase in the fact errorsin spite of the fact thatalmostall childrenwere addingandtradingon almost all problemson the posttest. Table 2 Numberand Kinds of Pretest and Posttest Additionand SubtractionErrors in Study1 Preaddition/ presubtraction Column add/sub Fact error Trading error Tests Pre Post Pre Post Pre Post Pre Post AdditionTen-Digit Test SubtractionTen-Digit Test 527 135 28 4 837 650 18 14 109 0 79 96 57 8 45 22 Note. There were a possible 1859 errorsin additionand 825 errorsin subtractioncalculatedby multiplyingthe numberof digits in the answer (11) by the numberof subjects (N = 169 for the Addition Test, N = 75 for the SubtractionTest). KarenC. FusonandDianeJ. Briars 193 SubtractionMultidigitComputation On the pretests 2 of the 75 children participatingin the subtractionlearning/ teaching approachshowed some evidence of trading;on the posttests72 of the 75 children showed such evidence, a very large and statistically significant change (McNemar's test chi-square= 70, p < .0001). Five of these 72 childrendemonstratedsuch tradingfor two- or three-digitproblemsbut not for largerproblems. Pairedt-testanalysesof pretest-posttestdifferenceson the digit scores for each test for each class separatelyrevealedsignificantimprovementfor every test for every class, p < .001 in all cases. Mean digit scores for each test for each class are given in Table 3. Performance by the high/averageclass was excellent on all tests, and for the othertwo classes performancewas good on the TimedTest andthe UntimedMinitest.Scoresfor the average and average/low classes on the Ten-DigitTest and on the Zeros Test revealed weaker performancethat was nevertheless above the level of consistent tradingerrors(36% and 33%, respectively).Teachersreportedthat some children knew subtractionfacts or used thinkingstrategiesto determinedifficultdifferences but that most counted up with one-handedfinger patternsto determinefacts they did not know. Table 3 SubtractionComputationPosttest Class Means by AchievementLevel in Study1 Achievementlevel Tests n Percentageof correctdigits in answers Untimed Minitest Ten-Digit Test Timed Test Zeros Test Mean numberof correctdigits completed in 3 minutes on Timed Test High/av Av Av/low 29 23 23 ng 95 95 92 89 72 84 78 87 75 84 (49) 22 15 16 Note. Percentageof correctdigits in the answeris out of all digits in the Untimed Minitest, Ten-Digit Test, and Zeros Test and out of the columns attemptedin the Timed Test. The Zeros Test for the Av/low class is in parenthesesbecause this class only began work on zero problems. ng means the test was not given. The erroranalysespresentedin Table2 indicatean almost completeelimination on the posttest of the large numberof presubtractionand column subtractionerrors made on the pretest.Substantialnumbersof tradingerrorswere made on the posttest,but most posttesttradingwas correct(over 80%of the tradesinvolved no errorin either column). Few fact errorswere made on the posttest, only half as many fact errorsas were made in addition. Place-Valueand MeaningfulMultidigitAdditionWrittenTests Results of the writtentest measures of place-value and meaningfulmultidigit additionaregiven in Table4. Almost all childrentakingthese tests were misled by 194 Base-TenBlocks Learning/TeachingApproach these items on the pretest(except for the Circle the LargerNumberTest). On the posttest, children in both classes showed very considerablegains on the placevalue tests, all children correctly aligned problems, and most children traded2 when they had 20-some ones or tens. Table 4 Percentage Correcton Place-Value and MeaningfulAdditionWrittenTest in Study1 Grade/achievementlevel 2 High/av Pre Tests Place-valuetests 3 2 Mixed Wordsto NumeralsTest TradedWord/NumeralTest Choose the LargerNumberTest Circle the largernumber Insert> and < symbols in the numberpairs AlignmentTest Trading2 Insteadof 1 Test 50 0 Meaningfuladditiontests 0 0 2 Av Post Pre Post 98 90 8 3 83 72 96 96 34 44 84 98 100 100 5 12 100 73 Note. The Class 2 High/av pretestswere given early in the year, and the Class 2 Av pretestswere given midyear. Understandingof Place-Value,Addition,and Subtraction Every interviewedchild correctlyclassified all four additionproblemsas having been solved correctlyor incorrectly,94% correctlyclassified the subtraction problems with no zeros, and 94% of the children completing instructionon the subtractionproblemswith zeros classified such problemscorrectly.Results of the interviewmeasuresare given in Table5. Every child but one identifieda numeral in the tens place as x tens at least once duringtheirexplanations.Similaridentification of a hundredsnumeralwas done by 92% of the second gradersbut by only 50%of the first graders.Almost every child explainedthe ten-for-onestradingand identifiedthe traded1 as a ten for both additionand subtraction;three-fourthsof these explanations were spontaneous without any prompts.The problems with errorswere muchmoreeffective thanwere correctproblemsin eliciting spontaneous explanations,indicatingthat the childrenwere not just repeatingmemorized verbalexplanationsfor correctproblems.Forthe hundredsconceptspromptswere requiredfor aboutthree-fourthsof the responses,but this seemed to stem as much from the fact thatonly a correctproblemwas given for the hundredtradeas from hundredsbeing more difficult. Childrenin the second-gradeaverage/low-achieving class and especially in the average-achievingfirst-gradeclass showed more limited understandingof the ten/hundredtradethan did the childrenin the other threeclasses. Most childrenfailing to identifythe traded1 as a hundredidentified it as a ten, and most of these identified that 1 as coming from the "8 tens plus 8 tens is 16 tens." Thus, they had learneda general aspect of multidigittrading,to tradethe tens digit from any two-digit sum, but they could not simultaneouslyfit this generalview of tradingwithin the named-valueplaces to name the new value 195 KarenC. Fuson and Diane J. Briars Table 5 Percentage of StudentsDemonstratingUnderstandingof Place Value,Addition,and Subtraction in Studyi Grade/achievementlevel 2 Hi/av Tests Identifythe tens and hundreds values of writtennumerals 2 Av 2 Av/lo 1 Hi 1 Av Ten Hun Ten Hun Ten Hun Ten Hun Ten Hun Place-value understanding 100 100 100 100 100 (13) 75 Addition and place-valueunderstanding 100 88 100 100 88 50 Explain writtenprocedureas (13) (25) (50) trading10 ones for 1 ten or 10 tens for 1 hundred 100 88 100 88 100 63 Identifythe traded 1 as a ten or a hundred (38) (25) (38) Subtractionand place-value understanding 100 100 100 100 100 75 Explain writtenprocedureas (13) trading 1 ten for 10 ones or 1 hundredfor 10 tens 100 100 100 100 100 75 Identifythe traded 1 as a ten or a hundred 88 38 38 Explain the double tradingover 100 100 75 two top zeros: hundreds to tens and tens to ones 88 (25) 25 100 100 88 (13) (13) 13 100 100 88 (13) (25) 13 100 75 ng ng ng ng ng ng ng ng ng ng ng ng in parentheses to thinkaboutthe arechildrenwhoresponded Note.Percentages onlyaftertheywereprompted blocks. ng means the test was not given. of the traded 1. Not a single interviewedchild identifiedthe traded1 as a one, in sharpcontrastto childrenreceiving traditionalinstruction. Discussion The second gradersandhigh-abilityfirstgradersshowedmultidigitadditionand subtractioncomputationperformancethatwas very considerablyabove thatshown by third graders receiving traditionalinstruction (cf. Kouba et al., 1988). The errorthat is so common in multidigitsubtraction subtracting-smaller-from-larger was almost completely eliminated. These children also showed competence far above that usually demonstratedby third gradersin verbally labelling tens and hundredsplaces, in changing words to numeralsand vice versa even when these were given in mixed orderor requiredtrading,in choosing the largernumber,in aligning unevenproblemson the rightratherthanon the left, in showing the quantitative meaning of tens and ones, and in identifyingthe traded 1 in additionand subtraction as a ten or as a hundred rather than as a one (cf. Cauley, 1988; Ginsburg,1977; Kamii, 1985;Kamii & Joseph, 1988; Labinowicz, 1985;Resnick, 1983; Resnick and Omanson, 1987; Ross, 1986, 1989; Tougher,1981). Kamii (1985; Kamii & Joseph, 1988) and Ross (1986, 1989) reportedthat on digit correspondencetasks most second gradersandmanythirdandfourthgraders 196 Base-TenBlocks Learning/TeachingApproach receiving traditionalinstructionalshow no understandingthatthe tens digit means ten things (these childrenshow one chip-rather than ten chips-to demonstrate what the 1 in 16 means), or they are misled by nontengroupingsand show only a groupingface-value meaning (for 13 objects arrangedas threegroupsof four objects and one left-over object, they say thatthe 3 means the threegroupsand the 1 means the one left-over object). These tasks were not available at the time this studywas carriedout, but reviewersraisedthe questionof whetherchildrenin the study would have demonstratedplace-valueunderstandingon these tasks.At that time two teacherswere still carryingout reasonablefacsimiles of the instruction with their above-averageand average-achievingsecond-gradeclasses. In an attempt to provide some information on this issue, half the children from each achievement-levelgroupingwithin each class were randomlychosen to be individually interviewed(n = 22). They were given these two tasks and a subtraction problemwith zeros in the top number. On the Kamii task (showing with chips what the 6 and the 1 in 16 mean), 12 children immediately showed ten chips as the meaning of the 1, another4 first showed one chip but showed ten chips when asked to show with the chips "what else could this part(the 1) mean?"anotherchild showed ten chips when given the task again after working the four-digitsubtractionproblem, and 3 children first showed one chip but showed ten chips when asked to "look at the places" in 16 (tens and ones were not mentioned).Thus, more than half of these children had tens andones availableas theirfirst meaningfor a two-digitnumeraland four others had it readilyavailableas a second choice, while four more first showed their unitarymeaningbut showed a tens and ones meaning when a multidigitcontext was elicited for them;overall, 91% of the interviewedchildrenshowed that the 1 meantten objects.Not a single child showeda groupingface-valuemeaningon the Ross task;performancewas the same as performanceon the Kamii task. Thus, on these tasks also, second gradersusing the base-ten blocks showed performance considerablyabove thatordinarilyshown by second gradersreceiving traditional instruction. STUDY2 Method Subjectsand Teachers Potentialsubjects were all second gradersin the 132 second-gradeclassrooms in the PittsburghPublic School system. A 2V2-hourin-service trainingsession on using base-ten blocks to teach multidigitadditionand subtractionwas offered to all second-gradeteachersin August. This in-service session was voluntary;teachers were paid salary to attend.The workshopwent throughthe teacherplans for the learning/teachingapproach,focusing particularlyon using the blocks andlinking actionson the blocks to steps in the writtenmultidigitadditionand subtraction procedures.In November,a follow-up 2V2-hoursession focusing more intensely on subtraction(includingthe new trade-firstalgorithm)was given to these teachers, and a 2V2-hoursession on both additionand subtractionwas given for teach- KarenC. Fuson and Diane J. Briars 197 ers who had not attendedthe August session. These sessions were given by the second author,who is experiencedin using the base-ten blocks to teach the multidigit algorithms.A math supervisorwho had no previous experience with the base-tenblocks gave another2?-hour in-servicesession in Decemberfor those not able to attendearliersessions. Most second-gradeteachers(91%)attendedat least one of these sessions. Teacherswere urged to use the base-ten blocks and lesson plans to teach the multidigitalgorithms.Three elementarymathematicssupervisorswere available as questionsarose, thoughthey also had many otherduties concerningteachersat other grade levels. The supervisorsencouragedteachersto try the approach,but because the goals went considerablybeyond the districtsecond-gradegoals, participation was voluntary.Many teachers startedteaching multidigitadditionand subtractionsomewhatlate in the yearandexpresseddoubtsthatthey would be able to finish all of the units. In orderto increase the numberof teachersfinishing at least the additionand subtractionwork,the supervisorssuggestednot covering the meaningful addition and place-value units but finishing the subtractionwork at least up to the problems with zeros. The number of teachers and children who participatedin variousaspects of the studyarediscussed in the final section of the methods section. Instruction Teacherlesson plans and a class set of studentworksheetsin individualstudent booklets (both as describedin Study 1) were sent to each second-gradeteacherin the district. At the in-service sessions some teachersexpressed a preferencefor using the blocks to show subtractionas take-awayinstead of as comparisonbecause the take-awaymethod fitted bettertheir conception of subtractionas takeaway.Teacherswereallowed to use take-awayif they wished: The top number(the minuend) was made with blocks and blocks were taken away for the bottom number.One class set of base-tenblocks (the EducationalTeachingAids neutralcolored blocks, metric version) was availablein each school. Testing Tests. The addition and subtractioncalculation tests and the place-value and meaningful-additionwrittentests used in Study 1 were used in this study.All tests were given as pretestsat the beginning of the year.The same tests were given as posttests as each phase of the learning/teachingapproachwas finished (e.g., the additioncalculationtests were given at the completion of the additionteaching). Teachers graded all tests according to written directions. They returnedto the central district office the pretests accompaniedby a class list containingpretest scores. Posttestsaccompaniedby a class list with posttest scores were returnedto the centraloffice as teachersgave them.For both the pretestsand the posttests,the tests of four childrenin each classroom, two boys and two girls, were randomly selected and gradedby researchstaff membersin orderto check the teachergrading. The few teacherswith systematicgradingerrorshad their scores corrected. 198 Base-TenBlocks Learning/TeachingApproach Criterionscores and errorclassification. Criterionscores were adoptedfor the additionand subtractionUntimedMinitests,the additionand subtractionTen-Digit Tests, and the subtractionZeros Test. These were based on the digit scores describedfor Study 1. The tradingcriterionscore was 8 or more for the additionand subtractionUntimedMinitestsandthe additionandsubtractionTen-DigitTestsbecause a child makingtradingerrorsthatwere incorrectin only one column could obtain scores of 6 out of 10 on the UntimedMinitestsand 4 out of 11 on the TenDigit Tests.A score of 8 requireda child to make at least two correcttradeswith no fact errorson the UntimedMinitestsand four correcttradeswith no fact errors on the Ten-DigitTests.For the subtractionZerosTest,a criterionscore of 9 (of the 12 digits correct)was selectedbecausethis scoremeantthatthe child demonstrated correcttradingfor at least two of the three zero aspects tested. Erroranalyses were carriedout on four Ten-DigitTests drawnat randomfrom each of 30 classroomsrandomlyselected for each test and time (pretest,posttest). Errorswere classified into the four categories used in Study 1. The classification was done by the same two coders used in Study 1; coder agreementwas 96%. The Pretest and Posttest Samples Of the 132 teachers,125 (95%)returnedpretestsfor 2723 children.Pretestswere returnedfrom at least one classroom for every school in the district.Across all of the tests the numberof completed pretests ranged between 2531 and 2378. To ascertainwhetherthe pretestsrepresentedthe whole sample of childrenwith one or more returnedpretests,on each test the scores of children who had complete dataon all tests were comparedto scores of childrenwho hadone or moremissing scores on other tests. Therewere no significantdifferencesbetween these groups on any tests. Only part of the potential sample of classrooms completed the work with the learning/teachingapproachandreturnedthe posttests.The numberof childrenwith returnedposttests is given for each test in Table 6. The numberof teachers who returnedadditioncalculation,subtractioncalculation,andplace-value/meaningful additionposttests was 42, 35, and 16, respectively.These teacherscame from 18, 18, and 9 different schools, respectively. This partialreturnraised the obvious questionof whetherthe childrenfor whom posttests were returneddifferedfrom the childrenwithoutreturnedposttests.The additioncalculationpretestswere the focus of the difference analyses because all other pretests showed floor effects. Several aspects of the additionpretestsfor the childrenwith no returnedposttests were comparedto pretestsfor the childrenwith returnedposttests.The percentage of childrenwith pretestscores on the UntimedMinitestat or above criterionwas a bit higherfor the childrenwith no postteststhanfor those with posttests,the mean digit scores on the UntimedMinitestand the Timed Test were aboutthe same for both groups, and children with no posttests showed somewhat more advanced errorsthan did the childrenwith posttests (more of the formermade at least one tradingerrorwhile moreof the lattermadepreadditionor column additionerrors). Thus, the posttest sample childrenwere, if anything,initially a bit worse at multi- KarenC. FusonandDianeJ. Briars 199 digit addition calculation than the children not participating, and all children showed floor effects on the multidigitsubtractioncalculationand place-valueand meaningfuladditionpretests;thatis, both groupshad the same low level of initial knowledge. Most (90%)of the posttestteacherscame from a school in which all the secondgrade teachersreturnedposttests. Teacherswithin a given school almost always returnedexactly the same posttests. Thus, the performancedata to be reported come from all achievementlevels of second graders.The schools with all teachers participatingwere distributedacross the whole range of schools in the city with respect to location, ethnicity,and socioeconomic level. Participatingteachersdid not seem to differ much from nonparticipatingteachersin theirrate of attendance at the in-service sessions: 76%, 15%, and 10% of the participatingteachers attended two, one, and zero sessions, respectively,while these percentagesfor the nonparticipatingteacherswere 68%, 23%, and 9%. Two classes from a magnet school were droppedfrom the additionsample because more than half the children were above criterionon the pretest, indicating previous addition instruction. One of these classes was also dropped from the subtractionsample for the same reason. Results AdditionMultidigitCalculationPerformance On the pretest 10%of the instructedsample met the criterionon the Untimed Minitest, and on the posttest 96% of the childrenmet this criterion.This shift for the Ten-Digit Test was from 5% on the pretest to 90% meeting criterion on the posttest. Both of these changes were significant, McNemar's test of correlated proportionschi-square= 674 and 659, p < .0001. The childrenwere quite accurate adders,with digit scores on the threetests showing thatthey solved between 89% and 96% of the columns correctly(Table6), and they solved a mean of 24.3 columns of multidigitproblemscorrectlyin 2 minutes. These children showed the same large reduction in preaddition and column additionerrorsfrom the pretestto the posttestas shown by the childrenin Study 1 (see Table7). Tradingerrorswere also reducedconsiderably,even thoughalmost all childrenwere tradingon the posttest. SubtractionMultidigitCalculationPerformance Hardlyany childrenmet the tradingcriteriaon the subtractionpretests(2%, 1%, and 0.4% on the Untimed Minitest,Ten-DigitTest, and Zeros Test, respectively), but 84%, 70%, and 81%of the instructedchildrenmet the criterionon the respective posttests. These changes were all significant, McNemar's chi-square= 580, 487, 486, p < .001. Children obtained digit scores on the various tests ranging between 80%and90%correct(see Table6). Childrensolved subtractionproblems more slowly than addition problems, solving a mean correct 18.4 columns in 3 minutes. 200 Base-TenBlocks Learning/TeachingApproach Table 6 Percentage Correcton the Additionand SubtractionComputationPosttests and the Place-Value and MeaningfulAdditionPosttests in Study2 Additioncomputationtests Untimed Minitest Ten-Digit Test Timed Test Subtractioncomputationtests Untimed Minitest Ten-Digit Test Timed Test Zeros Test Place-valuetests Mixed Wordsto NumeralsTest TradedWord/NumeralTest Choose the LargerNumberTest Circle the largernumber Insert> and < symbols in the numberpairs Meaningfuladditiontests AlignmentTest Trading2 Insteadof 1 Test n % Correct 783 776 780 96 89 92 707 705 669 602 90 80 85 85 360 360 88 53 360 363 67 65 300 278 85 80 Note. The %correctfor the additionand subtractioncomputationtests are the percentageof correctdigits out of the total digits in the UntimedMinitests,Ten-Digit Tests, and Zeros Test and out of the digits attemptedby a given child in the Timed Tests. Table 7 Numberand Kinds of Pretest and Posttest Additionand SubtractionErrors in Study2 Preaddition/ presubtraction AdditionTen-Digit Test SubtractionTen-Digit Test Column add/sub Trading error Fact error Pre Post Pre Post Pre Post Pre Post 341 282 7 8 798 984 3 26 79 6 45 187 11 1 83 58 thenumberof digitsin the Note.Therewerea possible1320errorsforeachtestcalculated bymultiplying answer(11)by thenumberof subjects(N = 120). The subtraction-error analyses indicateda substantialmovementfrom the presubtractionand column subtractionerrorsto the more advancedtradingand fact errors(see Table 7). The percentagesof posttest errorsfalling within each error categoryare similarfor Study 1 and Study 2. Place-Valueand MeaningfulAdditionTests The pretestscores on most of the place-valueandmeaningfuladditiontests were very low, indicatingthatchildrenwere respondingto the misleadingnatureof the items. For example, on the AlignmentTest, most childrenaligned the numberson the left, recopied the problems horizontally,or treated each digit as a separate numberand formednew problems(e.g., 67 + 1385 was writtenverticallyas 67 + 13 + 85). On the test giving mixed orderwords, 38% ignoredthe words andwrote the numeralsin theirgiven orderand 39% left blanksor wrote seemingly random Karen C. Fuson and Diane J. Briars 201 responses; only 23% showed even any partialknowledge. About a sixth of the childrendid get three of the five items correcton the Choose the LargerNumber Test and anothersixth got four or five items correct,indicatingsome pretestability to comparemultidigitnumbers. Performanceon the posttestMixed Wordsto Symbols Test,the AlignmentTest, andthe Trading2 Insteadof 1 Testwas good, rangingfrom 80%to 88%(see Table 6). Individualclass means on these tests rangedfrom lows of 59%to 66%to highs of 100%.Performanceon the Choose the LargerNumberTest improvedto moderatelevels of accuracy,with little differencebetween scores obtainedby circling the largernumberor inserting< or > between the numbers(67% and 65%). Class means on the TradedWord/NumeralTest were extremely variable,rangingfrom 3% to 88%, with an overall mean performanceof 53% of the items correct. Discussion Informalteacherreportsvia the supervisors and direct communicationto the districtmathematicsdirectorindicatedconsiderableenthusiasmandenjoymentof the learning/teachingapproachby both teachersand children.Being able to solve largeproblemsseemed to empowerchildrenandmakethemfeel good aboutthemselves and about mathematics. Children learned multidigit addition quite well, thoughthey still made some additionfact errorsandoccasionaltradingerrors.The subtractiontest scores and erroranalyses indicatedthatmost childrencould trade correctly and that few continued to make the presubtractionand the subtracterrorsso commonon the pretests.However,manychildrendid smaller-from-larger not completely mastersubtractioncomputationand continuedto make some trading andfact errors,especially on the ten-digitproblem.Both additionand subtraction performancewas considerablyabove that ordinarilyreportedfor thirdgraders, as was performanceon the AlignmentTest,the Mixed Wordsto Symbols Test, and the Choose the LargerNumberTest. Childrenshowed more limited ability to generalize tradingto the new TradedWord/NumeralTest. There were obvious limitations to this study. Because systematic classroom observationswere not made, it is not clear how closely the work with the blocks followed the lesson plans. Thus, no inferencescan be made aboutwhich features of the learning/teaching approachmight be crucial and whether any might be expendable.It is not clear why teachersin some schools participatedwhile those in other schools did not. Informalreportsto field supervisorsindicatedthat the school-baseddecisions to participatewere sometimesinitiatedby the principaland sometimes by the teachers.The field supervisorsreportedthat some teachersexpressed skepticismthat second graderscould learnmaterialso much above grade level even though the success of the approachwith the children in Study 1 was discussed in the in-service sessions; this skepticismmay have contributedto decisions not to use the approach.The partialparticipationby teachersdid not seem to bias the samplewith respectto initialknowledgeof the participatingchildren.The teacherassignmentandtransferpolicies of the districtmakeit unlikelythatthe best teachersare heavily concentratedin certainschools (i.e., only in the participating 202 Base-TenBlocks Learning/TeachingApproach schools), but there still might have been some bias towardparticipationby the better teachers in the district. Finally, although the scores on the addition and subtractioncomputationtests and the shifts in errorsfrom pretestto posttest were similarin Study 1 and Study 2, the lack of interviewdatain Study 2 means thatit is not clearwhetherthe childrenin Study2 understoodandcould explainmultidigit additionand subtractionas well as could the childrenin Study 1. GENERAL DISCUSSION On all tests and interview measures, performance by second graders of all achievementlevels considerablyexceeded thatreportedin the literaturefor third gradersreceivingusualinstruction.Most childrenlearnedto tradein four-digitaddition and subtractionproblems,columnerrorsfrequentlyresultingfrom usual instructionwere virtuallyeliminated,and childrenshowed considerablegeneralization of multidigitadditionand subtractionto multidigitproblemslargerthanfour digits. Most childrenaligned uneven additionproblemson the right, traded2 insteadof 1 when necessary,and could translatefrom mixed words and numeralsto multidigit numerals. Children in Study 1 showed in the interview quantitative understandingof writtenmultidigitnumeralsand used this understandingto explain one/tenandten/hundredtradingproceduresin both additionand subtraction. These results indicatethat second-gradeclassroom teacherscan use the learning/teachingapproacheffectively to supporthigh levels of meaningfullearningin many of theirchildren.Childrenfrom a small city/suburbanheterogeneouspopulation and childrenfrom a wide range of schools in a large urbanschool district demonstratedsuch learning,so the learning/teachingapproachcan be used successfully with a fairlywide rangeof children.The successful learningin both studies indicatedthatthe learning/teachingapproachcould be implementedon a broad scale with a moderateamountof in-service time, materials,and teachersupport. Many participatingteachersin Study 2 did ask for their own set of blocks for the coming year, so one set of blocks per building is clearly not ideal. In particular, more sets of blocks may facilitatethe use of place-valueunits in the crowdedendof-the-yearschedule. The approachdid not result in maximal learning in all areas by all children. Some childrencontinuedto make occasional tradingand fact errors,particularly in subtractionwith the ten-digitproblem.Some childrenwere not able consistently to choose the largerof two three-digitthroughseven-digit pairs of numbers,and manychildrenin Study2 did not generalizetradingto all of the items on the Traded Word/NumeralTest.Whetherthese limitationsare inherentin this approachor are due to inadequateimplementationof certainfeaturesof the approachor simply to insufficienttime with the approachfor some childrenis not clear.In the first study of the approach(Fuson, 1986a), telling childrento "thinkabout the blocks" was sufficient for most of them to self-correcterrorsthey were still making after the initial learningor to self-correcterrorsthat began to appearon delayed posttests after correct initial learning. Thus, the blocks can be a powerful support for children's thinking, but many children do not seem spontaneously to use their KarenC. Fuson and Diane J. Briars 203 knowledgeof the blocks to monitortheirwrittenmultidigitadditionor subtraction. This suggests thatfrequentsolving of one multidigitadditionor subtractionproblem accompanied by children's thinking about the blocks and evaluating their written-marksproceduremightbe a powerfulmeansto reducethe occasionaltrading errorsmade by children. A limitationof both of these studiesis thattheirdesigns did not permitan evaluation of any of the specific features of the learning/teachingapproach.The approach had many features, not all of which may be crucial to its success. These features stemmed from the need to provide children an opportunityto construct conceptual structuresfor the mathematicallydifferentEnglish named-valuesystem of numberwords and the positionalbase-ten system of writtenmarksand to think about how these systems work in multidigitaddition and subtraction;how the featuresrelateto children'slearningare discussed in Fuson (in press a). These studies are also limited because they were not intended to provide a complete addition and subtractionor place-value experience. Obviously importanttopics were omittedthatrelateto the goals of understandingmultidigitadditionand subtraction(e.g., estimation,alternativemethods of adding and subtracting).Future work might explore how well the learning/teachingapproachcould supportthese more extensive goals. These two studies raise several issues for futureresearchconcerningthe use of embodimentsin learningmultidigitadditionand subtractionand place value (see also Baroody,in press; Fuson, in press b). First, we took no position concerning whether the teacher or the children moved the blocks or whether learning proceeded within a total class approach,within simultaneoussmall groups,or within serial small groups. In Study 1 differentteachers used all of these, and they all seemed to be effective. Otherpossible outcomes of these differentapproaches(for example, beliefs thatsuccess dependson effort,attemptsto understand,andcooperation with peers as reportedin Nicholls, Cobb, Wood, Yackel, & Patashnick, 1990, for a small-group problem-solving classroom organization)might be explored. Second, relative benefits of using the learning/teachingapproachto support prechosen multidigit addition and subtractionprocedures,as in the present studies, versus using the approachto supportproceduresinvented by children, might be examined. Thus, the focus of the present studies on computation as meaning (on understandingmultidigit additionand subtractionand place value) mightbe contrastedwith computationas problemsolving (Labinowicz,1985). The latterdoes not necessarily result in more competence (for example, only 34% of the third graderswho had reinvented arithmetic without traditionalinstruction solved 43 - 16 correctly,Kamii, 1989), but the supportof the learning/teaching approachin Figure 1 might help childreninvent multidigitadditionand subtraction procedures.Third,several aspectsof a more gradualuse of base-tenblocks as proposedelsewhere do not seem to be necessaryfor high levels of skill andunderstanding,because they were not implementedin our approach.These includeprolonged work with two-digit numbers,followed considerablylater by work with three-digitand even laterby four-digitnumbers;ratherextensive experiencewith 204 Base-Ten Blocks Learning/Teaching Approach tradingbefore tradingis set within additionproblems;extensive practicejust with the blocks with no recording;pictorialrecordingbefore recordingwith base-ten written marks; and use of the blocks to count on by tens and hundreds (e.g., Baroody, 1987; Davis, 1984; Labinowicz, 1985; Wynroth,1980). Futureresearch may establishthat these aspects do bringparticularbenefits, but it seems wise to undertakesuch researchratherthanmerely to assertthese benefits. The resultssuggesta gradeplacementfor multidigitadditionandsubtractionand place-value concepts with this approach.Even though many average-achieving first graderswere able to learn the multidigitadditionalgorithm,their relatively poorerperformanceon some aspects of the interview suggests that the approach in these studies risks pushing childrenbeyond their comfortablelearningrange. Some of these childrenmay still requireperceptualunit items for thinkingabout single-digit numbers and thus may have trouble using the blocks to construct conceptualten-unit,hundred-unit,and thousand-unititems made out of collected ones. Therefore, for first graders of average and below-average mathematics achievement and perhapseven for many high-achievingfirst graders,it may be betterto concentratein the first grade on helping childrento build and use their unitarysequence/countingconceptualstructuresfor addingandsubtractingsingledigit numbers(i.e., sums and differences to 18). Trying to build simultaneously these unitaryconceptualstructuresand the multiunitnamed-value/base-tenconceptual structuresneededfor multidigitadditionand subtraction,especially given the interferencethe irregularEnglishnumberwords createfor this task (cf. Fuson & Kwon, in press), may be too difficultfor manyfirst graders.The learning/teaching activitiestested in these studiesdo seem to be developmentallyappropriatefor second-gradechildrenof all achievementlevels except perhapsthose with special difficulties. Teachersreportedthat second-gradechildrenin both studies enjoyed the learningactivities and felt good aboutthemselves and their ability to do such problems with understanding.Thus, the typical textbook extension of multidigit additionandsubtractionproblemsover Grades2 through4 or 5, addingone or two digits each year (Fuson, in press c), underestimateswhat our childrencan learn. The conceptualbases for generalmultidigitadditionandsubtractionalgorithmsare well within the capacity of most second gradersif they are learnedwith the supportof physical materialsthatembody the relative size of the base-tenplaces and demonstratethe positional natureof the multidigitwrittenmarksand if the focus of such learningis understandingand not just proceduralcompetence. REFERENCES Baroody,A. J. (1987). Children'smathematicalthinking: A developmentalframeworkfor preschool, primary,and special educationteachers. New York: TeachersCollege Press. Baroody,A. (in press). 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Children'scountingtypes: Philosophy, theory,and application.New York: PraegerScientific. Tougher,H. E. (1981). Too many blanks! What workbooksdon't teach. ArithmeticTeacher,28(6), 67. Underhill,R. (1984). Externalretentionand transfereffects of special place value curriculumactivities. Focus on LearningProblemsin Mathematics,1 & 2, 108-130. Wynroth, L. (1969/1980). Wynrothmath program: The natural numberssequence. Ithaca, NY: WynrothMath Program. AUTHORS KAREN C. FUSON, Professor,School of Educationand Social Policy, 2003 SheridanRoad, NorthwesternUniversity,Evanston,IL 60208-2610 DIANE J. BRIARS, Directorof the Division of Mathematics,PittsburghPublic Schools, 850 Boggs Avenue, Pittsburgh,PA 15211

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