Using a Base-Ten Blocks Learning/Teaching

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). How and when shouldplace-valueconcepts and skills be taught? Journalfor
Research in MathematicsEducation.
Bell, M. S., Fuson, K. C., & Lesh, R. A. (1976). Algebraic and arithmeticstructures: A concreteapproachfor elementaryschool teachers. New York: The Free Press.
Cauley,K. M. (1988). Constructionof logical knowledge: Studyof borrowingin subtraction.Journal
of EducationalPsychology, 80, 202-205.
KarenC. Fuson and Diane J. Briars
205
Cobb, P:, & Wheatley,G. (1988). Children'sinitial understandingsof ten. Focus on LearningProblems in Mathematics,10, 1-28.
Davis, R. B. (1984). Learningmathematics:Thecognitivescience approachto mathematicseducation.
Norwood, NJ: Ablex.
Davis, R. B., & McKnight,C. C. (1980). The influence of semanticcontenton algorithmicbehavior.
Journal of Children'sMathematicalBehavior, 3, 39-87.
Dienes, Z. P. (1960). Building up mathematics.London:HutchinsonEducationalLtd.
Fuson, K. C. (1986a). Roles of representationand verbalizationin the teachingof multi-digitaddition
and subtraction.EuropeanJournal of Psychology of Education,1, 35-56.
Fuson, K. C. (1986b). Teaching childrento subtractby countingup. Journalfor Research in Mathematics Education,17, 172-189.
Fuson, K. C. (1987). Researchinto practice: Adding by countingon with one-handedfinger patterns.
ArithmeticTeacher, 35(1), 38-41.
Fuson, K. C. (1988a). Children'scountingand concepts of number.New York: Springer-Verlag.
Fuson, K. C. (1988b). Research into practice: Subtractingby counting up with one-handedfinger
patterns.ArithmeticTeacher,35(5), 29-31.
Fuson, K. C. (in press a). Children'sconceptual structuresfor multidigitnumbers: Implicationsfor
additionand subtractionand place-value learningand teaching.Cognitionand Instruction.
Fuson, K. C. (in press b). Issues in place-valueand multidigitadditionand subtractionlearning.Journalfor Research in MathematicsEducation.
Fuson, K. C. (in press c). Researchon learningand teachingadditionand subtraction.In G. Leinhardt
& R. T. Putnam(Eds.), Cognitiveresearch: Mathematicslearningand instruction.
Fuson, K. C., & Kwon, Y. (in press). Chinese-basedregularandEuropeanirregularsystems of number
words: The disadvantagesfor English-speakingchildren.In K. Durkin& B. Shire (Eds.),Language
and MathematicalEducation.Milton Keynes, GreatBritain:Open UniversityPress.
Fuson, K. C., Richards,J., & Briars,D. J. (1982). The acquisitionand elaborationof the numberword
sequence. In C. Brainerd(Ed.), Progress in cognitive development: Children'slogical and mathematical cognition, Vol. 1 (pp. 33-92). New York: Springer-Verlag.
Fuson, K. C., & Secada, W. G. (1986). Teachingchildrento add by countingon with finger patterns.
Cognitionand Instruction,3, 229-260.
Fuson, K. C., Stigler, J. W., & Bartsch,K. (1988). Gradeplacementof additionand subtractiontopics
in China,Japan,the Soviet Union, Taiwan, and the United States.Journalfor Research in Mathematics Education,19, 449-458.
Fuson,K. C., & Willis, G. B. (1988). Subtractingby countingup: Moreevidence.Journalfor Research
in MathematicsEducation,19, 402-420.
Ginsburg,H. (1977). Children'sarithmetic: How they learn it and how you teach it. Austin, TX: ProEd.
Kamii, C. (1985). Youngchildrenreinventarithmetic.New York: TeachersCollege Press.
Kamii, C. (1986). Place value: An explanationof its difficulty and educationalimplicationsfor the
primarygrades.Journal of Research in ChildhoodEducation,1, 75-86.
Kamii,C. (1989). Youngchildrencontinueto reinventarithmetic-2nd grade: ImplicationsofPiaget's
theory.New York: TeachersCollege Press.
Kamii,C., & Joseph,L. (1988). Teachingplace value 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). Learningfrom children: New beginningsfor teaching numerical thinking.
Menlo Park,CA: Addison-WesleyPublishingCompany.
Miura,I. (1987). Mathematicsachievementas a functionof language.Journalof EducationalPsychology, 79, 79-82.
Miura, I. T., Kim, C. C., Chang, C., & Okamoto, Y. (1988). Effects of language characteristicson
children'scognitive representationof number:Cross-nationalcomparisons.ChildDevelopment,59,
1445-1450.
206
Base-TenBlocks Learning/TeachingApproach
Miura,I. T., & Okamoto,Y. (1989). Comparisonsof Americanand Japanesefirst graders'cognitive
representationof numberandunderstandingof place value.Journalof EducationalPsychology,81,
109-113.
Nicholls, J. G., Cobb, P., Wood, T., Yackel, E., & Patashnick,M. (1990). Assessing students'theories
of success in mathematics:Individualand classroomdifferences. Journalfor Research in Mathematics Education,21, 109-122.
Resnick, L. B. (1983). A developmentaltheoryof numberunderstanding.In H. P. Ginsburg(Ed.), The
developmentof mathematicalthinking.New York: Academic.
Resnick, L. B., & Omanson,S. F. (1987). Learningto understandarithmetic.In R. Glaser (Ed.), Advances in instructionalpsychology (Vol. 3). Hillsdale, NJ: Erlbaum.
Ross, S. H. (1986, April). The developmentof children'splace-value numerationconcepts in grades
two throughfive. Paper presented at the annual meeting of the American EducationalResearch
Association, San Francisco.
Ross, S. H. (1989). Parts,wholes, and place value: A developmentalview. ArithmeticTeacher,36(6),
47-51.
Song, M., & Ginsburg,H. P. (1987). The developmentof informaland formalmathematicalthinking
in Koreanand U.S. children.Child Development,57, 1286-1296.
Steffe, L. P., & Cobb, P. (1988). Constructionof arithmeticalmeanings and strategies. New York:
Springer-Verlag.
Steffe, L. P., von Glasersfeld,E., Richards,J., & Cobb,P. (1983). 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