Parts, Wholes, and Place Value: A Developmental View
Author(s): Sharon H. Ross
Reviewed work(s):
Source: The Arithmetic Teacher, Vol. 36, No. 6, FOCUS ISSUE: NUMBER SENSE (February
1989), pp. 47-51
Published by: National Council of Teachers of Mathematics
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Parts, Wholes,
and Place Value:
A Developmental View
By Sharon H. Ross
Whena studentmentally
computes
32 + 59 bythinking
30 4-50 is 80,9 +
2 is 11, and80 + 11 is 91, thatstudent
has had to call on some well-developed concepts of numericalpartwholerelationships
and place value.
Peoplewithgoodnumbersensemake
and flexibleuse ofthesetwo
frequent
relatedconceptsto performmental
and numericestimates.
computations
Studentsfindtheseconceptsdifficult;
theirunderstanding
develops slowly
overa periodof severalyears.To be
successfulat teachingnumbersense,
we mustdesigninstruction
that restudents'
need
to
construct
spects
theirownknowledge.
mentallyto compose wholes from
theircomponentparts, decompose
wholequantitiesintoparts,and perthe partsand recomhaps rearrange
all
confident
pose thewholequantity,
the while that the quantityof the
wholehas notchanged.
Place value and part-whole
relationships
of wholequanA naturalpartitioning
used by those
titiesthatis frequently
withgood numbersense is that of
"tens and ones"- or higherpowers
of 10withnumberslargerthan99. To
describedin
use thekindof thinking
the exampleof 32 + 59, a student
a firm
constructed
musthavementally
Part-Whole
of
the
of
understanding
concept place
value. Yet classroomteachersand
Relationships
observe that
A pupil'snumerical
awarenessdevel- researchersrepeatedly
students'
understandingof place
ops gradually.Firstgradersexplore valueis
poor.
the varietyof ways in whichsmall
Understanding
place valuerequires
7 can be
numbers
can be partitioned;
the
student'semergan elaboration
of
as 3 + 4 as well as 1 + 6
partitioned
of a part-whole
ing understanding
and2 + 5. As youngpupilsmaturein
can be foundto
Many
ways
concept.
number
senseand skills,theybecome
ofa nummoreflexible.Theycan solve a wide forma two-waypartitioning
+
beras largeas 52: 51 1, 50 + 2, 49
ofverbaladditionandsubtracvariety
+ 3, 48 + 4, .... Onlyone ofthese,
tion problems(Riley, Greeno, and
+
linkedto themeanstrat- 50 2, is directly
Heller1983)anduse related-fact
the
individual
of
digitsin our
ings
egies to recallbasic facts(Cobb and
notationalsystem.This
conventional
Merkel,in press; Thornton1978).
is knownas a standard
allowsthem representation
theirthinking
Eventually
Nonstandard
place-valuepartitioning.
like 40 +
tens-and-ones
partitionings
the
12 are useful in understanding
SharonRoss is an assistantprofessorin the
aspect of computational
regrouping
Caland statistics,
ofmathematics
department
algorithms.
Chico, Chico, CA
iforniaState University,
Ourrelatively
elaboratenumeration
95929-0525.
She teachescoursesforpreservice
educators.She is
mathematics
and in-service
systemis characterized
bythefollowalso codirector
oftheChicositeoftheCalifor- ingfourproperties:
nia Mathematics
Project,a summerinstitute
1. Positionalproperty.
The quantiand leadership
forteachers.
training
program
February1989
tiesrepresented
digbytheindividual
itsaredetermined
bythepositionthey
holdin thewholenumeral.
The valuesof
2. Base-tenproperty.
thepositionsincreaseinpowersoften
fromrightto left.
3. Multiplicativeproperty.The
valueofan individual
digitis foundby
thefacevalueofthedigit
multiplying
by thevalue assignedto itsposition.
.The quantity
4. Additiveproperty
numeralis
the
whole
by
represented
thesumof thevaluesrepresented
by
theindividual
digits.
To understand
place value thestua
andsynthesize
dentmustcoordinate
variety of subordinateknowledge
aboutourculture'snotationalsystem
for numbersand about numerical
part-wholerelationships.A student
who understands
place value knows
not onlythatthe numeral52 can be
used to represent"how many"fora
collectionoffifty-two
objectsbutalso
thatthe digiton the rightrepresents
two of them,the digiton the left
of them(fivesets of
fifty
represents
52
that
is the sum of the
and
ten),
by theindividquantitiesrepresented
ual digits.
A Research Study Using
Digit-Correspondence
Tasks
withgoodnumber
We expectstudents
sense to have constructed
meanings
for numeralsin the initialyears of
school.In a studyI conductedofthe
meaningsthat studentsattributeto
numerals,however,I found
two-digit
that many of them were still constructing
meaningsfortheindividual
47
digitsas late as fifth
grade (Ross 1985,
1986). In mystudyof sixtystudentsin
second throughfifthgrade, randomly
selected fromfivediverse elementary
schools, students were individually
administeredthe followingtask:
Fig. 1
task materials
Digit-correspondence
The studentwas instructedto spilla
sticks out
collection of twenty-five
of a bag, and I asked, "How many
sticks are there?" Aftercounting,
nearly all gave a correct oral response of "twenty-five." I then
Many studentsare still
constructing
meaningforplace
value infifthgrade.
asked the child to writedown how
many sticks had been counted.
Nearly all correctlywrote the numeral25. 1 thenencircledfirstthe 5
and thenthe 2 and each timeasked,
"Does thispartof yourtwenty-five
have anythingto do withhow many
sticksyou have?"
Of the sixty studentsinterviewed,
twenty-sixwere successful- theyexplainedin a varietyof ways thatthe 5
in 25 representedfiveof the sticksand
thatthe 2 representedthe othertwenty. Twelve studentsthoughtthe individual digits had nothingto do with
how many sticks were in the collection; fourteendescribed inventednumericalmeanings,such as that the 5
meant"half of ten," thatthe 5 meant
that groups contained five sticks, or
that the 2 meant "count by twos."
Eight students thought that the 2
meanttwo sticksand thatthe 5 meant
eitherfivesticksor had nothingto do
withthe numberof sticks in the collections.
In the stickstaskjust described,the
collectionof stickswas notgroupedin
any way. When the task was altered
the number52 witha
by representing
standard place-value partitioningof
base-ten blocks (five ten-blocks and
two unit-blocks)many more of the
students(44 out of the 60) were successful. But when 52 was represented
usingfourten-blocksand twelve unitblocks, the numberof successful students dropped to twenty.Similar re48
sults were found in tasks in which a
collection of forty-eightbeans was
partitionedinto cups. Figure 1 displays the materials used in the six
tasks used in the
digit-correspondence
study.
Whydid childrenfindthe tasks that
used a standardplace-value partitioning to be so much easier? In both
standardtasks, the representationsof
the groups of ten were very prominent. When, for example, I asked if
the 2 and 5 in the numeral 52 had
anythingto do with how many baseten blocks were on the table, the student was looking at five long purple
ten-blocksand two tan unit-blocks.It
is easy to see how the studentmight
have proposed thatthe 5 represented
the five purple blocks and yet have
had no thought of tens or fiftyin
mind- just fivepurple blocks.
In a follow-up study designed to
examinewhethersome studentsdid in
fact use this face-value interpretation
to assign meaning to the individual
digits, thirty third-grade students
were individuallyasked to count a
collectionof twenty-sixobjects and to
"write down how many." They were
thenasked to sortthe objects intosets
of four ("candies into party-favor
ArithmeticTeacher
objectspartitioned ^
Fig.2 Twenty-six
ä /~'
intosets offour
^
f~'
^
^
' QOJ
cups" or "wheelsto maketoycars").
The resultingpartitioned
collection,
infigure
2, containedsix sets
pictured
of fourobjectsand two "leftover."
The interviewer
then encircledthe
individual
digitsin 26 and asked,for
each digit,"Does thispartofyour26
have anything
to do withhow many
youhave?"
The prominentgroupingsin this
a correct
task,insteadof facilitating
an incorrectreresponse,facilitated
sponse,wherestudentsreversedthe
of thedigitsand wherethe
meanings
werenot in base ten. The
groupings
incorrect
response,however,wasconsistentwiththeface-valueinterpretation.Nearlyhalfof thethirdgraders
incorrectly
respondedthatthe2 in 26
stoodfortwooftheobjects,whereas
the6 in the 26 represented
the "six
cupsofcandies"or "the six cars."
^^
AX)
00
Stage 1, wholenumeral.As pupils
in our cultureconstructtheirknowledge about quantitiesup to ninetynine and theirsymbolicrepresentation as two-digitnumerals, their
of the whole
cognitiveconstruction
- the numeral52 reprecomes first
sentsthewholeamount.Theyassign
no meaningto the individualdigits.
For related work that reportsobserveddifferences
betweenpupilsin
the U. S. and thosefromAsian cultures,see the workof Miura (1987)
and ofMiuraet al. (in press).
f
Table 1
Stages intheDevelopmentofStudents'
of Two-Digit
Understanding
Numerationby Grade in School
Stage of development
Grade*
2
3
4
5
Total
1
Π
Ш
8
0
10
0
9
0
2
3
5
6
12
3
16
IV
4
6
17
5
16
V
0
2
7
16
*n = 15foreachgradelevel.
trulyrepresent
groupsof tenunitsto
studentsin stage 3; studentsdo not
Studentsin theirearlystagesof
recognize that the numberreprevalue
of
sentedbythetensdigitis a multiple
understanding
ofplace
to
ten.
appear understandmorethan
do.
zone. Stutheyactually
Stage 4, construction
dents know that the leftdigitin a
numeralrepresentssets of
two-digit
ten
2,
Stage positionalproperty.
Pupils
objects and that the rightdigit
in
numeral
the
know
that
a
two-digit
representsthe remainingsingleobStages of Interpreting
the
in
the
"ones
on
is
digit
right
jects, butthisknowledgeis tentative
Two-Digit Numerals
in
and
the
left
is
task
the
on
andis characterized
digit
place"
byunreliable
the
"tens
of
Their
knowledge
A five-stage
place."
performances.
modelof the interpretahowevdigitsis limited,
tionschildrenassignto two-digit
nu- theindividual
Students
Stage 5, understanding.
meralswas developedbased on data er, to the positionof the digitsand know thatthe individualdigitsin a
fromthe originalstudyand findings does notencompassthequantitiesto two-digit
numeralrepresenta partifromrelatedresearch(Ashlock1978; whicheach corresponds.
tioningof the whole quantityintoa
Baroody et al. 1983; Barr 1978;
Stage3yface value. Studentsinter- tenspartand a ones part.The quanBednarzand Janvier1982;Flournoy pret each digitas representing
the tityof objectscorresponding
to each
numberindicatedby its face value. digitcan be determined
evenforcol1967;Heibertand Wearne 1983; M.
Kamii 1980, 1982; NationalAssess- The set of objectsrepresented
in
by the lectionsthathave been partitioned
mentof EducationalProgress1983; tensdigit,however,maybe different nonstandard
ways.
Rathmell1972; Resnick 1983; Rick- fromthe objects represented
by the
man 1983; Scrivens1968; C. Smith onesdigit.Theymayverballylabelas
Ages and Stages
to
of "tens" theobjectsthatcorrespond
1969;R. Smith1973).Descriptions
thetensdigit,buttheseobjectsdo not Each of the sixtystudentsin therethestagesfollow.
February1989
49
for many
portedstudywas assignedto one of fractions,is inappropriate
the five stages accordingto perfor- students.
mances on six digit-correspondence How do so manystudents
reachthe
tasksanda positional-knowledge
task middlegrades with so littleunderinwhichstudents
wereaskedto iden- standingof two-digit
numbers?One
in
in
tify, a two-digitnumeral,which reasonis thatpupils stages2 and 3
morethan
digitwas in the "tens place" and
mayappearto understand
whichwas in the "ones place." The
theyactuallydo. Withstage-2knowlnumberof studentsin each stageis
edge of theleft-right
positionsof the
shown,by gradelevel,in table 1.
Althoughall the studentsin this
studywereat leastat stage1- able to
Each pupil mustbe provoked
countcollectionsof as manyas fiftyhis or her own
two objects and writethe two-digit intoconstructing
numbers
and the
numeralthat correspondedto the
knowledgeof
count twelveof themdemonstrated relationsamongthem.
no quantitative
of the
interpretation
individual
digits.
Sixteenstudents
succeededonlyon
digits,pupilsare able to succeedon a
those digit-correspondence
tasks in
varietyof tasks typicallyfound in
whichnumberswererepresented
us- theirtextbooksand on standardized
inga standard
place-valuepartitioning tests,suchas thefollowing:
of a collectionof objects- that is,
In 27, whichdigitis in the tens
those tasks in whichtheycould be
place?
successfulusinga stage-3face-value
How manytensare in 84?
interpretation.
Sixteenstudentssucceededon all
35 =
tensand
ones.
six digit-correspondence
=
tasks,dem7 tens+ 5 ones
a stage-5understanding
of
onstrating
Studentswho use a stage-3,facethenumbersrepresented
by theindivalue
of digitssucceed
interpretation
vidualdigitsin two-placenumerals.
on
an
even
wider
varietyof tasks,
No second-grade
pupildemonstrated
that
use manipulative
many
Even among including
stage-5understanding.
materials.
In
instructional
tasks
many
the fifthgraders,only halfwere at
students
are
asked
to
make
correstage5.
spondencesbetweendigitsand materials.Ifa collectionis alreadygrouped
intoa standardplace-valuepartitionImplications forthe
ingoftensand ones,a studentwhois
Classroom
asked to make correspondences
for
the
in
for
need
52,
digits
example,
More instructional
supportfocusing
and
onlylook for"fiveof something
on two-digit
numeration
is neededin
themiddlegrades.Studentsdo notall
developat thesamerate,nordo they
all have identicalexperienceswith Whena teachershowsstudents
numbers
and numerals.The data disstudentsdo nothave
something,
in
played table 1 show thateven in
to think;theysimplyfollow
fourth
and fifth
grades,onlyhalfthe
directions.
studentsintervieweddemonstrated
of the individual
good understanding
numerals.Others
digitsin two-digit
who have used digit-correspondencetwo of something
else." Using this
tasksto assess students'understand- face-value strategy,
a studenteasily
- beans in
ing of place value reportsimilarre- adapts to new materials
sults(C. Kamii1986;M. Kamii1980, cups, linkingcubes, base-tenblocks,
1982).Thetypicalplace-valueinstruc- or coloredchips,forexample.Only
tionin the middlegrades,whichfo- whenthecollectionsare groupedinto
cuses on symbolicexpressionsfor nonstandardpartitionings
does stumuchlargernumbersand on decimal dents' face-value interpretation
fail
50
with
them;whentheyare confronted
tasks,students'faultyunregrouping
becomesapparent.
derstanding
Further
studyneedsto be done,but
even extensiveexperiencewithembodimentslike base-tenblocks and
otherplace-valuemanipulatives
does
notappeartofacilitate
an understandingofplace valueas measuredbythe
digit-correspondencetasks (Ross
1988).If we introducematerialsthat
have been designedto embodybaseten groupingsbeforestudentshave
constructedappropriatequantitative
fortheindividual
meanings
digits,we
mayunwittingly
provokeor reinforce
a stage-3(face-value)interpretation
of
digits. With these materials the
teacherand manufacturer
may have
"embodiedtheten," butthe student
neednot.
materials
can serveas
Manipulative
a usefulcommunication
toolbetween
- they give us
teacher and student
to
talk
about.
something
Throughthe
use of such materialswe can often
learn a great deal about students'
thinking,
includingtheirmisconceptions(Labinowicz1985).We can also
use thematerials
to showand explain
whatwe wantstudents
to do. Careful
instruction
withplace-valuemanipulativescan facilitatethe acquisitionof
the proceduralknowledgerequired
forfacilitywithcomputational
algorithms(Fuson 1986; Resnick 1983).
We should not mislead ourselves,
however,thatas a resultof such instructionstudentsconstructunderstandingofeitherthecomplexplacevalue numerationsystem or the
is the
algorithm.If understanding
goal,itprobablydoesn'tmatterifthe
teacher shows childrenprocedures
with paper and pencil, beans and
cups, or base-tenblocks. When a
teacher shows studentssomething,
studentsdo not have to think;they
simplyfollowdirections.
comes only from
Understanding
each
thinking; pupilmustbe provoked
into constructinghis or her own
knowledgeof numbersand the relationsamongthem.Pupilsneedto entasks that
gage in problem-solving
them
to
think
about
useful
challenge
to
and
ways partition composenumbers.The additionand subtraction
of
numbersthroughninety-nine
are apArithmetic
Teacher
propriate
topicsfor second graders,
butwe shouldencouragepupilstofind
in theirown
sums and differences
ways.
For studentswho have not been
taughta conventionalpaper-andthe sum of
pencilprocedure,finding
32 + 59 is a nonroutineproblem
whosesolutionstrategy
is notreadily
The
can
apparent. problem be solved
usinga varietyof strategiesand a
- counters,finvarietyof materials
and
or
gers, paper
pencil.If theyare
in
working cooperativegroups,studentscan be encouragedto compare
methodsand tryto convinceone andoes or doesn't
otherthata strategy
use
"work"; theymight a calculator
to verifythe solution.In the larger
group discussion,studentscan be
theirsuccessful
askedto demonstrate
strategies.
Teaching place-value concepts
to doubleas a prerequisite
separately
is inefdigitadditionand subtraction
fectiveand unnecessary.
Experimental instructional
studieshave shown
thatpupilsin firstand second grade
can inventtheirown efficient
algoanddo so without
rithms,
place-value
manipulatives(Kamii and Joseph
1988;Cobb and Merkel[inpress]).In
materials
fact,manipulative
mayactudetract
from
thinkingbecause
ally
tasks are too easy to do with the
materials.
As teachers of mathematicswe
forall
need to provideopportunities
students
to developa strongsense of
number.To accomplishthatgoal we
can help each studentto construct
as she or he matures,inmentally,
elaborateconceptsof nucreasingly
andplace
relations
mericalpart-whole
value. If we challengestudentswith
for makingestimoreopportunities
and
matesand mentalcomputations
show them the conventionalalgorithmsonly aftertheyhave experienced a fertileperiod of inventing
theirown efficientproceduresfor
like32 + 59,we can
solvingproblems
expectmoreyoungstudentsto demflexthenumeric
onstrate
partitioning
of those
ibilitythatis characteristic
withgood numbersense. If students
thenlearna conventional
algorithm,
theywillviewitas one ofmanyways
to finda sumor difference;
they'llbe
February1989
able to choose fromand use, as apmentalcompuestimation,
propriate,
tation,and calculators,as well as inventedand conventional
algorithms.
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