Developing Ratio and Proportion Schemes: A Story of a Fifth Grader
Author(s): Jane-Jane Lo and Tad Watanabe
Source: Journal for Research in Mathematics Education, Vol. 28, No. 2 (Mar., 1997), pp. 216236
Published by: National Council of Teachers of Mathematics
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Journalfor Researchin MathematicsEducation
1997, Vol. 28, No. 2, 216-236
Developing Ratio and Proportion
Schemes: A Story of a Fifth Grader
Jane-JaneLo, Cornell University
Tad Watanabe,TowsonState University
There is a growing theoretical consensus that the concepts of ratio and proportiondo not
developin isolation.Rather,theyarepartof the individual'smultiplicativeconceptualfield, which
includesotherconceptssuch as multiplication,division,andrationalnumbers.The currentstudy
attemptedto clarify the beginning of this developmentalprocess. One fifth-grade student,
Bruce,was encouragedto schematizehis trial-and-error-based
method,which was effective in
solving so-called missing-valuetasks.This studydescribesseveraladvancementsBrucemade
during the teaching experiment and analyzes the challenges Bruce faced in attemptingto
schematizehis method.Finally, the mathematicalknowledge Bruceneededto furtherdevelop
his ratioandproportionconcepts is identified.The findings provideadditionalsupportfor the
view thatthe developmentof ratioandproportionconceptsis embeddedwithinthe development
of the multiplicativeconceptualfield.
Accordingto PiagetandInhelder(1975), proportional
reasoningis a second-order
relationshipthatinvolves an equivalentrelationshipbetweentwo ratios.Forexample, the statement"Fourcandybarscost five dollars"describesa ratiobetweenan
amountof money andthe amountof candythatcan be boughtwith thatamountof
money. Yet, proportionalreasoningis needed to figure out how an increasefrom
5 dollarsto 15 dollarsaffects the numberof candy barsone can get.
Ratioandproportionareimportantconceptsin currentmathematicscurricula.Very
oftenmultiplicationanddivisiontasksin lowergradesarepresentedin unit-rateform,
which is a specialformof ratioandproportion.Forexample,"Sallypaid $ 1.25 for
each muffin [unitprice].How muchdid she pay for a dozen muffins?"In the middle grades,wordproblemsinvolvingequivalentfractionsandfractioncomparisons
can also be thoughtof as ratioandproportionsituations.Forexample,considerthe
task "GroupA has 4 pizzas and6 girls. GroupB has 6 pizzas and8 boys. Who gets
morepizza,theboys or the girls?"(adaptedfromLamon,1993a).To solve thisproblem, some studentsmay drawpicturesto figure out thatin GroupA, each member
gets 2/3of a pizza, whereasin GroupB, each membergets 3/4of a pizza. They can
then comparethese two fractionswith the pictures.Otherstudentsmay use ratio
andproportionreasoning:"IfI add2 pizzasto GroupA, I wouldneed to add3 more
The researchin this paperwas supportedby the ArizonaState UniversityWest Research
Grant.The opinions expresseddo not necessarilyreflect the views of the funding agency.
An earlyversionof thisarticlewas presentedatthe annualmeetingof theAmericanEducation
ResearchAssociation in 1994. This articlewas writtenwhile the first authorwas a Visiting
Fellow of the Departmentof Educationat CornellUniversity.The first authorwishes to thank
Dr. JereConfreyandDr. David Hendersonfor the opportunityto participatein the activities
of the CornellMathematicsEducationResearchGroup.
Jane-JaneLo and Tad Watanabe
217
people.So GroupA is likehaving6 pizzasand9 members.So, eachmemberin Group
B gets more pizza."The ability to recognize structuralsimilarityand the sense of
covariationandmultiplicativecomparisonsillustratedin such a reasoningprocess
areat the core of algebraandmoreadvancedmathematics(Confrey& Smith,1995).
Because of the importanceof this topic in school mathematics,children'sconeducationresearch,
havelongbeena focusof mathematics
ceptsof ratioandproportion
and much has been learnedaboutstudents'errorsand difficulties in solving ratio
andproportiontasks(Hart,1984;Karplus,Pulos,& Stage, 1983) as well as different
taskvariablesthataffectstudents'choicesof strategiesandperformance(Harel,Behr,
Post, & Lesh, 1991; Kaput& West, 1994).
But what arethe roots of these difficulties?Whatarithmeticknowledgemay be
useful in developingthe conceptsof ratioandproportion?Vergnaud(1988, p. 141)
used the term "multiplicativeconceptualfield" to refer to "all situationsthat can
be analyzedas simpleor multipleproportionproblems."Mathematicalconceptsthat
are tied to those situationsinclude, as Vergnaudpointedout, multiplication,diviandlinearfunctions.He suggestedthatstudentsdevelop
sion,fraction,ratio,proportion,
these concepts not in isolation but in concertwith each otherover long periodsof
time throughexperiencewith a largenumberof situations.Therefore,researchstudies on children'sratioandproportionconceptsalso need to considerthe otherconcepts thatare a partof children'sdeveloping multiplicativeconceptualfield.
In this article,we reportthe findingsfrom such an attempt.The analysisis based
on datagatheredfromfifteen 70-minuteteachingsessions with one fifth-gradestudent, Bruce, over a period of 6 months.The purposeof this teachingexperiment
was to documentthe schematizingprocessof his self-generatedstrategiesandhow
the developmentof these strategiesmightinfluenceor be influencedby otherconstructsof the multiplicativeconceptualfield. Specifically, we askedthe following
researchquestions:
1. How did Bruce's informalstrategieswith ratio and proportiontasks develop
as he gained knowledge of otherconcepts in his multiplicativeconceptualfield?
2. Whatchallengeswould Bruceface when he was encouragedto schematizehis
informalstrategies?
3. Whatmathematicalknowledgewas usefulfor Brucein his developmentof ratio
and proportionconcepts?
We wereawarethata longitudinalteachingexperimentwas neededto answerthese
thatcould
questions,andwe hopedthiscase studyof Brucewouldprovideinformation
be used in a muchlargerresearchprogram.The currentanalyses,althoughsubjectto
haveprovidedus witha paradigmatic
furtherrefinementandrefutation,
case,havechaland
have
raised
more
for
our
own
notions
of
proportion,
questions futureresearch.
lenged
BRUCE AND HIS INFORMALSTRATEGIES
Brucewas 10 yearsand 10 monthsold at thebeginningof theteachingexperiment.
He wasjudged to be a top mathematicsstudentby his teacherandwas qualifiedfor
an enrichmentprogramofferedeveryThursdaymorningby the schooldistrict.Even
218
DevelopingRatio and ProportionSchemes
thoughhe was good at school mathematics,he did not particularlylike it. Whenhe
had sparetime, he liked to ride his bicycle, play basketball,and read science ficas he describedit,includedlisteningto theteacher's
tion.His wayof doingmathematics,
in
and
class
homework.
Mathematicsfor him meant either
doing
explanation
which
he
solved
computationalproblems,
by applying the proceduredemonstratedby the teacher,or wordproblems,which he solved by choosing the correct
operationhe had learnedin class.
The mathematicsinstructionin Bruce's classroom was the typical teacherexplanation/student-practice
approach.A textbookthat containedmore than 100
mini-unitswas followedfaithfullyby theteacher.Therewas alwaysone correctanswer
for every mathematicsproblemand a right way to reachthatanswer.The foci of
the fifth-grademathematicstextbookincludedmultidigitmultiplicationand divito decimals,anda littlegraphing.Although
initialintroduction
sion,fractionoperations,
and
measurementat the end of the textbook,the
therewere a few unitsof geometry
teacherdid not have time to cover them.
On the basis of his classroomexperience,Bruce did not see the need to provide
explanationorjustificationsat the beginningof the teachingexperiment.He rarely
madeanycommentsvoluntarily.He frequentlydescribedhis methodas a luckyguess
andwas quickto say he couldnot see alternativesolutionsor methods.Nevertheless,
he was cooperativeand triedto do his best to answerthe interviewer'squestions.
With more experience,Bruce improvedhis abilityto explain his solutionmethod
andbeganto see the explanationas an integralpartof his problem-solvingactivity.
Bruce was chosen to participatein the teachingexperimentbecause of an interesting strategyhe used to solve ratioandproportiontasks duringa previousstudy
(Lo& Watanabe,1993a,1993b)whenhe wasin thefourthgrade.His strategy,although
not widely documentedin the existing literature,was also used by four other
fourthgradersand one second graderwe interviewed.One sampleis as follows:
Task:Yesterday,I bought28 candieswith 12 quarters.Today,if I go to the
same store with 15 quarters,how many candies can I buy? [Note: Bruce had
12 quartersand 28 Unifix cubes availablefor his use.]
Bruce's SolutionStrategy(2 June 1992)
Brucefirstdivided28 "candies"(Unifixcubes)intosevengroupsof 4. Whenasked
why, he answered, "Because 4 could be divided into 28." Then he startedto
divide the quartersinto groupsof two, countedthe numberof groups,and found
thattherewere six groupsratherthanseven groups.Then he went on to find other
ways to groupthe quarters.Whennone of those groupingswas satisfactory,he then
regroupedthe candies.
Afterseveraltries,he madefourgroupsof sevencandies.Thenhe startedto divide
12 quartersinto groupsof 4, ratherthantryingto formfourgroups.Whenit did not
seem to work,Brucefinally formedfour groupsof 3 quartersand claimedthatthe
15 as 12 and3 more,
relationshipwas "3 quartersfor 7 candies."He reconceptualized
thus adding7 to 28 to get 35 candies.
Jane-JaneLo and Tad Watanabe
219
Significanceof thisstrategy.Initially,we were surprisedto find thattherewas no
recordof sucha strategyin the existingliterature.We conjecturedtwo possiblereasons forthe lackof literatureon thisparticular
typeof solutionmethod:(a) Mostexistolder
studies
worked
with
students
who
tended
to begin computingrightaway,
ing
with or withoutanyreasonsbehindthosecomputations;(b) The availabilityof physical objectsmighthave encouragedthe manipulationof objects.In the remainderof
this article,we referto Bruce'sstrategyof findinga useful"xquartersfory candies"
relationship,thenusing this relationshipas a countableunitto find the answeras a
method.We use the word"unit"becauseBrucewas able to conratio-unit/build-up
siderthe ratio"3 quartersfor 7 candies"as if it were a unitin a whole-numbersense,
such as "3 quartersfor 7 candies, 6 quartersfor 14 candies, 9 quartersfor 21 candies,..." (Lamon,1993a, 1993b).This strategyhas two advantages.First,it avoids
any fractionalor decimalcomputationswith which Bruce could not deal comfortably at thattime. Second,it has the potentialto become a powerfulmethodfor solving all the missing-valuetasks withinthe contextsthatmake sense to Bruce.
Bruce's strategyseemedto featurea conceptmost researchersconFurthermore,
sideran importantelementof proportionconcepts,homogeneity.His gesturesindicated thathe had the intentionto "even out" the quartersand candies. Therewas
an implicit notion thata relationshipexisted between the numberof quartersand
the numberof candiesin the given condition,andthisrelationshipneededto be preserved between certainsubsets of the quartersand certainsubsets of the candies.
Thus,3 morequartersrequired7 morecandies.Not all studentswe hadinterviewed
previouslyhad this sense of homogeneity.Forthose students,therewas no conflict
in thinkingthatone groupof 2 quartersgot 6 candiesandanothergroupof 2 quarters got 4 candies when given the initial conditionof "4 quartersfor 10 candies."'
However, at thattime we did not know whetherBruce would be able to schematize this strategy.
Bruce'sratio-unit/build-up
methodwas nota schemebecauseit was notyet a deliberateandreflective activity.Accordingto von Glasersfeld(1989), Piaget's notion
of scheme consists of threeparts:(a) the child's recognitionof an experientialsituation as one thathas been experiencedbefore; (b) the specific activity the child
has come to associate with the situation;and (c) the resultthatthe child has come
to expect of the activity in the given situation.To schematizethis method,Bruce
neededto (a) articulatemathematicallythe goal of his trial-and-error-based
actions;
(b) give mathematicalmeaningto theseactions,thusmakingthe wholeprocessmore
systematic;(c) internalizehis physicalactions so thathe could execute them mentally withoutthe sensory-motoractions;and (d) generalizehis actions acrosssimilar ratio and proportionsituations.
1Therealso were studentswho questionedthe validity of buying two candiesfor half of the price on
the basis of theirown daily experience.They arguedthatthe shop might chargea higherpricefor buying a few; for example, "Onefor 35 cents and threefor a dollar."With these studentswe triedto negotiate thatthis particularstore would indeed sell half the numberof the candiesfor half the price.
220
DevelopingRatio and ProportionSchemes
RESEARCHMETHODOLOGY
In this study we operatedfrom a constructivistview of knowledge as the epistemologicalbasis for examiningchildren'smathematicalthinking.We believe that
knowledgeoriginatesin a learner'sactivityperformedon mentalconstructsthatare
directlyrelatedto the action and experienceof thatlearner(Wheatley, 1991) and
that learningoccurs when an individualadaptshis or her schemes to cope with a
problematicsituation(Steffe, 1990). Therefore,as researchers,we had two highly
relatedtasks,(a) to identifyanddevelopmathematical
tasksthatarelikelyto be problematic to individual students and have the potential to advance their current
mathematicalknowledge,and(b) to listento childrencarefullyso thatwe canunderstandtheirmathematicsandfind ways to discuss it with them (Confrey, 1991). In
the process,we were preparedto examineandrevise ourown mathematicalunderstandingas we triedto understandthe meaningindividualstudentsgave to different ratio andproportiontasks.
Duringeach teachingsession, selectedmathematicaltaskswere presentedto the
students.Tools such as paper,colored pens, rulers,and a calculatorwith fraction
operationswere availablefor theiruse. The studentswere encouragedto verbalize
and reflect on theirthinkingprocesses. All sessions were videotaped.Preliminary
analyseswereconductedaftereach sessionto developnew tasksfor subsequentsessions. At the sametime we also revisedourtentativemodelof these students'developmentof ratioandproportionschemesin light of othermathematicsconceptsthat
arose during the sessions (e.g., unit construction and coordination,arithmetic
operations,measurement,and rationalnumberconcepts).
We emphasizethatthe word teaching here means providingpotentiallearning
for studentsto constructmathematical
meaningoveran extensiveperiod
opportunities
of time. The objectives of the teachingexperimentwere to (a) identify constructive mechanisms that children use in establishing ratio and proportionknowledge, (b) studythe dynamicsof children'sconstructionsof these concepts,and (c)
identifyinstructionalactivitiesconsistentwith the children'sconstructiveprocesses
(Hunting,1983).
The dataanalyseswere heavily influencedby ourown understandingof the various concepts in the multiplicativeconceptualfield and by our knowledge of the
researchliteratureon children'sknowledge of these concepts. Neitherof these is
static. Some recentstudieshave influencedour analyses.Among the most significant areVergnaud's(1988) workon multiplicativestructures,Schwartz's(1988)
work on intensive quantity, Confrey and Smith's (1995) Splitting Conjecture,
Steffe's (1988) workon multiplicationanddivision, and Streefland's(1984, 1985,
1991) work on fraction,ratio,and proportion.
RESULTS
Throughoutthe teaching experiment, we used many different mathematical
tasks to help Bruce develop his ratio and proportionschemes. As much as possible we triedto let Bruce's actionsandexplanationsguide the choices of new tasks.
Jane-Jane Lo and Tad Watanabe
221
In the following sections, we firstpresentan overview of the teachingexperiment,
including sample activities and brief descriptions of significant events. The
overview is followed by detailedanalyses of these events.
Overviewof the TeachingExperiment
We startedthe firstsession(8 December1992)withthecandy-buyingtaskbecause
Brucehad success with it duringthe previousinvestigation.To help Brucefocus on
thenumberrelationship,
we usedtaskswithdifferentratios,progressingfromeasy ones
=
=
2:6
8:?
to
?:21) hardones (e.g.,9:12= 21:?= ?:40).Duringthissession,Bruce
(e.g.,
usedmultiplication
andmissing-multiplicand
approachesto solve taskswithratiosthat
canbe simplifiedto 1:2and1:3.Otherwise,he eitherusedtheratio-unit/build-up
method
describedabove or he guessedthe numberof candiesperquarter(e.g., I and 1/2per
quarter,I and 1/3per quarter)to solve more difficult tasks. We conjecturedthat
Brucewas tryingto avoidthe divisionoperationandfractions.We also conjectured
thatremovingthe physicalobjectsfor manipulationmighthelpBrucefocus moreon
the numberrelationshipwhentherewere difficultratios.
To test these conjectures,we continuedto use the candy-buyingtaskswith difficult ratiosbut did not providephysicalcounters(14 December 1992). In addition,
we also introducedcomparative-ratio
tasksin a pizza-sharingcontext(e.g., "Group
A has 2 pizzasand 3 girls. GroupB has 3 pizzas and4 boys") in Session 2. Without
the physical objectsfor manipulation,Bruce startedto drawcircles to help him.
Duringthis session we observedthatBruce would assign values of mixed fractions to circles symbolizingquartersor candiesbut would physicallycut lines for
circles symbolizingpizzas.This observationmadeus wonderaboutthe role of context in Bruce's solutionstrategies.Therefore,Sessions 3 (10 February1993) and4
(17 February1993) hadidenticalmissing-valuetasks,butone used a candy-buying
situationandthe otheruseda pizza-sharingsituation.TheresultconfirmedthatBruce
conceptualizedthese situationsdifferently.During these two sessions, we also
chose the numbersin the taskcarefullyto furtherstudyBruce'sconceptsof fractions.
For example,the task "12 studentsshare16 pizzas equally,so how muchpizza can
3 studentsget?"was givento see if Brucewouldnoticethefractionrelationship
between
"12 students"and "3 students"because he had used the multiple relationship
between "3 students"and "12 students"to help him solve a similar problem
before. But Bruce paid little attentionto the relationshipand solved the task with
his usual trial-and-errorapproach.
The next five sessions were moreexploratory.Based on ouranalysesof the first
four sessions, we wonderedif the multiple/fractionrelationshipmight be formed
more easily with quantities in continuous contexts than in discrete contexts.
Therefore,we introducedtwo tasks,Magic LiquidandFish Feeding,which hadthe
following basic formats:
Magic-liquidtask:"A house is 9 feet tall and has a window which is 6 feet
abovetheground.Thishousebecame18 feet tallafterapplyinga certainamount
of magic liquid.How tall would the window be above the groundafterapplying the magic liquid?"
222
Developing Ratio and ProportionSchemes
Fish-feedingtask (adaptedfromPiaget, Grize, Szeminska,& Bang, 1977):
"Fish A is 9 centimeterslong and Fish B is 4 centimeterslong. If Fish A
needs 9 pieces of food every day, how many pieces of food does Fish B need
every day?"
Notice thatin the magic-liquidtask,the heightof the window is a fractionalpart
of the house, andboth arecontinuousquantities.The idea of "magicliquids,"liquidsthatcanenlargeor shrinkeitherhorizontallyor vertically,was presentedthrough
a book Anno's Math GameIII (Anno, 1991) in Session 5 (February24, 1993).
The fish-feedingtask was presentedwith a pictureof two lines to symbolize the
lengths of the two fish. These pictureshelped to establisha linearrelationshipfor
this particulartask. In this task the length of the fish is a continuousquantity,but
the numberof food pieces is a discretequantity.These two tasks, along with the
candy-buyingtask (discretevs. discrete),were used repeatedlywith variousnumber sizes and ratiosthroughoutthe rest of the teachingexperiment.2
Two majorteachinggoals for the last six sessions graduallyemergedat the end
of the ninthsession. The firstgoal was to help Brucebuild a deeperunderstanding
of numberstructuressuch as multiplesanddivisors.This knowledgewas essential
for Bruce to develop a strategyfor identifyingthe ratiounitin his ratio-unit/buildup methodsystematically.The secondgoal was to help him constructandintegrate
his knowledge of division, which includedinterpretingthe meaningof a division
operation,recognizingtheneedfor a divisionoperation,developingmeaningsbehind
his division algorithm, and integratinghis understandingof division with his
existingknowledgeaboutnumbersandoperations.This kindof understandingwas
crucialwhenBrucetriedto solve taskswith largernumbers.We also decidedto pay
special attentionto those of Bruce's mentalimages that seemed to have an effect
on his choice of strategies.The following area few examplesof the taskswe used
to achieve these goals.
Coveringtask (adaptedfrom Reynolds, 1993): "Ihave lots of small 2-by-2
squares.How many of these small squaresdo I need to have in orderto cover
a big rectanglethatis 30 by 40 with no gaps and overlappings?"
Cuttingtask:"A wood blockis 36 centimeterslong, 48 centimeterswide, and
60 centimeterstall. I wantto cut this block into cubes of exactly the same size
withnothingleft over.Whatis the biggestsize of cubeI can cut, andhow many
cubes will I get?"
task:"Ifa rectangular
pizza(2 inchesby 10 inches)is largeenough
Pizza-sharing
to serve 4 people, how many people can be served with a pizza 5 inches by 8
inches?"
In the subsequentsections,we addressthe threeresearchquestionsthroughexamples of Bruce's experiencewith differentmathematicaltasks.
2Werecognize the artificialfeaturesof many of the tasks we used and believe that classroomadaptationof these activities would requiremuch carefulthought.
Jane-JaneLoandTadWatanabe
223
TheDevelopmentof Bruce's Unit-Ratio/Build-UpMethod
To encourageBruceto focus on the mathematicalmeaningof his actions,we first
removedthe physicalobjectsfromthe candy-buyingtask.Brucerespondedby drawing circles to representquartersandcandies(see Figure1 for an example).The fact
thatBrucecould not move these pictorialcirclesfreelyhadindeedcreatedthe need
for him to reflect on the purposeof his actions, thatis, to find the equivalentrelationshipbetweenthe numberof quartersandthe numberof candies.It also created
the need to develop a more systematictrial-and-error
approach-"one quarterfor
two candies,""twoquartersfor threecandies, "onequarterfor one and a half candies," and so on. He was more aware of the requirementof coordinatinghis
actions with the numberof quartersand the numberof candies.
Figure 1. Bruce's drawingfor the candy-buyingtask
Episode 1
The pictures were an important part of Bruce's thinking process. On 17
February1993 (the fourth session), we asked Bruce to solve the following task
without drawingcircles:
Yesterday,I bought8 candieswith 12 quarters.Today,if I go to the same store
with 9 quarters,how many candies can I buy?
Withoutpictures,Brucelost the meaningof this situation.He concludedthatthe
answerwould be 5 because 8 is 4 less than 12 and 5 is 4 less than9. This "additive
error,"which was documentedby Hart(1984) in her studywith olderstudents,had
224
Developing Ratio and ProportionSchemes
not appearedin Bruce's reasoningwhen he was allowed to use physicalobjectsor
to drawpictures.Brucelaterrealizedhis errorwhenhe triedto use the sameapproach
to answerthe question,"Howmanycandiescan six quartersbuy?"Brucethenwrote
down numerals1 to 12 in one column and 1 to 8 in anothercolumn and used the
numeralsas circles to solve the task (Figure2). Bruce's initial action of finding a
rightway to groupquarterssuggestedto us the need to find a factor(divisor)of the
total number of quarters.To coordinate with the grouping of candies, Bruce
needed to find an appropriatecommonfactor3of the total numberof quartersand
the total numberof candies. Once the ratiounit was established,Bruce could use
the coordinatedbuild-upapproachto find the solution.He could furthercurtailthis
processby the abbreviatedbuild-upapproach,which requiredan understandingof
the division operation.
Figure 2. Bruce's solution when asked not to drawcircles
Thus,we providedactivitiesthatwouldencouragehimto makesucha construction.
It took fourmore sessionsbeforewe saw any evidencethatBrucewas awareof the
idea of "commonfactor"and able to provideverbalexplanationsof his method.
Episode 2
On 7 April 1993 (theeighthsession),Brucewas askedto solve thefollowingtask:
becauseBrucewas not thinkingaboutreducingthe ratio
3Wedeliberatelyused the word"appropriate"
to the smallestintegerratio,butjust a reducedratiothatwouldhelphim achievehis goal, "avoidingcomputationsinvolving fractionsor decimals."For example,on 2 June 1993, Brucewas askedto solve the
following task:
ABC Toy Storesells 400 toy carsfor 640 dollars.(1) How muchdoes it cost to buy 160
toy cars?(2) How many toy cars can a personbuy with 128 dollars?
Brucefoundthe commonfactorof 10 between400 and640. Thenhe usedthe following
table to answerboth questions.
120
160
40
80
256
128
192
64
Jane-Jane Lo and Tad Watanabe
225
A house was 24 feet tall andhad a window thatwas 12 feet above the ground.
Thishousebecame18 feet tallaftera certainamountof magicliquidwas applied.
Howtallwouldthewindowbe abovethegroundafterthemagicliquidwasapplied?
Brucesaid"9feet"ratherquickly,yet his explanationrevealeda rathercomplicated
thoughtprocess.He explained,"BecauseI divided6 into 24, came up with 4, 6 into
18, came up with 3, so I divided somethinginto 12, and made it into fourth,and I
took the third."The interviewer asked Bruce to explain again. Bruce said, "I
divided 6 into 24, came up with 4, 6 into 18 came up with 3. I took the 3 for ... the
3 for 24 is18, 3 for 12 is 9. Nine is one half of 18. Twelve is one half of 24." Still,
the interviewerwas not sure wherethe six came from. Brucethen wrotedown the
followinglist andelaborated:"Thehousewentdownone fourth,so thewindowwent
down one fourth."
4
3
8
6
12
9
16
12
20
15
24
18
Bruce's solution of this task clearly indicatedhis intentionto find the multiplicative relationshipbetweenthe new heightof thehouseandthe old heightof thehouse,
which he knew neededto be preservedbetween the new height of the window and
the old height of the window. He identified6 as a common factorbetween 24 and
18. The number 24 was reconceptualizedas 4 sixes, and the number 18 was
reconceptualizedas 3 sixes. Then it appearedthat there was another level of
reconceptualization,similarto what Lamon (1993a) describedas "norming":the
4 sixes were reconceptualizedas "one,"but the 3 sixes were reconceptualizedas
"threefourths."Bruce's last statements,"Nineis one half of 18. Twelve is one half
of 24," did not appearto be partof his originalthinking,butratheranotherway to
explain his solution.Implicitly,Bruce seemed to be saying, "See, all these equivalent relationshipsprove thatmy answeris correct."
In summary,Bruceidentifiedthe "went-down-one-fourth"
relationshipbetween
the old height of the house and the new height of the house, which he knew
needed to be preservedbetween the old height of the window and the new height
of the window. More significantly,once he had constructedthis relationship,his
methodwould work whetherthe magic liquid enlargedor shortenedthe house.
Episode 3
Duringthe next session (21 April 1993), Bruce developed a new interpretation
of this numericalre-presentation.He was asked to solve the following task:
Fish A is 18 centimeterslong andFish B is 12 centimeterslong. If Fish B needs
60 piecesof foodeveryday,how manypiecesof foodwill FishA needeveryday?
Priorto this session, Bruce had solved similartasks eitherby his ratio-unit/buildup method or by finding how many pieces of food a 1-cm-longfish would need.
Both approacheswere still basedon trial-and-error.
Forexample,he would try 11/2
pieces of food for 1 cm, 11/3pieces of food for 1 cm, and so forth,until he found
somethingthatworked.
DevelopingRatio and ProportionSchemes
226
To solve this particular task, Bruce first did two computations,
18 x 4 = 72 and 18 x 3 = 54, on paper.We inferredthatBruce wantedto know if
a "niceratio"existed between 18 and 60. He found out throughthese two computationsthat60 was not a multipleof 18. (Brucedid not realizethat60 pieces of food
was whatFish B, but not whatFish A, needed.Nevertheless,the rest of this thinkHe thenstaredat the result
ing processwas still valid withinhis own interpretation.)
andkept silentfor about50 seconds.It appearedthathe was tryingto find an alternative approach.He then wrote down the following:
3
6
9
12
15
18
21
24
27
30
x2
60
He laterexplainedthathe madethis list by firstidentifying3 as a numberthatcould
be divided into 18 and 60 evenly, andhe knew if he wrote down all the multiples
of 3, the list wouldincludeboth 18 and60, andthatlist was whathe wantedto have.
Afterlookingat the resultfor 10 seconds,he pointedsix timesfrom3 to 18. Then
he stoppedand thought,while tappinga few times on the 18. He then wrotedown
the following row of numbersunderneaththe numbersabove:
2
4
6
8
10
12
12
14
16
18
x2
40
Bruce then announced,"Fortypieces of food."
His explanation
indicatedthathe wantedto findanotherlist of multiplesthatincluded
12 in the same position in the sequence as 18 in the originalsequence,thatis, the
sixthposition.Thenwhatevercorrespondedto 60 in the secondsequencewouldbe
the answer.He was able to come up with this list by findingthe multiplicativerelationshipbetween3 and 18, anddecidedthe samerelationshipneededto be preserved
between the numberhe was seeking and 12. This is a very complicatedsolution,
andit demonstratesa genuineattemptto coordinatethreeknownquantitiesandone
unknownquantityto preserveone relationship.At this time,Brucewas alreadyable
to take the table he created from the build-up process as a mental object and
could build the whole table (multiplesof 2 and 3) directlyfrom partof it (the relative position of 18, 60, and 12).
Episode 4
EventhoughBruceintendedto find a commonfactor,he hadnot developeda systematicway to do it, especially when the numberswere large. He also continued
to experiencedifficultyin his attemptto curtailthe build-upprocess throughmultiplicationanddivisionwhen the numberswere large.Forexample,on 5 May 1993
(the eleventh session), Bruce was given the following task:
A carof thefuturecantravel8 milesin 3 minutes.How farwill it travelin 5 hours?
Bruce picked up the pencil and wrote down "60 + 3 = 20. 20 x 5 = 100."Then
he claimedthe answerwas 100 miles. The intervieweraskedBruceto explainthe
meaningof his computation.Bruce was not able to do so. The interviewerasked
him to reconsiderhis computation.Bruce thoughta while and wrote down "20 x
8 = 160"butwas not sureif he got it right.Whenhe was askedto explainthis again,
Bruce changedto the build-upapproachand made the following table:
Jane-JaneLo and Tad Watanabe
8
16
24
32
227
40
48
56
64
72
80
12
6
15
18
21
24
27
3
9
30
He thenadded10 eightiestogetherandgot 800. Brucecommentedthatit was a long
way andappearedto be a littlerestless.The interviewerencouragedBruce.She said
thateven thoughthis methodtook a long time, it was still a good methodbecause
it was meaningfulto him. Also, Bruce was more confidentaboutthe result.After
the interviewer'scomment,Brucesaid, "Oh,I know how, I know I hadto do these"
(pointingto his originalcomputations).The interviewerencouragedBruce to follow through.Bruce said, "Twentytimes 5 is right,thenI had to time 8." The interviewer then asked, "If you computedthis first (20 x 8), then what did you need to
do next? Whatdoes 160 mean?"Bruce then realizedthat 160 was the distancefor
1 hour;thus he needed to multiply 160 by 5 to get the distancefor 5 hours.Bruce
continuedto requiresimilarprobingsto applythe abbreviated
build-upapproachduring the next threesessions.We believe thathis difficultycame largelyfromhis lack
of experiencewith givingmeaningto his multiplicationanddivisioncomputationsfor example, formingan intensive quantitylike "xpieces of food per centimeter."
Bruce also continuedto avoid using fractionsor decimals in any computations
otherthanrepeatedaddition.Nevertheless,his uneasinesswith fractionsanddecimal numbersseemed to decreaseas he "discovered"the fractionfunctionkeys on
the calculatorandfiguredout how to use those keys fromreadingthe manual.(The
role of the calculatorin Bruce's developmentof variousconcepts like multiplication, division,fractions,anddecimalnumberswas one areathatwe wished we had
had time to explore more.)
Episode 5
On 2 June 1993 (the last session), Bruce was given the following task:
FishA is 108 centimeterslong andFishB is 48 centimeterslong.If FishA needs
45 piecesof foodeveryday,how manypiecesof foodwill FishB needeveryday?
As usual, Bruce went for the calculatorimmediately.He entered108 + 48 = 2.25,
then 108 + 45 = 2.4. Again,he did not like the results,becauseneitherof thesecomputationsgave him whole-numberquotients.The interviewerencouragedBruceto
thinkaboutwhateach computationmeant.Brucethoughta while, entered45 + 2.25
into the calculator,and got an answer,20. He was quite sureabouthis answer.He
explained,"BecauseFish B is 2.25 pieces of Fish A. So he could eat 2.25 pieces
of food of Fish A." (Note:We believedthatBrucehadtherightidea,butthathe had
just mislabeledthese fish). This was the first time Bruce accepteda nonintegeras
a divisor,andhe was now muchmorearticulate
aboutthemeaningof his multiplication
and division operationseven when fractionsand decimals were involved.
Challengesto Bruce's Attemptto SchematizeHis Ratio-Unit/Build-UpMethod
From the five examples above, we could identify the following challenges
Brucehadwhen developinghis ratio-unit/build-up
method.The firstchallengewas
228
DevelopingRatio and ProportionSchemes
to distancehimselffromthe actionof doing,to reflecton the meaningof his actions,
and then finally to construct a mental object of these activities that he could
manipulate.The processtook place gradually.Brucefirstdrewpicturesof quarters
and candies to representthe physical objects (Figure 1). These pictureswere static, primarilyfor the purposeof signifyinghis actions(Figure2). At the sametime,
he also developed a numericaltable to keep trackof the build-upprocess afterhe
identifiedtheratiounit.The needfor a tablearosewhenlargenumberswereinvolved
in the tasks. Finally, the table in Episode 2 became an object.Bruce knew how to
createthe entirelist (multiplesof 4 vs. multiplesof 3) from one element of it (24
vs. 18). By creatingan appropriatetable, Bruce was able to identify the ratiounit
(whichwas less importantfor him now) andthe answerhe was seekingwithinone
framework.In the followingweek, Brucestartedto curtailthe tableprocessby using
multiplicationto replacea seriesof repeatedadditions(Episode3). His originalratiounit/build-upmethodhad takenan entirelydifferentform,which came very close
to being called a scheme. Unfortunately,we did not have enoughtime with him to
see its furtherdevelopment.
The secondchallengewas to distinguishbetweenpartitiveandquotitivedivision
andto determinewhichdivisionwas usefulfora givensituation.Whenanalyzedcarefully,Bruce'sinitialstrategyrequireda shiftbetweenquotitiveandpartitivedivision.
In his initialstrategy,he firsttriedto put all quartersinto groupsof certainamounts
of his choice (quotitivedivision),andthenhe neededto groupcandiesaccordingly
so thatthenumberof candygroupswouldbe the sameas thenumberof quartergroups
(partitivedivision).The switchfromquotitivedivisionto partitivedivisionwas difficult for him. Whenhe successfullyput 12 quartersinto fourgroupsof 3 (quotitive
division),thenextthinghe wantedto do was to put28 candiesintogroupsof 3 (quotitive division)ratherthantryingto put28 candiesintofourgroups(partitivedivision).
Withoutthe shifting,he couldstillget whathe wantedby trialanderror.We suspected
thatwas why his pictureswere so importantto him. Later,Brucedid makethe shift
successfullywhen creatinghis table (Episode3). Identifying3 as a commonfactor
of 18 and60 involveda quotitivedivision.Then,when he countedhow manysteps
between 3 and 18, he was essentiallytryingto decidehow many groupshe needed
to makewith 12, which was a partitivedivision.
When the numbersinvolved in a task were large, Bruceknew thatit would take
a long time to make his table. Thus, he createdthe thirdchallenge for himself: to
curtailthe build-upprocess.Normally,he attemptedto achieve this goal by either
making"jumps"withinthe table,as in Episode 3, or using multiplicationanddivision, as in Episode4. Brucehadless troublewiththe firstapproachbecausethe table
was meaningfulto him. When using multiplicationand division with large numbers, Bruce experiencedboth technical andconceptualdifficulties.
The technicaldifficultycame fromthe fact thatin school he hadnot learnedhow
to divide with a multidigitdivisor,how to write a quotientin fractionalformat,or
how to compute with fractions. At the beginning of the teaching experiment,
Bruce had only learned how to divide a multiple-digitnumber by a one-digit
number.He statedthathe couldnot solve a problemsuch as "1000dividedby 105"
(3 March 1993) because he had not learnedthatin school. As for fractions,Bruce
Jane-Jane Lo and Tad Watanabe
229
a fractionwitha circularorrectangular
hadlearnedthenamingof a fraction,representing
shape,and comparingunit fractionslike 1/2and 1/5.However,we soon realizedthat
these technical difficulties were not the main obstacle in curtailinghis build-up
process.Withsome help,Brucewas ableto solve computationaltaskslike 21/3x 30
by calculatingthe whole-numberpartandthefractionalpartseparately.By theend of
thefourthteachingsession(17 February1993),Brucehadlearnedin classhow to representa quotientin fractionformat(e.g., converting126 + 8 = 15 R 6 to 153/4).
Bruce'sconceptualdifficultywithcurtailingthebuild-upprocesswas muchgreater
and more complicated.Ourdatashowed thathis confusion frequentlycame from
a limitedunderstandingof divisionandmultiplicationoperations.Forexample,the
following episode, which occurredon 28 April 1993 (1 week priorto Episode 4)
sheds some light on his limited understandingof multiplicationoperation.
Episode 6
A helicopterflies 16 miles fromthe airportto a downtownhotelin 10 minutes.
At this rate, how far could the helicopterfly in 2 hours?"
Bruce solved this problemby makinga long table like the following:
10
16
20
32
30
48
...
...
110
176
120
192
He used a calculator,on which he repeatedlyentered"+ 16"each time, to help him
writedownthebottomrow.TheintervieweraskedBruceif he couldthinkof a quicker
way to do it, thinkingthathe might use multiplicationand division. Surprisingly,
Bruceanswered,"NotanythingI knowof' ratherquickly.Theinterviewerthenasked
Bruce to close his eyes, and they had the following dialogue:
Interviewer: A helicopteris flying16 milesper10 minutes.Canyoutellmehowfarit
canfly for50 minutes?
I canmultiply16times5.
Bruce:
Interviewer:
Why?
Bruce:
Interviewer:
Bruce:
Interviewer:
Because5 times10is 50.
Closeyoureyes.Howabout2 hours?
Multiplyby 120.
Yes, 2 hoursare120minutes.Whatdo youreallywantto multiply?
Brucestillwantedto multiplyby 120, even thoughtheinterviewerkeptemphasizing
that 16 miles is for 10 minutes.So, the interviewergave Brucea new piece of paper
and asked him to write down exactly what he wantedto do to solve this question.
Quickly,Brucewrotedown "120x 16 = 1920."Whenconfrontedwith the answer
he got previously, Bruce said, "Oh,I need to take away the zero" and proceeded
to cross the 0 out of 1920. However,when askedwhy he wantedto cross the 0 out,
Brucesaidfranklythathe hadno idea.The intervieweraskedBruceto close his eyes
again, and posed anotherquestion,"Sixteenmiles 10 minutes, 100 minutes...?"
230
Developing Ratio and ProportionSchemes
Bruce:
One-hundred
andsixty.Because100isjustadding0 to 10,so youadd0 to 16.
Interviewer: So howabout110minutes?
Bruce:
Interviewer:
Add 16 to 160.
Whatmultiplication
Okay,orI cando what?If I wantto use one multiplication.
can I use?
Brucewas confused.He said it's not possible to do just one multiplication.Then
the interviewerasked Bruce to open his eyes, and she wrote down "16 x 10 + 16
= 16 x ?" and asked Bruce to fill the blank.Bruce solved this task by first calculatingthe left side andgot 176. He thenused the missing-multiplicand
approachand
11
he
his
Then
that
confirmed
with
a
calculator.
the interwork;
guess
guessed
might
viewer gave Bruce a similartask, "16 x 10 +16 x 3 = 16 x ?"Again, Bruce solved
this taskwith the same approach,butthis time, he hadto guess threetimes, first 16,
then 14, then 13. Bruce still did not see any patternsin these two questions.So the
interviewerasked Bruce to close his eyes again.
Interviewer: Canyouthinkaboutstringsof beads?Everystringhas 16beads.If I have
10 strings,thenadd3 strings.
Bruce:
Interviewer:
Say thatagain?
If I have 10 stringsof beads. Each stringhas 16 beads on it. Now I add 3
morestrings.Theyall have16 on eachof them.So howmanybeadsdo I
havealtogether?
Bruce thoughta while. Suddenly,he smiled, openedhis eyes, and said, "It's 13 x
16. " The interviewerthen directedBruce's attentionto the originalquestion,and
Bruce figuredout he could multiply 16 by 12 to get 192.
as repeated
Thisandseveralpreviousinstancessuggestedto us thatthe"multiplication
addition"model made sense to Bruceonly when the multiplicandwas less than 10.
Moreover,althoughBrucecouldcarryouttwo-digitby two-digitmutliplicationwith
the learnedprocedureeasily, he was not surehow andwhy the procedureworked.
Bruce's understandingof division operations and the division algorithmwere
similarlyproceduralwhen the divisor was largerthan 10. Thus, he had to relearn
each type of multiplicationand division as the number increased. Like Steffe
(1988), we do not think "multiplicationas repeated addition"was a primitive
model for Bruce.The mentaldemandin buildingan iterableunitlargerthan 10 was
of multiplication
Brucehada gapin his understanding
great.Withoutthisconstruction,
and division.
Mathematical Knowledge Needed for Bruce to Develop His Ratio and
Proportion Schemes
The observationsdiscussedin the previoussectionsled us to concludethatthe folknowledgewas usefulfor Bruce'sdevelopmentof his ratioand
lowingmathematical
proportionschemes:(a) the structureof numbers,suchas divisorsandmultiples;(b)
anddivisionthroughmultidigitnumbers;(c) famila conceptualbasisof multiplication
of
with
a
variety multiplicationanddivision situations,includingbothquotiiarity
tive and partitivedivision; (d) meaningfulmultiplicationand division algorithms;
Jane-Jane Lo and Tad Watanabe
231
and (e) integrationof the above with the developmentof rational-number
concepts.
Anotherimportant
observationconcernedtheinfluenceof mentalimageson Bruce's
strategies (Lo & Watanabe, 1993a). For example, Bruce changed his strategy
from the ratio-unit/build-upapproachto the "how manyper one" approachwhen
we changedthe settingof the tasksfromcandiesto pizzas,even thoughall the numbers stayed the same. This suggests thatmany of the tasks encouragedthe use of
drawings,which tendedto be geometricallyrich (Streefland,personalcommunication). We were surprisedto see Bruce's difficultywith some geometricallyrich
tasks.Forexample,we used the following questionto understandBruce's concepts
of multiplicativestructure(adaptedfrom Reynolds, 1993):
I have lots of small2-by-2 squares.How manyof these smallsquaresdo I need
to cover a big rectanglethatis 30 by 40 with no gaps and overlappings?
Immediately, Bruce added 30 and 40, got 70, divided 70 by 2, and got 35,
which he claimedwas the answer.To help Brucereevaluatethe situation,the intervieweraskedhim to drawa picturefor an easiertask,"Coveran 8-by-8 squarecompletely with 2-by-2 squares"(see Figure3a). The activityof drawinghelpedBruce
form a more elaboratedimage of the situation,even thoughit was still not correct.
After being remindedthatthe big squareneeded to be completely coveredby the
small squares,Bruceredrewanotherpicture(Figure3b). In this picture,Brucefirst
drew four squaresacrossfrom the left to the right,addedthreemore down, added
threemore across from the right to the left, then addedthreemore up to complete
the outline.Finally,he drewa "cross"in the middleto cover the "hole"in the middle. Whenaskedhow manysquareshe needed,Brucecounted"one,two, three,four"
across fromthe rightto the left, then "five, six, seven, eight"fromthe rightagain.
Whenhe realizedthathe hadfoursquaresin eachrows,he recountedthe whole picture as "four,eight, twelve, sixteen."
(A)
(B)
Figure 3. Bruce's initial drawingsfor covering tasks
232
DevelopingRatio and ProportionSchemes
However,this experiencestill did not help Brucesolve the originaltask.He now
claimedtheanswershouldbe "140,"whichhe gotby adding30, 30, 40, and40 together.
He was not ableto makefurtheradjustmentwhenthe interviewerpointedout to him
thateachsmallsquarewas already2 inchesby 2 inches.Theinterviewerconjectured
thatthe confusionmightbe due to the factthatsquareshave equallengthandwidth.
So, she changedthe questionto "...2-by-3 rectangles,to cover 6-by-9 rectangle."
Brucethoughtaboutthenew question.He carefullydrewtheoutlineof therectangle
by drawing"oneunit"at a time.He thencarefullydrewthe outlineof the 2-by-3rectangle.He startedto dividethe sides into twos andthrees.He thenproceededto subdividethe whole rectangleinto 9 smallrectangles(Figure4). The interviewerasked,
"Istherea way you candecidehow manyyou will get on thisside withoutdrawing?"
Brucesaid"yes"immediatelyandexplained,"Divide2 into 6, you will get 3." The
interviewerrepeatedthe same questionon the otherside. Bruceansweredcorrectly.
Thenthe interviewersaid,"Onceyou decidedthatyou got 3 on this side and3 on the
otherside,will yoube ableto knowhow manyrectanglesyouwill need?"Brucereplied,
"Yeah.Multiply3 by 3."It was clearto us thatBrucehaddevelopeda mentalimage
of this type of taskanda procedureto solve it with confidence.We believe thatit is
very importantfor studentsto work on this type of geometricallyrich task to build
meaningfulandflexible mentalimagesthatcan facilitatethe developmentof multiplicationanddivisionconceptsat a muchdeeperlevel.
Figure 4. Bruce's drawingfor the covering task:Use 2-by-3 rectanglesto cover 6-by-9 rectangle
DISCUSSION
Throughoutthe teachingexperiment,Bruce had developedmethodsotherthan
method.Thosemethodsweresimilarto whatVergnaud(1988)
theratio-unit/build-up
identified as scalar and functional methods. However, because task variables
were notthefocus of ourstudy,we can only offerthefollowinglimitedobservations
for consideration.
Jane-JaneLo and Tad Watanabe
233
Compatible with the previous findings of Kaput and West (1994), Bruce's
choice of methodappearedto be influencedby threemajorfactors:(a) the size of
the numbers,(b) the type of ratio, and (c) the situationpresentedin each task. For
example,when askedto solve the task, "If2 quarterscan buy 6 candies,how many
candies can 8 quartersbuy," Bruce determinedthe unit price, threecandies for 1
quarter,thenmultipliedthe unitpriceby 8 to solve this task (functionalreasoning).
When he was asked to solve anothertask, "If 4 quarterscan buy 10 candies, how
many candies can 12 quartersbuy,"Bruce figuredout the multiplicativerelationship between4 quartersand 12 quarters,thenmultiplied10 candiesby 3 (scalarreasoning). Also, he used a scalarrelationshipto solve the magic-liquidtask (Episode
2) andused functionalreasoningto solve the fish-feedingtask (Episode3). When
the ratiosinvolved could not be reducedto either 1:Kor K:1 (with K being a positive integer)or the numbersinvolvedbecamelarge,Brucepreferredto use theratiounit/build-upmethod.Bruceseemedto havethe mostconfidencethatthisparticular
methodwould give him the correctanswereventually.Generallyspeaking,Bruce
became more and more flexible and effective in choosing an efficient methodfor
a particulartask.
of multiplication,diviIn this study,we identifiedBruce's limitedunderstanding
sion, and fractionand decimal concepts as the root of his difficulties in developing ratioandproportionconcepts.Do these findingsimply thatinstructionon ratio
andproportionshouldwait untila moreintegratedunderstandingof multiplication,
division, and fractionand decimalconcepts is achieved?Ouransweris "no."Our
analyses demonstratedthatratio and proportiontasks were accessible to younger
students(see also Schorn,1989;Van den Brink& Streefland,1979),andthese tasks
have the potentialto encouragestudentsto examine theirknowledge of multiplicationanddivisionandto recognizethe need for havingnonintegernumbers.Being
exposed to the routinemultiplicationanddivisiontasksalone will not help students
build a deeperunderstandingof these concepts. When asked to solve a varietyof
ratioandproportiontasks,studentswill developtheirratioandproportionconcepts
with othertopics in theirmultiplicativeconceptualfields, just as Bruce did.
The presentstudy has suggested several possible roots of students'difficulties
with ratio and proportiontasks. These findings provide additionalsupportfor the
view thatthe developmentof ratioandproportionconceptsis embeddedwithinthe
developmentof the multiplicativeconceptualfields. The observationswe reported
here suggestthatthe interrelationships
andinterdependencyof the topics withinthe
fields
are
multiplicativeconceptual
complex.Bruce'sconceptsof ratioandproportion
were influencedby his understandingof such topics as multiplicationanddivision
operations.However,his experienceswith ratioandproportiontasksalso provided
contextswithinwhichhe was ableto developmorecomplexunderstanding
of those
operations.Additionalinvestigations,longitudinalin nature,areneededto answer
many questionsraisedor left unansweredby this preliminaryresearch.
Consideringthe growingliteratureon elementaryschoolteachers'difficultieswith
divisionandrational-number
concepts(Ball, 1990;Graeber,Tirosh,& Glover,1989;
Simon, 1993),it shouldnotbe surprisingthatthesedifficultiesexistfor middleschool
or high school students,thusinterferingwith theirlearningof ratioandproportion
234
Developing Ratio and Proportion Schemes
concepts.Futureresearchstudieson ratioandproportion,or even algebra,will be
enhancedby the study of students'constructionsof those topics usually included
in the elementaryschool curriculum.It is clearthatwe need morelongitudinalstudies to examine the developmentof the multiplicativeconceptualfield as a whole.
ouranalysisquestionsthe claim thatfor a wordproblemthatcan be
Furthermore,
modeledby two quantities,the appropriate
operationto performis invariantoverthe
numbersinvolved in the problem(Greer,1994). Ourdataindicatethateven within
whole numbers,numbersthatareless than10feel differentfor Brucefromthosethat
arelargerthan10 whenhe uses multiplicationanddivision.A similarpointhasbeen
raisedby CobbandWheatley(1988) throughtheirstudyon children'sconstructions
of 10 as a unit.Greer(1994) pointedout the importanceof payingspecial attention
to the transitionfromintegersto fractionsto decimalswhenextendingthe meanings
of multiplicationanddivision. Ourstudy showed thatthe processof extendingthe
anddivisionoperationsfromsingle-digitnumbersto mulmeaningsof multiplication
and
to
then
rationalnumbers,shouldalso be takenseriously.
tidigitnumbers,
Our analysis has pointed out the importanceof providingstudentswith a variety of mathematicallyrich activitiesas the foundationfor mathematicscurriculum
andinstructionin any topic area.Most important,studentsneed to have moreexperiencewith tasksinvolving geometryandmeasurement,becausebothproviderich
contextsfor developingconceptsof numbersandoperationsat all gradelevels. But
we also wantto be clear aboutourgoal of using a varietyof mathematicaltasksto
provide opportunitiesfor students to construct their mathematics.We do not
believe thataddinga set of routineratio and proportiontasks to the mathematics
curriculum,classified by eithermathematicalor semanticstructure,is the way to
introduceratio and proportionconcepts in early gradelevels. Rather,we believe
that the instructionshould startwith situationsthatare meaningfulto students.
In this study,we did not have the time to examineBruce's constructionof fraction concepts carefully.Nevertheless, our limited data suggest that Bruce had an
intuitiveknowledgeof fractionsthatcouldbe used to dealwith a wide rangeof ratio
andproportiontaskswhen he neededto use fractions.If the goal for teachingratio
andproportionis morethanintroducingthe cross-multiplicationalgorithm,we do
not see any reasonto wait to introduceratiosandproportionsuntil afterinstruction
on fractions,as is typically done.
Throughoutthe teachingexperiment,we used manyothertasks. Because of the
scope of this article,we reportedonly those occasions where we felt Bruce made
significantchanges. But the questionremains,Whatkinds of mathematicalexperiences would help childrendevelop more sophisticatedunderstandingof proportion?Ourstudypointsto the potentialrichnessof geometryandmeasurement(e.g.,
magic-liquidtask)andthe importanceof encouragingstudentsto verbalizethe meanings of theiractions. Additionalstudies may shed new light on this issue.
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Authors
Jane-Jane Lo, Visiting Scholar,Departmentof Mathematics,CornellUniversity,Ithaca,New York,
14850-7901; e-mail:janejanel@aol.com
Tad Watanabe, AssistantProfessor,Departmentof Mathematics,Towson StateUniversity,Towson,
Maryland,21252-7097; e-mail: tad@midget.towson.edu