Understanding the Concepts of Proportion and Ratio Constructed by Two Grade Six Students
Author(s): Parmjit Singh
Source: Educational Studies in Mathematics, Vol. 43, No. 3 (2000), pp. 271-292
Published by: Springer
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PARMJITSINGH
UNDERSTANDING THE CONCEPTS OF PROPORTION AND RATIO
CONSTRUCTED BY TWO GRADE SIX STUDENTS
ABSTRACT. The purposeof this study was to constructan understandingof two grade
six students'proportionalreasoningschemes. The data from the clinical interviewsgives
insight as to the importanceof multiplicativethinking in proportionalreasoning. Two
mental operations,unitizing and iteratingplay an importantrole in student'suse of multiplicative thinking in proportiontasks. Unitizing a composite unit and iteratingit to its
referentpoint enables one to preservethe invarianceof a ratio. Proportionsinvolved the
coordinationof two numbersequences,keeping the ratiounit invariantunderthe iteration.
In the iterationprocess, one needed to explicitly conceptualizethe iterationaction of the
composite ratio unit to make sense of ratioproblemsand to have sufficientunderstanding
of the meaning of multiplicationand division and its relevance in the iterationprocess.
One needed to have constructedmultiplicativestructuresand iterationschemes in orderto
reasonproportionally.
KEY WORDS:mathematicslearning,mathematicsunderstanding,multiplicativereasoning, proportion,proportionalreasoning,ratio
BACKGROUND
Numerousstudieshave shownthatearlyadolescentsandmanyadultshave
difficulty with the basic concepts of fractions, rates and proportionand
with problemsinvolvingthese concepts(Noelting, 1980a;Vergnaud,1983;
Hart, 1988; Lesh et al., 1988; Kaputand West, 1994). Studiesof student's
thinkingindicatethatthey often use additivereasoningin solving problems
wheremultiplicativereasoningis required(Hart,1981, 1988; Karplus,Pulos and Stage, 1983b; Noelting, 1980b; Vergnaud,1988; Lamon 1993a;
Resnick and Singer, 1993). Since studentshave difficultyreasoningmultiplicatively and reasoning multiplicativelyis necessary for proportional
reasoning,it is importantto find ways to help studentsreasonproportionally.
Verylittle is known aboutthe manifestationof multiplicativestructure
in proportionalreasoning. The identificationof mathematicalideas that
contributeto proportionalreasoningappearsto be an essential,butomitted,
contributionto a domain definition. Although researchhas documented
children'sdifficultiesin learningabout ratio and proportion,results tend
#
EducationalStudiesin Mathematics 43: 271-292, 2000.
T ? 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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PARMJIT
SINGH
to be at the level of identifying lists of things that studentscan and cannot do (Hart, 1988; Robitaille, 1989; Kaputand West, 1994). We need to
move beyond the level of identifyinga litany of tasks variablesthat affect
problemdifficulty,towardthe identificationof componentsthatoffer more
explanatorypower for children'sperformancesin the domain.
There is a wide agreement(Hiebert and Behr, 1988) that if research
is to informinstruction,it is importantto analyze mathematicalstructures
and children'ssolutionprocesses in light of the developmentalprecursors
(or, sometimes, prerequisites)to the knowledge needed to function competently in a domain.These precursorsor cognitive buildingblocks have
been called by manynames:key cognitiveprocesses(HiebertandWearne,
1988), key informal strategies (Hiebert and Behr, 1988), and theorems
in action (Vergnaud,1983). They are all names for mental constructions
considerednecessaryfor meaningfullearning.
MULTIPLICATIVE SCHEMES IN PROPORTIONAL REASONING
The goal of this analysiswas to determinewhatknowledgewas criticalfor
understandingratio and proportion.The two main issues to be discussed
will be the multiplicativeschemes and the natureof students'proportional
reasoning.
Studies of multiplicativestructuresby researcherssuch as Vergnaud
(1983), Confrey(1995) and Steffe (1994) in recentyears have given multiple perspectiveson this structure.However,their theoreticalmathematical and semanticanalysesarebased on the assumptionthatthe curriculum
provideslimitedperspectiveof the multiplicativeconceptualfield andthat
understandingwill requirea broaderrangeof experiences.
Vergnaudsees the main problem in building multiplicativestructures
as consisting of building dimensionalrelationshipsin simple and multiproportionalsituationand in the extension of concepts and operationto
ever more complex situations.His approachis based on the analysis of
studentswork and teaching situation.He does not discuss explicitly the
mentalconstructionof the child or in otherwordsthe child's voice.
Confrey's approachto multiplicativestructureare based on the idea
of splitting. She defines splitting as "an action of creatingequal partsor
copies of an original..., an operationthat requires only recognition of
the type of split and the requirementthat the parts are equal" (1994, p.
294). In other words her splitting structureis an action of duplictingand
constructingsimultaneoussplits of an original.
Steffe (1988) has arguedthat the key to students'meaningfuldealings
with multiplicationis the ability to iterateabstractcomposite units. In us-
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
273
ing children'snumbersequences as a startingpoint, Steffe (1994) argues
that: "For a situation to be established as multiplicative,it is necessary
to at least coordinatetwo composite units in such a way that one of the
composite units is distributedover elements of the othercomposite units"
(p. 19). This involves taking a set as a countableunit while maintaining
the unit natureof its elements.Example,"Ifthereare 6 groupsof 4 blocks,
how manyblocks are there?".The coordinationof two numbersequences,
establishesan iterablecompositeunit. Thatis, the child counts: 1 groupis
4, 2 groupis 8, 3 is 12, 4 is 16 and5 is 20. Iteratingis a formof distributing
one quantityacrossanother.
The most distinctdifferencesbetweenConfrey'sandSteffe's approaches
are that Confrey observes multiplicativeactions as independentof addition ideas, whereas Steffe's approachconsidersearly multiplictiveacts as
making naturaluse of counting based mental structures.His concept of
multiplicationis not based entirely on counting as he sees multiplication
as the coordinationof units. That is, in a rectangulararray,an element
is simultaneouslyin a row and columns. There has to be the thinkingin
units otherthanone as well. I do not see a majorconflict between Steffe's
and Confrey's approachesto multiplicativestructure.Confrey's splitting
idea came fromher workon exponentialfunctionswhile Steffe's approach
startedfromcountingandnumbersequences.This researchis basedon the
constructthatproportionis coordinationof two ratiosandthereforetaking
this into account,Steffe's multiplicationschemes aremorerelevantfor this
study.
As childrenmove from additiveto multiplicativereasoningwith whole
numbers,there are two significantrelatedchanges. There are changes in
what the numbersare and changes in what the numbersare about. Steffe
(1988) tracedchildren'sconstructionof numbersfrom the constructionof
single entities as units to the constructionof composite and iterableunits.
It is the shiftfromoperatingwithsingletonunitsto coordinatingcomposite
unitsthatsignals the onset of multiplication.It is not a trivialshift, because
it representsa changein the meaninggiven to a number.Littleresearchhas
dealt with the centralrole of multiplicativeschemes in proportionalreasoning. I believe multiplicativereasoningis an entrypoint to the world of
ratio and proportion.In this study,I will elaboratehow students'thinking
of compositeunit schemes can develop into proportionalreasoning.
1. RESEARCH
METHODOLOGY
This study was conductedusing a constructivisttheoryof learning.It was
designedwith the intentionof focusing on a child's constructionof propor-
274
PARMJITSINGH
tional reasoning.That is, I, as the researcher,could closely monitoreach
student'sthinkingand the meaninghe or she gave to the concepts of ratio
andproportion.I set out to explorehow two studentsconstructproportional
reasoningwith the conjecturethateach child would have a uniqueway of
accomplishingthis task.
Clinical interviews
Clinical interviewswere the methodologyadoptedfor this study.The assumptionsunderlyingtheir use are: (1) A source of mathematicalknowledge can be found in the ways that studentsact to resolve situationsthat
they findto be genuinelyproblematic;and(2) Taskscan be designedwhich
providethe researcheran adequateopportunityto make inferencesbased
on the interviewsabout students'cognitive processes. Two main advantages of utilizing clinical interviews are, first, allowing for interventions
wherestudentswere encouragedto elaborateon theirstatementsandjudgements; this provides an opportunityto make valid inferences about students' covert intellectualprocesses (Opper,1975). Second, this approach
to gatheringdata provides for a continualinteractionbetween inference
and observation(Cobb, 1986b). Hence, the researchercontinually tests
conjecturesaboutthe students'thinkingandinterveneswheneverthe problem solving activity of the solver cannot be adequatelyexplainedby the
model.
The use of interviewingas a successfultool of researchmustbe accompaniedby appropriatelearningtasks. Accordingto Cobb (1986b), appropriateproblemsolving tasks are those which serve as genuine challenges
for students,allowingthemto become 'task-involved'duringthe interview.
Specifically,
deIf, for example,the child'sproblemis to inferthe behaviorsthe interviewer
sires,he orsheis engagedin socialratherthanmathematical
problemsolving.The
interviewis essentiallyworthless.Itis thereforeimperative
thatthechildconstrue
interviewtasksas challengesratherthanas opportunity
to fail andappearstupid.
Inthelanguageof achievement
thechildmustbe task-involved
rather
motivation,
thanego-involved
(Cobb,1986b,p. 99).
In order to analyze the ways learners constructproportionalreasoning
knowledge from the problem solving activity,a set of learningtasks developed by the researcherand adaptedfrom the literaturewas used. The
task situationsthatwere used in the intervieware given below:
-
A situationthatconcernsmoney (quarters)and candies.
A situationof sharingpizzas amongfriendsthatconcernsthe number
of people and the numberof pizzas.
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
-
275
A stretchersand shrinkerssituationof a rectanglethat concerns the
length of a verticalside and the length of a horizontalside.
There are threereasonsfor choosing these situations.Firstly,all threeare
typical situationsin which proportionalreasoningis used andall situations
deal with intensive quantities,relationshipbetween quantities,(e.g. candies per quarteror pizza per person) which are difficult to express by
using picture illustrations.Usually, intensive quantitiesare not measures
or counts but rather,are generatedthroughthe act of division. Secondly,
these situationsare both familiarand less familiarto students.The first
two situationsare expected to be more familiarthan the third situation.
And thirdly,the stretchersand shrinkerssituationis commonlyacceptedby
researchersas one of the most difficultproportionsituationsfor students
and in view of this, it is ideal for the investigationof students' mental
constructions.
Thestudents
Alice and Karenwere the two beginning grade six girls (11 years), who
agreedto participatein the study.Althoughthey were in differentclasses,
they had a common mathematicsteacher.
Alice was consideredby her teacherto be a very bright student,who
would surely get an A in the Nationalexam, as she had always been an A
student.
Karenwas a hardworking,responsiblegirl who alwaysgot thingsdone.
Accordingto her parents,she taughther youngerbrothermathematicsand
ensuredthat he did his school work. She was in the second from the top
class of the school for sixth graders,as classes are ability grouped.The
teacher believed that she was also good at mathematicsalthoughnot as
good as Alice.
Contentknowledgeof students
Duringmy discussionwith the schoolteacher,she said thatAlice had been
taughtthe ratiotype of questionsin a section on the topic of Money though
no terminologyof ratio or proportionalitywas used. She also stipulated
that the unit strategywas taughtto studentsas it was the easiest way to
solve this type of problem!Below areexamples,as seen fromAlice's work
sheet
Example 1
The weight of 4 anchovytins cost RM18, whatis the cost for 6 tins? (RM denotes
for Malaysiancurrency,Ringgit Malaysia)
276
PARMJIT
SINGH
The steps shown: 4 anchovy tins - RM 18
1 anchovy tin is 18/4 = 4.5
6 anchovy tins is 4.5 x 6 = RM27
Example2
Ah Chong spends RM 140 to buy 8 pairs of shoes. How much does he needs in
orderto buy 12 pairs?
Solution: 8 pairs- RM 140
1 pair-140 = RM 17.5
12 pairs17.5 x 12 = RM 210
In two other similar examples as seen from Alice's work sheet, all the
tasks used the unit method, finding the rate for one and multiplying to get
the rate for many.
It is interesting to note that, from the discussion with the teacher, Alice
had studied this method in class while Karen had not as her class was
slower than Alice's class.
The episodes below gave a descriptive analysis of the meanings that the
two students Karen and Alice gave to tasks and the conceptual advances
they made as they dealt with ratio and proportion tasks.
ANALYSISAND RESULTS
Karen's scheme's of proportional reasoning
Episode I with Karen
Karen was given the task "If 4 cents can buy 6 sweets, how many can 10
cents buy?" and below is the description of the schemes she utilized in
solving the task.
Karen tried to solve this task utilizing the formal method of unit analysis, (find the rate for one and multiplying to get the rate for many). She
divided 4 by 6 and obtained 0.66 which did not make sense to her. She
seemed confused about this outcome and abandoned it. She then used a
method which was more meaningful to her. She spent quite some time writing on her paper and used her fingers occasionally in doing calculations.
After some time working on it, she said:
("R"standsfor Researcherand "K"for Karen)
K:
R:
K:
R:
15 sweets
How did you do that?
4 cents buy 6 sweets and 8 cents buy 12 sweets and 6 cents buy 9 sweets.
How do you know 6 cents buys 9 sweets?.
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
277
K: Yousee,6 is between4 and8, so 4 centsyoubuy6 sweets,6 cents9 sweets,
8 cents12sweets,10cents15sweets,12cents18 sweets.Thereis a pattern
3,6,9, 12...
R: So, howmanysweetscanyoubuywith 10cents?
K: 15 sweets.
Initially,Kareniteratesthe compositeunit 4 cents buys 6 sweets to 8 cents
buys 12 sweets. Then she was able to constructanothercompositeunit of
6 cents buys 9 sweets as it was between the earliertwo composite units.
Karenwas then able to coordinatethe compositeratiounit of 4 cents buys
6 sweets to 6 cents buys 9 sweets, 8 buys 12 sweets and 10 cents buys 15
sweets, 12 cents buys 18 sweets. It was unclearif she could coordinatethe
two ratios because it seems that she used a patternin her build up. I was
not sure if she had constructedan invariantrelationshipin her iteration
because this is an importantcomponentin understandingproportionality.
My focus in the following episodes was to investigatewhethershe in fact
had coordinatedthe ratiounits and if she understoodinvariance.
Episode 2 with Karen
Question: To bake donutsMariahneeds 8 cups of flourto bake 14 donuts.
Using the same recipe, how many donutscan she bake with 12 cups
of flour?
In about20 seconds she said 21 withoutany recording.
R: Howdidyouget it so fast?
K: 4, 8, 12 you get 7, 14,21
R: Can you please explain?
K: 4 cups7 donuts,8 cups14donutsand12cups21 donuts
R: Howdidyouknowthat4 cupsmake7 donuts?
K: I divided8 cupsand14donutswith2 to get 4 cupsand7 donuts
She decomposedor unitizedthe composite ratio unit 8 cups to 14 donuts
to find a ratio unit of 4 cups to 7 donuts and then iteratedit to its referent
point. She simultaneouslycoordinatedtwo numbersequences4, 8, 12 with
7, 14, 21, and because of this I inferredthat she was able to preservethe
relationshipnot because of the patternbut because of the constructionof
the unit ratios4 to 7. Iterationof ratiosimplies that there are two number
sequencesof 4, 8, 12 and 7, 14, 21 thatare coordinated.Unitizingappears
to be an importantdevelopmentin proportionalreasoning.In solving problems, it is useful to view a ratioas a unit,the resultof multiplecomposition
of composite units.
278
PARMJITSINGH
Episode 3 with Karen
Question: John used exactly 15 cans to paint 18 chairs.How many chairscan he
paintwith 25 cans?
She managed to solve this problem mentally in which she derived 5 cans
to 6 chairs as equivalent to 15 cans to 18 chairs. Then she coordinated both
ratios simultaneously as 5 to 6, 10 to 12, 15 to 8, 20 to 24 and 25 to 30, and
said "with 25 cans you can paint 30 chairs". I inferred that she coordinated
the two number sequences of 5, 10, 15, 20, 25 with 6, 12, 18, 24, 30. This
again shows her ability to coordinate ratio units.
Episode 4 with Karen
Question: To bake 6 cakes, you need 15 eggs. Using the same recipe, how
many eggs do you need to bake 4 cakes?
She quickly solved it mentally and gave the answer 10 eggs.
R: Can you explain?
K: In 2, 4, 6 therearethreeand all aretimes 2. Then 15 divides 3 because 1,2,3,
got three2 cakes - 5 eggs, 4 cakes - 10 eggs, and 6 cakes you need 15 eggs.
R: How do you know thatfor 2 cakes you need 5 eggs?
K: Because I divided 6 cakes and 15 eggs by 3.
R: Why did you divide by 3?
K: Because in 2, 4, 6 thereare one, two, three,so I dividedby 3
Initially, she looked at the relationship between 6 and 4 and then said in 2,
4, 6 there are three. This was an intriguing construction because "in 2, 4, 6
there are three so I divided by 3" seems to indicate that she constructed a
scheme in which the composite unit consisted of three ratio units. It means
that:
1: 2 cakes - 5 eggs
2: 4 cakes - 10 eggs
3: 6 cakes - 15 eggs
I inferred that this shows that she was able to coordinate unit (1st ratio) of
unit (cake) of unit (eggs) simultaneously. She treated 2 to 5 as a ratio unit
and iterated it to obtain her response.
R:
K:
R:
K:
How many eggs do you need to bake 8 cakes?
20 (Veryfast)
How come you got it so fast?
Likejust now, 6 gives 15 so 8 needs 20
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
279
Using her schemes from her previous composite unit 6 cakes to 15 eggs
she was able to iterate it to the next ratio of 8 to 20. This suggests that
in iteratingfrom the composite unit of 6 gives 15, Karenknew that the
composite units are themselves composed of units and thereforedid not
have to runthroughthe iteratingactivitythatis impliedbecause the initial
segment of Karen'scompositeiteratingis symbolizedby "6 gives 15".
R: Whatif oneneededto bake100cakes?
She replied250 eggs, in seconds.
K: Theyare50 timesmore(100dividedby 2) and50 times5 equals250 eggs.
She utilizedmultiplicativethinkingin that 100 cakes is 50 times morefrom
the ratio unit of 2 cakes to 5 eggs and was able to iterateit 50 times, not
by addingbut by multiplyingit by 50. When the numbersinvolved in the
task were large, Karenknew that it would take a long time to figureit out
mentally.Thus, she multipliedratherthaniterated.
I was curious abouther statementfrom the last episode when she said
"Becausein 2, 4, 6 there are three so I divided by 3". I tried to probe her
thinkingfurther,as reflectedin the following episodes.
Episode 5 with Karen
R: Patriciauses 12 cupsof flourto make21 donuts.Usingthe samerecipe,
howmanycupsof flourareneededto make14donuts?
She did it mentally(therewas a briskmovementof threefingers)and said
8 cups. Her explanationwas "14 and 21 aretimes 7 thatmeans in 7, 14, 21
there are 3 numbers,so divides 12 by 3". Her explanationhere "in 7, 14,
21 thereare 3 numbers"indicatesthatthe compositeunit consists of three
ratiounits. She constructedthe compositeunit and iteratedthe unit as:
1: 7 d - 4 cups
2: 14 d - 8 cups
3:21 d- 12cups
I believe her finger movement played an importantrole during the 20
seconds she took in solving this problem.She used her fingersin the process of coordinatingthe two numbersequences.I believe thatshe was able
to iteratethe composite units, keeping track of each composite with her
fingerpatternuntil she reachedthe thirdratiounit. Althoughshe coordinated two numbersequences,she did not treatthe units separatelyas 7, 14,
21 and 4, 8 and 12, they were linked by the ratio unit. To be considered
280
PARMJITSINGH
additive,there would have been a pause in each finger in orderfor her to
count mentally.Again, her explanationshows thatshe was able to preserve
the relationshipof each ratio in keeping them invariant.It is significantto
note that althoughthe finger patternhelped her in unit-coordination,this
is not to be mistakenfor a perceptualfactor.In other words, she did not
always rely on her fingersto iteratethe compositeunits since she was able
to determinethe answer.
Karendid not draw any picturesor diagramsto representthe physical
objects of quartersor candies and this shows abstractnatureof her thinking. She developed a scheme for keeping the ratio unit invariantmentally
in which she was able to deal meaningfullywith compositeunits and also
able to take a ratio as a composite unit, maintainingthe ratio unit of its
element.In otherwords, she formedthe intentionof keepingthe ratiounit
invariantunderthe iteration.
My next objectivewas to see if she was consistentin utilizing her unit
coordinationscheme across differentsettings following von Glasersfeld's
(1980) theoryof scheme in which he said that one should be able to generalize her actions across similarproportionalitytasks.
Question: These two rectanglesare the same shape.Find h
3 cm
h
4 cm
16 cm
Again mentally,she iteratedthe composite unit as 4 to 3, 8 to 6, 12 to 9
and 16 to 12 and got the answer 12 cm. This coordinationof the ratios
commencedfrom the ratiounit of 4 to 3. She seemed comfortablewith her
methodas it workedwell for her and most importantlyit was meaningful
to her.
In anothersimilartask, she was looking confidentand was smiling and
after a while became quite serious. In the earliertasks, the ratio of 4 and
16 producedan integerwhile in this task, the ratio9 and 24 does not.
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
6cm
281
h
9 cm
24 cm
She was uttering something to herself but I was not able to figure it out.
After a while I asked:
R: What are you thinking?
K: This is tough
She asked if she could use paper to work on and after working on it for a
while, she said:
K: 16
R: How did you get that?
K: 9 divide by 3 is 3 and 6 divide by 3 is 2
She divided the ratio of 9 cm to 6 cm by 3 to obtain the ratio unit of 3 cm
to 2 cm.
R: Why did you divide by 3?
K: Because 9 can't go into 24
This, I believe indicates that she tried to use multiplicative reasoning as
she said, "9 can't go into 24". Her belief indicates that in problems, the
numbers should divide evenly. Then she did the following on her working
paper:
3 cm - 2 cm
6-4
9-6
12-8
15-10
18-12
21-14
24- 16
Again, she unitized the composite unit to get the ratio unit and iterated it
to 24. Unitizing the composite unit in which the ability to construct a ratio
282
PARMJITSINGH
unit and to interpretthe situationin termsof thatunit played an important
role in her proportionalreasoning.
Interpretation of Karen's activity
Karen'sstrategiesprovidedme with meaningfulinsightson multiplicative
schemes in proportionalreasoning.She had constructediterableratiounits
and was coordinatingthese units such that one ratio was distributedover
the next ratio,which Steffe (1994) maintainsis basic to the constructionof
multiplication.In the analysisof an iterableunit,RichardandCarter(1982)
stated that for childrento use ten as an iterableunit the child "mustbe
able to stripit of its compositequality"(p. 18) This was echoed by Steffe
when he said "Ibelieve this is preciselywhatis startedwhen a child makes
an iteratingunit".Utilizing a similarapproachin proportionalreasoning,
Karenwas able to unitize the units in a composite and furthermorewas
able to deal meaningfullywith composite units. In short, she was able to
take a ratioas a compositeunit and maintainthe ratiounit of its elements.
I believe that in the analysis of the iterableratio unit, the child must be
able to strip it of its composite quality. The key foundationin Karen's
meaningfuldealing with proportionalreasoningwas the ability to iterate
composite units. As shown in episode 4, she was able to coordinateunit
(1st ratio ) of unit (cake) of unit (eggs) simultaneously.She treated2 to 5
as a unit and iteratedthe ratiounitby coordinatingthe ratiosas 2 cake to 5
eggs and 4 cake to 10 eggs. This involvedthe coordinationof two number
sequences with the objective of the proportiontask, to keep the value of
the ratiounit invariantunderthe iteration.
I refer to Karen'sscheme of findinga useful 'x quartersfor y candies'
relationship,then using this relationshipas a countable unit to find the
answer a ratiounit method.I use the word 'unit' because Karenwas able
to consider the ratio '3 quartersfor 7 candies' as if it were a unit in a
whole-numbersense, such as '3 quartersfor 7 candies, 6 quartersfor 14
candies, 9 quartersfor 21 candies'. This methodhas two advantages,first
it avoids any fractionalor decimal computation,and secondly, it has the
potentialto become a powerfulmethodfor solving all missing value tasks
within the contextsthatmade sense to Karen.
I believe thatKaren'smethodwas a scheme if one goes with the definition given by von Glasersfeld.According to von Glasersfeld(1980), it
consisted of three parts:a) the experientialsituationas perceivedby the
child that he/she has experiencedbefore;b) the child's specific activityor
proceduresthe child associateswith the situationand c) the resultthatthe
child has come to expect of the activityin the given situation.To schematize this method,Karenwas able to articulatethe goals of her actions and
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
283
give mathematicalmeaningto the proceduresor reasoningshe undertook.
Secondly, she was able to internalizeher actions so that she could execute
them without pattern(sensory-motor)actions. Although I noted that she
used her fingers on several occasions, this should not be mistaken with
sensory motor actions. This is because she re-presentedpatternwith her
fingers to keep track of the number sequences without the intention of
monitoringthe countingact. Lastly,she was able to generalizeher actions
across similarproportionalitytasks.
Alice's schemesof proportionalreasoning
The following sections provides a reporton Alice's solutions of similar
tasks.
Episode I
R: Simon worked 3 hours and earned $12. How long will it take him to earn
$36?
A: 9 hours (mentally)
R: Can you explain how you got it?
A: I divided 12 by 3 and got 4, 36 dividedby 4 is 9 as 4 times (multiplied)9 is
36.
R: Why did you divide 12 by 3?
A: To get 1 hourhow much it is.
R: Verygood
She utilized the unit strategyto solve this problem and did it very efficiently in producingthe answer.I then gave her a task that I hoped would
perturbher because the unit ratio does not give a whole number.I wanted
to know whethershe could use her methodflexibly.
Episode 2
R: Chin earns $63 in 6 weeks. If he earns the same amount of money each
week, how much can he earnin 4 weeks?
She sat quietly for a long six minutes.
R: What are you thinking?
A: Can I use the paper?
R: Okay
After about4 minutes,
SINGH
PARMJIT
284
A:
R:
A:
R:
$42
Can you explain how you got that?
I divided63 by 6 and I got 10.5 and then multipliedit with 4.
If you don't find for 1 week, can you solve the problem?
After a little while (3-4 minutes)
A: I don't know.We are taughtthis way in school.
R: Whatway?
A: Find for one week and then multiply.(unit method)
She seemed very persistent with the formal method although I thought that
I had perturbed her by giving her a task where the ratio does not produce
a whole number. I believe that because I allowed her to use paper, she
continued with the method, which was successful for her.
Episode 3
In anticipating her usage of a formal method in the earlier episodes, I then
asked her to do it mentally without using paper and the task given was also
one where the ratio did not produce a whole number.
Question: To bake donuts Mariah needs 8 cups of flour to bake 14 donuts.
Using the same recipe, how many donuts can she bake with 12 cups
of flour?
After a long while (5 minutes)
R:
A:
R:
A:
What are you thinking?
I am counting
Whatare you counting?
I am tryingto divide 14 by 8... I got 1.75 and I times (multiplied)by 12
I was surprised at the amount of time she spent thinking of algorithmic
procedures i.e., on how to divide 14 by 8, in which she was successful !
She was trying to use her algorithm mentally.
R:
A:
R:
A:
Then what did you do?
53 donuts
How did you get that?
1.75 x 12 you get...350 + 175
She was mentally multiplying 1.75 by 2 (350) and 1.75 by 1 (175) She
estimated 35.0 + 17.5 = 53.
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
285
R: Can you do it any otherway, withoutfindingfor one cup?
She was quiet for some time and I tried to perturb her with the following
task.
How many donutscan you bake with 16 cups of flour?
Can I write?
Okay
28 and said 8 gives 14 so 16 gives 28,
She wrote on her paper 14 - 8 = 1.75, and then 1.75 x 16 = 28.
R:
A:
R:
A:
R:
A:
R:
A:
R:
What about 12 cups?
1.75X 12=21
Just now you told me 53
I countedit wrongly
Can you show me where you were wrong?
Since she knew that 1 cup gives 1.75 donuts, she then tried to use a method
evolving from the unit method and said:
A: For 9 cups you add 1.75 that gives 15.75 (14 + 1.75) andjust keep adding
until you get 12 cups.
She did not try to construct a relationship between 8 cups 12 donuts as
11/2 times more, which is an important criterion in multiplicative reasoning. She relied heavily on algorithmic procedures and utilized additive
reasoning.
Episode 4
Although the earlier episodes were successful in eliciting some information about her thinking in proportional reasoning, I tried new tasks with
the hope that she would give me insight into her thinking.
R: Johnused exactly 15 cans to paint 18 chairs.How many chairscan he paint
with 25 cans?I wantyou to try solving it, withoutusing yourschool method.
A: I will try
Again she insisted on using the paper and after about seven minutes
A:
810 chairs
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PARMJITSINGH
Here, she tried to carry out a procedure mentally
R: Can you explain?
She showed me the algorithmic procedures she did on her working paper.
15- 18
1617- 18 + 18 + 18= 54 cans
20 cans - 162 chairs(she got this by multiplyingit by 3)
25 cans - 810 chairs (The algorithmicprocedureson her working paper were:
162+162+162+162+162= 810)
Alice was not able to construct a composite unit consisting of 15 to 18. Her
reasoning seemed to be based on additive reasoning rather than multiplicative reasoning where each subsequent ratio is added on to the previous
one. She was not able to unite a sequence of counting acts into composite
units. Her 'adding' iteration was schematized as a ratio unit of 1 can to 18
chairs, where for each additional can, she added 18 chairs.
R: Is it possible?
A: I will do it in anotherway
She reverted to the unit strategy, 18 divided by 15 = 1.2 and then multiplied
it by 25 and got 30.
A:
R:
A:
R:
A:
30
Which is correct,810 chairsor 30 chairs?
30 is correct
What is wrong with the 810?
My calculationis wrong.
She could not show me her mistakes and just stated that 30 was the answer. Alice did not identify the invariant elements of the situation and did
not construct a ratio preserving relationship. In other words, she did not
construct a relationship between the number of cans and numbers of chairs
and furthermore she did not coordinate the units.
Episode 5
R: In baking a cake, for every 8 eggs, one needs 18 cups of flour.If Mary has
4 cups of flour,how many eggs does she need?
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
287
She used the unit method and computed 18 divided by 8 to match 1 cup
with the numberof eggs. She did it mentallyandobtained2.25 (which was
actually 1 egg with 2.25 cups).
When she multiplied 2,25 by 4 she found the result of 18 puzzling.
"Thatcan't be!" she said. Then she utilized the scalar method and computeda division of 18 by 4 which was 4.5.
She wrote:
4.5 cups with 2 eggs
4 cups with 1.5 eggs (as both cups and eggs are lesser by 1/2)
She utilizedthe additivestrategy.This seemedto indicatethatoverreliance
on the unit strategytended to confuse Alice. When she tried to reason
ratherthanfollow a procedure,she thoughtadditively.
Proportions in geometrical settings
a)
6cm
h
15 cm
When the relationshipbetween 15 and 8 was not an integer,she utilized
the additive strategy and responded that 15 - 8 = 7 and so 6 + 7 = 13 cm.
36 cm
b)
20 cm
27 cm
p
Withina minuteshe responded11 cm. She foundthe differencebetween36
-27 = 9 cm, p = 20 - 9 = 11 cm. Again, when the relationship between 36
and 27 was not an integer,she used the additivestrategy.In other words,
since 36 could not be multiplied by a number,she subtracted.Thus, it
seems that since the reason for a search for a multiplicativerelationship
was not understood,she looked for any relationshipin which the pair
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PARMJIT
SINGH
matched, and in this case it was additive reasoning. In short, when the
relationwas non-integral,she accepteda matchbased on additivereasoning.
Interpretationof Alice's activity
Alice's conceptualizationin proportionalreasoningis solely based on the
unit method, a memorizedprocedureratherthan a conceptual one. She
was able to use the unit methodto solve varioustasks to get the answers.
However,she was not able to describeher reasoningin a meaningfulway,
other than describingthe proceduresshe used. She saw mathematicsas
utilizinga taughtmethodin producinganswersratherthanmakingsense of
the activity.She was not able to thinkin termsof the compositeratiounit,
which explicitlyconceptualizesthe iterationactionof the compositeunitto
make sense of ratioproblems.I believe thatAlice's proceduralorientation
influencedher action in dealingmeaningfullywith ratioandproportion.
DISCUSSION
Different levels of multiplicationreasoningcan be seen in the strategies
utilizedby these two students.While Karenhadconstructedmultiplication
schemes in solving proportionalreasoning,Alice had not, thoughshe was
successful in obtainingsome answers.Alice's methodis proceduralrather
thanconceptual.Her orientationin proportionalreasoningis based solely
on the unit method,a memorizedprocedure.When asked to solve/reason
in anotherway (other than the unit method) she was unable to construct
an explanationfor whether her strategieswere or were not viable. Her
thinkingon some of the tasks was based on additivereasoningratherthan
multiplicativereasoningin which each subsequentratio was addedto the
previous one. She was unable to unite sequences of counting acts into
composite units. For example, her 'adding' iterationwas schematizedas
a ratio of 1 can to 18 chairs,where for each additionalcan, she added 18
chairs.
Karenon the otherhand,unitizedthe compositeunit to find a ratiounit
and then iteratedit to its referentpoint. She simultaneouslycoordinated
two numbersequences (example:5, 10, 15, 20, 25 with 6, 12, 18, 24, 30)
and was able to preservethe relationshipin the iteration.She was able to
unitize the units in a composite unit and was able to deal meaningfully
with composite units, in which she was able to take a ratioas a composite
unit while maintainingthe ratiounit of its element.
Alice's activity suggested that she was not able to think in terms of
composite ratio units, and furthermorewas not able to make a decision
UNDERSTANDINGTHE CONCEPTSOF PROPORTIONAND RATIO
289
about which unit to use, especially in utilizing the unit strategy.As mentioned earlier,her procedurehad been to use one unit on ALL occasions.
Althoughshe was successfulin obtainingthe answersutilizingthis method
on some tasks, she was not able to use it to solve a problem requiring
qualitativeproportionalreasoning, for example as in the paint problem.
It is not enough for students to recognize the change in the nature of
composite unit. In order to deal effectively with problem situation,they
must anticipate the unit structure of the situation.
Reflecting on these data, I believe that there are varying degrees of
sophisticationbetweenKaren'sandAlice's solutionsto the problemsbased
on whetherthey formedcomposite ratio units and reinterpretedthe problems in termsof those units, as well as the units they chose as some units
were more efficient than others.For example, 8 cups of flour are needed
to make 14 donuts.Using the same recipe,how many donutscan one bake
with 12 cups of flour?Conceiving the unit as 1 cup or 1 donut, does not
help to solve the problem. In contrast,thinking of 4 cups per 7 donuts
as a unit and iteratingit to its referentof 12 cups is much more efficient
than finding 1 cup as 14/8 and operatingon it. It is as if the solution process is revealedin the anticipationof an appropriateunit. In other words,
multiplicativesituationsin proportionalreasoningrequirean appropriate
schema of the unit before solutionprocedurescan be implemented.
Karen'smethodof iteratingcomposite units seems to avoid the additive strategythat Alice used. The usage of additivereasoningis possible
due to the manipulationof the scalaroperators(betweenratios)and functional operators(within ratios), which deals primarilywith searchingfor
a relationshipbetween the two ratios, utilizing the formal multiplicative
reasoning of these students.This relationshipbetween the two ratios is
somewhat similarto a trial and errormethod where studentsfirst see the
relationof functionand 'if it does not fit', they then try the scalarmethod
and 'if this does not fit' they then resort to the additive strategy.However, in Karen'sunit coordinationscheme, she coordinatedtwo composite
units where one composite unit was distributedover the ratio of the other
composite unit.
This iterationscheme is based on a simple action scheme that can underlie the solutionto proportionality.This scheme has severalfeaturesthat
are key to its nature.The firstis the seeing of the ratiounit as a set of units.
The second is the distributionof this ratiounit across another.And finally
these schemes have a recursivenature.The replicatingact is appliedto the
resultof a previousreplicatingact.
The findings also suggest that teaching studentsto use the unit ratio
strategy(findthe ratefor one and multiplyingto get the ratefor many)as a
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PARMJIT
SINGH
standardapproachto proportionalproblemsmay not help studentsdevelop
multiplicativereasoning.Thatis, in orderto deal with unit structuresmore
meaningfully,one must anticipatethe unit structureof the situation.For
example, how many donutsare needed for 24 people, if 7 donutsare for 6
people? Conceivingthe units as 1 donutor 1 persondoes not help to solve
the problemmeaningfully.In contrast,constructinga ratiounit of 6 people
with 7 donuts and then interpretinga situationof that unit may help to
understandthe natureof composite units thatchildrencan createand how
they mightreasonusing those units,whetheradditivelyor multiplicatively.
The unit analysis methodwithouta meaningfulunderstandingof multiplicativereasoningis a procedurallyorientatedmathematicaloperation
that exists independentlyof young childrenwho are to learn them, rather
than questions about the fundamentalcomposite units that children can
constructto give meaning to phrasessuch as If "8 cups of flour are used
to bake 14 donuts,how many donutscan one bake with 12 cups of flour?"
Utilizing the unit cost of algorithmproceduresis disembodiedfrom children'sinitial sense makingof proportionalreasoning.This notionmay not
help to understandthe natureof the compositeunitthatchildrencan create
and how they might reason using those units (Steffe et al., 1988). The
unit methodbecomes essentially instrumentalfor the childrenandposes a
seriousthreatto the developmentof insight(Freudenthal,1979), especially
in multiplicativereasoning.
CONCLUSION
One of the goals of this study was to investigatethe ways in which the
studentsbeganto developproportionalthinking.It is essentialto determine
what enables studentsto make a transitionfrom solving ratioproblemby
iteratingcomposite ratio units to using multiplicationand division. The
ability to use operationwith composite units seemed to involved three
essential components.First, one needs to explicitly conceptualizethe iterationaction of the compositeratiounit to make sense of ratioproblems.
Second, one needs to have sufficientunderstandingof the meaningof multiplicationand division so that one can see theirrelevancein the iteration
process. Third,and finally,one needs to have sufficientlyabstractedthe iterationprocess so one could reflecton it, and subsequentlyreconceptualize
it in termsof thatunit.
This researchindicates that the unit method should not be taught to
students until they have a good grasp of the unit coordinationschemes
(composite units, ratio units, unitizing, iteratingof ratio units and iterable composite units).This could be accomplishedby posing problems,in
UNDERSTANDING
THECONCEPTS
OFPROPORTION
ANDRATIO
291
which the constructionof unit coordinationschemes (iteratingratiounits)
is likely. The problem of teaching multiplicativereasoning is compounded by the ways in which we teach proportionsin the school setting. Too
often, we ignore the child's experiencewith ratio and proportionsoutside
of formal mathematicslessons and teach childrenalgorithmswhich utilize techniquesthat are alien to them. These techniquesmay be useful in
getting the answer for a problem, but they do not provide rich learning
opportunitiesfor childrento make sense of ratiosand proportions.
Furtherresearchis needed to elaboratehow children'scomposite unit
schemes of multiplicativestructurecan developinto proportionalthinking.
Indicationsare that many concepts within the domain of multiplicative
structureare not well taughtand thereforenot well learned.Determining
what experience might be importantto foster understandingrequires a
thoroughanalysisof the quantities(bothunitsof measureandmagnitudes)
germaneto multiplicativesituations.
ACKNOWLEDGEMENTS
This datawas collected in a school in Malaysiaas partof the researcher's
dissertationat a universityin the USA. It was funded by the Education
Ministryof Malaysia.
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