CHILDRENS SOLUTIONS TO MULTIPLICATION AND DIVISION PDF

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CHILDREN'SSOLUTIONSTOMULTIPLICATIONAND

DIVISIONWORDPROBLEMS:ALONGITUDINALSTUDY

JoanneMulligan,MacquarieUniversity

70children

fromYear problemstructures,five formultiplicationandfivefordivision, wereclassifiedonthebasis ofdifferencesinsemanticstructure.

Therelationshipbetweenproblemcondition(i

e.smallorlarge numbercombinationsanduse ofphysicalobjectsorpictures),on

Theresultsindicatedthat

75%ofthechildrenwereableto

solvetheproblemsusingawidevariety ofstrategies eventhough generallyincreased foreachinterviewstage,butfewdifferences forCartesian andFactorproblems. anddivisionproblemsatthreelevels: (i)directmodellingwithcounting; strategies; counting-all,skipcounting anddoublecounting.Analysisof

1981;Lesh&Landau,

DivisionWordProblems25

alsobeenimportantresearchdevelopments inyoungchildren'sacquisitionof

Glasersfe1d,Richards

&Cobb,1983;Steffe,Cobb&Richards,1988),

1982;Carpenter&

(Hunting,

1989)areexamplesofthisresearch.

largelyinfluencedbytheconstructivistview oflearning.Whilethe constructivistmovement knowledge definitions knowledge constructiveprocessesfirsthand(Steffeet aI.,1983)oftenthroughtheuseof theconstnlctivist by

InformalandFormalStrategies

toinstruction (Carpenter,Hiebert&Moser,

1981;Fuson,1982;Gelman&Gallistel,1978;

Deri,Nello

instructionitcannot ronnalmathematicalideas,orthattheir ownstrategiesmatchthoseencouraged byinstruction.

Itappearsthen,that

meaninglessones(Hiebert, theystoppedanalysing theadditionandsubtractionproblemstheyhad previouslybeenable tosolve.

AdditionandSubtractionWordProblems

conceptsandprocesses inthepastdecadehasfocussedonanalysing children'ssolutionstrategies

26Mulligan

Greeno&Heller,

asmodellingandcounting, thatreflectedthesemanticstructure oftheproblem(Carpenter&Moser,1984; toamorecoherentpicture usingacognitiveapproachhasbeenadvanced. ofthe al.,1983).Thedevelopment ofthesemodelshasindicatedthatthereare butthese tennsofthe processes.

MultiplicationandDivisionWordProblems

1989;DeCorte,

Verschaffel&VanCoillie,

1988).Earlier,theConceptsinSecondary

1977;Brown,

Attempts

ofquantitiesused,and pupil'sintuitivemodels(Belletal.,

1989;FischbeinetaL,1985;Nesher,

1988;Schwartz,

problems(Anghileri,

1984;Kouba,1989;

Steffe,1988).Thesestudieshaveprovided

development process. ofhow

DivisionWordProblems27

problemstructuresforyoungchildren, abstractness,. andstrategyuse,and (iv)providesevidence division.

Methodology

(1984)longitudinalstudy of35children

2-yearperiod,anda

studybecauseitallowedtheresearcher todirectlyexaminesolutionstrategies andhowthesechangedovera2-yearperiod.

Sample

8 ability,asindicatedbythe

ACERPrimaryReadingSurveyTest,were

Procedures

instructionin ofthefinalinterviewall facts.

Subjectswereinterviewedbytheresearcher

inaroomseparatefromthe Each

Eachinterviewlastedfrom

15to55minutes.

28Mulligan

Partition

theproblemwaspresented tothechild.Theseproblemswereselectedfor provideamodeltoassistthechild infindingasolution.Responseswere recordedon andthechild'sbehaviourwerealsonoted.

TableI

WordProblems(SmallNumbers)

MultiplicationDivision

RepeatedAddition

(a)Thereare2tablesinthe classroomand4childrenare seatedateachtable.Howmany childrenaretherealtogether? (b)Peterhad2drinksatlunchtime everyday for3days.How manydrinksdidhehave altogether? (c)Ihavethree5cpieces.How muchmoneydoIhave? Rate

Ifyouneed5ctobuyonesticker

howmuchmoney doyouneed tobuytwostickers?

Factor

Johnhas3booksandSuehas4

timesasmany.Howmany booksdoesSuehave? Array

Thereare4linesofchildren

with3childrenineachline.

Howmanychildrenarethere

altogether?

CanesianProduct

Youcanbuychickenchipsor

plainchipsinsmall,mediumor largepackets.Howmany differentchoicescanyoumake?

Partition(Sharing)

(a)Thereare8childrenand2tables intheclassroom.Howmany childrenareseatedateachtable? (b)6drinksweresharedequally between3children.Howmany drinksdidtheyhaveeach? Rate

Peterbought4lollieswith20c.

Ifeachlollycostthesameprice

howmuchdid onelollycost?

Howmuchdid2lolliescost?

Factor

Simonehas9booksandthisis

3timesasmanyasLisa.How

manybooksdoesLisahave?

Quotition

(a)Thereare16childrenand2 childrenareseatedateachtable.

Howmanytablesarethere?

(b)

12toysaresharedequally

betweenthechildren.Ifthey eachhad3toys,how many childrenwerethere?

Sub-division

Ihave3applestobeshared

evenlybetweensixpeople.

Howmuchapplewilleach

personget?

ProblemStructure

types(Table (Anghileri,

DivisionWordProblems29

foundinsolutionstrategies inthepilotstudy.Therewere14smallnumber problemsand lliargenumberproblemsaskedintotal.

ProblemCondition

only. for performanceandstrategyusecould beattributedtootherfactors.

Results

that accordingtothedifficulty oftheproblemstructureandsizeofnumber problems. in of childrenwereunabletosolvetwoormore oftheeasiest11smallnumber problemsatanyinterviewstage.Many ofthesechildrenreliedonimmature ofthenumbersto chooseanoperation, range &Moser(1984).

InterviewStages

additive

30Mulligan

exclusivelyfordivision. ormoreofthe typerepresents morethan50% order tohere.Most structure orrelationshipdescribed intheproblem.1

Table2

NumberSize:InterviewsIto4

PROBLEMSMALLNO.LARGENO.PICTURESTRUCTURE

IntclViews

IntelVicwsIntelVicws

12342342 34

MULTIPLICATION

Repeated(a)5077 7992274554805020208

Addition(b)5174849527526865

Rate72828998

Factor112944570163547

Array46778492397076 783916115

Cartesian318210

DIVISION

Partition(a)6669747523332955

(Sharing)(b)61808197346464831414 142

Rate51546685

Factor43 61700010

Quotition(a)3458558526364472

(b)4764699334455073

Subdivision4160738210233543

DivisionWordProblems31

Whilesomeconsistencyin

primarystrategiesisfoundacrossinterview stages,theuse

RepeatedAddition

solvedwithasimpleadditionfact.

Table3

MultiplicationSmallNumberProblems

Repeated(a)

Addition

counting-alJb counting-alladditionfact rep.additionmulti.factskipcounting additionfactadditionfactadditionfact addition skipcountingskipCOW1tingfact multi.fact multi.fact skipcountingmulti.factskipcounting counting-allskipcOW1tingcounting-all totalcorrectstrategies bItalics typeindicatesDirectModelling here,hencethepredominance oftheadditionfact.Thestrongemergenceof

32Mulligan

strategiesforotherproblemstructures. to

Table4

SmallNumberProblems

Quotition(a)

(b) doublecountskipcounting skipcounting doublecountdoublecountmulti.fact skipcounting

Sub-division

halving halving halvinghalving total·correctstrategies bItalicsindicatesDirectModelling

DivisionWordProblems33

facts strategiesbased fingers; strategiesbased modelling; and (ii) modellingidentified as: (i) (iii) levels wherechildrenfonnedequivalentsetsrepresentingthequantity given inthe andnumber

Levellbutwereidentified

describingtheirvisualisation ofthemodeloftheproblem.Thisshowedmuch moreadvancedmentalprocessing. inthegroupand thenfannedgroupsofequalsize.

Inmostcases,counting-all(dividend),skip

sharingone-by-onewasrarelyused.

Ifchildrenwereunsuccessfulintheir

fanned problemsbecause largerdividends.Children

Quotitionproblemsandrelied

multiplicationfacts stage 4. anduseofknownfacts,werefound tobeanalogoustotheadditionand subtractionstudy(CarpenterandMoser,

1984).Theuseofadditiveand

subtractivestrategies,revealed inbothmultiplicationanddivisionproblem

34Mulligan

quotesdbs_dbs47.pdfusesText_47

[PDF] CHILDRENS SOLUTIONS TO MULTIPLICATION AND DIVISION

CHILDREN'SSOLUTIONSTOMULTIPLICATIONAND

DIVISIONWORDPROBLEMS:ALONGITUDINALSTUDY

JoanneMulligan,MacquarieUniversity

70children

Therelationshipbetweenproblemcondition(i

Theresultsindicatedthat

75%ofthechildrenwereableto

1981;Lesh&Landau,

DivisionWordProblems25

Glasersfe1d,Richards

1982;Carpenter&

1989)areexamplesofthisresearch.

InformalandFormalStrategies

1981;Fuson,1982;Gelman&Gallistel,1978;

Deri,Nello

Itappearsthen,that

AdditionandSubtractionWordProblems

26Mulligan

Greeno&Heller,

MultiplicationandDivisionWordProblems

1989;DeCorte,

Verschaffel&VanCoillie,

1988).Earlier,theConceptsinSecondary

1977;Brown,

Attempts

1989;FischbeinetaL,1985;Nesher,

1988;Schwartz,

1984;Kouba,1989;

Steffe,1988).Thesestudieshaveprovided

DivisionWordProblems27

Methodology

2-yearperiod,anda

Sample

ACERPrimaryReadingSurveyTest,were

Procedures

Subjectswereinterviewedbytheresearcher

Eachinterviewlastedfrom

15to55minutes.

28Mulligan

Partition

TableI

WordProblems(SmallNumbers)

MultiplicationDivision

RepeatedAddition

Ifyouneed5ctobuyonesticker

Factor

Johnhas3booksandSuehas4

Thereare4linesofchildren

Howmanychildrenarethere

CanesianProduct

Youcanbuychickenchipsor

Partition(Sharing)

Peterbought4lollieswith20c.

Ifeachlollycostthesameprice

Howmuchdid2lolliescost?

Factor

Simonehas9booksandthisis

3timesasmanyasLisa.How

Quotition

Howmanytablesarethere?

12toysaresharedequally

Sub-division

Ihave3applestobeshared

Howmuchapplewilleach

ProblemStructure

DivisionWordProblems29

ProblemCondition

Results

InterviewStages

30Mulligan

Table2

NumberSize:InterviewsIto4

PROBLEMSMALLNO.LARGENO.PICTURESTRUCTURE

IntclViews

IntelVicwsIntelVicws

12342342 34

MULTIPLICATION

Repeated(a)5077 7992274554805020208

Addition(b)5174849527526865