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Knowledge Connectedness in Geometry Problem SolvingAuthor(s): Michael J. Lawson and Mohan ChinnappanSource: Journal for Research in Mathematics Education, Vol. 31, No. 1 (Jan., 2000), pp. 26-43Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749818 .

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Journalfor Researchin MathematicsEducation

2000, Vol. 31, No. 1, 26-43

Knowledge Connectedness inGeometry Problem Solving

Michael J.Lawson,Flinders Universityof SouthAustralia

MohanChinnappan,UniversityofAuckland,New Zealand

Ourconcernin thisstudywastoexaminetherelationshipbetweenproblem-solvingperformance

andthequalityof theorganizationof students'knowledge.We reportfindingsontheextent towhich content and connectednessindicatorsdifferentiatedbetweengroupsof high-achieving

(HA) andlow-achieving(LA)Year 10 studentsundertakinggeometrytasks. TheHA students'

performanceon the indicatorsof knowledgeconnectednessshowedthat,comparedwiththe LA

group,theycould retrievemoreknowledgespontaneouslyandcould activatemorelinksamong

givenknowledgeschemas andrelatedinformation.Connectednessindicatorswere moreinflu-

ential than content indicatorsin differentiatingthe groups on the basis of their success in

problemsolving.The tasks usedin thestudyprovidestraightforwardways for teachersto gaininformationabouttheorganizationalqualityof students'knowledge.

Key Words:Assessment;Geometry;Knowledge;Problemsolving; Secondarymathematics;

Teachingpractice

A majoraimof mathematicseducationis to devisewaysof encouragingstudents

to take more active roles in acquiring,experimentingwith, andusing the mathe-

maticalideas andproceduresthat are included in the school curriculum.Hiebert

et al. (1996) haverecentlyinterpretedthisaim as meaningthat studentsin math-

ematicsclasses "shouldbe allowedandencouragedtoproblematizewhatthey study,to defineproblemsthat elicit their curiositiesandsense-makingskills"(p. 12). In

statementson mathematicsteaching,

teachershave been asked tohelp

studentsto

"developmultiplerepresentationsandconnections,and constructmeaningsfrom

new situations"(NationalCouncilof Teachersof Mathematics,1989, p. 125). It

is arguedthat the betterthe qualityof the students'problematizingandof their

knowledgeconnections,the morepowerfulwill be theknowledgerepresentationsthat can be calledupon duringa problem-solvingepisode.

Researchon the use of self-explanationprocessesduringthestudyof new infor-

mationin the areasof computing,mathematics,and science providessupportfor

theseviews, emphasizingthekey roleof encodingprocessesininfluencingsubse-

quent problem solving (e.g., Bielaczyc, Pirolli,&

Brown, 1995;Chi, Bassok,

Lewis,Reimann,&Glaser,1989;Chi,De Leeuw,Chiu,&LaVancher,1994;Renkl,

This researchwas supportedby grantsfrom the researchbudgets of the Queensland

Universityof TechnologyandFlindersUniversityandtheAustralianResearchCouncil.The

authorsacknowledgethe cooperationand assistanceof the staff and studentsof St Peters

College, Brisbane.The comments of reviewersof theinitial version of the articlearegrate-

fully acknowledged.



Michael J. LawsonandMohanChinnappan 27

1997).These studieshave shownthatthe moredetailed andfocused theconstruc-

tive activity generated by the students' ongoing self-explanations, the moresuccessful have been theirlaterproblem-solvingactivities.What this self-expla-nation researchdoes not provideis informationabout the organizationalstateof

theknowledgerepresentationthatcan be drawnuponduringproblemsolving.The

focus of this article is an investigationof the relationshipbetween success in

problemsolvingandthequalityof theknowledgeconnectionsthathavebeendevel-

opedby studentswhentheyhave triedto make senseof theirmathematics.Inpartic-ularwe developedproceduresto illustratethe connectednatureof students'knowl-

edge andexaminetherelationshipbetweentheextent of knowledgeconnectedness

andproblem-solvingperformance.In contemporarystudies of cognition,the connectednatureof memoryis well

established (e.g., Anderson, 1995; Derry, 1996). Although there are different

views of the basis for the connectionsin memory,thereis substantialacceptanceof the view thatmemoryis an associative structure.In Anderson's(1990) two-

concept theoryof memory, componentsof this associative structureareconcep-tualizedasvaryingin bothstatesof activationandstrength.Activationrefersto the

momentaryavailabilityof aknowledgecomponent,whereasstrengthdescribesthe

durabilityof theknowledge componentover thelong term(Anderson,1990).The

two characteristicsof a particularknowledge componentarearguedto varyinde-pendently.Knowledgecomponentsthatarein ahigh-activation,high-strengthstate

are expected to be readily accessed during problem solving. Although high-activationcomponentsshould be easily accessed, high-strengthcomponentsthat

arelow in activationmightnot be accessed so readily. Componentsthat are low

in both activationandstrengthare not likely to be accessed. Anderson'sdiscus-

sion of strengthand activationis useful here because it suggests that researchers

attemptingto gain informationabout the state of organizationof the knowledgebase need to probethe natureof the connectionsamong knowledge components

in a mannerthatprovidesopportunitiesfor the studentto access componentsthatarelow in activationlevel. If such probingdoes not occur, statementsabout the

state of connectednessof knowledge may be based on incomplete information

becauseknowledgethatis availableto the studentmayremaininert and is likelyto be regardedas missingfrom the student'sknowledgebase.

The failure of some studentsto access availableknowledge atthe appropriatetimeduringthe solutionattempthas been discussedby Bransford,Sherwood,Vye,and Rieser (1986) and by Prawat (1989), and our own work (Lawson &

Chinnappan,1994)providedanexampleof its existence in mathematicalproblem

solving.We showed thatagroupof less successfulproblemsolvers athighschoollevel failed to use a substantialbodyof theiravailableknowledge duringattemptsto solve geometryproblems,yet theycould access thatknowledgewhenpromptedto do so. Importantcomponentsof knowledge remainedinert in these students

untiltheyweregiven cues by the researcher.This failure to access relevantavail-

able knowledge was less common among the successful problem solvers we

observed.



28 KnowledgeConnectedness

We contended(Chinnappan& Lawson, 1994) that failureto access available

knowledge might arise from three aspects of students'processing activity: thestudents'dispositionalstates,thestrategicnatureof theirmemory-searchactivity,and the qualityof organizationof the knowledgerelevant to the problemsbeingconsidered.Thus,by way of illustration,access failuremightresult fromone or

more of thefollowing problems:lack of persistencewith the solutionattemptdue

to low self-efficacy, ineffective use of cues providedin theproblemstatement,or

lack of stronglyconnectedknowledgerelevantto theproblem.Inthis researchwe

set out to gatherevidence that could be used to examine the last of our three

contentions.We reportthe results of a studythat was designedto providefurther

informationaboutthe relationshipbetween indicatorsof the knowledge states ofstudents and theirproblem-solvingperformance.This relationshipis examined

through comparisonof the performanceof groups of high-achievingand low-

achievingstudents.Inthedesignof thestudywe have made a distinctionbetween

performancemeasuresthatindicatewhatproblem-relevantknowledgeis available

to the student and measures that allow us to draw inferences aboutthe state of

connectednessof thatknowledge.We referto these measuresas content indica-

torsandconnectednessindicators,respectively;we nextprovidearationalefor the

use of these indicators.

ContentIndicators

The mostcommonlyused indicatorsof the states of students'knowledgebases

are what studentsdo and what they say duringa problem-solvingepisode. The

students' writtenand verbal actionsprovideinformationabout available knowl-

edge, andclassroomteachersandresearchersuse theseactionsto make inferences

aboutknowledgestates.At all levels of education,teachers'analysesof problem-

solving behavior depend heavily on evidence gatheredfrom students' written

actions.If studentsmakeappropriate

movesintheirwrittenactions,

we makejudg-ments that"theyknow this" or "theycan use this procedure."Thesejudgments

mightbe made aboutrelativelysimpleknowledgeschemas such as a right-angleschema,for whichwe mightuse themarkingof arightangleas an indicatorof the

schema,orthey mightbe aboutmorecomplex schemas.

A student's verbal statementsduringa solution attemptcan also be used as

evidence of availableknowledge, thoughthese statementsare often not available

tothe teacher.A teacherwhois markinga student's homeworkor examinationscript

mightat times wish thatthe studentwerepresenttoexplainaparticularmove("Why

did youdo

this?")because the verbal

explanationmightreveal

somethingmore

about the student'sknowledgestate than can be identifiedin the writtenactions.

However,when mostmarkingis being done, studentsare absent.

Researcherscan moreeasily gainaccess to students'verbalactionsby requiringstudentsto talk while theysolve problems.Althoughuse of suchthink-alouddata

is not unproblematic(see Payne, 1994), these data do providea rich source of

knowledge for makingjudgmentsaboutknowledge states(e.g., "Becausethat is



MichaelJ. Lawson and MohanChinnappan 29

a right-angledtriangleandI know the lengthof side BC, I can now calculate the

lengthofAC"). Inferencesaboutcontentarealso based on students'recallorrecog-nition of particularknowledge components,either without assistance from the

teacher or in responseto a cue or hint. A studentmightbe asked to freely recall

what is known about a theoremor a proof in geometryor to identifykey terms,

partsof a diagram,orpossible solutionpaths.

Although these tasks provide informationabout the knowledge that can be

accessed by the student,they are of limited use as indicatorsof how thatknowl-

edge is organized.Usually, the recall or recognitiontasksprovideevidence onlyaboutthe student'sknowledgein a discreteformand do notrequirethe studentto

showrelationshipsbetweenaknowledgecomponentandotherrelatedcomponents.In this situationthe researcheragainlacks directandextensive evidence of how

the student'sproblem-relevantknowledge is organizedanddifferentiated.More

sensitiveprocedures,throughwhichone canexamine theorganizationalrelation-

ships among knowledge componentsand knowledge schemas, are needed. We

discuss examplesof these in the next section.

Connectedness Indicators

Arange

ofprocedures

have been usedto representthe structurednatureof

knowledge,andthese havedemonstratedthepositiverelationshipbetweenknowl-

edge organizationandproblem-solvingperformance.Deese (1962) reviewedthe

use of word-associationproceduresthathad beenused to illustratetheassociative

structureof verbalmemory.The majorinterestin this workwas to representthe

frequencyandthepatterningof verbalassociations.Patternsof responsewererepre-sentedby Deese throughuse of factoranalysis.Inthis workthere was anexplicitconcerntorepresenttheorganizationalstructureof verbalmeaning,andthesesame

procedureswere usedby others(e.g., Johnson,1965) to examine therelationship

between knowledge organizationand problem-solving performance.Johnson'sresearchhasrelevanceherebecausetheorganization-performancerelationshipwasexaminedwithin the domainof physics problemsolving and,althoughtherewassome variationin thepatternof results,word-associationperformancewas relatedto the level of problem-solvingperformance.

A numberof mappingprocedureshave also been used to representfeaturesof

knowledge organization.Concept-mappingprocedures,such as those developedby Novak andGowin (1984) andby McKeownand Beck (1990), have been usedfor thispurpose,principallyto establish the existence of, andlabels for, the links

that studentshave establishedamongknowledgecomponents.Attemptsto repre-sent the structureof these concept mapsin quantitativetermshave not been verysuccessful(Lawson, 1994).Othermappingprocedureshavemorereadilyyieldedquantitativeinformation.Shavelson(1972) used digraphsas the basis for gener-atingdistancematricesthatwere used in the analysisof changesin relatednessofstudents'cognitivestructuresfollowinginstruction.Naveh-Benjamin,McKeachie,Lin,andTucker(1986) arguedagainsttheuse of distancematricesanddeveloped




arepresentationof cognitivestructureusingan ordered-treeprocedurethatgener-

ated measuresof organization,levels of organization,andsimilarity.In all thesemapping procedures,measures of organizationwere positively associatedwith

students'achievement.

Naveh-Benjaminet al. (1986) criticizedthe use of distancematricesonthe basis

thatthey requiredthe use of measuresthatwere somewhatremovedfrom thedata

generated by the student, ignored the inert-knowledge problem, and failed to

reflectthedynamicnatureof knowledge organization.However,it is not clearthat

use of the ordered-treerepresentationprovidesan adequatesolutionfor all these

problems.InNaveh-Benjaminet al.'s procedure,the dataweregeneratedfromuse

of a set of discreteconcepts.This choice seems to limit thepotentialfor the tech-nique to provide rich informationabout the dynamic structureof knowledge,

particularlythe structureof knowledgewhile it is beingused. Inaddition,how the

ordered-treeproceduredirectly addresses the problem of inert knowledge is

unclear, inasmuch as the student's control over activation of knowledge was

removedwhenthe studentwas providedwith a list of concepts.Forthese reasons

otherindices of knowledgeconnectednesswereused in this study.

Responsetimehas a long historyof use as an indicatorof knowledgeorganiza-tion. In generalit is assumedthat the longerthe responsetime, the less strongly

related and the less accessible is the problem-specificknowledge (Anderson,1990).Inthissense,thismeasurecanbe arguedtoprovidemorespecificinformation

aboutthe state of particularknowledge componentsthanthe accessible/notacces-

sible informationprovidedby the contentindicators.Response-timeinformation

does allow for moresensitivecomparisonsbetweencharacteristicsof knowledge

components,and aresponse-timemeasurewas developedfor thisstudy.Thefocus

of this measurewas on time needed for recognitionof knowledge components

presentedin diagrams,such contextsbeing centralto the students'use of knowl-

edgeingeometry.However,on theirown,response-timemeasurescanprovideonly

an incomplete indicationof knowledge organization.Although response timesprovideinformationaboutthe ease of access to schemasand so move us beyond

absence/presencejudgments,theydo not focus directlyon the detailsof relation-

ships in particularknowledge configurations.Othermeasures can be used to

providemore informationaboutthe connectionsbeingconstructedby the learner.

Eventhoughthecontentindicatorsdescribedabovecanprovideinformationof

whatknowledgeis in a high stateof activation,they areunlikelyto indicatethe

extent of knowledge thatis not highly active, thatmight remaininert.It is this

knowledge that may be accessed by students during systematic prompting.

Campione,Brown, andFerrara(1982) developeda gradedhintingtask thatwasdesignedtoprovidea measureof whichknowledgeschemascouldbe accessedand

usedon transfertasks.We (Lawson&Chinnappan,1994)adaptedthis taskforuse

in examiningthe levels of connectednessof knowledgeschemasused in solving

simplegeometryproblems.By providingstudentswithincreasinglevels of cueing

support,one can use the hintingtask to index the level of what Mayer (1975)referredto as the internalconnectednessof a schema.A schemawithcomponents



MichaelJ. LawsonandMohanChinnappan 31

thatareeffectivelyorganizedis onefor which minimallevels of cueingarerequired

for activation.When a greaterlevel of hintingsupportis needed for access, wearguedthattheknowledgeschema is either less extensive or less well-connected.Use of this measureprovidesinformationabout the relatednessof componentsinaknowledgeschema,informationthatis notavailableusing presence/absenceindi-cators.Whenstudentsin ourstudyweresystematicallypromptedwithgradedhintsafter a solution attempt, they accessed a furthersubstantialbody of problem-relevantknowledgethat had not been accessedduringthe solutionattempt.

In ourstudyof geometry problemsolving (Lawson& Chinnappan,1994), we

developedtwo other tasks to examinerelationshipsamongknowledge schemas,

or what Mayer (1975) referredto as external connectedness. Both tasks weredesignedto examine students'use of relevantschemas.The firstrequiredstudentsto usea schematodevelopa sampleproblem,thesolutionof whichwouldcalluponuse of that schema. This taskrequiredstudentsto move beyondsimpleaccessingof a schemato embedthecompleteschemain anappropriateproblemframework.This applicationtask did not, however,requirethe studentsto directlyrelatethe

targetschemato otherrelatedschemas.Evidenceof connectionsamonggeometry-relatedschemaswasrequiredin theelaborationtask.With thistaskwe investigatedhow differenttheoremschemasthatwere relevantto a particularproblemcould

be related,one to the other. In both these external-connectednesstasks, studentswere requiredto work on their own to make connections within and amongschemas.

Therecognition,hinting,application,andelaborationtasksprovideinformationthatmoredirectlyindexesthestateof organizationof knowledgethandothecontentindicators generatedfrom observations of problem-solving and recall perfor-mance.The fourformertypesof tasks canbe usedtoprovideindicatorsof theeaseof access of differentknowledge components,of the amountof supportrequiredto facilitatethataccess, and of the internalandexternalconnectednessof knowl-

edge schemas. We contend thateffectively organizedknowledgewill be readilyaccessed andmorerichlyconnected,internallyandexternally.If this assertionis

correct,then differencesbetween less successful and more successful problemsolvers on organizationindicatorsare likely to be substantial.We designed this

studyto provideevidence relevantto a test of these arguments.Inthisstudywe comparedtheperformanceof groupsof high-andlow-achieving

studentson setsof contentandconnectednessindicators.Thecomparisonbetween

groupsdifferingin level of problem-solvingperformancewas set up to facilitatethe investigationof theinfluence of thetwo sets of indicatorson problem-solving

performance.On the basis of previous research(e.g., Lawson & Chinnappan,1994),we predictedthatthegroupswoulddifferonbothsets of indicatorsbutthatthedifferencesbetweenthemwould be greateron the connectednessindicators.




METHOD

Participants

Theparticipantswere36 Year10malestudentsfromaprivatecollege in metro-

politanBrisbane,Australia;these studentsvolunteeredto participatein the study.In this college, studentswere streamedinto differentclasses on the basis of their

performancein Year 9 and Year 10 mathematicstests. The college curriculum

requiredthatall studentscompletea topic involving trigonometryandgeometryduringYears8, 9, and10. At the time of thisstudy,all thestudentshadcompletedthis topic. High-achievingstudents(HA: n = 18) came fromthe uppertwo Year

10 streams.Thelow-achievingstudents(LA:n = 18) came fromthethreeclassesof the lower streams.

Procedure

All studentsparticipatedindividuallyintwo 60-minutesessions.Duringthefirst

session, studentswere requiredto completefourtasks:the Free Recall Task,theProblemSolvingTask,theGeometryComponentsTask,andtheHintingTask.Thestructureof these tasks andtheproceduresused to scorestudents'responseswere

the same as those we hadused earlier(Lawson

&Chinnappan,1994).

The Free

Recall Task(Recall)requiredstudentsto identifyknowngeometrytheoremsand

formulas.Studentswere askedtorecallanygeometrytheoremsthattheyknew,and

they were told thatthey could identifythe theoremsby verbal andwrittenstate-

ments orthroughuse of diagrams.If high-achievingstudentsdevelopmoreeffec-

tively organizedgeometricschemas,we shouldexpectthem to be able to retrieve

more extensive bodies of within-schemaknowledge in a free-recall situation.

Performanceon this task will not, however, isolate the reason for this outcome:

Better recall performancecould reflect the existence of either more extensive

availableknowledge

or more effective recall ofavailableknowledge.The scorefor this task was thenumberof theoremsrecalled ordemonstrated.

TheProblemSolving Task consisted of fourplanegeometryproblemsthat can

be solved by the use of theoremsandformulasthat aretaughtin the first 3 yearsof the high school mathematicscurriculum.One of the problemsis shown in

Figure 1. The taskprovideda sampleof students'problem-solvingperformance

duringwhich theiraccessingof problem-relevantknowledgewould be cuedbythe

problemstatementsandby theirown problem-solvingactions. This observation

of performancewasnecessarytoprovidean estimateof students'knowledgeacti-

vation whenthey

worked unaidedontypical problems.

A student'sperformanceon the problemswas scoredusing a 3-point scale (2, 1, or 0 points scored);the

middlescorereflectedpartialcreditfor a solutionattemptthatinvolvedappropriatemoves but was incomplete.

The GeometryComponentsTaskwas developedto examine students' knowl-

edge of partsof geometricfiguresandof thetheoremsor rulesthatarerepresented

by these figures. Studentswere shown figures related to the problemshown in



Michael J. Lawsonand MohanChinnappan 33

AE is a tangentto the circle,centreC.

AC is perpendicularto CE,andangleDCEB

has a measure of 300.

Theradiusof the circleis equalto 5 cm.

FindAB.C E

Figure1. Problem4 usedin theProblemSolvingTask.

Figure 1 and were requiredto identify the parts of the figure (Forms) and to

producea ruleor theoremthatwas illustratedby thefigure(Rules).Studentssaw

onefigure

at a time and were shown fivefigures during

thistask,

one of which is

shown in Figure 2. The rule or theorem associated with this figure was "The

tangentto a circle is perpendicularto theradiusat its pointof contact."The score

for this taskwas the numberof correctidentificationsof forms and theorems.

A

B ABCis a straightlinetouchingthecircleatB. 0 is the centre.

Figure2. Figureusedin theGeometryComponentsTask.

In theHintingTask studentswereprovidedwith a sequenceof gradedhints on

the basis of a commonly adoptedsolutionpathfor three of the problems.(Hintswere not given for the otherproblembecause the elements of thatproblemwere




used as cues in the programfor the RecognitionTask.)When studentsfailed to

producethe complete solutionfor one of these problems,they were requiredtoattemptto solve thatproblemwiththehelpof hintsgivenby theinvestigator.Eachhintwithina sequenceprovideda studentwith anincreasedlevel of assistance.The

initialhintsdrewthestudent'sattentionto apartof theproblem,andtheresearcher

waited to see if that would lead the student to generate any furtherproblem-relevant information.If studentsneeded furtherhints, they might be asked to

attendto the markspresenton specific lines in a diagram;the final hintwould bea "give-away"hintthatshoweda methodof solutionfortheproblem.The number

of hintsrequiredconstituteda student'sscore on this task.Any studentwho was

providedwiththegive-awayhintwas regardedas nothavingfunctionalaccess tothatparticularknowledgecomponent.An exampleof a sequenceof hints is givenin Table 1.

Table1Exampleof a Sequenceof Hints Used in theHintingTask

Level Hint

1. WhatdoyounoticeabouttriangleABC?

2. WhatdoyounoticeaboutlinesACandBC?3. WhatdoesthequestionstatementtellyouaboutlinesACandBC?4. LinesACandBCareof equallength.5. Whatcanyou sayabouttriangleABC?6. TriangleABCis isosceles.AnglesBACandABCareequal.

Duringthe secondsession, studentswere requiredto completethree tasks: the

RecognitionTask,theGeometryApplicationTask(Application),andtheGeometry

ElaborationTask(Elaboration).TheRecognitionTaskdevelopedfor thisstudywascomputer-presentedandcomputer-controlledand was based on HyperCardsoft-

ware that recordedthe time takenby a studentto correctly identify a particular

geometricformorrelationship.Detailsof theprogramareprovidedby Chinnappan,Lawson,andGardner(1998).A samplescreenfromtheprogramis showninFigure3. This task involved students in identifyingthe names of selected features of

geometricalformsthatweredisplayedon thecomputerscreen. Studentsindicated

recognitionof acomponentby clickingon thatcomponent,and thentheytypedin

the nameof thecomponent.Thetime takenfortypingwas notincludedin therecog-

nition time. Studentswere instructedto work quicklyandaccuratelyin makingrecognitiondecisions. Eachdisplayin thisHyperCardprogramwas developedto

representa geometricschema commonly taughtin the classroom,for example,

right-angledtriangleand its properties.Figureswith multiple componentswere

cycled throughthepresentationformat,with therecognitiontimebeingrecorded

from time of presentationof thefigureuntil the studentsignaledrecognition.Onlytimes forcorrectrecognitionswere usedin theanalysisfor this article.The scores




Inthe ElaborationTask,theinvestigatorpresentedpairsof theoremsor formulas

to the students,one pairat a time. Studentswere requiredto generatea problemthatinvolved useof boththeorems.Onepairof problemsusedin this task is shown

in Figure5. This taskwas designedto providean estimate of the extent to which

students could establish and exploit connections among related schemas. The

score forthistaskrangedfrom 0 to 4: A scoreof 1was givenfor a partiallycorrect

connection,2 for a correct "basic"connection,3 for a correctnovel connection,and a score of 4 was awardedif more than one correctnovel connection was

providedby the student.

Theorem1: Theperpendiculardrawnfromthe centre of acircleto its chord bisects the chord.

Theorem2: Pythagoras'theorem.

Figure5.Figureusedin ElaborationTask.

RESULTS

Scoresfor the two groupsof studentson each of thetasks areshown in Table2.

As was expected, given the design of the study,the groupsdifferedsignificantlyin performanceon the ProblemSolving Task.

Theperformancesof the two groupson eachof thesets of contentandconnect-

ednessindicatorswerecomparedusingseparateone-waymultivariateanalysesof

variance.Because theF valuesforbothsets of indicatorsweresignificantbeyondthe .05 level, the initial analyseswere followed up with univariatet tests of the

differencesbetween groupmeans on each indicatorwithin a set. Injudging the

significance of the individual univariatecomparisons,we made a Bonferroni

adjustment,so thatthealphalevels set forsignificancewere .017 and.012 for the

contentand connectednessindicators,respectively(Stevens, 1996, p. 160). The t

values and effect sizes for each comparisonare also shown in Table 2.

ContentIndicators

The differencebetween thegroupsforthe set of contentindicatorswas statisti-

cally significant(MultivariateF(3, 32)= 3.72,p < .03),suggesting,ingeneralterms,thatthe HA groupwas able to spontaneouslyaccess a wider rangeof problem-relevantknowledge.Theunivariatecomparisonssuggestthatit was performanceon the RulesTask thatcontributedto the multivariatesignificantdifferencefound



Michael J. Lawson and MohanChinnappan 37

Table2DescriptiveStatisticsand UnivariateTest Resultsfor All Indicators

Highachieving Lowachieving UnivariateTask (n= 18) (n= 18) t value

(Possiblescore) M SD M SD (pvalue) Effectsize

Problems 4.72 2.02 2.39 1.91 3.55(8) (.001)

Contentindicators

Recall 10.83 4.48 7.06 5.77 2.19 0.65(Open) (.036)Forms 14.17 2.23 12.78 3.59 1.39 0.39(24) (.172)Rules 4.67 0.76 3.56 1.25 3.22 0.89(5) (.003)

Connectednessindicators

Hinting 13.06 9.74 21.06 10.25 3.06 -0.78(Open) (.004)Application 20.67 5.57 16.89 3.80 2.38 0.99(25) (.024)Elaboration 8.06 4.09 4.39 2.89 3.10 1.27(12) (.004)Recognition 7.87 2.50 11.63 5.80 2.52 -0.65time (.017)(seconds)

betweenthe groupson these indicators.Comparisonof the effect sizes indicates

thatthedifferencebetweenthegroupsontheRulesTaskwasgreaterthanthateither

for theirfree recallperformanceor fortheaccuracyof recognitionof geometricalforms.

Thepatternof performanceon these taskssuggestedthatthe differencebetween

the groupsin termsof content was not simply in abilityto recognize the simple

geometricalforms thatprovide the basis of knowledge relevant to this area of

problem solving. Instead,differences between the groupswere moreapparentin

the morecomplex relationshipsrepresentedin the RulesTask.

ConnectednessIndicators

The multivariatetest of differencebetweenthegroupsonthe connectednessindi-

catorswas alsosignificant(MultivariateF(4, 31) = 4.52,p < .01).Weinterpretthis

differenceas pointingto a superiorityin organizationof theknowledgeof theHAgroup.This superiorityin organizationwas reflected in relativeperformanceson

each of the indicatorsdesignedto reflect the facility andextentof connectedness

of the students'knowledgebases relevantto this area of geometricalknowledge.The effect sizes relatedto the comparisonsof thegroupson theseindicatorswere

generallylargerthanthoseforthe contentindicators.Whentheindividualunivariate

comparisonswere considered,the t values for both the Hintingand Elaboration




comparisonswere significant at the adjustedalpha level. The t value for the

Recognitioncomparisonwas slightlyoutsidethisadjustedalphalevel. Ineach casethe HA groupperformancereflected use of a knowledge base that was charac-

terizedbybetterqualityknowledgeconnections.The HAgrouprequiredless assis-

tance in the form of graded hints to access relevant knowledge. This group

requirednotonly fewer hintsto access suchknowledgebut also fewergive-awayhints (HA: 0.2 hints;LA: 1.9 hints), thoughthis difference was not statistically

significant(t(34) = -1.96, p > .05).TheHA studentsalsoshowedgreaterevidenceof externalconnectednessamong

schemas in the ElaborationTask. Theirperformanceon the RecognitionTask

suggestedthattheymightalsobe able to morequicklyactivateknowledgecompo-nentsthatwere relevantto theselected areaof problemsolving.The HAgroupnot

only had lowermeanrecognitiontimes butalso madea greaternumberof correct

recognitions.

DiscriminantAnalysis

A differentperspectiveontheinfluenceof thetwo sets of indicatorsontheperfor-manceof thegroupscan be gainedthroughuseof descriptivediscriminantanalysis.Inthis case the

purposeof the

analysiswas to

gaininformationaboutwhich indi-

cators were most importantin predicting the membershipof the two groupsobservedin this study.In particularthe focus of interesthere was in the relative

contributiontopredictionsof groupmembershipof the contentandconnectedness

indicators.Forthisanalysisall the contentand connectednessindicatorswereused

as predictorsand were enteredinto a directdiscriminant-analysisprocedureusingthe SPSS Discriminantprogram.Because theuse of the seven indicatorswiththe

availablesamplesize is close to the minimumrecommendedcase/variableratiofor

discriminantanalysis, the results should be takenonly as suggestive of possible

strengthsof influence of the indicators.Discriminantanalysis producedone significantdiscriminantfunction(Wilks'sLambda= 0.57,X2(7,N= 36) = 16.95,p < .02).The structurecoefficients(discrim-inantloadings)andstandardizedweightsforeach variableareshowninTable3. The

structurecoefficientsarecorrelationsbetweenthe variablesandthediscriminantfunc-

tion,similarinnatureto factorloadingsinfactoranalysis,and aregenerallyfavored

asindicatingthecontributionof each variabletothe discriminantfunction(Stevens,

1996). Thompson(1998) arguedthat both structurecoefficientsand standardized

weightsmust be inspectedto makeajudgmentaboutthecontributionof a variable

to the discriminantfunctionbecause a standardizedweightnearzero (see weightsforHintingandRecall)does notnecessarilyindicatethata variableis unimportant.Theresultsin Table3 suggestthateachof thepredictorsmadeacontributionindiffer-

entiatingthe two groupsof studentson thebasis of theirproblem-solvingperfor-mances,althoughthecontributionof theFormsscorewas lowestinthisanalysis.Apartfromthe Rulesscore,the connectednessindicatorscontributemorestronglytosepa-rationof thegroupsthando theremainingcontentindicators.



Michael J. LawsonandMohanChinnappan 39

Table3Discriminant

AnalysisCoefficients Discriminantanalysiscoefficients

Indicator Structurecoefficient StandardizedweightRules .64 .71Elaboration .62 .54Hinting -.61 -.02Recognition -.50 -.58Application .47 .15Recall .44 -.01Forms .28 -.48

CONCLUSIONAND DISCUSSION

Ourconcernin this studywas to develop a detailedpictureof the natureof the

knowledge representationsand connections that are developed by studentsas a

result of study in the areaof geometry. Such a descriptionshould be of use to

teachersand researcherswho are seeking furtherunderstandingof the reasons

behindeffective, andineffective,problem-solvingperformance.Thethrustof our

approachwas to seek to examinetherelationshipbetweenproblem-solvingperfor-mance andthe qualityof the organizationof students'knowledge. To studythis

relationship,we investigated the influence of a range of measures of content

knowledge and knowledge organizationon students' problem-solving perfor-mance. We were interestedin examiningtheextent to whichthese sets of content

andconnectednessindicatorswoulddifferentiatebetweengroupsof studentswho

differedin levels of achievementin high school geometry.Of particularinterest

was investigationof the predictionthat the groups would be differentiatedbystudents'performanceson the tasks designed to provide informationabout the

qualityof knowledge organization.In bothfree-recallandpromptedsituations,theHA studentswere able to access

a widerbodyof knowledgeof geometryfactsandtheoremsthanthe LAgroup.The

groupsdid not differ in theirrecognitionof geometricforms,but they did differ

significantlyintheirspontaneousaccessingof geometricrules.The lowerfrequencyof give-awayhintsprovidedtotheHAgroupsuggeststhat,relativetothe LAgroup,these studentshada widerbodyof problem-relevantknowledgeto callupon.This

resultwas also foundin thecomparisonof HA andLA groupsin ourearlierwork

(Lawson& Chinnappan,1994).

However, other resultssuggest that this difference in access was not the resultof just a differencein the extent of knowledgeavailableto thestudentsin the two

groups.The resultsof theHintingTask showed thattheLA studentsrequiredmore

assistanceto access relevantknowledgethathadnotbeenaccessedspontaneously.Consistentwith ourearlierfindings(Lawson& Chinnappan,1994),the LA group

appearedto have knowledge available that was not accessed until they were

providedwith cues thatfacilitatedmemorysearch.Thispatternof performanceis,




we argue,indicativeof aless effectively organizedandless well-managedknowl-

edge base. A reviewerof this articlearguedthat the differencein Hintingperfor-manceis a biasedindicatorinasmuchas the HA studentswerehighly likelytoneed

fewer hintsthan the LA studentsbecauseof their betterproblem-solvingperfor-mance.Althoughthis bias exists, the results of the HintingTask are not without

value. Use of this task allowed us to make two importantjudgmentsabout the

students.First,we couldprovideevidence thatsupportedtheexpectationthatthe

more successfulstudentswould sufferless from theproblemof inertknowledge.Second, and moreimportant,by probingsystematicallyfor students'knowledgein this HintingTask,we were able to makereasonableclaims about the quantum

of knowledge available to the studentsin this area of geometry. Without thisevidence,derivedfromthefrequencywith whichgive-awayhintswereneeded,we

would have less justificationfor claimingthat the differencebetweenthe groupswas not simplya matterof quantityof knowledge.

Theresponse-timedataprovidedfurtherevidencesuggestiveof the more effec-

tiveorganizationalstateof theknowledgebases of theHA students.These students

were abletocorrectlyrecognizerelevantknowledgecomponentsmorequickly.This

findingalso suggeststhat some featureof the stateof organizationof theirgeom-

etryknowledge, possibly strength,allowed morerapidaccess to this knowledge.

Othermeasuressupportthe view thatthismorerapidrecognitionis associatedwithmore effective connectednesswithin andamong knowledgeschemas.

The resultsof theApplicationandElaborationTasksaddressthe issue of knowl-

edge organizationmoredirectly.In these cases it is not so much the influence of

searchproceduresas the stateof connectednessof knowledgethatis of concern.

We contendthatstudentswithhigh scores on thesetasksshow evidence of beingable to activate wider networksof geometryknowledge.Theperformanceof the

HA studentsindicates thatthey had a richer set of connectionsamong schemas

relatedto this area of geometry,a findingthatin terms discussedearliersuggests

the presenceof more linksamongrelatedknowledgecomponents.The differencesinpatternsof connectionwithin andamongschemasforthe HA

and LA groupsthat we have attemptedto demonstrateheremightalso be related

to the diagram-configurationmodel of geometry-theoremprovingdeveloped by

Koedinger and Anderson (1990),1 who argued that expert geometry-problemsolversorganizetheirgeometryknowledgein clustersof facts"thatare associated

with a single prototypical geometric image" (p. 518). Although we have not

attemptedto developa formalmodelof specific schemas,theprocedureswe have

used in this studyseem to be ones that could access problemsolvers' geometric

"perceptualchunks."Ourresultssuggestthattheperceptualchunksof theHAgroupareof betterqualitythanthoseof the LA students.

The resultsof this studyprovidefurtherinformationaboutwhy high-achievingstudentsare abletoproducebettersolutionoutcomesthanlow-achievingstudents.

1Ananonymousreviewerof this articledrewtheworkof KoedingerandAnderson(1990)to ouratten-

tion.



Michael J. Lawsonand MohanChinnappan 41

We have arguedthat,amongotherfactors,theorganizationalqualityof students'

geometric knowledge is associated with better problem-solving performance.Superiorsolution attemptsof high-achievingstudentsappearto be drivenby a

geometric-knowledgebase thatis more extensive and better structuredthan that

of low-achieving students. The HA students'performanceon the indicatorsof

knowledge organizationused in this study showed that,comparedwith the LA

group,they (a)were able to retrievemoreknowledge spontaneouslyand(b)could

activate or establish more links among given knowledge schemas and related

information.Thus,the resultsof thisstudysuggestthatsuccessfulproblem-solving

performanceis associated with a knowledge base that is betterorganized and

more extended, supportingthe views expressed by Prawat(1989) and Larkin(1979).

The researchreportedhere has two importantimplicationsfor theway teachers

of high school mathematicsteach andassess students'understandingin the area

of geometry.First,our results showed thattheorganizationalqualityof geometry

knowledgeconstructedby the high achievers confers on them an advantageover

the low achieversin the solutionof problems.Thechallengeformathematicseduca-

tors and classroom teachers is to devise strategies for helping all students to

improvethe state of connectedness of theirknowledgebases, butparticularlyto

assist the less effectiveproblemsolverstoexploitmore of theknowledgetheyhaveacquired.More effective connectionsareimportantboth withinspecific schemas

andamongrelated schemas. In the termsdiscussedby Koedingerand Anderson

(1990), betterqualityconnections allow studentsto "thinkat a largergrainsize"

(p. 547). Given the active, constructivenature of students'study practices,we

believe thatclassroominstructiontime should be allocatedto displayanddiscus-

sion of the schemas that studentsdevelop for topics within their mathematics

programs.The findings of the present study suggest that less effective problemsolversmightneed extratimeand discussionto setupthetypesof connectionsand

representationsthat lead to effective accessingof knowledge.A second majorimplicationof this studyconcernsassessmentof school math-

ematics,especiallygeometry.Ina recentarticleon assessment,Senk,Beckmann,andThompson(1997) found thathigh school teacherstended to assess students'

understandingsfrom a narrowbase of standardizedtests andarguedfor the needto use moreopen-endedtasks.The tasks thatwe have developedand used in this

study,especiallytheElaborationandApplicationTasks,appeartoprovidea wider

andpossibly moreproductiveenvironmentin which studentscould displaytheir

geometricalknowledge.These tasksrequirethestudenttoretrieveanduseconnec-

tions thatmightnot be activatedin otherways. With access to this information,theteacheris likely to have a broaderpictureof the state of a student'sknowledgeon which to base decisionsaboutany difficultybeing experiencedby the student.

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Authors

Michael Lawson, Associate Professor,School of Education,FlindersUniversity,GPO Box 2100,Adelaide5001, Australia;[email protected]

Mohan Chinnappan,Lecturer,MathematicsEducationUnit,Departmentof Mathematics,Universityof Auckland,Auckland,New Zealand;[email protected]

geometry problem solving 2000

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