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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.
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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.
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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
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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
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30 KnowledgeConnectedness
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
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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.
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32 KnowledgeConnectedness
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
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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
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34 KnowledgeConnectedness
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
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36 KnowledgeConnectedness
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
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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
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38 KnowledgeConnectedness
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.
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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,
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40 KnowledgeConnectedness
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.
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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]