middle school student analogy process in solving algebra

IlkogretimOnline-ElementaryEducationOnline,2021;20(3):pp.128-139http://ilkogretim-online.orgdoi:10.17051/ilkonline.2021.03.13

Middle School Student Analogy Process in Solving AlgebraProblems based on Example Questions in terms of StudentsDivergentThinkingAbility

Dwi Priyo Utomo, University of Muhammadiyah Malang, [email protected],[email protected],ORCIDID:0000-0002-2741-6773

Abstract. The purpose of this study was to explore and describe the analogy process of students insolvingproblemsbasedonsamplequestionsintermsofstudents'differentthinkingabilities.Thisstudyusesaqualitativeapproachwithadescriptive-exploratorymethod.Thesubjectsof thisstudywere7thgradestudentsofSMPNegeri3Malang,Indonesia.Theresultsshowedthattheconvergentprocessofthestudents'analogywasencoding,concluding,mapping,andthenapplying.Convergentstudentsbegintheanalogyprocessbyunderstandingandcalculatingthevariablesposedbythetargetquestion.Convergentstudents are able to clearly communicate the analogy process in solving target problems and usemathematical language to expressmathematical ideas appropriately.Divergent students take the samestepsasconvergentstudentsincoding,concluding,butthestagesarestagesandapplications.Keywords:AnalogyProcess,ProblemSolving,DivergentThinking

Received:07.12.2020 Accepted:20.01.2021 Published:04.02.2021INTRODUCTIONMinister of Education and Culture Regulation No. 22 of 2006 concerning the objectives of learningmathematics states that students use reasoning on patterns and properties, perform mathematicalmanipulationsinmakinggeneralizations,compileevidenceorexplainmathematicalideasorstatements.Oneoftheprinciplesandstandardsofmathematicsinschoolsisthatstudentshavethepowertoreasoninductively(NCTM,2001).

Inductive reasoning begins with making certain observations and then drawing broaderconclusionsbasedonobservations(Remigioetal.,2014).Inductivereasoningisageneralthinkingskill,which is associated with almost all higher-order cognitive skills and processes, such as generalintelligence,problemsolving,knowledgeacquisitionandapplication,andanalogicalreasoning(Molnáretal.,2013).Theconceptofinductivereasoningreferstocognitiveactivitiesthatproduceconclusionsthatclassicallyfulfilltwocriteria:(a)thedirectionofinductivereasoningmovesfromobservingaparticularcasetotheformulationofmoregeneralrulesand(b)thelevelofconfidenceinductivereasoningisaformofreasoningunderuncertaintybecauseinvolvestheformationofhypothesesaboutrules(Molnáretal.,2013).ScienceteachersinthePhilippinesarelookingforteachingmethods,whichcanenablestudentstodevelopreasoningskillsinscience,onewayofachievingthisisthroughtheuseofanalogy(Remigioetal.,2014).

Theability toanalogy isnotonlyused in theapplicationofmathematics,butalmostall sciencesrequireanalogyskills,suchasinthefieldsofphysics,language,buildingdesigntechniquesandsoon.Theabilitytoanalogizesomethingcanbeusedasaguide insolvingproblemsthathaveasimilarstructure.Theabilityofanalogycanalsobeusedbypeopleatanytimetogainknowledge,thisisinaccordancewiththeopinionofLoc&Uyen(2014)whichstatesthat'analogyasalearningmechanismisacrucialfactorinknowledge acquisition at all ages'. The role of analogy in mathematics in particular is in formingperspectivesandfindingsolutionstoproblems(Isoda&Katagiri,2012).Themoreoftenstudentspracticeusing analogy in solving math problems, the students' analogy thinking skills in solving problems ineveryday lifewillbe formed.Analogymayplayan important role inproblemsolving,decisionmaking,creativity,explanationandcommunication(Lancor,2014).

In learning touseproblem-solvingstrategies, the teacher trainsstudentswithquestionsrangingfromeasytodifficultones.Inanefforttohelpstudentssolveproblems,theteacherprovidesexamplesofquestionsandtheirsolutionsatonce.Suchaproblemisusuallycalledan'exampleproblem'.Thesamplequestionshaveastructuresimilartothepracticequestionsandprovidevariousconceptsandproceduresneededbystudentsincompletingthepracticequestions.Becausethesamplequestionshaveseveral

129|DWIPRIYOUTOMO Middle School Student Analogy Process In Solving Algebra Problems Based On Example Questions In Terms Of Students Divergent Thinking Ability

conceptsandprocedures,thestudents'analogyskillswillgreatlyhelpstudentsinsolvingmathproblems.Tohelpstudentssolveproblems,itisnecessarytoknowhowtheanalogousthinkingprocessofstudentsin solving math problems. Analogy is used in mathematics learning to solve problems by means ofstudents applying known knowledge to solve unknown problems (English, 2004). This means thatanalogy is one way of solving mathematical problems. Cognitive abilities of creative people that areclosely related to problem discovery and problem solving are divergent thinking (Müller & Pietzner,2020).

Divergentthinkingandconvergentthinkingarenotcompletelyseparate,butarecloselyintegrated(Sunetal.,2020).Researchhasfoundthatgeneratingideasthroughdivergentthinkingismoredifficultthanchoosingideasusingconvergentthinking(Antink-Meyer&Lederman,2015;Sunetal.,2020).Giventhe importance of divergent thinking in creativity, training in divergent thinking skills has receivedgreater attention in creativity development (Ritter&Mostert, 2017; An et al., 2016; Forthmann et al.,2016; Sun et al., 2020). In contrast to convergent thinking, where people follow a set path, divergentthinkingisanimportantskillforcreativepeopletothinkcreatively(Müller&Pietzner,2020).Divergentthinkingrequiresmanydifferentsolutionstooneexistingproblem(Forthmannetal.,2016).Convergentthinkingreferstotheprocessofarrivingatasingleanswer,solution,orconclusion(Gallavan&Kottler,2012). Divergent thinking can be described as the process of regaining existing knowledge andassociatingandcombiningunrelatedknowledgeinnewandmeaningfulways(Sunetal.,2020).

One of the topics of learning mathematics in high school that is interesting to research is thesystemoftwo-variablelinearequations.Thistopicisinterestingbecauseitisverycontextualinstudents'dailyactivities. SPLDV isa socialarithmeticmaterial that starts to introduce theexampleofaquantityaskedby aquestionwith twovariables at once. In this topic, students are introduced tomathematicalmodeling and procedural knowledge related to this problem. Previous research found that only theanalogy group experienced an increase in the balance task, these findings support the use of analogylearningtoimprovebalanceinpreschoolchildren(Chatzopoulosetal.,2020).Inaddition,otherfindingsshow that teaching through analogical reasoning improves mathematics learning, because teachingmathematical concepts through analogical reasoning modifies misunderstandings and difficulties forstudentsinmathproblems(Mofidietal.,2012).

Some of the explanations above explained that research on the analogy process in solving flatshapeproblemsisinterestingtostudy.Therefore,thepurposeofthisstudywastoexploreanddescribetheanalogyprocessof seventhgrade juniorhigh school students in solvingSPLDVquestionsbasedonsamplequestionsintermsofstudents'divergentthinkingabilities.

THEORITICALREVIEW

SolvetheproblemSolvingaproblem isabasicactivity forhumans (Hadi&Radiatul,2014).Therefore to solveproblems,peoplemustlearnhowtosolvetheproblemstheyface.Meanwhile,NCTM(2001)statesthatlearningtosolveproblemsisthemainreasonforlearningmathematics.Insolvingmathematicalproblemsstudentsarerequiredtobeable tounderstandthemathematicalconcepts thatare learnedandbeable toapplytheminsolvingproblems(Nasution,2018).

Problem solving is one of the foundations of teaching mathematics (Tarim & Öktem, 2016).Problemsolvingisaprocessthatstartswithinitialcontactwiththeproblemandendswhenananswerisreceived based on the information provided (Olaniyan et al., 2015). In addition, problem solvingaccordingtoHadi&Radiatul(2014)isabasicabilitythatmustbemasteredbystudents.

Basedontheexplanationabove,problemsolvingisaplannedprocessthatneedstobedonetofindanswerstoaproblembyusingconceptsandideasthatmustbemasteredbystudents.In solving theproblem thereare several stages.Theproblemsolving stageused in this research is theproblemsolvingstageaccordingtoPolya.AccordingtoPolya, the fourstagesare:a)understandingtheproblem,b)planningastrategy,c) implementingtheplanandd)checkingagain(Hensberry&Jacobbe,2012;Tong&Loc,2017;Abdullahetal.,2015).AccordingtoPolya,theproblemsolvingstagewaschosenbecausethestagesofproblemsolvingproposedbyGPolyaweresimple,theactivitiesateachstagewereclear,andallowedstudentstogainexperienceusingtheirknowledgeandskillstosolveproblems.Suleiman(Olaniyanetal.,2015) foundthatproblemsolvingaccording toPolyawaspreferredover theproblemsolvingmodelsofGickandBransfordandStein.

AnalogyProcessinSolvingProblemsAnalogyisapowerfulcognitivemechanismusedtolearnnewabstractionsbystudentsandisoftenusedintheformoftext,images,videosandverbalexamplesintraditionalclassrooms(Remigioetal.,2014).


Potency Known Problem Known Problem

Mapping

Known Relationship Structure

Mapped Unknown Relationship

Structure

Known Settlement Procedure

Mapped Unknown Settlement Procedure

With possible adaptations

WilbersandDuit(Lancor,2014)arguethatlearningbyanalogyisnotonlyabouttransferringstructuralfeaturesbetweensourceandtargetdomains,butalsotheprocessofbuildingrelationships.Meanwhile,accordingtoMariah(Remigioetal.,2014)definesanalogyasaconcreterepresentationofthesuitabilitybetweenbasic concepts and targets.Analogy as a comparisonof somethingunfamiliarwith somethingfamiliartoexplainsharedprinciples,suchasabridgebetweenwhatyouwanttolearnandwhatstudentsalready know (Lancor, 2014). From the description above, analogy is a sequence of steps to drawconclusionsstartingfromcomparingtwothingsortwosymptomsthenlookingforsimilaritiesinaspects,propertiesandstructuresbetweenthingsthatarenotyetknownandthingsthatarealreadyknownfirst.

Analogy learning can limit the amount of verbal information that beginners access in the earlystagesof learning, leading to less relianceondeclarative informationandmotorplanning (Duijn et al.,2020). Based on the research results of Chui and Lin (Remigio et al., 2014) using analogy not onlyadvances in-depth understanding of complex science concepts, but also helps students correct theirmisconceptionsaboutconcepts.Analogyhasproventobeausefultoolinexpressingstudents'thoughtsandhelpingthemunderstandnewsituations(Lancor,2014).

Inusinganalogyskills,studentsmustbefamiliarwiththeconceptoftargetsandbeabletoreviewthe concept of analogy (Loc & Uyen, 2014). Wong (Lancor, 2014) argues that constructing analogiesserves to (a) make new situations familiar, (b) represent problems specifically from an individual'spreviousknowledge,and(c)stimulateabstractthinkingabouttheunderlyingstructuresorpatterns. Intheproblemsofanalogyreasoning, therearetwoquestions,namelythesourcequestionandthetargetproblem.Thesourceproblemhasthecharacteristics:a)givenbeforethetargetproblem;b)intheformofeasy andmediumproblems; c) canhelp solve the target problemor as initial knowledge in the targetproblem (English, 2004). The target problem is characterized by: a) a modified or extended sourceproblem; b) the target problem structure is related to the source problem structure; c) is a complexproblem(English,2004).

The use of known problems is as information in terms of linking and comparing them withunknownproblemssothatthestructureofthesourceproblemcanbeappliedtothetargetproblem.Thismeans thatknownproblemscanhelpsolveunknownproblems. In this case theknownproblem is thesource problem and the unknown problem is the target problem. The source problem is the problemgiven before the target problem. The source problem can be used as initial knowledge in solving thetargetproblem.Therelationshipbetweenthesecharacteristics incarryingouttheanalogyisthatwhenobtainingasourceproblem,studentsthenlookatandsolvetheproblemusesaconceptthathealreadyknows.Furthermore,insolvingthetargetproblemstudentsidentifytherelevantpropertiesofthesourceproblemasinitialknowledgetosolvethetargetproblem,thenmaptherelatedtraits.Thefollowingisanillustrationoftheanalogypresented(English,2004).

Source Target

Figure1.AnalogicalthinkingprocessinsolvingMathematicsproblemsFigure 1 explains that the analogy thinking process in solving mathematical problems is: 1)

identifying the information contained in the source problem and the target problem; 2) mapping thestructure of the relationship between the source problem and the target problem; and 3) map thestructureof the sourceproblemsolving to the targetproblem.Theprocessof reasoningusinganalogyincludesencoding,inferring,mapping,andapplying(English,2004).


Table1.StagesofAnalogyinSolvingProblems(English,2004).

AnalogyStages StagesofProblemSolvingEncoding(Encoding)Studentscanunderstandtheinformationcontainedinthesourceproblemandthetargetproblem.

Understandtheproblem

Inferring(Inference)Studentscandeterminethestructureandfindthesourceproblemsolvingrelationalinformation

Searchingforinformation

Mapping(Mapping)Studentscanmaprelationalstructures/makeplansforsolvingsourceproblemstotargetproblems.

Makeaplan

Applying(Application)Studentscanapplytherelationalwayofsolvingsourceproblemsinsolvingthetargetproblem

Carryouttheplan

The role of analogy specifically in mathematics is to form perspectives and find solutions toproblems (Isoda, M. & Katagiri, 2012). This means that analogy is one of the tools used in solvingmathematical problems, the more often students practice using analogy in solving mathematicalproblems, theanalogical thinkingprocessofstudents insolvingproblemsoutsideofmathematicsor ineverydaylifewillbeformedsothatitwillbenefitlifeandthedevelopmentofscience.otherknowledge.Mathematicalanalogy indicators forsourcequestionsand targetquestionsaccording toEnglish(2004)canbeseenintable2.

Table2.AnalogyIndicators

AnalogyStages indicatorEncoding Studentsknowwhatinformationisknownandwhatisbeingaskedabout

thetarget.Inferring Studentsareabletoidentifyinformationrelatedtosourcequestionsand

targetquestions.Mapping Shivaisabletomaporplanthecompletionofthetargetproblembased

onthesolutiontothesourceproblem.Applying Studentsareabletoapplythesourcequestionsolvingmethodtothe

targetquestionsandsolvethequestionscorrectly.

DivergentandConvergentThinkingDivergent thinking is defined as thinking that generates several ideas, solutions or products, whileconvergentthinkingleadstoonecorrectanswerasinthecaseoftraditionalabilitytests(Sanchez-ruizetal.,2015;Colzatoetal.,2017).Inparticular,creativityhasbeenconceptualizedasdivergentthinkingortheperformanceofcreativeexperts(Anetal.,2016).Divergentthinkingischosentorepresentathinkingstyle that allows many new ideas to be generated and more than one correct solution to a problem(Colzato et al., 2017).Divergent thinking is generally used to verify creative potential and capture theextenttowhichindividualscreatenovelty(Kleibeukeretal.,2013).Whereasconvergentthinkingrefersto analytical and evaluative thinking, and thinking quickly and focusing on the one best solution to aproblem (Gabora, 2010). Unlike divergent thinking where each student can provide a reasonable butuniqueanswer,studentsinconvergentthinkingareexpectedtorespondwiththesameanswergiventotheteacherwithlittleornopeer-to-peerinteraction(Gallavan&Kottler,2012).

In learning practice in schools it is important to distinguish between activities related todivergent abilities and activities related to convergent thinking skills. Divergent thinking is easy tocontrast with convergent thinking. Convergent thinking leads to conventional and correct ideas andsolutions,whiledivergentthinkingleadstooriginalideasandsolutions(Runco&Acar,2012).Inlearningpractice, convergent thinking is emphasized more than divergent thinking, where students often facemultiplechoiceteststofindtherightansweramongseveralalternativeanswerscomparedtoessayteststhatallowmanywaysofsolving(Colzatoetal.,2017).

The tendency for divergence or convergence of students' thinking can be measured usingstandardized tests. Divergent thinking tests have dominated the field of assessment of creativity fordecades (Runco & Acar, 2012; Sanchez-ruiz et al., 2015). The Torrance test is basically a divergentthinkingtestbasedontheGuilfordmodel,andhasbeenverywidelyusedasacreativitytest(Baer,2011).Guilford(Gallavan&Kottler,2012)identifiesthecharacteristicsofdivergentthinkingthatstrengthen


learning, namely ideational fluency, associational fluency, expressional fluency, spontaneous flexibility,adaptiveflexibility,elaboration,originality,andsensitivitytoproblems.Thebenefitsofdivergentthinkingare concept attainment, vocabulary development, depth exploration, breadth expansion, contextapplication, critical reasoning, conversational defense, classification labeling, example generation, andpossibilityadvancement(Gallavan&Kottler,2012).

METHODS

Thisstudyusesaqualitativeapproachwithdescriptive-exploratorymethods.Casestudyresearchdesign.Thisstudyrevealstheanalogousprocessofstudentsinsolvingalgebraicproblems.Throughathoroughandin-depthexaminationprocess,theresearcherdigsupstudentdataaboutwhatisbeingthought,done,writtenandsaidwhensolvingproblems.Thedataaredescribedastheyaretoobtainanaturalpictureoftheanalogyprocessofstudentsinsolvingalgebraicproblems.

The subjects of this studywere 6 grade 7 students of SMPNegeri 3Malang, Indonesia. Thedetermination of the research subject is based on the category of students' divergent thinking skillsthroughtheDT(DivergentTest)test.Twosubjectswereinthedivergentthinkingcategoryand2subjectswereConvergentThinking.TherespectivesubjectcodesareC1,C2,D1,andD2.

Datacollectiontechniquesinthisstudyweretestsandinterviews.Thetestusedistheanalogytestwhichisconductedtodeterminetheanalogyprocessoftheresearchsubject.Thetestinquestioncanbeseenintable3.Problemsourceisdonebytheteacherasanexampletohelpstudentsindoingpracticequestions.Thequestionsforthisexercisearecalledthetargetquestions.Theconceptsandprinciplesinthe source problem are the same as in the target problem. Themethod andprocedure for solving thesource questions are intended by the teacher to be a guide for students to solve the target questions.Mathematical analogy indicators for source questions and target questions can be seen in table 2.Interviews were conducted to confirm and complete data on student answers. This interview isunstructured because the questions asked depend on the students' written answers to the questionsgiven.

The data analysis technique used was the interactive technique of Miles and Huberman(Sugiyono,2011).Thedataanalysisprocessstartswithdatareduction,continueswithdatapresentation,and ends with drawing conclusions. Data reduction is the process of clarifying, classifying, removingunnecessarydata, and classifyingdataobtained from the field.Thedataweredescribed systematicallyaccording to the divergent thinking ability category group then analyzed based on the mathematicalanalogy indicator.Thepresentationof the analogyprocessdatawas compiledby the research subject,startingwiththepresentationofthestudent'swrittenanswerdatafollowedbytheresultsofinterviewswiththesubjectandthentheresultsoftheanalysisofeachanalogyindicator.Thedrawingof researchconclusions is supported by valid data, at this stage the conclusions presented by the researcher arebasedonalltheresultsofdataanalysisobtained.

Table3.ProblemSourceandProblemTarget

ExampleProblem/ProblemSource PracticeQuestions/TargetQuestions

Cahayabought4notebooksand3pens,shepaidIDR 19,500.00. If he buys 2 notebooks and 4pens, he will have to pay Rp. 16,000.00.Determinethepriceofanotebookandapen!

ParlanandSurtiworkatabagfactory.Parlancancomplete 3 bags every hour and Surti cancomplete 4 bags every hour. The number ofworkinghours forParlanandSurti is16hoursaday with the number of bags made by both ofthemis55bags.Iftheirworkinghoursaredifferent, determine their working hoursrespectively!

RESULT

ProcessanalogyC1Thesubjectcandescribealltheanswerstothetargetquestionsbyperformingtheanalogystageappropriately,thiscanbeseeninthestudents'answersandthefollowinginterviewresults.


Surti (Y) 3X + 4Y = 55 x1 3X + 4Y = 55 X + Y = 16 x3 3X + 3Y = 48

Example : Parlan (X)

Then :

Y = 7 Y = 7 X + Y = 16

X + 7 = 16 X = 16 – 7 X = 9

So, Parlan 9 and Surti 7

Figure2.AnswerstoC1targetquestionsFromtheanswerinFigure2,thesubjectstartstheworkbyconsideringwhatisbeingaskedbythe

questions,namelyxandy.Creatingamathematicalmodelbasedontheinformationunderstood(systemof linear equations). These two things show that the subject knowswhat is known andwhat is beingasked (encoding). Subjects are able to plan problem solving by making a system model of linearequations,namely3x+4y=55andx+y=16(mapping).Furthermore, thesubjectsolvesasystemoflinear equations using the elimination and substitutionmethod as the solution to the source problem.Subjects are able to correctly answer questions about the target (Applying). The following interviewresults will complement the information about the subject's analogy process in solving the targetquestions.P :"Whatdoyougetincommonindoingthisproblem"?C1 :"solvingthetwoproblems(sourceproblemandtargetproblem)usinganexample,makinga

systemoflinearequations,usingeliminationandsubstitution"P :"Howdoyousolvethetargetproblem?"C1 : "It is the same as used in the source problem, at first I suppose the x and y hours for the time

requiredbyParlanandSurti.Thenmakeasystemofequationsbasedontheinformationfromtheproblem.3x+4y=55andx+y=16.ThenIdidtheeliminationandsubstitutionuntilx=9andy=7wereobtained.

P:"Sohowdoyouconcludeyouranswer"?C1:"Parlantakes9hoursandSurti7hoursaday"

From the interviewabove, it appears that the subject understands the similarity of the concept,principleandmethodofsettlementbetweenthesourcequestionandthetargetquestion(inferring).Thesubjectcompletesthetargetproblem,thesubjectstartsmakingxandyassumptions,makesasystemoflinearequations, completes systemsof linearequationsusing theeliminationandsubstitutionmethod,thenanswersthequestionsonthetargetquestion.Thewaythesubjectsolvedthetargetproblemwasthesameasthatusedinthesourceproblem.

ProcessanalogyC2In the analogy process, the subject can describe all the answers to the target questions by doing theanalogystagecorrectly,thiscanbeseeninthewrittenanswersandthefollowinginterviewresults.

Example : Parlan working hours (X)

Surti working hours (Y) 3X + 4Y = 55 x1

X + Y = 16 x3

3X + 4Y = 55 x1 X + Y = 16 x4

3X + 4Y = 55 3X + 3Y = 48

Y = 7 3X + 4Y = 55 4X + 4Y = 64

X = 9 So, Parlan working 9 hours and Surti working 7 hours

Figure3.AnswerstoC2targetquestions


FromtheanswerinFigure3,thesubjectstartstheworkbyconsideringwhatisbeingaskedbythequestions,namelyxandy.Creatingamathematicalmodelbasedoninformationthatisunderstood.Thesetwothingsshowthatthesubjectknowswhatisknownandwhatisbeingasked(encoding).Subjectsareable to plan problem solving by making a system model of linear equations (mapping). To completeinformation about the subject's analogy process in solving the target questions and the results of theinterview.P:"Whatarethesamesolutionstothesourceandtargetproblems"?C2:"Solvethetwoproblemsusingtheequation,makeasystemoflinearequations,useelimination"P:"HowdidAnandasolvethetargetproblem?"C2:"Itisthesameasthatusedinthesourceproblem,atfirstIsupposeittookParlanandSurtitomake

bags with x and y. Then make a mathematical model based on the information from thequestions.3x+4y=55andx+y=16.ThenIdidanothereliminationtogetthevalueofyandIdidanothereliminationtogetx.

P:“Whynotusesubstitutiontogetthevalueforx?C2:"It'sthesameastheresult"

Fromtheresultsoftheinterviewabove,thesubjectcanidentifyandunderstandthesimilarityofconcepts, principles and methods of solving between the source question and the target question(inferring).Thesolutionprocessstartsbycalculatingxandy,makingamathematicalmodelofasystemoflinearequations,thenusingtheeliminationprocedure.Thesubjectusesthesamesolutionmethodasthesourceproblemsolutionmethod.Gettingthecorrectresultis9hoursneededbyParlanand7hoursforSurti(Applying).IfC1usestheeliminationmethodandthenthesubstitutionmethod,C2usesonlytheeliminationmethodtoobtainthexandyvalues.Thesubject issolvingthesametargetproblemasthatusedinthesourceproblem.

Table4.ConvergentSubjectAnalogyProcessinSolvingProblems

Step Subjectactivity

Encoding Understandtheinformationonthetargetquestionandunderstandwhatthetargetquestionasks.

Inferring Findinformationrelatedtosourcequestionsandtargetquestions.

Mapping Map/makeplansforsolvingtargetquestionsbasedonsolutionstosourcequestions.

Applying Applythesourcequestionsolvingmethodtothetargetproblemandsolvetheproblemcorrectly.

ProcessAnalogyD1In the analogy process, the subject can describe all the answers to the target question by performingdifferentanalogystageswithsubjectsC1andC2.SubjectD1usesadifferentsolutionmethod fromthemethodofsolvingthesourceproblem.Thiscanbeseeninthestudents'answersinFigure4.

Example : Parlan working hours (X) Surti working hours (Y)

3X + 4Y = 55 X + Y = 16

Y = 7 hours X = 16 – 7 = 9 hours

So, Parlan working 9 hours and Surti working 7 hours

Figure4.AnswerstoTargetQuestionsD1Thesubjectunderstandstheinformationandwhatisaskedbythetargetquestions.Subjecttakes

theworkhoursrequiredbyParlanandSurtiwithxandy(encoding).Thesubjectisalsoabletomakeasolutionplan,namelyasystemoflinearequations(mapping).Fromtheanswerabove,thequestionariseswhyy=7suddenlyappearsandwhythesubjectdoesnotusetheeliminationmethodtosolvethetargetproblem as in the source problem solution. The following interview results provide answers to thesequestions.


Parlan = 3 bag/hours Surti = 4 bag/hours Parlan + Surti = 16 hours = 55 bag

Parlan = 3,6,9,12,15,18,21,24,27 = 9 hours Surti. = 4,8,12,16,20,24,28 = 7 hours

P:“whyistherey=7onyouranswersheet?D1:"yesma'am.Icreatedtheequations3x+4y=55andx+y=16,thenIestimatedthatParlan'stime

comparedtoSurti4:3. If theSurtitimeis6thentheParlantimeis8.SincethetotaltimeforParlanandSurtiis16,thenmyestimateisthetimeforSurti7andthentheParlantimex=16-7=9"

P :"isitfulfilledfortheequation3x+4y=55?D1 :meetma'am.Icheckedearlier.Q :Whynotusetheeliminationmethodasthesolutiontothesourceproblem?D1 :Iwantanotherwayma'am.P :“Whyisn'ttheguess-checkmethodwrittenontheanswersheet?D2 :No,Mom.I'lljustwritedowntheresults.Iwrotetherestonopaquepaper.

Fromtheinterviewabove,itisclearthatthesubjectisabletoidentifysimilaritiesinconceptsandprinciplesbetween the target question and the sourcequestion.The subject alsounderstands that themethodofsolvingthesourceproblemcanbeusedtosolvethetargetproblem(inferring).AlthoughD1isable toperform the inferringprocess,D1 triesothermethods.D1uses theguessandcheckmethodorguessandcheck.Guessthattheratioofxandy is4:3becausethefirstequation is3x+4y=55.Thenguessthat8isforthevalueofxand6isfory.Afterchecking,itturnsoutthat8plus6doesnotequal16(secondequationx+y=16).D1'snextguessis7foryand9forx.ThenD1planstocheckwhetherx=9andy=7satisfytheequationsx+y=16and3x+4y=55(mapping).D1doesnotwritedownpartofthestagesofsolvingthetargetproblem,namelytheprocessofguessingy=7andcheckingtheguessintheequation3x+4y=55 (inferring).Thesubjectdidnotuse theeliminationmethodasused to solve thesourceproblem,thesubjectusedadifferentmethod,namelytheguess-checkmethod.Anotherthingthatwasrevealedfromtheanswersheetsandinterviewresultswasthatpartofthecompletionprocesswasnotwrittenontheanswersheets.Subjectstendtowriteshortanswers.

ProcessanalogyD2D2performsadifferentanalogyprocesswiththeprevioussubject.D2usesadifferentsolutionmethodbutcandecipheralltheanswerstothetargetquestioncorrectly,thiscanbeseenintheanswertoFigure5,below.

Figure5.AnswerstoquestionsonTargetD2WritingParlan=3bags/hour,Surti=4bags/hourandParlan+Surti=16hours=55bags

showsthatthesubjectunderstandstheinformationinthequestion.Bywritingaseriesofnumberswithup to 9 hours for Parlan and 7 hours for Surti, it shows that the subject also understands what theproblemasks(encoding).Seetheseriesofnumbers3,6,9,....etcand4,8,12,etc.ItappearsthatD2hasaplantosolvetheproblembywritingmultiplesof3andmultiplesof4.Thesubjectmakesplanstosolvedifferentproblemswithsolutionstothesourceproblem.Whythereisnoexample,linearequationsystemexpression, elimination or substitution method to solve the target problem, the following interviewresultswillexplainallofthat.P :"whyistherenoexplanation?D2 :"yesma'am.IjustwrotethatParlanproduces3bagsandSurti4bagsinonehour.P :"whynotmakealinearequationmodel?D2 :Noma'am,becauseIdon'tusetheeliminationmethod.Q :CanAnandausethesolutionasthesourceproblem?D2 :Yes,Mom.I'vefinisheditonanotherpaper.P:"TrytoexplainhowAnandasolvedtheproblem!D2 :Theamountof time it tookParlanandSurti tomake55bags is16hours. In the firsthourParlan

produced3bagsandSurti4bags,aftertwohoursParlan6bagsandSurti8bags.AndsoonsothatParlanproduced27bagsandSurti28bags(27+28=55).Parlantakes9hoursandSurti7hours.

P:“Whynotgiveanexplanationontheanswersheet?D2:No,Mom.Iwritebriefly.Iwrotetherestonopaquepaper


Encoding Information Asked

Information Known

Information Asked

Information Known

Concepts & Principles

Concepts & Principles

Based on the results of the interviewwithD2, it appears that the subject is able to identify thesimilarityofconceptsandprinciplesbetweenthetargetquestionsandthesourcequestions.Thesubjectalsounderstandsthatthemethodofsolvingthesourceproblemcanbeusedtosolvethetargetproblem(inferring). Even though the subjectwas able to perform the inferring process, the subject tried othermethods.D2usesthemakeanOrganizedlistmethod.ThefirstrowishowmanybagsParlangenerateseveryhour.ThesecondrowisthenumberofbagsthatSurtimakeseveryhour.Bothsequenceswillstopwhenthelasttermis55.Itoccursinthe9thtermintheParlanseriesandthe6thtermintheSurtiseries.SoittakesParlan9hoursandSurti7hours.Thesubjectdidnotusetheeliminationmethodasusedtosolve the sourceproblem, the subjectusedadifferentmethod,namely themethodof registering inanorganizedmanner.Another thing thatwas revealed from the answer sheets and interview resultswasthatmany completion processes were not written on the answer sheets. The subject's compulsion toproduce short answers results in erroneousmathematical expressions. The first and second "=" signsexpress"thenumberofbagsmadebyParlan".Thethird"="signexpresses"thetimetakenbyParlanandSurti"andsoon.

Table5.ConvergentSubjectAnalogyProcessinSolvingProblems

Step Subjectactivity

Encoding Understandtheinformationonthetargetquestionandunderstandwhatthetargetquestionasks

Inferring Identifyinformationrelatedtosourcequestionsandtargetquestions.

Mapping Makedifferentplanstosolvethetargetquestions.Actuallythesubjectisabletomakethesameplanastheoneinthesourcequestion.

Applying Applyadifferentsolutiontothesolutiontothesourceproblem.Thesubjectgetsthecorrectanswertothetargetquestionquestion.

DISCUSSIONANDCONCLUSIONSBased on the results of the research data analysis that has been stated above, the analogy process ofstudentsinsolvingproblemscanbepresentedinthefollowingchart. ProblemSource ProcessAnalogy ProblemTarget

Inferring

Mapping

Applying

Information: : next

: Provide Instructions

Settlement procedures used

Settlement procedures used

Settlement Pattern / Structure

Settlement Pattern / Structure


Figure6.AnalogicalThinkingProcess

ConvergentStudentAnalogyProcessConvergent students carry out the analogy process, namely encoding, inferring, mapping, and thenapplying. The convergent student begins the process of analogy by supposing x and y for the variableaskedby theproblem.Understand the information from the target questions asmaterial formaking acompletion plan. Both calculating and understanding the information about the questions are theencodingstages.Thentheconvergentstudentscompareinformationfromthesourcequestionswiththetarget questions. Things that are compared include concepts, principles, mathematical models, andmethodofsolvingproblems(Inferring).ThisisinaccordancewithRemigioetal.,(2014)whosayanalogyisamappingofknowledgefromthebasetothetarget.

Thenext stage, the convergent studentsmade amathematicalmodel in the formof a systemoflinear equations. The system of linear equations ismade based on the information obtained from theproblemusingthexandyvariablesas forexample in theencodingstage.Thestageofmakinga linearequationsystemmodelusingthexandyvariablesaccordingtoEnglish(2004)isthemappingstage.AttheMappingstagestudentsmustbeabletoanalyzeandsolvetargetquestionswiththecorrectmethod.Errorsatthemappingstagewillbefollowedatalaterstage.Intheapplyingstage,studentscarryouttheplan, which is to solve the target problem using themethod of elimination and substitution as in thesolutiontothesourceproblem.ThisisinlinewithGallavan&Kottler(2012)whichstatesthatstudentsinconvergentthinkingrespondtoanswersinthesamewayasthequestionsgiventotheteacher.Fromtheprocessofeliminatingthesystemofequations,thexvalueisobtained.Theyvalueisobtainedfromthesubstitutionprocess.Byobtainingthexandyvaluesthenthestudentsansweredthequestionsposedbythe target questions. This is consistent with what Lancor (2014) studied by analogy not only abouttransferring structural features between source and target domains, but also the process of buildingrelationships. Convergent students are able to clearly communicate the analogy process in solving thetargetproblem.

DivergentStudentAnalogyProcessIftheconvergentstudentperformstheencoding,inferring,mapping,andthenapplyinganalogyprocess,thedivergentstudentperformsdifferentstepsinthemappingandapplyingstages.Attheencodingstage,divergent students begin the analogy process by considering x and y for the variables asked by thequestions or detailing the relationship between time and the number of bags produced. This evidenceshowsthatthesubjectunderstoodtheinformationandwhatthequestionsasked.Divergentstudentsfindinformation related to the source questions and the target questions. The inferring process passed bydivergent students is still used to make plans for solving target questions. As has been explained bySternberg(English,2004)thatinferringisaprocessoflookingforrelationshipscontainedinthesourceproblemtosolvethetargetproblem.

At themapping stage, divergent studentsmakedifferentproblem-solvingplanswith convergentstudent solution plans. After getting inspiration from the solution to the source problem, divergentstudents try to make a different solution plan from the solution to the source problem. This is inaccordancewithColzatoetal.,(2017)thatdivergentthinkingmakesitpossibletofindmanynewideas.Thesubjectguessesthevalueofthevariableinquestionthenchecksthecorrectnessoftheguess.Thisisconsistent with Lancor (2014) that analogy has proven to be a useful tool in expressing students'thoughtsandhelpingthemunderstandnewsituations.

Attheapplyingstage,studentscarryouttheplanthatwasmadeatthemappingstage.Thestudentguessesthevalueofxandybasedontheinformationontheproblemthenchecksthevalueoftheguessinthesystemofequations.Iftheguessvaluesatisfiesthesystemofequations,thenthatvalueisusedtoanswer thequestionquestions.Likewise in themethodofmakinganorganized list,divergentstudentsmakealistofnumberswithacertainpatternforthefirstvariableandthesecondvariable.Thesequenceofnumbersstopsatthenthtermaccordingtotheinformationaboutthetarget.Byobtainingthexandyvaluesthenthestudentsansweredthequestionsposedbythetargetquestions.Theuseofanalogieshasbeen shown to be very effective in teaching students as it helps motivation and visualizes difficultconcepts(Nworgu&Otum,2013).

Besidestendingtousedifferentmethods,divergentstudentsalsotendedtogiveshortandshortanswers.Thesubject'sencouragementtomakeshortanswersresultedinmanyverbalexpressionsthatshouldhavebeennotwrittenontheanswersheet,sothattheproposedsolutionswerelesscleartotheteacher.


CONCLUSIONTheprocessofconvergingstudentanalogyisencoding,inferring,mapping,andthenapplying.Convergentstudentsstarttheprocessofanalogybyunderstandingandcalculatingthevariablesaskedbythetargetquestion.Thenextstage is tounderstandtherelationshipbetweenthesourcequestionsandthetargetquestions.Thenthestudentsmakeplansforsolvingattractiveproblemsbymodelingasystemoflinearequations.Intheapplyingstage,studentscarryouttheplan,whichistosolvethetargetproblemusingthemethodofeliminationandsubstitutionasinthesolutiontothesourceproblem.Convergentstudentsareabletoclearlycommunicatetheanalogyprocessinsolvingthetargetproblemandusemathematicallanguagetoexpressmathematicalideasappropriately.

Divergent students do the same steps as convergent students in the encoding, inferring, butdifferent stages of mapping and applying. At themapping stage, divergent students make a problem-solvingplanthatisdifferentfromthesolutionplanforthesourceproblem.Studentsusethecheck-guessmethodandmakeanorganizedlist.Furthermore,studentscarryouttheplansthathavebeenmadeatthemapping stage until they find the x and y values. By obtaining the x and y values then the studentsansweredthequestionsposedbythetargetquestions.Attheapplyingstage,studentscarryouttheplanbyguessingandcheckingandmakinganorganizedlist.

Thetendencyofdivergentstudentstogiveshortanswersresultedinstudentsmakingmistakesinexpressingsolutionideasandignoringverbalexpressionsthatshouldbewrittenontheanswersheet.

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