University of Groningen

Fostering Indonesian Prospective Mathematics Teachers' Geometry Proof CompetenceAnwar, Lathiful

DOI:10.33612/diss.195236562

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Introduction

Chapter 1

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Introduction

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Introduction

For years, I have personal experience in teaching mathematical proof to prospectivemathematicsteachers(PMTs),ages18to19years, inthemathematicseducationdegreeprogramat thepublic university inMalang in Indonesia. Basedonmyobservations andinterviewswiththePMTs,amathematicalproofingeometry,algebraoranalysisisdifficultfor them, especially for those students who learn proof and proving for the first time.AlthoughmostPMTswillnotteachmathematicalproofatthesecondaryschool,theyneedtolearntocomprehendandconstructmathematicalproofatuniversity.ThePMTsseethatprovingisdifficultforthemasitinvolvesformalandabstractthinking.Itismyexperiencethattheylearnedproofbymemorizingtheproofdemonstratedbythelecturerorwritteninthetextbook,buttheydonotunderstandwhichstrategytouse,howtostarttheproof,howtoconnectpremisestoconclusionlogically,andwhatthegoaloftheproofis.Mostofthegeometryproofstheylearnduringthefirstsemesterarepresentedinthetwo-columnproofformatintheircoursetextbookorlecturenotes.Itismyimpressionthatthetwo-columnproofsdonotprovideenoughscaffoldforstudentstocomprehendtheproof,particularlytoidentifycomponentsofproofandtherelationshipbetweenpremisesandconclusion.

This situationmotivated me as a lecturer to design a learning trajectory that supportsPMTs’ development of proof competences, such as proof comprehension and proofconstruction.Thisdesignwasthecontexttoinvestigatestudents’learningtrajectories,andwas implemented inan introductoryEuclideangeometryproofcourseas IassumedthatEuclideangeometryproofisagoodstartingpointtolearnmathematicalproof(Tall,1998,1995).Thegeometryproofisnotasabstractasmathematicalproofinalgebra,analysis,andnumbertheory,becauseinmostgeometryproofproblems,thepremisesaregivenasverbalorsymbolicinformation(i.e.,conceptualinformation)togetherwithageometricfigure(i.e.,visualinformation)(Uferetal.,2009).

Research Context

I conducted this research in Indonesia where geometry proof is a part of a specificmathematicscourse inthemathematicseducationdegreeprogramatMalanguniversity.Inthiscourse,BasicMathematics III,PMTs learnaboutEuclideangeometryproofaimingtobeabletounderstandandconstructgeometryproofs.Duetothespecificnatureofthecourse,wewill describe the Indonesian education systemfirst and the differenceswithgeometryproofcoursesinothercountries.Inthisthesis,weusetermslecturerandschoolteachertorefertotheteacheratuniversitylevelandtheteacheratsecondaryschoollevel,respectively.

Chapter 1

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IndonesianEducationSystem

TheeducationalsysteminIndonesiaisunderauthorityoftwoministries:TheMinistryofEducationandCulture(MoEC)andtheMinistryofReligiousAffairs(MoRA).Therearearound250,000schoolswithabout50millionstudentsand2.8millionteachers (PusatdatadanStatistikPendidikandanKebudayaan,2017).84%ofschoolsaresecularschools includinggeneralandvocationalschools,authorizedbytheMinistryofEducationandCulture(MoEC),andtheMinistryofReligiousAffairs(MoRA)managestheremaining16%religiousschools(madrasah) (Pusat data dan Statistik Pendidikan dan Kebudayaan, 2017). Indonesia hasimplemented a 2-6-3-3-4 school-based educational structure consisting of two years ofkindergarten,sixyearsofprimaryschool,threeyearsofjuniorsecondaryschool,threeyearsofseniorsecondaryschoolandfouryearsofundergraduateuniversitylevel,seeTable1.1.

Table 1.1. OverviewoftheIndonesianeducationsystem(MinistryofEducationandCulture,2020,2021;RepublicofIndonesia,2010)

Level Grade/degree(regularages)

Typeofschools

Pre-school A(4-5yearsold)andB(5-6yearsold)

•Generalschools(TamanKanak-kanak/TK)•Religiousschools(Raudatul-Atfal/RA)

PrimarySchool 1-6(7–13yearsold) •Generalschools(SekolahDasar/SD)•Religiousschools(MadrasahIbtidaiyah/MI)

SecondarySchool 7-9(13–16yearsold) •Generalschools(SekolahMenengahPertama/SMP)•Religiousschools(MadrasahTsanawiyah/MTs)

10-12(16–19yearsold)

•Generalschools(SekolahMenengahAtas/SMA)•Vocationalschools(SekolahMenengahKejuruan/SMK)•Religiousschools(MadrasahAliyah/MA)

Higher/TertiaryEducation

Bachelor/Diploma(19–23yearsold)

•Academicprogram(bachelor’sdegree):educationalandnon-educationalprogram

•Vocationalprogram(Diplomadegree)

Master(23–25yearsold)

•Academicprogram(master’sdegree):educationalandnon-educationalprogram

•Professionalprogram•Specialistprogram(afterProfessionalprogram)•Appliedmasterprogram(aftervocationaleducation)

Doctoral(25–28yearsold)

•Academicprogram(Doctoraldegree):educationalandnon-educationalprogram

•Applieddoctoralprogram(afterappliedmasterprogram)

To support the teaching and learning in school, the Center of Books and Curriculum(PUSKURBUK)oftheIndonesianMinistryofEducationandCulturedevelopsandpublishescompulsoryelectronicschooltextbooksforelementaryandsecondaryschool.Thetextbooksarebasedonthesyllabusofthenationalcurriculum.Theschoolscanuseadditionaltextbooksfrom other publishers that are recommended by PUSKURBUK to support their teaching

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andlearning.Personally,Iaminvolvedindevelopingthecompulsoryschoolmathematicstextbookforgrade12(As’arietal.,2018).

The compulsory school textbooks formathematics in elementary and secondary schoolconsistofastudenttextbookandateachermanual.Thestudenttextbookisusedasaguideforlearningactivities(activity-basedlearning)inclassthatsupportsstudentsindevelopingcertain competencies. It contains explanationof basic concepts, flow-charts of concepts(conceptmaps),learningactivitiesdesignedtodevelopthebasiccompetenciesthroughanapproachwithfivebasiclearningtasks(i.e.,observation,asking,gatheringinformation,andcommunication),andexercises.Theteachermanualisusedbytheteacherasguidanceforteaching.Itcontainsinformationabouthowtousethestudenttextbookincludingcoreandbasiccompetencies,howtoconductlearningactivitiesandtoassess.Itprovidesassessmentinstrumentsandteachingmethods.

InIndonesia,therearetwotypesofexamination,namelytheNationalExamination(UN)andtheNationalStandardizedSchoolExam(USBN).ThesetwoexaminationsaimtomeasureandevaluatetheIndonesianstudents’competencesattheendofeacheducational level(i.e., grades 6, 9 and 12). The UN is administered by the Board of National EducationStandard(BadanStandardNasionalPendidikan,BSNP),andisalsousedasaninstrumentinmonitoringschools’performanceandidentifyingschoolsinneedofsupport.Incontrast,theUSBNisdevelopedbytheschoolwithreferencetothenationalstandardsprovidedbythegovernment.TheUSBNisusedtodeterminewhetherstudentscancontinuetheireducationtothenextlevelornot.Theschoolteachersareexpectedtosupporttheirstudentstopassinbothexaminations.AsindicatedbyastudybyCreeseetal.(2016),thismightleadtotheconsequencethatschoolsandteacherswouldfocusontest-orientedteachinginsteadofthelearninggoalsrequestedbythecurriculum.

GeneralMathematicsandGeometrySchoolCurriculum

Allelementary,junior,andseniorsecondaryschoolstudentslearnmathematics.However,atelementaryschoolmathematicsistaughtasanintegratedandthematicsubject.Atjuniorsecondaryschool,mathematicsistaughtwithatotalallocationtimeofabout200minutesperweek.Forstudentswhoselectthesciencetrackatseniorsecondaryschool,mathematicsistaughtwithatotalallocationtimeofabout180minutesperweek.

Geometry,asapartofmathematics,isalsotaughtinelementary,junior,andseniorsecondaryschool. As shown in Table 1.2, at junior secondary school, students learn properties,measurementof two-and three-dimensionalfiguresand theirapplications in contextualproblems.And,atseniorsecondaryschool,thegeometrycontentsfocusonpropertiesofperpendicularandparallellines,spacediagonalsandmeasurement(e.g.,distance,areaoftrianglesviatrigonometry).Inthecontextofthisresearchproject,itisimportanttonotethatgeometryproofisnotanofficiallearninggoalinelementaryandsecondaryschool.

Chapter 1

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Table 1.2.Topicsofgeometryinsecondaryeducationbasedoncurriculum2013(MinistryofEducationandCulture,2013a,2016)

Level Grade 2013 2016JuniorSecondarySchool

7 Areaandperimeteroftwo-dimensionalfigures(square,rectangle,rhombus,trapezoid,kite,parallelogram)andirregularshapes

Perimeterandareaoftrianglesandquadrilaterals(square,rectangle,rhombus,trapezoid,kite,parallelogram)

Transformation(rotation,dilatation,translation,reflection)ofobjects

8 Circle:components,area,perimeter/circumference,arc,sector,segment,centralangles,inscribedangle

Pythagorastheorem,triplePythagoras,anditsapplicationincontextualproblems

Pythagorastheorem,triplePythagoras,anditsapplicationinpatternofnumbers

Circle:components,area,perimeter/circumference,arc,sector,segment,centralangles,inscribedangle

Areaandvolumeofrectangularcuboids(cube,prism),pyramids

Areaandvolumeofrectangularcuboids(cube,prism),pyramids

Areaandvolumeofirregularthree-dimensionalshapes

9 Congruenceandsimilarityanditsapplicationincontextualproblems

Transformation(rotation,dilatation,translation,reflection)ofobjectgeometry

Areaandvolumeofcylinder,cone,andsphere

Congruenceandsimilarityanditsapplicationincontextualproblems

Approximateareaandvolumeofirregularthree-dimensionalshapes

Areaandvolumeofcylinder,cone,andsphere

SeniorSecondarySchool

10 Distanceandangleofpoints,lines,andspace

Ratiooftrigonometryinarighttriangle

Ratiooftrigonometryinarighttriangle Trigonometryfunctionviaacircle

11 Perpendicularandparallellines

Areaoftrianglesviasineandcosine

12 Three-dimensionalproperties(spacediagonal)

Distancesinspace(points,pointandline,pointandplane)

ProspectiveMathematicsTeacherpreparationandEducation

In Indonesia, a teacher education institution (TEI), known as LPTK (Lembaga PendidikanTenagaKependidikan),isaninstitutionthatpreparesteachersandeducationsupportstaff,suchaslaboratorystaff,schooladministrationstaff,andlibrarystaff.Therearetwotypesof teachereducation institutions (TEI):privateandpublic institutions.Also, two typesofpublicTEIs.ThefirsttypeareuniversitieshistoricallyfoundedasteachertrainingcollegesfocusingonteacherpreparationandknownasIKIP(InstitutKeguruandanIlmuPendidikan).Thesecond typeofpublicTEIsaregeneralpublicuniversities thathistorically includedasmall teachereducationprogramsuchas FKIP (FakultasKeguruandan IlmuPendidikan).TheprivateTEIsareteachereducationinstitutionswhichareprivatelyownedandmanaged.TheprivateTEIsalsohavetwotypes.ThefirstaretheTEIswhichweretheex-privateIKIP,such asUniversitas AhmadDahlanwhichwas IKIPMuhammadiyah Yogyakarta. And the

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secondprivateTEIsareprivateTEIswhicharesmallerandnewerthanthefirstprivateTEIs.Recently,intotal,thereare380privateTEIsand40publicTEIsinIndonesia.

Since the end of 1990s, most of LPTKs became universities that provide two types ofprograms,namelyeducationalstudyprograms,andnon-educationalstudyprograms.Forinstance,StateUniversityofMalang(UniversitasNegeriMalang)wasknownasIKIPMalangand,since1999,hasgainedawidermandatetoprovidenotonlyeducationalstudyprograms.Sincethen,thedepartmentofMathematicsoffersstudyprogramsinmathematicsandinmathematicseducation.

Since2005,the Indonesiangovernmenthas implementedaTeacherLawNo.14/2005toselectandrecruitgraduatesfromeducationalstudyprogramsinpublicorprivateuniversities(TEI graduates) or from non-educational study programs in public or private university(non-TEI)asschoolteacher.TheTeacherLawNo.14/2005gaveminimumrequirementstobecomeacertifiedprofessionalschoolteacher.Itmeansthatprospectiveteachersshouldfollowat least fouryearsofapre-service teacherprogramandoneyearofapre-or in-serviceteacherprofessionalprogram.Oneofadvantagesoftakingtheprofessionalprogramisthatthecertifiedprofessionalteacherwillgainadoublesalaryasaschoolteacherandasaprofessionalteacher.

Students who graduate from senior secondary school (general, vocational, or religiousschool) can enroll in a pre-service teacher educationprogram in a public or private TEI.Figure 1.1 depicts the general paths of teacher selection and recruitment. To enter thepublicTEIs,studentcandidateshavethreeoptions,namelyselectionvia(1)evaluationofacademic records fromsenior secondary school (SNMPTN), (2)national collegeentranceexam (SBMPTN), or (3) local entrance test (Ministry of Education and Culture, 2020).The first two selection options aremanaged nationally by LTMPT under theMinister ofEducationandCulture,andthelastoneisfacilitatedbytheuniversity.Thepercentagesofstudentswhoareacceptedviathesethreeselectionsareatleast20%viaSNMPTN,40%viaSBMPTNandatleast30%vialocalselection.ThenumberoffemalestudentsatIndonesianundergraduatelevelishigherthanthenumberofmales.Forinstance,in2019,thenumberofnewfemaleentrantsinundergraduateprogramsisslightlyhigherthanthemaleentrants(430,974vs420,378).At thestateuniversityofMalang, thenumberofmalestudents in2019is11,105and17,794forfemalestudents(BadanPusatStatistikKotaMalang,2020).ThenumberoffemalestudentsattheFacultyofMathematicsandSciencein2019ishigherthanthenumberofmales(2,999vs794).

Chapter 1

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Figure1.1.Theroutesofmathematicsschoolteacherselectionandrecruitment

AftergraduatingfromtheTEIsoreducationalstudyprogramsatuniversity,theprospectiveteacherscan(1)applyasateacheratthesecondaryschoolthroughoneofthreetypesofselectiontests:nationaltest,selectionbydistrictauthoritiesorselectionbyschool,or(2)applyforatwo-semesterprofessionaleducation(PPG)tobecomeacertifiedprofessionalteacher.Studentswhograduatefromanon-TEIornon-educationstudyprograminuniversityshouldtakePPGtoapplytobeaschoolteacher.Theinfluxofthisprofessionalprogramarethose studentswho graduate fromTEI (LPTK), thus having educational background, andthosestudentswhograduatefromnon-TEIornon-educationalstudyprograminuniversity(Abadi&Chairani,2020).Consequently,thePPGcurriculumdesignedforTEIandnon-TEIdiffersforeachsemester,coveringtheacademicandprofessionalcompetencies.Table1.3presentscurriculumstructuresoftheprofessionalprogramforeachofthegroups.

Table 1.3.Curriculumstructureoftheprofessionalprogram(PPG)(Abadi&Chairani,2020;MinistryofEducationandCulture,2013b)

Semester TEIinflux Non-TEIinflux

Courses Credits Courses Credits

1 Pedagogicalknowledge 5 Pedagogicalknowledge 10

Contentknowledge 8 Contentknowledge 8

TeachingMaterialsandmicro-teaching

6 TeachingMaterialsandmicro-teaching

9

Researchcompetence:Actionresearch

1 Researchcompetence:Actionresearch

1

2 Teacherinternship 16 Teacherinternship 16

Totalcredits: 36 Totalcredits: 44

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TheGeometryCourseinTheMalangMathematicsCurriculum

In Indonesia, the Higher Education curricula are developed by the university, and as aconsequence they might be different from other universities. The universities preparetheircurriculumbyfollowingtheguidelinesprovidedbythreenationalpolicydocumentswithnational policies and then coordinate thiswithotheruniversities. For instance, thecurriculumdevelopersoftheMathematicsEducationstudyprogramoftheStateUniversityofMalang,whereIcollecteddataforthisPhDresearch,preparedanewcurriculumin2017and then communicated their plans with othermathematics education study programsnationwideinaforummanagedbytheIndonesianMathematicsSociety(IndoMS)(Abadi&Chairani,2020). In this forum, thedevelopersdiscussed theminimumcurriculumthatshouldbe included inallmathematicseducationstudyprograms including theminimumlearningoutcomesonknowledgeandspecificskills.

The discussion in the forum was used to formulate the curriculum offered by theMathematicsEducationstudyprogram.Forinstance,ageometrycoursewasrecommendedtobeincludedinthecurriculum,andthemathematicseducationstudyprogramatStateUniversityofMalangofferedtwogeometrycourses:Basic Mathematics III andAnalytical Geometry.OneoftheintendedlearningoutcomesoftheBasic Mathematics III,whichisanintroductiontoEuclideangeometryproof,isthatstudentsareabletounderstand,constructandevaluategeometryproof.

ThecourseBasic Mathematics III isheldinthefirstsemesteroftheacademicyear.Intotal,there are sixteen coursemeetings of around 150minutes. The prerequisite knowledgeincludesbasicgeometryconcepts,includingspecialtypesofpolygons(rectangle,triangle,isosceles,righttriangle,parallelogram,etc.)andtheirproperties,andbasicconceptsfromlogicsincludingtypesofmathematicalstatements(e.g.,conditionalstatements)andtheirtruth. These concepts are taught in secondary school, and prerequisite courses are notneededtotakethiscourse.ThefollowingTable1.4presentsthedescriptionoftheBasic Mathematics IIIcourse.

Chapter 1

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Table 1.4. DescriptionofthecourseofBasicMathematicsIII(DepartmentofMathematicsUM,2016)

Course BasicMathematicsIIIIntendedlearningoutcomes

Studentsareableto:•understandgeometricalconcepts•thinklogically•comprehendthereadingofgeometryproof•constructgeometryproof•evaluategeometryproof•determinetheareasofpolygonsandcircle•determinesurfaceareaandvolumeofaprism,cylinder,andcone

Topics •definitionofgeometricterms•geometrypropositionintermsofaconditionalstatement•basicaxiomsandtheoremsinEuclideangeometry•directproof•polygons•congruenttriangles•perpendicularity•similarity•circles•areaandvolumeofpolygonandcircle

Prerequisite -Reference Lewis,H.(1971).Geometry: A Contemporary Course. Second Edition. London:VanNostrand.Credits 3

Duringthefirstsixmeetings,thecoursefocusesonteachingandlearningaboutdefinitionsofgeometricalconcepts,suchaslinesegments,angles,midpointofalinesegment,polygons(particularlytriangles),congruenceoftriangles,axiomsorpostulatesandtheoremsrelatedto these concepts, and proof and proving related to congruent triangles. This course isa prerequisite course for other compulsory courses in mathematics education degreeprogram,suchasMathematics Instructional Media,Assessment in Mathematics Instruction,andMathematics Learning Materials.

Research Purposes

It is widely accepted that comprehending and constructing mathematical proof is anessentialtopicatanylevelofmathematicseducation,includinghighereducation.Indonesianstudentsstarttolearnformalmathematicalproofatuniversitylevel,particularlywhentheyenrollinamathematicseducationprogram.Proof,includinggeometryproof,isamaingoalforsomecourses,andaprerequisitecompetenceforothercompulsorycourses.ThePMTslearnmathematicalproofbecauseproofcompetencecouldhelpthemtounderstandandexplainmathematicalconceptstotheirstudents(Heinze&Reiss,2007;Knuth,2002a).Atpresent,proofandprovingarenotthemaingoalsofmathematicslearninginIndonesianhighschool. Itmeans that thePMTswillnot teachgeometricalproof in theschool.But,inthefuture,IbelievethatmathematicalproofshouldbeoneofthemainmathematicallearninggoalsinIndonesiansecondaryschoolmathematics.Consequently,understandingandconstructingaproofisanessentialmathematicalcompetenceforthePMTs.

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Above, IdescribedmyobservationthatmanyofPMTsfaceddifficulties inunderstandingandconstructingproof,particularlyinthecourseBasic Mathematics IIIinwhichtheylearnproofforthefirsttime.Geometricalproofistaughtinthefirstsemesterasastartingpointforstudents,whichispursuedinotherareassuchasproofinalgebra,analysis,andnumbertheory.Withacourseongeometricalproofbasedonfindings fromresearch literature, Iwanttocontributetotheresearchareaby investigatingtheeffectofan interventionforPMTsanddescribingtheprogressoftheirdevelopmentofcompetenciesofgeometryproof.Therefore,themainresearchquestionofthisstudyis:

How does a course on geometry proof support prospective mathematics teachers’ proof competences?

The term competence, here, refers to the knowledge and skills related tomathematicalproof,particularlygeometryproof,suchasconjecturing,proofreadingcomprehensionandproofconstruction.Thus, the intended learningoutcomesof thegeometryproofcourse,seeTable1.4,describemoredetailedwhatstudentsshouldbeabletodointhecontextofproofandproving.

ResearchDesign

ThepurposeofthisresearchistosupportPMTs’proofcompetence.Todoso,wedesigneda teaching intervention and investigated how this intervention could support PMTs indevelopingproofunderstanding.Therefore,ourmethodologicalframeworkisEducationalDesignResearch,becausedesignandresearchareintertwinedinthisproject.Bakker(2018)statedthatdesignresearchprojectshaveanoveralladvisoryaim,butoften includesub-studies with a descriptive, comparative, or evaluative aim. Indeed, this design researchprojectincludessub-studieswithdescriptiveandevaluativeaims.Inthisresearch,Idescribethe pedagogical potential of the teaching intervention to support PMTs in developinggeometry proof competence based on my literature study (see Chapter 2) as well asempiricalstudieswhereIdescribetheinterventionandevaluateitseffectsinconjecturinginChapter3,readingcomprehensioninChapter4andproofconstructioninChapter5.

Myresearchprojectfollowedthephasesofeducationaldesignresearchandalignedwiththeprinciplesofdesignresearchasaninteractivecyclicalprocessindesigningateachinginterventiontosupportstudents’geometryproofcompetence(Bakker,2018).Thephasesof the research include (1) a preparation and design phase, (2) a teaching experiment,and (3)a retrospectiveanalysis.Thedetaileddescriptionof thetimelineof this researchis shown in Table 1.5. I used the first cycle to evaluate the first version of the teachingmaterials.Theseevaluationswereusedtorevisetheteachingmaterials.However,regardingtheexperimentalstudieselaboratedinchapters3,4and5,Ifocusedonandusedthedatafromtheteachinginterventioninthesecondcycle.

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Table 1.5. Designresearchtimeline

Cycle Phase Period Purposes I preparationand

designphaseOctober2017–July2018

•Reviewingextantresearchliteratureonteachingandlearninggeometryproof

•DevelopinginitialHypotheticalLearningTrajectory(HLT),includingaims,learningsequence,tasks

•Developingresearchinstruments:rubricforevaluatingstudentsanswerstotasks,andinterviewscheme

teachingexperiment

August–September2018

•ConductingteachinginterventioninBasic Mathematics III intwoclasses

•Collectingdataduringinterventionandinterviewsessionsretrospectiveanalysis

October2018–March2019

•Analysingdatafrominterventionandinterviewsessions

II redesignphase March–August2019

•RevisingtheinitialhypotheticalLearningTrajectory(HLT)basedontheresultsofdataanalysisandreviewingextantliteratureinthe1stcycle

•Revisingresearchinstruments:rubric,andinterviewschemeteachingexperiment

August–October2019

•ConductingteachinginterventioninBasic Mathematics III intwoclasses

•Collectingdataduringinterventionandinterviewsessionsretrospectiveanalysis

October2019–February2021

•Analysingdatafrominterventionandinterviewsessions

In this research project,we designed an intervention for PMTs following the interactivecycles which are commonly conducted in design research (Plomp, 2013). To do so, wedevelopedaHypothetical LearningTrajectory (HLT)basedonourexperience in teachinggeometryproofinmyuniversityandaliteraturereview(seeTable1.5).TheresultsofourliteraturereviewarepresentedintheChapter2ofthisthesis.TheHypotheticalLearningTrajectoryconsistsofthreecomponents:(a)thelearninggoals,(b)thelearningactivities,and (c) thehypothetical learningprocessasapredictionofhowthestudents’ reasoningandunderstandingwillevolveinthecontextofthelearningactivities(Simon,1995).Duringthe design phase, the HLT guides the design of instructional means including teachingmaterials,lessonplans,studenttasks,instrumentsforassessmentsandthedevelopmentofinstrumentsfordataanalysis,forinstance,aconjecturingmodelwhichisusedasaprotocolforthelecturertopredictstudents’conjecturingactivitiesandfortheresearchertoanalysestudents’ conjecturing.During the intervention, theHLT functions as a guideline for theteachingbythelecturer,andforinterviewingandobservingbytheresearcher.

Structure Of This Thesis

One of the learning outcomes of the geometry proof course under study (see Table1.4) is that students are able to comprehend and construct geometry proof. Proofcomprehension means reading comprehension of proof written in textbooks or lecturenotes,ordemonstratedduring lectures (Mejia-Ramosetal.,2012).Proofconstruction isthereasoningfromprovenfacts(premises)byusingappropriateproperties(e.g.definition,axioms,proventheorems)andlogicallyconnectedstepstoarriveataconclusion(Knuth,2002b).So,asastartingpointtointroduceprooftostudents,weconsiderthatstudents’

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proofcomprehensioncouldbenefitfromattemptsinproofconstruction.Also,conjecturingcouldencouragestudentstoformulateaconjectureinanopenproblem,andthismayhelpthemunderstandandconstructaproofofthatconjecture(Fernández-Leónetal.,2020).Therefore,inthisresearchIfocusedonsupportingPMTs’competenceinconjecturing,proofcomprehensionandconstruction.

Chapter 2 discusses literature about the pedagogical aspects of understanding proof,assessmentof thatunderstandingand the roleofaDynamicGeometrySystem (DGS) inproofandproving.Inanextstep,IimplementedsuggestionsfromliteratureinateachinginterventionsupportingPMTsincomprehendingandconstructinggeometryproof.Chapters3, 4 and 5 report on the intervention with a specially designed course in conjecturinggeometricalpropositiontobeproven,proofcomprehensionandproofconstruction.Thedesign of the intervention, the task-based interviews, and the conjecturingmodel usedtopredictstudents’processesofconjecturing,togetherwithadescriptionandanalysisofPMTs’processesofconjecturingwillbepresented inChapter3.Chapter4will showtheresult of our quasi-experimental study regarding the effect of the use ofmultiple proofformatsusedinourteachinginterventiontosupportPMTs’proofcomprehension.Chapter5will showadetaileddescriptionof thedesignand implementationof the interventionfocusingonproofconstructionandadescriptionofPMTs’progressionofunderstandingthestructureofproof,andtheirabilitiestoconstructtheirowngeometricalproof.Chapter6summarizesalltheresultsandpresentstheoreticalandpracticalimplications.Theappendixcontainsthefulllessonmaterialsincludinglessonplan,tasks,instrumentsofresearchandtherevisedHLT.