Reasoning in geometry How first learning to appreciate the generality of arguments helps students

280 Pages·2012·11.17 MB·English

Reasoning in geometry
How first learning to appreciate the generality of arguments helps
students come to grips with the notion of proof
MScThesis
writtenby
OlgaGrigoriadou
underthesupervisionof Dr. WolterKaper,andsubmittedinpartialfulfillmentofthe
requirementsforthedegreeof
MScinMathematicsandScienceEducation
attheUniversityofAmsterdam.
February2012

Abstract
Oneoftheaimsofteachinggeometryatschoollevelistohelpstudentsunderstandwhatcounts
as an acceptable argument in mathematics (a ‘proof’) and to move from using inductive ar-
guments to using deductive ones for supporting mathematical statements. The research study
described in this thesis focuses on students’ reasoning when they start abandoning inductive
arguments for deductive ones. More specifically, we propose a teaching module (partly based
onexistingliterature)whichassistsstudentsinthislearningprocess,andwepresenttheresults
ofimplementingthismoduleonasampleof20studentsinaGreekhighschool. Weexaminethe
changeinthekindsofreasoningusedbystudentsateachpointoftheinterventionandtheeffect
ofthischangeonthebeliefsofstudentsaboutmathematicsandproof. Ourresultssuggestthat
layingemphasisonmakingcleartothestudentsthedistinctionbetweendeductiveandinductive
argumentsatthebeginningofthelessonscycle,canhelpthemtounderstandbettertheconcept
ofmathematicalproofandtoproduceproofs.

Acknowledgements
First and foremost I would like to thank Wolter, my supervisor. I still have vivid memories
of the moment when I first talked to Wolter about my ideas. He was so enthusiastic that we
kept exchanging long emails about the topic for a few days after, sharing our ideas and our
enthusiasm. This was a great push forward for me, and made me even more excited to put
all my ideas to work. I feel very lucky to have worked with such an enthusiastic, helpful and
hardworkingsupervisor. Wolter’sfeedbackandsupportalwayscamerightwhenIneededthem,
and we exchanged productive criticism at all times. Without him, the way to the end of this
projectwouldhavebeenmuchmoredifficult.
Iwouldalsoliketothankthestudentsthatparticipatedintheteachingintervention. These
studentsofferedwithenthusiasmmuchoftheirfreetimeinreturnfornewexperiencesinmath-
ematics. Igreatlyappreciatetheefforttheyputin,allthewaythroughtheintervention,andtheir
feedback. It was my first teaching experience, and I feel lucky that this was with such lovely
andmotivatedstudents. IthankthemallwholeheartedlyandIwishthemgoodluckinwhatever
they do in their lives! Of course, I would not have met these students had it not been for the
cooperatingteacher,MakisPanteliadis,whogavemepermissiontoworkwiththestudentsfrom
hisclassesforthisproject. Hiscomments,helpandsupportwereatalltimesuseful,forwhich
Iwouldliketothankhim.
Finally,Iwouldliketothankallthepeoplethatsupportedmeduringthisproject: myteach-
ers and fellow students at the AMSTEL institute who guided me and shared their experiences
withme;myfriendsallovertheworldforsupportingmeandhelpingmeforgetaboutthisproject
duringmyholidays;Olga,Rodi,VassilisandGiannisfortakingpicturesandvideosduringthe
intervention;myprofessorAnastasiosPatronisfromtheUniversityofPatrasforintroducingme
to the world of mathematics education and for the fruitful discussions we had during the first
stagesofthisproject;myparentsforteachingmetoneverstoplearning;andmypartnerTikitu
forproofreadingandhelpingmestructurethisthesisandforbeingthereformewhateverIdo.

Contents
Contents i
ListofFigures v
ListofTables vii
1 Introduction 1
2 TheoreticalFramework 3
2.1 Themeaningofproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Deductiveversusinductivereasoning . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Geometryasadeductivescience . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Previousresearchonstudents’geometricalreasoning . . . . . . . . . . . . . . 6
2.4.1 TheVanHieletheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4.2 Schoenfeldonstudents’beliefsaboutproof . . . . . . . . . . . . . . . 7
2.4.3 TheHarel&Sowderproofschemes . . . . . . . . . . . . . . . . . . . 8
2.4.4 TheprojectJustifyingandProvinginSchoolMathematics(JPSM) . . . 15
2.5 VanHielelevelsandHarel&Sowderproofschemes: Acorrespondence . . . . 16
2.6 PreviousresearchontheVanHieletheory . . . . . . . . . . . . . . . . . . . . 17
2.7 TeachingstrategiesassistingthemovethroughtheVanHielelevels . . . . . . 18
2.8 Conclusionsandhowourresearchbuildsontheliterature . . . . . . . . . . . . 22
3 AimsandResearchQuestions 25
3.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 ResearchQuestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 ResearchSetting 29
4.1 Theplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 TheGreekeducationalsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 GeometryintheGreekcurriculum . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 ThegeometrybooksoftheGreekGymnasium . . . . . . . . . . . . . . . . . . 30
5 Methodology,TeachingDesignandAnalysisFramework 35
5.1 Identifyingthestudents’VanHielelevels . . . . . . . . . . . . . . . . . . . . 36
5.2 Identifyingthestudents’kindsofreasoning . . . . . . . . . . . . . . . . . . . 36
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CONTENTS
5.2.1 AdaptingtheinstrumentsoftheJPSMprojecttoourresearch . . . . . 36
5.2.2 Classifyingthestudents’kindsofreasoningbasedontheHarel&Sow-
derproofschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Identifyingthestudents’beliefsaboutmathematicsandproof . . . . . . . . . . 40
5.3.1 TheadaptedSchoenfeldquestionnaire . . . . . . . . . . . . . . . . . . 40
5.3.2 Ourquestionnaireaboutwhatthestudentsbelievetheylearned . . . . . 40
5.3.3 AnadaptedversionofcertainJPSMprojecttasks . . . . . . . . . . . . 41
5.4 DeterminingthecorrelationbetweenproofschemesandVanHielelevels . . . 41
5.5 Designingandevaluatingateachingstrategy . . . . . . . . . . . . . . . . . . 41
5.5.1 Themovefromthedescriptivetothetheoreticallevelinthreestages . . 42
5.5.2 ThefiveVanHielephasesforeightconceptsinourteachingsequence . 44
5.5.3 Methodforevaluatingourteachingstrategy . . . . . . . . . . . . . . . 49
5.6 Otherdatacollectionmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 TheIntervention 51
6.1 Populationandsample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Studentparticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 Diaryoftheintervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 Results 65
7.1 TheUsiskinVanHieletestresults . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 TheHarel&Sowderproofschemescodingprocess . . . . . . . . . . . . . . . 69
7.3 CorrelationsbetweenVanHielelevelandHarel&Sowderproofscheme . . . . 78
7.4 Theroleofourteachingstrategy: Aqualitativeanalysis . . . . . . . . . . . . . 82
7.4.1 Movingthroughthethreestagestowardsdeductiveproofschemes . . . 82
7.4.2 Moving through the five Van Hiele teaching phases towards the theo-
reticalVanHielelevel . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.5 Students’beliefsaboutmathematicsandproof . . . . . . . . . . . . . . . . . . 120
7.5.1 ModifiedSchoenfeldquestionnaireresults . . . . . . . . . . . . . . . . 120
7.5.2 Students’ownideasonhowtheirbeliefschanged . . . . . . . . . . . . 124
7.5.3 Students’ understanding of the validity, convincingness and generality
ofarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8 ConclusionsandDiscussion 149
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.1.1 AnsweringSQ.1andH.1 . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.1.2 AnsweringSQ.2andH.2 . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.1.3 AnsweringSQ.3andH.3 . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.1.4 Answeringourmainresearchquestion . . . . . . . . . . . . . . . . . . 152
8.2 Discussionandrecommendations. . . . . . . . . . . . . . . . . . . . . . . . . 153
References 159
FurtherReading 165
ii
CONTENTS
A Pre-andpost-testsandquestionnaires 167
A.1 TheoriginalUsiskinVanHieletest . . . . . . . . . . . . . . . . . . . . . . . . 168
A.2 TheVanHieletestasusedinourresearch . . . . . . . . . . . . . . . . . . . . 181
A.3 ThemodifiedSchoenfeldquestionnaire . . . . . . . . . . . . . . . . . . . . . 190
A.4 Beliefs&reflectionsquestionnaire . . . . . . . . . . . . . . . . . . . . . . . . 192
B Taskrelatedworksheets 193
B.1 TRIANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
B.2 ANGLESUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
B.3 CHOOSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
B.4 CONCEPTIONSABOUTPROOF . . . . . . . . . . . . . . . . . . . . . . . . 205
B.5 QUAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
B.6 OPPANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.7 BIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.8 BIS-GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.9 EXT-GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
B.10 TRIMID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
B.11 TRIMID-GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
B.12 QUADMID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
B.13 IFTHEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
C Handouts 215
C.1 Invitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
C.2 Introductiontothelessonscontentandexpectations . . . . . . . . . . . . . . . 218
C.3 Theprosandconsofarguments . . . . . . . . . . . . . . . . . . . . . . . . . 219
C.4 Hownewtruthsarediscoveredingeometry . . . . . . . . . . . . . . . . . . . 222
C.5 Whatwediscoveredinthegeometrylessons . . . . . . . . . . . . . . . . . . . 223
C.6 Wheredoesitallbegin? Themapofaxioms . . . . . . . . . . . . . . . . . . . 240
C.7 ThestoryofEuclideangeometry . . . . . . . . . . . . . . . . . . . . . . . . . 241
D Tablesandresults 243
D.1 VanHielepre-test: Analysisandcomparisonwithotherstudies . . . . . . . . . 244
D.2 DistributionofanswerstotheVRandEPquestionsinTaskCHOOSING . . . 256
D.3 VRandEPscoresbeforeandafterourintervention . . . . . . . . . . . . . . . 258
D.4 Moreresultsofthebeliefs&reflectionsquestionnaire . . . . . . . . . . . . . . 258
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iv