Computer Aided Assessment of Mathematics

196 Pages·2013·1.689 MB·English

by  Chris Sangwin

COMPUTER AIDED ASSESSMENT OF MATHEMATICS
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Computer Aided Assessment of Mathematics
CHRIS SANGWIN
SchoolofMathematics,UniversityofBirmingham
3
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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©ChrisSangwin2013
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CONTENTS
ListofFigures viii
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Multiple-choicequestions 2
1.2 Assessmentcriteria 4
1.3 Chapters 7
1.4 Acknowledgements 8
2 Anassessmentvignette. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
2.1 Thestudent’sperspective 9
2.2 Assessinganswerstosimplequestions 14
2.3 Furtherintegrals 16
2.4 Discussion 18
3 Learningandassessingmathematics . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Thenatureofmathematics 19
3.2 Termsusedinassessment 21
3.3 Purposesofassessment 22
3.4 Learning 23
3.5 Principlesandtensionsofassessmentdesign 25
3.6 Learningcyclesandfeedback 33
3.7 Conclusion 35
4 Mathematicalquestionspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1 Whyrandomlygeneratequestions? 38
4.2 Randomlygeneratinganindividualquestion 39
4.3 Linkingmathematicalquestions 42
4.4 Buildingupconceptions 44
4.5 Typesofmathematicsquestion 46
4.6 EmbeddingCAAintogeneralteaching 49
4.7 Conclusion 51
5 Notationandsyntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1 Anepisodeinthehistoryofmathematicalnotation 54
5.2 Theimportanceofnotationalconventions 56
5.3 Ambiguitiesandinconsistenciesinnotation 60
5.4 Notationandmachines:syntax 61
5.5 Otherissues 65
5.6 TheuseoftheAiMsystembystudents 66
vi | CONTENTS
5.7 Proofandarguments 67
5.8 Equationeditors 68
5.9 Dynamicinteractions 70
5.10 Conclusion 71
6 ComputeralgebrasystemsforCAA. . . . . . . . . . . . . . . . . . . . . . . . 73
6.1 Theprototypetest:equivalence 75
6.2 AcomparisonofmainstreamCAS 76
6.3 TherepresentationofexpressionsbyCAS 78
6.4 Existenceofmathematicalobjects 82
6.5 ‘Simplify’isanambiguousinstruction 86
6.6 Equality,equivalence,andsameness 88
6.7 Formsofelementarymathematicalexpression 91
6.8 Equations,inequalities,andsystemsofequations 94
6.9 Othermathematicalpropertieswemightseektoestablish 96
6.10 Buggyrules 97
6.11 GeneratingoutcomesusefulforCAA 99
6.12 Sideconditionsandlogic 100
6.13 Conclusion 101
7 TheSTACKCAAsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1 Background:theAiMCAAsystem 102
7.2 DesigngoalsforSTACK 103
7.3 STACKquestions 106
7.4 ThedesignofSTACK’smulti-parttasks 107
7.5 Interactionelements 111
7.6 Assessment 112
7.7 Qualitycontrolandexchangeofquestions 113
7.8 ExtensionsanddevelopmentoftheSTACKsystembyAalto 114
7.9 UsagebyAalto 117
7.10 Studentfocusgroup 121
7.11 Conclusion 125
8 Softwarecasestudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.1 Someearlyhistory 127
8.2 CALM 129
8.3 Pass-IT 132
8.4 OpenMark 138
8.5 DIAGNOSYS 140
8.6 Cognitivetutors 146
8.7 KhanAcademy 147
8.8 Mathwise 148
8.9 WeBWorK 150
8.10 MathXpert 154
8.11 Algebratutors:AplusixandT-algebra 157
8.12 Conclusion 160
CONTENTS | vii
9 Thefuture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.1 Encodingacompletemathematicalargument 162
9.2 Assessmentofproof 166
9.3 Semi-automaticmarking 169
9.4 Standardsandinteroperability 170
9.5 Conclusion 172
Bibliography 173
Index 183
LIST OF FIGURES
1.1 Incorrectlongmultiplication....................................... 2
1.2 AsamplequestionfromtheSTACKCAAsystem. ...................... 4
2.1 AsimpleSTACKCAAquestion. ................................... 10
2.2 Validationofasimpleanswer. ..................................... 11
2.3 Feedbackbasedonproperties. ..................................... 12
2.4 Invalidityformathematicalreasons.................................. 12
2.5 Incompleteanswers:anopportunityforpartialcredit. ................... 13
2.6 Acorrectanswer. ............................................... 13
2.7 Ageneraltestforindefiniteintegrationproblems. ...................... 17
4.1 Asequenceofsimplequestions .................................... 43
4.2 DynamicmathematicswithGeoGebra. .............................. 50
5.1 TheDragMathequationeditor. .................................... 68
5.2 Dynamicinteractions. ........................................... 70
5.3 SketchingthederivativeinWeBWork................................ 71
6.1 Mathematicalobjectsandtheirrepresentation. ........................ 83
6.2 Feedbackabouttheformofananswer................................ 92
6.3 Ananswerwhichisasystemofequations. ............................ 95
7.1 AnAiMquestion,asseenbyastudent................................ 103
7.2 Manyparts,oneassessmentalgorithm................................ 108
7.3 Manyparts,independentalgorithms. ................................ 108
7.4 Manyparts,manyalgorithms. ..................................... 109
7.5 AnexampleofdisplayfromtheAaltoSTACKsystem. ................... 115
7.6 Muti-partquestionwithrandomlygenerateddiagrams. .................. 116
7.7 AquestionincludingaGeoGebraapplet. ............................. 117
7.8 Thegradingsystemonthecoursediscretemathematics.................. 119
7.9 Studentscoresfromexaminationsandexercises........................ 120
8.1 FirstphaseoftheCALMCAAsystem................................ 130
8.2 APass-ITquestion,beforestepsarerevealedtothestudent. .............. 133
8.3 APass-ITquestion,afterstepshavebeenrevealedtothestudent. .......... 133
8.4 APass-ITquestioninwhichpartoftheanswerisagraph. ................ 134
8.5 ASCHOLARquestioninwhichpartoftheanswerisapie-chart............ 135
8.6 ProgressivefeedbackfromOpenMark................................ 139
8.7 MathinputpanelfromtheDIAGNOSYSsystem........................ 141
8.8 TheDIAGNOSYSsystem......................................... 141
LISTOFFIGURES | ix
8.9 DIAGNOSYSalgebraskills........................................ 143
8.10 DIAGNOSYSskills.............................................. 144
8.11 Skilllatticeforsubtraction......................................... 145
8.12 AnexampleoftheCognitiveTutor. ................................. 147
8.13 Mathwisevectorsmodule:thedivergencequiz. ........................ 150
8.14 TheuserinterfaceintheMathXpertsystem. .......................... 155
8.15 TrackingsideconditionsintheMathXpertsystem. ..................... 156
8.16 TheAplusixsystem. ............................................. 158
8.17 BacktrackinginAplusix. .......................................... 159
9.1 ProofintheEASysystem. ........................................ 167