TABLE OF CONTENTS

DECLARATION ……………………………………………………………………………………………. i
ACKNOWLEDGEMENTS ……………………………………………………………………………… ii
ABSTRACT ………………………………………………………………………………………………… iv
TABLE OF CONTENTS ……………………………………………………………………………….. vi
LIST OF FIGURES ……………………………………………………………………………………….. x
LIST OF TABLES ……………………………………………………………………………………….. xi
CHAPTER 1: INTRODUCTION AND BACKGROUND ………………………………………. 1
1.1 THE RESEARCH PROBLEM ……………………………………………………………… 1
1.1.1 Background: South African early grade classrooms …………………………. 2
1.1.2 Number concept development: learning mathematics for teaching……… 5
1.2 RESEARCH QUESTION ……………………………………………………………………. 7
1.3 THE AIM AND OBJECTIVES OF THE STUDY ……………………………………… 8
1.3.1 Objectives ………………………………………………………………………………….. 8
1.4 THEORETICAL BACKGROUND …………………………………………………………. 9
1.5 METHODS AND DESIGN ………………………………………………………………… 10
1.5.1 Research design ……………………………………………………………………….. 10
1.5.2 Sampling ………………………………………………………………………………….. 11
1.5.3 Data collection ………………………………………………………………………….. 11
1.5.4 Data analysis ……………………………………………………………………………. 12
1.6 RESEARCH ETHICS ………………………………………………………………………. 13
1.7 TRUSTWORTHINESS …………………………………………………………………….. 14
1.8 STRUCTURE OF THE STUDY …………………………………………………………. 15
1.9 SUMMARY …………………………………………………………………………………….. 16
CHAPTER 2: ASPECTS OF YOUNG CHILDREN’S EARLY MATHEMATICAL
CONCEPT DEVELOPMENT ………………………………………………………………………… 17
2.1 INTRODUCTION …………………………………………………………………………….. 17
2.2. INITIAL THEORIES OF CHILDHOOD COGNITIVE DEVELOPMENT …….. 20
2.3 CORE COGNITION OF NUMBER …………………………………………………….. 24
2.3.1 Core system 1: the core representation of numerical magnitude –
approximate number system (ANS) ………………………………………………………… 25
2.3.2 Core system 2: the precise representation of distinct entities or small
quantities – object tracking system (OTS) …………………………………………………. 26
2.3.3 Symbolic learning of number……………………………………………………….. 27
2.4 A CONCEPTUAL MODEL OF NUMBER DEVELOPMENT …………………… 30
2.5 TEACHING MATHEMATICAL CONCEPTS AND PEDAGOGY ……………… 42
2.6 LANGUAGE AND THE TEACHING OF MATHEMATICS ……………………… 47
2.7 CONCLUSION ……………………………………………………………………………….. 50
CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY ……………………………. 52
3.1 INTRODUCTION …………………………………………………………………………….. 52
3.2 RESEARCH DESIGN TYPE …………………………………………………………….. 52
3.2.1 The MARKO-D test ……………………………………………………………………. 53
3.2.2 Interviews and observations ……………………………………………………….. 54
3.3 METHOD: DATA COLLECTION PROCESS ……………………………………….. 54
3.4 METHOD: SAMPLING …………………………………………………………………….. 57
3.5 METHOD: COLLECTION OF DATA…………………………………………………… 59
3.5.1 MARKO-D test ………………………………………………………………………….. 59
3.5.2 Observations: MARKO-D test (a) and Classroom (b) ……………………… 62
3.5.3 Interviews …………………………………………………………………………………. 64
3.6 DATA ANALYSIS ……………………………………………………………………………. 65
3.7 ETHICAL CONSIDERATIONS ………………………………………………………….. 66
3.8 CHAPTER SUMMARY …………………………………………………………………….. 67
CHAPTER 4: RESEARCH RESULTS AND DISCUSSIONS OF FINDINGS ON
GRADE 1 TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE AND THE
TEACHING OF MATHEMATICS IN THE CLASSROOM …………………………………. 69
4.1 INTRODUCTION …………………………………………………………………………….. 69
4.2 DATA SETS 1 AND 2 – RESULTS OF THE MARKO-D TEST ………………. 72
4.2.1 Observations during the MARKO-D test ……………………………………….. 88
4.3 DATA SETS 3 AND 4 – TEACHER PEDAGOGY: QUALITATIVE DATA …. 95
4.3.1 Data set 3: Classroom observations …………………………………………….. 96
4.3.2 Data set 4: Interviews with the Grade 1 teachers ………………………….. 103
4.4 ANALYSIS OF THE RAW DATA FROM QUANTITATIVE AND
QUALITATIVE METHODS ………………………………………………………………………. 117
4.5 CATEGORIES THAT WERE DERIVED FROM THE DATA …………………. 117
4.6 FINAL THEMES ABSTRACTED FROM THE RAW DATA …………………… 120
4.6 CONCLUSION ……………………………………………………………………………… 121
CHAPTER 5: DISCUSSION OF FINDINGS, RECOMMENDATIONS AND
CONCLUSION …………………………………………………………………………………………. 122
5.1 INTRODUCTION …………………………………………………………………………… 122
5.2 DISCUSSION OF FINDINGS ………………………………………………………….. 123
5.2.1 The clear usage of language to teach and explain mathematics
throughout Grade 1 is essential for learning ……………………………………………. 123
5.2.2 Well-trained teachers use different strategies and evaluation
procedures, to ensure maximum learning in mathematics lessons …………….. 127
5.2.3 Concepts at different levels of mathematical cognitive development
are taught throughout schooling and some specifically in Grade 1 …………….. 133
5.2.4 Children’s approach to mathematics activities highlight the difficulties
they encounter ……………………………………………………………………………………. 137
5.2.5 The use of concrete materials fulfils an important role in early grade
mathematics learning ………………………………………………………………………….. 140
5.3 THE LIMITATIONS OF THIS STUDY……………………………………………….. 141
5.4 RECOMMENDATIONS ………………………………………………………………….. 142
5.5 SUMMARY …………………………………………………………………………………… 143
REFERENCES …………………………………………………………………………………………. 145
APPENDIX A: ETHICS CLEARANCE …………………………………………………………. 160
APPENDIX B: PARENT/GUARDIAN CONSENT LETTER (PRE-TEST) ………….. 166
APPENDIX C: PARENT/GUARDIAN CONSENT LETTER (POST-TEST) ………… 167
APPENDIX D: MARKO-D RAW SCORE PRE-TEST RESULTS……………………… 168
APPENDIX E: MARKO-D RAW SCORE POST-TEST RESULTS …………………… 176
APPENDIX F: PRE-TEST RESULTS – LEVELS …………………………………………… 198
APPENDIX G: POST-TEST RESULTS – LEVELS ………………………………………… 204
APPENDIX H: CLASS A – PRE- AND POST-TEST LEVEL PERCENTAGE …….. 212
APPENDIX I: CLASS B – PRE- AND POST-TEST LEVEL PERCENTAGE………. 217
APPENDIX J: SCRIPTS FOR MATHEMATICS LESSONS…………………………….. 222
APPENDIX K: INTERVIEW QUESTIONS WITH GRADE 1 TEACHERS ………….. 230
APPENDIX L: TRANSCRIPT OF INTERVIEW QUESTIONS WITH GRADE 1
TEACHERS ……………………………………………………………………………………………… 231
APPENDIX M: CODES TEACHERS’ INTERVIEWS – TEACHER A ………………… 240
APPENDIX N: CODES TEACHERS’ INTERVIEWS – TEACHER B ………………… 243