Buchtitel	1
	Abstract	7
	Acknowledgements	11
	Prologue	13
	Table of contents	17
	1 A prospective pathway for meeting mathematics education challenges	19
	1.1 Mathematical knowledge	19
	1.1.1 Towards a framework	20
	1.2 Theoretical framework	21
	1.2.1 Theory of conceptual fields	21
	1.2.2 Educational measurement	23
	1.3 Problem statement	24
	1.3.1 Global concern over mathematics education	24
	1.3.2 Perceived factors influencing under-performance	25
	1.4 Research focus	27
	1.4.1 Research questions	28
	1.4.2 Research design	30
	1.4.3 Literature review	30
	1.4.4 Investigation of the multiplicative conceptual field	32
	1.5 Summary: A prospective pathway	34
	2 Threshold concepts in the unfolding number systems	35
	2.1 From intuitive notions into explicit knowledge	35
	2.1.1 Research questions	37
	2.2 Epistemological context	37
	2.3 Unfolding number systems	38
	2.3.1 From number sense to a number system	39
	2.3.2 Natural number systems	41
	2.3.3 Integers	42
	2.3.4 Rational number system	42
	2.3.5 Real number system	43
	2.3.6 Complex number system	43
	2.3.7 Algebra	44
	2.4 Summary: Central factors in mathematical development	44
	3 Theory of conceptual fields: Essential domains informing teaching and learning	46
	3.1 Embracing the complexity in learning mathematics	46
	3.1.1 Components of the theory	47
	3.1.2 Research questions	48
	3.2 Conceptual domain	49
	3.2.1 Mathematical concept as a “triple of sets”	49
	3.2.2 Conceptual fields	50
	3.2.3 Some factors in development of mathematics knowledge	51
	3.3 Cognitive domain	52
	3.3.1 The subject and the external world	52
	3.3.2 Operational-structural relations	54
	3.3.3 Threshold concepts	55
	3.3.4 From schemes and situations to generalisable concepts	55
	3.3.5 An integration of key ideas	57
	3.4 Didactic domain	58
	3.4.1 Nurturing the learning process	58
	3.4.2 The teacher’s role	59
	3.5 Semiotic domain	59
	3.5.1 The status of knowledge	59
	3.5.2 Developmental stages towards greater abstraction	60
	3.5.3 Language, an elaborated social system	60
	3.5.4 Summary: Language precision and mathematics	61
	3.6 Evaluative domain	61
	3.6.1 Assessment for learning	62
	3.7 Summary: Consequences for educational research and measurement	62
	4 Assessment and measurement: A discussion of core requirements	65
	4.1 From mathematics to measurement	65
	4.1.1 Research questions	65
	4.1.2 Large-scale assessment and learning	67
	4.2 A theory of mathematics assessment	68
	4.2.1 Conceptions of mathematics	68
	4.2.2 Critical elements for the formulation of an assessment programme	69
	4.2.3 Core notions for assessment	72
	4.3 Measurement and the Rasch model	72
	4.3.1 Measurement	73
	4.3.2 Mathematical models	75
	4.3.3 The development of the Rasch model	76
	4.3.4 Validity	81
	4.3.5 Reliability	82
	4.3.6 Core ideas underpinning the Rasch model	82
	4.4 Validity of assessment practices	83
	5 The multiplicative conceptual field	85
	5.1 Mathematical structure and developmental consequences	85
	5.1.1 Research questions	86
	5.2 Multiplication and division	87
	5.2.1 Problem situations	87
	5.2.2 Extension to rational numbers	89
	5.2.3 Multiplicative structures	90
	5.2.4 Building the base for rational number	97
	5.3 Rational number	97
	5.3.1 Rational number sub constructs	97
	5.3.2 Operations on fractions	103
	5.3.3 Synthesis of rational number	104
	5.3.4 Proportional reasoning	105
	5.3.5 Functional relationship and link to calculus	108
	5.3.6 Considering salient features	109
	5.4 Percent	110
	5.4.1 Mathematical Structure	111
	5.4.2 The language of percent	114
	5.4.3 Tasks and problems	115
	5.4.4 A concise language with important consequences	116
	5.5 Probability	117
	5.5.1 Mathematical structure	117
	5.5.2 Historical factors	118
	5.5.3 The acquisition of probabilistic concepts	118
	5.5.4 A distinctive reasoning	118
	5.6 Proficiency in the multiplicative conceptual field	118
	5.7 Summary: Didactic implications, assessment and research	120
	6 Exploration of data within the Rasch measurement framework	122
	6.1 Understanding complexity through application of the Rasch model	122
	6.1.1 Research questions	122
	6.2 Methodology for the empirical investigation	122
	6.2.1 Test development within a Rasch measurement framework	123
	6.2.2 Participants	123
	6.2.3 Test formulation	124
	6.2.4 Test situation, administration and scoring	126
	6.2.5 Data Analysis	127
	6.3 Analytic framework for item analysis	135
	6.3.1 Contextual factors	136
	6.3.2 Type of situation	136
	6.3.3 Mathematical structure	137
	6.3.4 Mode of representation	138
	6.3.5 Number range and value	138
	6.3.6 Response processes and procedures	139
	6.4 Item analysis	140
	6.4.1 Item by strand analysis	142
	6.5 Fraction item analysis	143
	6.5.1 Critical findings: Fraction items at Levels 1, 2, 3 and 4	146
	6.6 Ratio, proportion and rate item analysis	148
	6.6.1 Critical findings: Ratio, rate and proportion items at Levels 1 to 7	150
	6.7 Percent item analysis	153
	6.7.1 Critical findings: Percent items at Levels 2, 3, 4, and 7	155
	6.8 Probability item analysis	157
	6.8.1 Critical findings: Probability items at Levels 2, 3 and 4	159
	6.9 Pre-Algebra item analysis	161
	6.9.1 Critical findings: Pre-Algebra items at Levels 2, 3, 4 and 5	163
	6.10 Summary descriptions at Levels 1 to 7	165
	6.10.1 Critical points and threshold concepts	170
	6.10.4 Reflections and further insights	171
	7 Identifying threshold concepts in reasoning behind item responses	172
	7.1 Tracking learner competences	172
	7.1.1 Research questions	173
	7.2 Research method	173
	7.3 Framework for interview analyses	177
	7.4 High proficiency learners	182
	7.4.1 Levels 6 and 7: Adele (School A), Anna (School B)	182
	7.4.2 Level 5: Kelly, Jane, Angela, Carla (School A), Prinella (School B)	186
	7.4.3 Proficiency exhibited at Levels 5, 6 and 7	194
	7.5 Middle-high proficiency	196
	7.5.1 Level 4, Thembani and Sipho (School B)	196
	7.5.2 Level 4: Shiluba, Carola, Linda and Kate (School A)	199
	7.5.3 Proficiency exhibited at Level 4	205
	7.6 Middle-low proficiency	206
	7.6.1 Level 3, Phaphama, Maria, Mpho (School B)	206
	7.6.2 Level 3: Cheryl and Zanele (School A)	211
	7.6.3 Proficiency exhibited at Level 3	215
	7.7 Low proficiency	216
	7.7.1 Level 1: Mishack, Amukelani and Mahesh (School B)	216
	7.7.2 Proficiency exhibited at Level 1	218
	7.8 Overview of four proficiency levels	219
	7.9 Theoretical insights from the theory of conceptual fields	221
	7.10 Recommendations for the instrument	222
	7.11 Reflections on the interviews	223
	8 Addressing complexity: Implications for curriculum, teaching and assessment	224
	8.1 Answering Poincaré	224
	8.2 Insights from the theory of conceptual fields	224
	8.3 Insights from the perspective of assessment and measurement	225
	8.3.1 Rasch analysis and the theory of conceptual fields	226
	8.3.2 Person-item map	226
	8.3.3 Cognitive and pedagogical insights	227
	8.4 Implications for curriculum, teaching and research	227
	8.4.1 Levels of development	228
	8.4.2 Identifying threshold concepts	230
	8.5 Reflections and limitations	232
	8.5.1 Instrument development recommendations	232
	8.5.2 Limitations of the study	233
	8.6 Future Research	235
	8.7 Conclusion	236
	9 References	238
	List of Abbreviations	245
	List of Figures	246
	List of Tables	247