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Learning Pathways within the Multiplicative Conceptual Field - Insights Reflected through a Rasch Measurement Framework
von: Caroline Long
Waxmann Verlag GmbH, 2015
ISBN: 9783830982890 , 248 Seiten
Format: PDF, OL
Kopierschutz: frei
Preis: 35,99 EUR
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Learning Pathways within the Multiplicative Conceptual Field - Insights Reflected through a Rasch Measurement Framework
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
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