Long Learning Pathways within the Multiplicative Conceptual Field

Insights Reflected through a Rasch Measurement Framework
1. Auflage 2015
ISBN: 978-3-8309-8289-0
Verlag: Waxmann Verlag GmbH
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 248 Seiten

The transition from whole numbers to rational numbers and the associated mastery of the multiplicative conceptual field constitute an important development in lower secondary schooling. This study draws primarily on the theory of conceptual fields as a framework that is mathematical and enables a cognitive perspective by identifying the concepts- and theorems-in-action that lead to underlying concepts and theorems.
Application of the Rasch model configures the location of both item difficulty and learner proficiency on one scale. Diagnostics explore the validity of the instrument for measurement. The ordering of items enables the analysis of hierarchical conceptual strands and additional insights into the mastery of concepts by subsets of learners at particular levels. The resulting matrix, of interactions of learner proficiency and item complexity, provides an overview of the concepts attained and not yet mastered. These insights permit teacher interventions specific to each learner subset at a shared common current zone of proximal development along the scale.

Caroline Long has received her doctorate in Mathematics Education from the University of Cape Town in 2011 and is Senior Lecturer in the Faculty of Education at the University of Pretoria, where she is responsible for teaching mathematics education courses, and modules on assessment. She is also Deputy Director at the Centre for Evaluation and Assessment. Her primary research foci are mathematics education, professional development, teacher agency and assessment. Current work relies on collaboration with researchers at other South African institutions, and in Australia, Canada, England, Germany, India, the Netherlands, Scotland and the USA.

Autoren/Hrsg.

Long, Caroline

Inhaltsverzeichnis

1;Buchtitel;1
2;Abstract;7
3;Acknowledgements;11
4;Prologue;13
5;Table of contents;17
6;1 A prospective pathway for meeting mathematics education challenges;19
6.1;1.1 Mathematical knowledge;19
6.1.1;1.1.1 Towards a framework;20
6.2;1.2 Theoretical framework;21
6.2.1;1.2.1 Theory of conceptual fields;21
6.2.2;1.2.2 Educational measurement;23
6.3;1.3 Problem statement;24
6.3.1;1.3.1 Global concern over mathematics education;24
6.3.2;1.3.2 Perceived factors influencing under-performance;25
6.4;1.4 Research focus;27
6.4.1;1.4.1 Research questions;28
6.4.2;1.4.2 Research design;30
6.4.3;1.4.3 Literature review;30
6.4.4;1.4.4 Investigation of the multiplicative conceptual field;32
6.5;1.5 Summary: A prospective pathway;34
7;2 Threshold concepts in the unfolding number systems;35
7.1;2.1 From intuitive notions into explicit knowledge;35
7.1.1;2.1.1 Research questions;37
7.2;2.2 Epistemological context;37
7.3;2.3 Unfolding number systems;38
7.3.1;2.3.1 From number sense to a number system;39
7.3.2;2.3.2 Natural number systems;41
7.3.3;2.3.3 Integers;42
7.3.4;2.3.4 Rational number system;42
7.3.5;2.3.5 Real number system;43
7.3.6;2.3.6 Complex number system;43
7.3.7;2.3.7 Algebra;44
7.4;2.4 Summary: Central factors in mathematical development;44
8;3 Theory of conceptual fields: Essential domains informing teaching and learning;46
8.1;3.1 Embracing the complexity in learning mathematics;46
8.1.1;3.1.1 Components of the theory;47
8.1.2;3.1.2 Research questions;48
8.2;3.2 Conceptual domain;49
8.2.1;3.2.1 Mathematical concept as a “triple of sets”;49
8.2.2;3.2.2 Conceptual fields;50
8.2.3;3.2.3 Some factors in development of mathematics knowledge;51
8.3;3.3 Cognitive domain;52
8.3.1;3.3.1 The subject and the external world;52
8.3.2;3.3.2 Operational-structural relations;54
8.3.3;3.3.3 Threshold concepts;55
8.3.4;3.3.4 From schemes and situations to generalisable concepts;55
8.3.5;3.3.5 An integration of key ideas;57
8.4;3.4 Didactic domain;58
8.4.1;3.4.1 Nurturing the learning process;58
8.4.2;3.4.2 The teacher’s role;59
8.5;3.5 Semiotic domain;59
8.5.1;3.5.1 The status of knowledge;59
8.5.2;3.5.2 Developmental stages towards greater abstraction;60
8.5.3;3.5.3 Language, an elaborated social system;60
8.5.4;3.5.4 Summary: Language precision and mathematics;61
8.6;3.6 Evaluative domain;61
8.6.1;3.6.1 Assessment for learning;62
8.7;3.7 Summary: Consequences for educational research and measurement;62
9;4 Assessment and measurement: A discussion of core requirements;65
9.1;4.1 From mathematics to measurement;65
9.1.1;4.1.1 Research questions;65
9.1.2;4.1.2 Large-scale assessment and learning;67
9.2;4.2 A theory of mathematics assessment;68
9.2.1;4.2.1 Conceptions of mathematics;68
9.2.2;4.2.2 Critical elements for the formulation of an assessment programme;69
9.2.3;4.2.3 Core notions for assessment;72
9.3;4.3 Measurement and the Rasch model;72
9.3.1;4.3.1 Measurement;73
9.3.2;4.3.2 Mathematical models;75
9.3.3;4.3.3 The development of the Rasch model;76
9.3.4;4.3.4 Validity;81
9.3.5;4.3.5 Reliability;82
9.3.6;4.3.6 Core ideas underpinning the Rasch model;82
9.4;4.4 Validity of assessment practices;83
10;5 The multiplicative conceptual field;85
10.1;5.1 Mathematical structure and developmental consequences;85
10.1.1;5.1.1 Research questions;86
10.2;5.2 Multiplication and division;87
10.2.1;5.2.1 Problem situations;87
10.2.2;5.2.2 Extension to rational numbers;89
10.2.3;5.2.3 Multiplicative structures;90
10.2.4;5.2.4 Building the base for rational number;97
10.3;5.3 Rational number;97
10.3.1;5.3.1 Rational number sub constructs;97
10.3.2;5.3.2 Operations on fractions;103
10.3.3;5.3.3 Synthesis of rational number;104
10.3.4;5.3.4 Proportional reasoning;105
10.3.5;5.3.5 Functional relationship and link to calculus;108
10.3.6;5.3.6 Considering salient features;109
10.4;5.4 Percent;110
10.4.1;5.4.1 Mathematical Structure;111
10.4.2;5.4.2 The language of percent;114
10.4.3;5.4.3 Tasks and problems;115
10.4.4;5.4.4 A concise language with important consequences;116
10.5;5.5 Probability;117
10.5.1;5.5.1 Mathematical structure;117
10.5.2;5.5.2 Historical factors;118
10.5.3;5.5.3 The acquisition of probabilistic concepts;118
10.5.4;5.5.4 A distinctive reasoning;118
10.6;5.6 Proficiency in the multiplicative conceptual field;118
10.7;5.7 Summary: Didactic implications, assessment and research;120
11;6 Exploration of data within the Rasch measurement framework;122
11.1;6.1 Understanding complexity through application of the Rasch model;122
11.1.1;6.1.1 Research questions;122
11.2;6.2 Methodology for the empirical investigation;122
11.2.1;6.2.1 Test development within a Rasch measurement framework;123
11.2.2;6.2.2 Participants;123
11.2.3;6.2.3 Test formulation;124
11.2.4;6.2.4 Test situation, administration and scoring;126
11.2.5;6.2.5 Data Analysis;127
11.3;6.3 Analytic framework for item analysis;135
11.3.1;6.3.1 Contextual factors;136
11.3.2;6.3.2 Type of situation;136
11.3.3;6.3.3 Mathematical structure;137
11.3.4;6.3.4 Mode of representation;138
11.3.5;6.3.5 Number range and value;138
11.3.6;6.3.6 Response processes and procedures;139
11.4;6.4 Item analysis;140
11.4.1;6.4.1 Item by strand analysis;142
11.5;6.5 Fraction item analysis;143
11.5.1;6.5.1 Critical findings: Fraction items at Levels 1, 2, 3 and 4;146
11.6;6.6 Ratio, proportion and rate item analysis;148
11.6.1;6.6.1 Critical findings: Ratio, rate and proportion items at Levels 1 to 7;150
11.7;6.7 Percent item analysis;153
11.7.1;6.7.1 Critical findings: Percent items at Levels 2, 3, 4, and 7;155
11.8;6.8 Probability item analysis;157
11.8.1;6.8.1 Critical findings: Probability items at Levels 2, 3 and 4;159
11.9;6.9 Pre-Algebra item analysis;161
11.9.1;6.9.1 Critical findings: Pre-Algebra items at Levels 2, 3, 4 and 5;163
11.10;6.10 Summary descriptions at Levels 1 to 7;165
11.10.1;6.10.1 Critical points and threshold concepts;170
11.10.2;6.10.4 Reflections and further insights;171
12;7 Identifying threshold concepts in reasoning behind item responses;172
12.1;7.1 Tracking learner competences;172
12.1.1;7.1.1 Research questions;173
12.2;7.2 Research method;173
12.3;7.3 Framework for interview analyses;177
12.4;7.4 High proficiency learners;182
12.4.1;7.4.1 Levels 6 and 7: Adele (School A), Anna (School B);182
12.4.2;7.4.2 Level 5: Kelly, Jane, Angela, Carla (School A), Prinella (School B);186
12.4.3;7.4.3 Proficiency exhibited at Levels 5, 6 and 7;194
12.5;7.5 Middle-high proficiency;196
12.5.1;7.5.1 Level 4, Thembani and Sipho (School B);196
12.5.2;7.5.2 Level 4: Shiluba, Carola, Linda and Kate (School A);199
12.5.3;7.5.3 Proficiency exhibited at Level 4;205
12.6;7.6 Middle-low proficiency;206
12.6.1;7.6.1 Level 3, Phaphama, Maria, Mpho (School B);206
12.6.2;7.6.2 Level 3: Cheryl and Zanele (School A);211
12.6.3;7.6.3 Proficiency exhibited at Level 3;215
12.7;7.7 Low proficiency;216
12.7.1;7.7.1 Level 1: Mishack, Amukelani and Mahesh (School B);216
12.7.2;7.7.2 Proficiency exhibited at Level 1;218
12.8;7.8 Overview of four proficiency levels;219
12.9;7.9 Theoretical insights from the theory of conceptual fields;221
12.10;7.10 Recommendations for the instrument;222
12.11;7.11 Reflections on the interviews;223
13;8 Addressing complexity: Implications for curriculum, teaching and assessment;224
13.1;8.1 Answering Poincaré;224
13.2;8.2 Insights from the theory of conceptual fields;224
13.3;8.3 Insights from the perspective of assessment and measurement;225
13.3.1;8.3.1 Rasch analysis and the theory of conceptual fields;226
13.3.2;8.3.2 Person-item map;226
13.3.3;8.3.3 Cognitive and pedagogical insights;227
13.4;8.4 Implications for curriculum, teaching and research;227
13.4.1;8.4.1 Levels of development;228
13.4.2;8.4.2 Identifying threshold concepts;230
13.5;8.5 Reflections and limitations;232
13.5.1;8.5.1 Instrument development recommendations;232
13.5.2;8.5.2 Limitations of the study;233
13.6;8.6 Future Research;235
13.7;8.7 Conclusion;236
14;9 References;238
15;List of Abbreviations;245
16;List of Figures;246
17;List of Tables;247