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VISUAL REASONING IN SOLVING MATHEMATICAL

PROBLEMS ON FUNCTIONS AND THEIR DERIVATIVES AMONG

MALAYSIAN PRE-UNIVERSITY STUDENTS

HALIZA ABD HAMID

THESIS SUBMITTED IN FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

INSTITUTE OF GRADUATE STUDIES

UNIVERSITY OF MALAYA

KUALA LUMPUR

2017

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ABSTRACT

The main objective of this study was to develop a framework and assess the visual

reasoning process adopted by pre-university students when integrating Cartesian graphs

to solve mathematical problems on functions and derivatives. The study identified the

usage levels of graphs, method of preference and graph-reasoning ability, and

subsequently, the correlation among them. The study also investigated the

misconceptions and difficulties faced by the students. The study employed a 3-phase

descriptive quantitative method. The development of framework in Phase 1 involved a

three-stage process: the document analysis on theories and models on visual reasoning

and Cartesian graphs, focus group discussion among experts in the domain content and

visual reasoning, and a 3-round Delphi method to confirm the framework. In Phase 2,

three instruments were prepared; the Visual Representation Usage Level on four

categories of using graphs, the Mathematical Visuality Test to measure the students’

preference method and the Graph-Reasoning Test measuring their graph-reasoning

ability. Phase 3 involved the collection and analysis of data on 194 pre-university

students. The developed framework consisted of seven categories of encoding and five

categories of decoding processes. Results indicated between 41.75% to 84.02% of the

students were very positive towards the use of graphs and diagrams in the teaching and

learning of mathematics although between 56.70% and 78.87% said they faced

difficulties in constructing and interpreting them. Students exhibited fluctuating patterns

of visual reasoning ability. In the encoding process, the students were categorised into

three types of mathematical visuality; 26.8% were visual, 16.5% were partially-visual

and 56.7% were non-visual. This exhibits their reluctance to sketch graphs although

they managed to obtain the correct solutions which indicates that they had a

predominant preference for algebraic method as compared to visuals. Responses in the

decoding process were based on the three levels of graph reasoning. At least 68% of the

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students managed to get correct solutions for all the read graph items. More than 70%

of the students managed to obtain 75% and 43% of the items correct for the read

between graph and read beyond graph respectively. These indicate that as the tasks get

harder visually, more cognitive load is needed for the students to read and interpret

graphs. Strong positive correlation values of at least 0.91 were obtained among the three

decoding levels. They perform fundamental, operational and subjective types of errors

and encountered the non-use of graph, generic and idiosyncratic difficulties when

relating the algebraic forms of functions and derivatives to their visual representations

on graphs and vice versa. The results of the study were significant in providing reliable

and important ideas depicting the development of visual reasoning that is useful in the

students’ thinking and understanding, to guide the development of instructional

materials to improve students’ understanding and reasoning, for educators and

curriculum developers to enhance the learning outcomes and teaching strategies to

challenge students’ thinking and reasoning skills, and to reduce gap in the literature and

knowledge on visual reasoning in the Malaysian educational system.

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ABSTRAK

Tujuan utama kajian adalah untuk membangun rangkakerja dan mentaksir proses

penaakulan visual yang digunapakai oleh pelajar pra-universiti dengan mengintegrasi

graf Cartesian dalam menyelesaikan masalah fungsi dan terbitan. Kajian mengenalpasti

tahap penggunaan graf, kaedah pilihan dan keupayaan penaakulan graf, korelasi di

antara ketiga-tiganya dan menganalisa kesukaran dan kesilapan yang dihadapi oleh

pelajar. Rekabentuk kajian adalah berdasarkan kaedah 3-fasa deskriptif kuantitatif. Fasa

1 melibatkan proses 3-peringkat pembangunan rangkakerja: analisa dokumen ke atas

teori-teori dan model-model berkaitan penaakulan visual dan graf Cartesian,

perbincangan kumpulan fokus melibatkan pakar-pakar dalam domain kandungan dan

penaakulan visual, dan kaedah Delphi 3-pusingan bagi proses pengesahan rangkakerja.

Tiga instrumen utama telah disediakan dalam Fasa 2: Visual Representation Usage

Levels bagi mengukur empat kategori pelajar dalam pengggunaan graf, Mathematical

Visuality Test bagi mengukur kaedah pilihan dan Graph-Reasoning Test bagi megukur

kebolehan penaakulan visual pelajar. Fasa 3 melibatkan proses pengumpulan dan

penganalisaan data kek atas 194 orang pelajar pra-universiti. Rangkakerja yang

dibangunkan mengandungi tujuh kelas pengekodan dan lima kelas penyahkodan.

Seramai 41.57% hingga 84.02% pelajar adalah sangat positif terhadap pengunaan graf

dan gambarajah dalam pembelajaran dan pengajaran matematik walaupun seramai

56.70% hingga 78.87% pelajar menghadapi kesukaran dalam membina dan mengtafsir

graf. Pelajar mempamerkan corak kebolehan penaakulan visual yang turun naik. Dalam

proses pengekodan, pelajar dapat dikelaskan dalam tiga kumpulan: 26.8% visual, 16.5%

separa-visual dan 56.7% bukan-visual. Ini mempamerkan keengganan pelajar untuk

melakar graf tetapi mereka mampu untuk menyelesaikan masalah tersebut. Ini

menunjukkan bahawa kaedah algebra adalah menjadi pilihan pelajar berbanding kaedah

visual. Analisis proses penyahkodan adalah berdasarkan tiga tahap penaakulan graf.

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Sekurang-kurangnya 68% pelajar berjaya menyelesaikan masalah bagi semua item baca

graf. Lebih dari 70% pelajar berjaya menyelesaikan 75% dan 43% masing-masing

daripada item baca antara graf dan baca luar graf. Ini menunjukkan bahawa beban

kognitif yang lebih diperlukan apabila tahap visual yang dipaparkan semakin sukar

untuk dibaca dan ditaksir. Ketiga-tiga tahap decoding mempunyai korelasi positif yang

kukuh dengan nilai sekurang-kurangnya 0.91. Pelajar telah mempamerkan jenis

kesilapan asas, operasi dan subjektif, dan menghadapi kesukaran dari segi tidak

menggunakan graf, generik dan pelbagai bila mengaitkan hubungan di antara bentuk

algebra dan graf serta sebaliknya. Hasil kajian adalah sangat signifikan dalam

menyediakan ide ketara dan yang boleh dipercayai bagi menggambarkan pembangunan

penaakulan visual yang berguna dalam pemikiran dan pemahaman pelajar, membimbing

untuk pembangunan bahan pengajaran dan pembelajaran bagi membaikpulih proses

pemahaman dan penaakulan pelajar, para pendidik dan penyedia kurikulum boleh

memperkuatkan hasil pembelajaran dan strategi pengajaran yang boleh mencabar proses

pemikiran dan penaakulan pelajar, dan membantu merapatkan jurang dalam tinjauan

literatur dan pengetahuan tentang penaakulan visual setiap peringkat dalam sistem

pembelajaran Malaysia.

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ACKNOWLEDGEMENTS

There are many deserving thanks for all their efforts in helping me along my

journey toward earning my doctoral degree.

I would like to give my heartfelt gratitude to my supervisor, Professor Dato’ Dr

Noraini Idris who has been unreservedly supportive in her supervisory role, while

providing constructive critique and challenge that has helped shape my thinking. When

the journey started, it was a foggy road. But her thought provoking comments shed light

that paved way to this destination. Her influences, the words of encouragement and her

trust in me have been the key ingredients to producing this thesis.

I wish to acknowledge my colleagues for their willingness to participate, their

openness, and honesty were central to informing the research, not to forget their

generous support and encouragement. To my students, thank you for the opportunity to

investigate some passages to your success in understanding your learning styles. I

always feel good to discover that our interaction has indeed changed you for the better.

My family provided untold support, consistently thinking the best of me. My

husband, Bahri, has been an inspiration to me during times when my motivation was

lacking. He never gave up on me even when the stress was overwhelming, instead

turned me from frustration and discouragement toward enthusiasm and productivity. I

am so blessed to be married to such a wonderful man. My kids - Aisya, Aida, Aimi &

Ali, gave up precious time with me and supplied regular stress relieving moments and

laughter. I dedicate this thesis to my mother whom I know that the pain of my father’s

loss always pricks like a thorn. Despite this, she has been there for me. To the memory

of my father, no one can fill up the empty space you have left in my heart and soul. This

piece of work has been produced to ease the unbearable pain of losing you. I love you,

Ayah.

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TABLE OF CONTENTS

Page

Title Page i

Original Literacy Work Declaration Form ii

Abstract iii

Abstrak v

Acknowledgements vii

Table of Contents viii

List of Figures xiv

List of Tables xvii

List of Symbols and Abbreviations xx

List of Appendices xxii

CHAPTER 1 : INTRODUCTION

1.1 Background of the Study 1

1.1.1 Learning of Functions and Derivatives 2

1.1.2 Graphs as Visual Tools 4

1.1.3 Visual Reasoning in Learning Differentiation 6

1.2 Statements of the Problem 10

1.3 Objectives of the study 14

1.4 Research Questions 15

1.5 Definition of terms 16

1.6 Significance of the Study 19

1.7 Conclusion 22

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CHAPTER 2 : LITERATURE REVIEW AND CONCEPTUAL FRAMEWORK

2.1 Introduction 23

2.2 Review of Literature 23

2.2.1 Defining Visual Reasoning 24

2.2.1.1 Visual Reasoning in Mathematics Education 27

2.2.1.2 Visual consideration in problem solving 31

2.2.2 Conceptual Understanding of Functions and Derivatives 34

2.2.2.1 The derivative function 42

2.2.2.2 The Concepts of Limits 45

2.2.2.3 The Application of Derivative - Rate of Change 46

2.2.3 Defining Graphs 48

2.2.3.1 Making Sense of Graphs 50

2.2.4 Empirical support for graphs as visual tools in functions and

derivatives 54

2.2.5 Visual Reasoning Models 56

2.3 Conceptual Framework 61

2.3.1 The Visual Reasoning Constructs 61

2.3.2 The Encoding Process 63

2.3.3 The Decoding Process 65

2.3.4 Knowledge and Scheme 68

2.3.4.1 Making sense of graphs 68

2.3.4.2 Performance standard 70

2.3.4.3 Conceptual knowledge 71

2.3.4.4 Perceptual knowledge 72

2.3.5 Framework for assessing visual reasoning in this study 73

2.4 Summary 74

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CHAPTER 3 : METHODOLOGY

3.1 Introduction 76

3.2 Research Design 77

3.3 Participants 81

3.4 Instrumentation 84

3.4.1 Visual Representation Usage Level (VRUL) 84

3.4.1.1 Description of VRUL 84

3.4.1.2 Validity and reliability of the VRUL 85

3.4.2 Mathematical Visuality Test (MVT) 88

3.4.2.1 Description of MVT 88

3.4.2.2 Validity and reliability of the MVT 93

3.4.3 Graph Reasoning Test (GRT) 96

3.4.3.1 Description of GRT 96

3.4.3.2 Validity and reliability of the GRT 110

3.5 Data Collection 113

3.6 Data Analysis 116

CHAPTER 4 : ANALYSIS OF RESULTS

4.1 Introduction 118

4.2 Phase 1 : Development of the framework to assess visual reasoning 119

4.2.1 Stage 1 : Initial development of the framework 119

4.2.1.1 Step 1 : Planning of synthesis 120

4.2.1.2 Step 2 : Synthesis 123

4.2.1.3 Step 3 : Refinement of synthesis 128

4.2.2 Stage 2 : Refining the framework 133

4.2.3 Stage 3 : Development of the final framework 137

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4.3 Usage levels of visual representations 145

4.3.1 Frequencies and percentages 145

4.3.1.1 Analysis on the usage levels on using graphs or

diagrams in their daily learning behaviour 146

4.3.1.2 Analysis on the usefulness of graphs or diagrams

in solving mathematical problems 148

4.3.1.3 Analysis on the difficulty on the use of graphs or

diagrams in solving mathematical problems 151

4.3.1.4 Analysis on the teacher’s behaviours in using graphs

or diagrams in solving mathematical problems 153

4.3.2 Analysis on VRUL based on gender, race and major 156

4.3.3 Correlations among the categories in VRUL 156

4.4 Mathematical Visuality Test (MVT) 157

4.4.1 Frequencies and percentages 157

4.4.1.1 Analysis on the mathematical visuality for item 1 158

4.4.1.2 Analysis on the mathematical visuality for item 2 160

4.4.1.3 Analysis on the mathematical visuality for item 3 161

4.4.1.4 Analysis on the mathematical visuality for item 4 162

4.4.1.5 Analysis on the mathematical visuality for item 5 163

4.4.2 Analysis on visuality measure 164

4.4.3 Analysis on visuality measure based on gender, race and major 165

4.5 Graph Reasoning Test (GRT) 166

4.5.1 Frequencies and percentages 166

4.5.1.1 Read the graph 167

4.5.1.2 Read between the graph 168

4.5.1.3 Read beyond the graph 169

4.5.2 Correlations among the overall GRT and the decoding scales 170

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4.5.3 Analysis on visual reasoning ability based on gender, race

and major 171

4.6 Correlation among the results of the instruments 172

4.7 Analysis on misconceptions and difficulties 176

4.7.1 Mathematical Visuality 176

4.7.2 Graph Reasoning 183

4.8 Summary 191

CHAPTER 5 : MAIN FINDINGS, DISCUSSION AND CONCLUSION

5.1 Introduction 193

5.2 Main findings of the study 193

5.2.1 Development of the framework 193

5.2.2 Usage levels of graphs 195

5.2.3 Mathematical Visuality 197

5.2.4 Visual Reasoning 198

5.2.5 Correlations among the instruments 199

5.2.6 Difficulties and misconceptions 199

5.3 Discussion 201

5.3.1 Usage Level of Visual Representation 201

5.3.2 Preference on the Use of Graphs 204

5.3.3 Graphs as Communication Tools 207

5.3.4 Ability to employ graphs as visual information 210

5.3.5 Misconceptions and difficulty in sketching and employing graphs 212

5.4 Limitations of the study 224

5.5 Implications for Practice 225

5.5.1 Mathematics Teaching and Learning 225

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5.5.2 Assessing Techniques 229

5.5.3 Curriculum Development 231

5.6 Recommendations and future directions 234

REFERENCES 236

APPENDICES 270

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LIST OF FIGURES

Figure 2.1 The concepts of chord and tangent 40

Figure 2.2 The concept of slope represented visually 41

Figure 2.3 Illustration of the idea of tangent 42

Figure 2.4 Illustration on the properties of graphs 43

Figure 2.5 Illustration on the changes in the values of the first derivative 44

Figure 2.6 Illustration of the idea of limits 46

Figure 2.7 Graph of distance-time car traveling at constant velocity 47

Figure 2.8 Illustration on the average and instantaneous rate of change 48

Figure 2.9 The Visual Reasoning Model 57

Figure 2.10 Conceptual framework of the study 74

Figure 3.1 Flowchart of the phases in the research design 77

Figure 3.2 Flowchart of the development of framework for assessing visual

reasoning in Phase 1 79

Figure 3.3 An example of item on rate of change in the MVT 89

Figure 3.4 An example of item on slope and limits in the MVT 90

Figure 3.5 An example of item on properties of graph in the MVT 91

Figure 3.6 An example of item on graph of function and its derivative in

the MVT 92

Figure 3.7 An example of item on applications of derivatives in the MVT 93

Figure 3.8 An example of item on slope in the GRT 101

Figure 3.9 An example of item on tangent in the GRT 102

Figure 3.10 An example of item on properties of functions in the GRT 104

Figure 3.11 An example of item on graphs of functions and their derivatives

in the GRT 105

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Figure 3.12 An example of item on applications of gradient of derivative

in the GRT 107

Figure 3.13 The framework of the instrumentation 115

Figure 4.1 Document analysis to locate the framework to assess visual

reasoning 121

Figure 4.2 The usage levels in using graphs or diagrams in daily learning

Behaviour 148

Figure 4.3 The usefulness on using graphs or diagrams in solving

mathematical problems 150

Figure 4.4 The difficulty in using graphs or diagrams in solving

mathematical problems 153

Figure 4.5 The teachers’ behaviour in using graphs or diagrams in solving

mathematical problems 154

Figure 4.6 Sample of student’s work that was assigned to IGIS 158

Figure 4.7 Sample of student’s work that was assigned to IGCS 158

Figure 4.8 Distribution of the encoding process for item 1 of the MVT 159

Figure 4.9 Distribution of the encoding process for item 2 of the MVT 160

Figure 4.10 Distribution of the encoding process for item 3 of the MVT 161

Figure 4.11 Distribution of the encoding process for item 4 of the MVT 163

Figure 4.12 Distribution of the encoding process for item 5 of the MVT 164

Figure 4.13 Sample of student’s work that was assigned to ICSIR 167

Figure 4.14 Sample of student’s work that was assigned to ISINR 167

Figure 4.15 Means for VRUL against means for MVT 173

Figure 4.16 Means for VRUL against means for GRT 173

Figure 4.17 Means for MVT against means for GRT 174

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Figure 4.18 Sample of wrong graph sketched and wrong definition and

explanation provided by students for item 1 of MVT 178

Figure 4.19 Sample of wrong graph sketched but with correct description

and explanation provided by students for item 2 of MVT 179

Figure 4.20 Samples of solutions provided by students for item 3(h)(ii)

of the MVT 180

Figure 4.21 Samples of wrong graphs sketched and wrong definition and

explanation provided by students for item 4 of the MVT 182

Figure 4.22 Sample of sign diagram drawn by students for item 5 of the MVT 183

Figure 4.23 Sample of wrong chord drawn by student for item 1 of the GRT 184

Figure 4.24 Samples of solutions by students for item 2 of the GRT 187

Figure 4.25 Sample of incorrect solution with incorrect reason by students for

item 3 of the GRT 188

Figure 4.26 Samples of various solutions by students for item 4 of the GRT 190

Figure 4.27 Samples of solutions by students for item 5 of the GRT 191

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LIST OF TABLES

Table 2.1 Samples of Key Ideas and Teaching and Learning Strategies 62

Table 2.2 Simon’s (1986) Diagram Drawing Sub-skills 63

Table 2.3 The Carlson’s (1998) Mental Actions of the Co-variation

Framework 64

Table 2.4 Krutetskii’ s (1976) categories of visual ability 65

Table 2.5 Levels of graph comprehension by Friel, Curcio & Bright (2001) 66

Table 2.6 Yumus’s (2001) levels of reasoning 67

Table 2.7 Sharma’s (2013) framework for interpreting graph 67

Table 3.1 Distribution of students’ demographic details 82

Table 3.2 Reliability coefficients of the VRUL and its categories 87

Table 3.3 The rubric for the MVT 95

Table 3.4 Framework for the Graph Reasoning Ability 98

Table 3.5 Scales of the decoding process for the items in GRT 99

Table 3.6 The descriptions of the final rubric for the GRT 111

Table 3.7 Division and nature of the instruments 112

Table 4.1 Theories identified for synthesis 122

Table 4.2 Comparison of theories/models/frameworks on visual reasoning

for points of convergence 129

Table 4.3 Taxonomy of skills on encoding process among theories/models/

framework for points of convergence 131

Table 4.4 Taxonomy of skills on decoding process among theories/models/

frameworks for points of convergence 132

Table 4.5 The initial framework for assessing visual reasoning 133

Table 4.6 Responses from the experts in the focus group discussion 135

Table 4.7 The refined framework for assessing visual reasoning 138

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Table 4.8 The distribution of 3-round emailing experts 140

Table 4.9 Analysis on the refined framework – Round 1 141

Table 4.10 Analysis on the refined framework – Round 2 142

Table 4.11 Analysis on the refined framework – Round 3 143

Table 4.12 The final framework for assessing visual reasoning 144

Table 4.13 The usage levels on using graphs or diagrams in their

daily learning behaviour 147

Table 4.14 The usefulness of graphs or diagrams in solving mathematical

problems 149

Table 4.15 The difficulty on the use of graphs or diagrams in solving

mathematical problems 152

Table 4.16 The teachers’ behaviours in using graphs or diagrams in solving

mathematical problems 155

Table 4.17 Correlation among the overall VRUL and the categories in VRUL 156

Table 4.18 The analysis on the Mathematical Visuality for item 1 159

Table 4.19 The analysis on the Mathematical Visuality for item 2 160

Table 4.20 The analysis on the Mathematical Visuality for item 3 161

Table 4.21 The analysis on the Mathematical Visuality for item 4 163

Table 4.22 The analysis on the Mathematical Visuality for item 5 164

Table 4.23 Distribution of mathematical visuality measure for the MVT 165

Table 4.24 The analysis on the items for the decoding scale : Read the graph 168

Table 4.25 The analysis on the items for the decoding scale : Read between

graph 169

Table 4.26 The analysis on the items for the decoding scale : Read beyond

graph 170

Table 4.27 Correlation among the overall GRT and the decoding scales 170

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Table 4.28 The linear regression and correlation coefficients among the

VRUL, MVT and GRT 175

Table 4.29 Distribution of errors for the Mathematical Visuality Test 177

Table 4.30 Distribution of errors for the Visual Reasoning 185

Table 5.1 Descriptions of the categories for Mathematical Visuality 197

Table 5.2 Descriptions of the categories for Visual Reasoning 198

Table 5.3 Descriptions of the categories for errors 200

Table 5.4 Descriptions of the categories for difficulties 201

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LIST OF ABBREVIATIONS

CDC Curriculum Development Centre

CGCS Correct graph with correct solution

CGIS Correct graph with incorrect solution

CSIR Correct solution with invalid reason

CSNR Correct solution with no reason

CSVR Correct solution with valid reason

CMI Communication of Mathematical Information

CR Cartesian Graph

GRT Graph Reasoning Test

IELTS International English Language Testing System

IGCS Incorrect graph with correct solution

IGIS Incorrect graph with incorrect solution

ISINR Incorrect solution with invalid reason or no reason

Int. International

JPA Jabatan Perkhidmatan Awam

Loc. Local

KBSM Kurikulum Bersepadu Sekolah Menengah

MARA Majlis Amanah Rakyat

MC Mathematical Content

ME Mathematic Education

MKSA Mathematical Knowledge and Skills and Their Application

MMP Mathematical Modeling and Problem Solving

MPI Mathematical Processing Instrument

MVT Mathematical Visuality Test

NA No answer / Not attempted

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NCTM National Council of Teachers of Mathematics

NGCS No graph with correct solution

NGIS No graph with incorrect solution

PETRONAS Petroleum Nasional

SACE South Australian Certificate of Education

SAM South Australian Matriculation

SPM Sijil Pelajran Malaysia

SPSS Statistical Package for the Social Science

VR Visual Reasoning

VR-G Visual Reasoning over Graph

VRUL Visual Representation Usage Levels

WISC-R Wechsler Intelligence Scale for Children-Revised

YTN Yayasan Tenaga Nasional

xxii

LIST OF APPENDICES

APPENDIX A Performance standards for Stage 2 Mathematical Studies

APPENDIX B Visual Representation Usage Levels

APPENDIX C Request to adopt questionnaire

APPENDIX D Mathematical Visuality Test

APPENDIX E Feedback from expert on MVT and GRT

APPENDIX F Graph Reasoning Test

APPENDIX G List of questions and probes for focus group discussion

APPENDIX H Feedbacks on the framework

APPENDIX I Feedback from expert on the framework

APPENDIX J Analysis based on gender, race and major