A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy in Education

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THE RELATIONSHIPS BETWEEN CONSTRUCTIVIST LEARNING ENVIRONMENT, MATHEMATICS ATTITUDE, AND ALGEBRA PERFORMANCE FROM

THE PERSPECTIVE OF FORM ONE STUDENTS IN SELANGOR, MALAYSIA

BY

HASNIZA BINTI IBRAHIM

A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy in Education

Kulliyyah of Education

International Islamic University Malaysia

SEPTEMBER 2021

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ii

ABSTRACT

Previous studies had shown that learning environment had an influence on students’

attitude and achievement. This study extends the previous studies by examining the relationships between students’ perceptions of their mathematics constructivist learning environment, mathematics attitude, and algebra performance in Selangor, Malaysia. Algebra was chosen in this study due to its importance in pursuance of higher-level mathematics, however at the same time, algebra is found to be of considerable challenge to the Form One students. Meanwhile, constructivism in the teaching and learning of mathematics today has been proposed to be the answer to the challenges above. Constructivist learning environment is a setting where learners work together to construct knowledge, receive support from teachers and classmates, apply their previous knowledge or experiences and solve given appropriate tasks using various approaches. This study employed a fully quantitative cross-sectional survey design. To achieve the aim of this study, two instruments namely the Algebra Test for examining students’ algebraic thinking and a Questionnaire Survey - Constructivist Learning Environment Evaluation and Mathematics Attitude (CLEEMA) were developed and evaluated. The respondents of this study were 336 Form One students who were selected using the cluster sampling method. Algebra Test was validated using Rasch Measurement Model and the hypothesized measurement models of Constructivist Learning Environment and Students’ Mathematics Attitude were validated using SEM-AMOS software. The result revealed that the majority of the respondents performed poorly in the Algebra Test, while the level of the constructivist learning environment and students’ mathematics attitude was at a moderate level. The results of Structural Equation Modelling analysis discovered that constructivist learning environments are significantly and positively influencing algebra performance and students’ mathematics attitude as the study had hypothesized.

However, students’ mathematics attitude does not significantly influence algebra performance as hypothesized in the study. In conclusion, it can be confirmed that the majority of Form One students in Selangor have low ability in the Algebra Test and hence their performance in algebra is also low. This is consistent with numerous findings in the previous studies which stated that algebra is very challenging for many students. Moreover, this study confirmed that some mathematics classrooms in Selangor have practiced the constructivist learning environment elements, but only at the moderate level.

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ثحبلا ةصلاخ

ABSTRACT IN ARABIC

ريشت تاساردلا ةقباسلا

ىلإ نأ ةئيبلا ةيميلعتلا اهل

ريثأت

ٍّ لك ىلع هاج تا نم

،ب لّطلا مهزاجنإو

دعُتو . هذه

ةساردلا اًدادتما كلتل

،تاساردلا كلذو

نع قيرط صحف تاقلّعلا نيب

ةثلّث

،رواحم

،اهلوأ تاروصت ب لّطلا

هاجت ةئيب مُّلعتلا ةيئانبلا يف تا يضايرلا ةدام

. يناثو

،اه مهتاهاجتا وحن

تا يضايرلا ةدام

،اًريخأو . مهؤادأ يف ةدام

،ربجلا كلذو يف ةيلاو

،روجنلّس ايزيلام

.

ٍّ مت دقو رايتخا ربجلا يف هذه

،ةساردلا اًرظن

هتي مهلأ يف ةسارد

تايوتسملا ىلعلأا

يف تا يضايرلا ةدام

. ديب هنأ يف تقولا

،هسفن ربتعُي ربجلا تاي دحتلا دحأ

ةريبكلا لا يت

هجاوت ب لّط ةلحرملا ة يوناثلا ةن سلا يف

ىلولأا . يفو نوضغ مَّدقُت ،كلذ

ةيرظنلا ةيئانبلا يف ميلعت م لعتو

تا يضايرلا هذه

،مايلأا اهرابتعاب

ًٍّلّح تاي دحتلل ةروكذملا

اًفنآ . ريدجو ركذلاب انه نأ ةئيبلا ةيميلعتلا

،ةيئانبلا

يه راطلإا يذلا لمعي نوم لعتملا هيف

ًٍّيوس ا نم لجأ نيوكت

،مهتفرعم لابقتساو

معدلا نيم لعملا نم

،ءلّمزلاو

فيظوتو مهتفرعم

مهبراجتو ةقباسلا

يف لح ماهملا ةلَكوُمـلا مهيلإ مادختساب بيلاسأ

ةع ونتم . نمو لجأ قيقحت

فدهلا نم هذه

،ةساردلا

ٍّ مت ريوطت ميوقتو نيتادأ عمجل

،تانايبلا امهو

رابتخا

،ربجلا صحفل ريكفت بلّطلا

،يربجلا ةنابتساو

مييقت ةئيب ملعتلا

،ةيئانبلا فقوملاو هاجت

ةدام تايضايرلا (CLEEMA).

،اذه دقو غلب

يلامجإ ددع دارفأ ةنيعلا 336

،اًبلاط مت مهرايتخا مادختساب

ةقيرط ةن يعلا ةيدوقنعلا .

ٍّ مت دقو ققحتلا نم

ةيحلّص رابتخا

ربجلا نم للّخ مادختسا جذومن

( شار ) ايقلل س (Rasch Measurement

Model).

امنيب مت ققحتلا نم ةيحلّص جذامنلا

ة يسايقلا ةضرتفملا ةئيبلل

ةيميلعتلا

،ةيئانبلا هاج تاو ب لّطلا

وحن

،تا يضايرلا ةدام مادختساب

جمانرب جذومن ةلداعملا ةيئانبلا ( سومأ -(SEM-AMOS.

دقو تراشأ

جئاتن ةساردلا ىلإ نأ ةيبلغأ نيكراشملا ناك

أ مهؤاد اًفيعض يف رابتخا

،ربجلا امنيب تناك تايوتسم ةئيبلا

ةيميلعتلا

،ةيئانبلا هاج تاو ب لّطلا وحن

،تا يضايرلا ةدام ةط سوتم

. امك رهظأ ليلحت جئاتن جذومن ةلداعملا

،ةيئانبلا نأ تائيبلا ةيميلعتلا ةيئانبلا

رثؤت

ٍّ لك ىلع نم ءادأ بلّطلا يف

،ربجلا مههاج تاو وحن

تا يضايرلا ةدام

لكشب سوملم

، يباجيإو اًقبط

تاضارتفلا ةساردلا

. عمو كلذ ، نإف فقوم بلّطلا نم ةدام تايضايرلا لا

رثؤي لكشب ريبك ىلع ءادأ ربجلا امك وه ضرتفم يف ةساردلا . يف

،ماتخلا نكمي ديكأتلا ىلع نأ ةيبلاغ

بلّط جذومنلا لولأا

يف روغنلّيس مهيدل

ةردق ةضفخنم يف

رابتخا ربجلا يلاتلابو نإف مهئادأ يف ربجلا

ضفخنم اضيأ

. اذهو قفتي عم ديدعلا نم جئاتنلا يف تاساردلا ةقباسلا

يتلا تركذ نأ ربجلا لكشي ايدحت

اريبك ريثكل نم بلّطلا . ةولّعو ىلع

،كلذ تدكأ هذه ةساردلا نأ ضعب لوصف تايضايرلا يف

روغنلّيس دق

تسرام رصانع ةئيب ملعتلا

،ةيئانبلا ٍّو نكل طقف ىلع ىوتسملا لدتعملا

ٍّ .

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APPROVAL PAGE

The thesis of Hasniza Binti Ibrahim has been approved by the following:

_____________________________

Madihah Khalid Supervisor

_____________________________

Mohd Burhan Ibrahim Co-Supervisor

_____________________________

Joharry Othman Internal Examiner

_____________________________

Munirah Ghazali External Examiner

_____________________________

Mazlini Adnan External Examiner

_____________________________

Radwan Jamal Yousef Elatrash Chairman

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DECLARATION

I hereby declare that this thesis is the result of my own investigations, except where otherwise stated. I also declare that it has not been previously or concurrently submitted as a whole for any other degrees at IIUM or other institutions.

Hasniza Binti Ibrahim

Signature ... Date ...

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COPYRIGHT PAGE

INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA

DECLARATION OF COPYRIGHT AND AFFIRMATION OF FAIR USE OF UNPUBLISHED RESEARCH

THE RELATIONSHIPS BETWEEN CONSTRUCTIVIST LEARNING ENVIRONMENT, MATHEMATICS ATTITUDE, AND ALGEBRA PERFORMANCE FROM THE PERSPECTIVE

OF FORM ONE STUDENTS IN SELANGOR, MALAYSIA

I declare that the copyright holders of this thesis are jointly owned by the student and IIUM.

No part of this unpublished research may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without prior written permission of the copyright holder except as provided below

1. Any material contained in or derived from this unpublished research may be used by others in their writing with due acknowledgement.

2. IIUM or its library will have the right to make and transmit copies (print or electronic) for institutional and academic purposes.

3. The IIUM library will have the right to make, store in a retrieved system and supply copies of this unpublished research if requested by other universities and research libraries.

By signing this form, I acknowledged that I have read and understand the IIUM Intellectual Property Right and Commercialization policy.

Affirmed by Hasniza Binti Ibrahim

……..……….. ………..

Signature Date

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DEDICATION

Bismillah.

To my beloved father and mother, Ibrahim Bin Abdullah & Hasnah Bt. Yaakub

You are the most beautiful thing that has ever happened to me.

Thank you for your prayers and for always being there for me.

To them I owe everything I have accomplished in my life.

My loved ones, dearest family, lecturers and friends.

Your support has made me a stronger person and I will forever be grateful.

Your efforts to raise me up, reaching the sky is the best definition of love.

Your trust on me makes me believe that I can be a better version of me.

Thank you so much from the bottom of my heart.

May Allah accept our love and sacrifice. Ameen.

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ACKNOWLEDGEMENTS

Praise be to Allah, the Lord of the Worlds. Allah (Glorified and Exalted be He) has bestowed upon me abundant blessings and by the will of Allah I have accomplished my Thesis; I pray this work of mine would be a benefiting deed for me in this world and in the here-after. I infinitely thank Allah for the blessings.

First and foremost, I would like to express my gratitude and special thanks to my dearest supervisor. The successful completion of this work would not have been possible without the expert guidance from her, Associate Professor Dr. Madihah Khalid. Her guidance, willingness to put extra efforts, insightful suggestions and invaluable advice have been most helpful. I really appreciate and express my humble gratitude to her for everything she did, thank you. Also, thanks to my co-supervisor Associate Professor Dr. Mohd Burhan Ibrahim who has been always kind and ready to offer his help whenever I needed.

I would also like to thank all of the outstanding professors, lecturers and mathematics teachers who have helped me in the content validation of the instruments. Furthermore, I would like to thank all the amazing lecturers I have had the honor of learning from during my time at IIUM. Also special thanks to Dr. Azmah Ghaus for being there at the right time, saving my life during the critical moment. My heartfelt gratitude goes to all the people who have been with me during the struggles.

Writing the thesis during the pandemic – alone - seriously was not easy. Your presence in my PhD journey is Allah’s answer to my desperate du’a. Thank you.

I would also like to express my gratitude to the Ministry of Education of Malaysia, the Selangor State Education Department, headmasters of the schools in Selangor and teachers for helping me to conduct the survey and the test in the schools.

Furthermore, special thanks to all the Form One students who were willing to participate in this study. Without your kind assistance and cooperation, it is impossible for me to complete the study. Honestly, I cannot repay your kindness, and may Allah reward you abundantly.

Finally, I would like to express my deep gratitude to my beloved family members who have remained very supportive and helpful to me in every step, who kept on praying the best for me without fail, who always believe in me – the potential in me. Thank you, mom (Hasnah Yakob) for your unconditional love and being there with me in every step of my life. Not to forget my late father (Ibrahim Abdullah). I love you and miss you so much. My siblings, my nieces and nephews - Alya, Ashraf, Amni, Aelldyra, Aisya, Addam, Adrian, Ayden and Aqif. You bring colours and happiness into my life. Thank you all for your endless support, love and understanding.

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CHAPTER TWO: LITERATURE REVIEW ... 28

2.1 Introduction... 28

2.2 Learning Theories ... 28

2.3 Constructivism ... 32

2.3.1 Social Constructivism ... 35

2.3.2 Constructivist Learning Environment ... 37

2.3.3 Five Constructivist Learning Environments ... 40

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2.3.3.1 Teacher Support ... 40

2.3.3.2 Student Centredness ... 42

2.3.3.3 Cooperation ... 43

2.3.3.4 Task Orientation ... 44

2.3.3.5 Personal Relevance ... 45

2.4 Algebra ... 46

2.4.1 Introduction ... 46

2.4.2 Algebra History ... 48

2.4.3 Algebraic Thinking ... 49

2.4.4 Challenges in Learning Algebra... 52

2.5 The Theoretical Framework ... 56

2.6 Algebra Performance ... 60

2.7 Mathematics Attitude... 61

2.8 Constructivist Learning Environment and Mathematics Attitude ... 65

2.9 Constructivist Learning Environment and Algebra Performance ... 67

2.10 Mathematics Attitude and Algebra Performance ... 69

2.11 Constructivist LEarning environment, students’ Attitude and students’ Performance ... 70

2.12 Chapter Summary ... 71

CHAPTER THREE: METHODOLOGY ... 73

3.2 Research Design ... 73

3.3 Participants ... 75

3.3.1 Population of the Study ... 75

3.3.2 Sampling Procedures ... 77

3.3.3 Sample Size ... 79

3.4 Instrumentation - Questionnaire Survey ... 80

3.4.1 Identifying the Constructs and Their Respective Operational Definitions ... 81

3.4.2 Preliminary Survey... 82

3.4.3 Refinement of Items ... 83

3.4.4 Content Validation Process via Expert Judgement ... 83

3.4.5 Translation Process ... 85

3.4.6 Pilot Study and Finalization of the Survey ... 88

3.4.7 Exploratory Factor Analysis ... 89

3.4.7.1 actor Analysis on CLEE ... 90

3.4.7.2 Factor Analysis on Mathematics Attitude ... 96

3.4.7.3 The Reliability ... 101

3.4.7.4 Summary of the EFA Results ... 102

3.5 Algebra Test... 102

3.5.1 Justification of The Rasch Measurement Model ... 106

3.5.2 Adequacy of the Algebra Test... 108

3.5.2.1 Item Polarity ... 108

3.5.2.2 Item Fit ... 109

3.5.2.3 Unidimensionality ... 112

3.5.3 Construct Validity ... 113

3.5.3.1 Continuum of Increasing Intensity ... 113

3.5.3.2 Summary of Item Difficulty and Student Ability ... 115

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3.5.4 Consistency of Result with Purpose of Measurement ... 115

3.5.4.1 Reliability and Separation ... 115

3.5.4.2 Precision of Measures ... 117

3.5.5 Revision of the Algebra Test... 118

3.6 Data Collection Procedures ... 120

3.7 Data Analysis ... 121

3.7.1 EFA and CFA Justifications ... 121

3.7.2 Assessment of the Measurement Models ... 125

3.7.3 Assessment of the Structural Models ... 127

3.8 Conduct of Survey and Ethical Considerations ... 127

3.9 Chapter Summary ... 128

CHAPTER FOUR: RESULTS AND FINDINGS ... 131

4.2 Screening and Data Cleaning... 131

4.2.1 Missing Data ... 132

4.2.2 Screening for the Outliers ... 132

4.2.3 Test of Normality ... 133

4.2.4 Assessing Linearity ... 134

4.2.5 Assessing Homoscedasticity ... 134

4.2.6 Multicollinearity ... 135

4.3 Descriptive Statistics on Demographics ... 136

4.4 Testing The Assumption ... 137

4.4.1 Gender ... 137

4.4.2 Respondents Distribution ... 137

4.4.3 Interest Toward Mathematics ... 138

4.4.4 The Grade for Mathematics Mid-Term Test ... 139

4.5 RQ1 - Algebra Test findings ... 139

4.5.1 Final Items According to the Strands of Algebra... 139

4.5.2 Preliminary Analysis of the Main Data ... 140

4.5.2.1 Validity of Test Items ... 141

4.5.2.2 Consistency of Result with Purpose of Measurement ... 143

4.5.3 Conclusion on Adequacy of Algebra Test ... 145

4.5.4 Students’ Perfomance in Algebra Test ... 146

4.5.4.1 Descriptive Analysis on Algebra Test Performance ... 146

4.5.4.2 Summary of Item Difficulty and Student Ability ... 149

4.5.4.3 Difficulties According to the Sub-Topic/Algebra Strands 151 4.6 Multivariate Analysis... 153

4.7 Adequacy of CLEE Measurement Model... 154

4.7.1 Assessing CLEE Measurement Model Fit ... 154

4.7.2 Reliability of the Constructs... 158

4.7.3 Construct and Convergent Validity ... 160

4.7.4 Discriminant Validity ... 162

4.8 Adequacy of Mathematics Attitude Measurement Model ... 163

4.8.1 Assessing CLEE Measurement Model Fit ... 163

4.8.2 Reliability of the Constructs... 166

4.8.3 Construct and Convergent Validity ... 167

4.8.4 Discriminant Validity ... 168

4.9 Addressing Common Method Bias ... 168

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4.9.1 Common Method Bias (CMB) for Constructivist Learning

Environment ... 169

4.9.2 CMB for Students’ Mathematics Attitude ... 170

4.10 Level of Constructivist Learning Environment and students’ mathematics attitude ... 171

4.10.1 Level of Constructivist Learning Environment... 171

4.10.1.1 Level of Teacher Support ... 172

4.10.1.2 Level of Cooperation ... 172

4.10.1.3 Level of Personal Relevance ... 173

4.10.1.4 The Level of Task Orientation ... 174

4.10.1.5 Level of Student Centeredness ... 174

4.10.2 Level of Mathematic Attitudes... 175

4.10.2.1 Level of Behavioral Engagement ... 176

4.10.2.2 Level of Cognitive Engagement ... 177

4.10.2.3 Level of Affective Engagement ... 177

4.11 The Assessment of Structural Model ... 178

4.11.1 Constructivist Learning Environment and Students’ Mathematics Attitude ... 179

4.11.2 Constructivist Learning Environment and Algebra Performance 181 4.11.3 Students’ Mathematics Attitude and Algebra Performance ... 184

4.12 The Assessment of Structural Model Based on Conceptual Framework 186 4.12.1 Constructivist Learning Environment and Students’ Mathematics Attitude ... 188

4.12.2 Constructivist Learning Environment and Algebra Performance . 189 4.12.3 Students’ Mathematics Attitude and Algebra Performance ... 189

4.13 Summary of the Results ... 189

CHAPTER FIVE: DISCUSSION AND CONCLUSION ... 235

5.2 Discussion of Findings ... 235

5.2.1 RQ1 - The Level of the Algebra Performance ... 236

5.2.2 RQ 2 - Adequacy of the Hypothesized Measurement Models ... 239

5.2.3 RQ 3 – The Level of Constructivist Learning Environment and Students’ Mathematics Attitude ... 244

5.2.4 RQ 4 to RQ 6 - Relationship between Factors ... 246

5.2.5 RQ 7 - The Structural Model Based on Conceptual Framework .... 251

5.2.5.1 Constructivist Learning Environment vs Students’ Mathematics Attitude ... 252

5.2.5.2 Constructivist Learning Environment vs Algebra Performance ... 253

5.2.5.3 Students’ Mathematics Attitude is not associated with Algebra Performance ... 255

5.3 Contributions/Implications of the Research ... 255

5.3.1 Theoretical Implications... 255

5.3.2 Practical Implications ... 257

5.4 Limitations and Further Research ... 259

5.5 Conclusion ... 262

5.6 Recommendations... 263

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REFERENCES ... 265

APPENDIX A: CLEEMA (ENGLISH) FOR PILOT TEST ... 288

APPENDIX B: CLEEMA (B.MELAYU) FOR PILOT TEST ... 290

APPENDIX C: ALGEBRA TEST ... 292

APPENDIX D: CLEEMA CONTENT VALIDATION ... 298

APPENDIX E: CLEE PRELIMINARY STUDY ... 301

APPENDIX F: PERMISSION FROM MINISTRY OF EDUCATION ... 302

APPENDIX G: PERMISSION FROM DEPARTMENT OF STATE ... 303

APPENDIX H: PERMISSION FROM IEA ... 304

APPENDIX I: PERMISSION FOR WIHIC AND CLE ... 306

APPENDIX J: ACTUAL RESULTS FOR THE ALGEBRA TEST ... 308

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LIST OF TABLES

Table No. Page No.

3.1 Distribution of Districts Amongst Respondents 79

3.2 Sample Item of Each of the Construct 87

3.3 KMO and Bartlett’s Test of CLES 91

3.4 Correlation Matrix of Constructivist Learning Environment 92 3.5 Results of the PAF, Factor Loadings (FL), Communalities

(Comm), Eigenvalues (EV), and Variance Explained (N=250) 94

3.6 KMO and Bartlett’s Test of MA 97

3.7 Correlation Matrix of Mathematics Attitude Scale 99

3.8 Results of the PAF, Factor Loadings (FL), Communalities (Comm), Eigenvalues (EV), and Variance Explained (VE)

(N=250) 100

3.9 Reliability estimates of the constructs 101

3.10 Topics and Algebra Strands Covered in Test Items 103

3.11 Content Validity Ratio 104

3.12 Item Polarity and Item Fit Based on MNSQ Value 109

3.13 Item Statistic: Misfit Order 111

3.14 Reliability of the Item Difficulty Estimates 116

3.15 Reliability of the Person Ability Estimates 117

3.16 Thresholds for GOF Applied in the Study 125

3.17 Threshold used in assessing convergent validity, discriminant

validity and reliability 126

3.18 Summary of the Data Analysis 129

4.1 Mean, SD, Skewness and Kurtosis of the Data 133

4.2 VIF and Tolerance Values of the Variables 136

4.3 Gender distribution amongst respondents 137

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4.4 Distribution of Districts Amongst Respondents 138

4.5 Distribution of Respondents’ Interest 138

4.6 Distribution of Respondents’ Performance in Mathematics Mid-

Term Test 139

4.7 Final Items According to the Strands of Algebra 140

4.8 Item Polarity Statistics: Correlation Order (Actual Data) 142

4.9 Person Reliability 144

4.10 Item Reliability 144

4.11 Students’ Performance by Levels 148

4.12 Students’ Performance by Levels 148

4.13 Items difficulty level based and the strands of algebra 153 4.14 Cronbach’s Alpha Reliability for Factors for CLEE 158 4.15 Reliability of the Extracted Constructivist Learning Environment

Dimensions 159

4.16 Standardized Factor Loadings, CR and AVE for CLEE 161

4.17 Discriminant validity test outcomes 162

4.18 Reliability of the Extracted Mathematics Attitude Constructs 166 4.19 Standardized Factor Loadings, CR and AVE for MA 167

4.20 Discriminant validity test outcomes 168

4.21 Mean and Standard Deviation of CLEE 171

4.22 Mean and Standard Deviation of Teacher Support 172

4.23 Mean and Standard Deviation of Cooperation 173

4.24 Mean and Standard Deviation of Personal Relevance 173 4.25 Mean and Standard Deviation of Task Orientation 174 4.26 Mean and Standard Deviation of Student Centeredness 175 4.27 Mean and Standard Deviation for Mathematics Attitudesmary 176 4.28 Mean and Standard Deviation for Mathematics Attitudes 276

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4.29 Mean and Standard Deviation of Cognitive Engagement 177 4.30 Mean and Standard Deviation of Cognitive Engagement 178

4.31 Path Coefficients of the Hypothesized Model 179

4.32 Path Coefficients of the Hypothesized Model 188

5.1 Hypotheses and Results of Research Question One 239

5.2 Final Items and Constructs for CLEE 242

5.3 Final Items and Constructs for MA 243

5.4 Hypotheses and Results of Research Question 247

5.5 Hypotheses and Results of Research Question Seven 252

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LIST OF FIGURES

Figure No. Page No.

1.1 Malaysian Grade 8 Scores in TIMSS 11

1.2 Proposed Conceptual Framework 20

2.1 Scaffolding and Zone of Proximal Development (ZPD) 56

2.2 Social Cognitive Learning Theory 58

2.3 Kaput Algebra Thinking Strands, 2008 59

3.1 Estimated Number of Children Population According to States in

Malaysia 76

3.2 Location of the Districts in Selangor 77

3.3 Cluster sampling method based on districts 78

3.4 Content Validation Process 84

3.5 Scree Plot CLEE 93

3.6 Scree Plot MA 98

3.7 PCA Analysis to Test Unidimensionality 112

3.8 Wright Map distribution 114

4.1 Homoscedasticity Scatter Plots 135

4.2 Results of the PCA of Standardized Residuals 143

4.3 The Distribution of the Marks and the Percentage of the

Frequency 147

4.4 Wright Map Showing the Estimates of Items Distribution with the

Most Difficult Item at the Top 150

4.5 Estimates of Items Distribution Map According to the Strands 152 4.6 Initial Constructivist Learning Environment Measurement Model 155 4.7 Full Revised Constructivist Learning Environment Measurement

Model 157

4.8 Initial Mathematics Attitude Measurement Model 164

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4.9 Full Revised Mathematics Attitude Measurement Model 165 4.10 One Factor Measurement Model for Constructivist Learning

Environment 169

4.11 One Factor Measurement Model for Students’ Mathematics

Attitude 170

4.12 The Structural Model of CLEE and MA 180

4.13 The Structural Model of CLES and Algebra 182

4.14 The Structural Model of CLEE and Algebra Revised Model 183 4.15 The Structural Model of Mathematics Attitude and Algebra

Performance 185

4.16 The Structural Model of Constructivist Learning Environment,

Mathematics Attitude and Algebra Performance 187

5.1 The Five Elements of Constructivist Learning Environment 241

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CHAPTER ONE INTRODUCTION

“A symbol is a sound, or something visible, mentally connected to an idea. This idea is the meaning of the symbol. Without an idea attached, a symbol is empty, meaningless” – Richard R. Skemp, 1971

1.1 THE IMPORTANCE OF LEARNING ENVIRONMENT

Life during the COVID-19 pandemic is difficult for all and children are also not excluded. Prepared or not, the world has left us with not many choices and the human beings must adapt to the new normal. Since the pandemic outbreak in February 2020, many things including teaching and learning scenarios around the world have been drastically changed. Many countries including Malaysia have closed schools as part of a measure to control the spread. To ensure the continuity of learning process, there has been a sudden increase in online education. As a consequence, online education platforms become crowded with users. Unfortunately, in poorer or more rural areas, the teaching and learning process has been limited due to inadequate technological facilities.

Therefore, it is not surprising that some teachers, students, and parents are becoming more stressed in dealing with the challenges of online teaching, while the effectiveness of online learning is also still questionable. To overcome the problems of online learning, some countries introduced other alternatives, for instance, blended learning as one of the solutions. Blended learning combines face to face or physical classroom learning with online or virtual classroom learning. The main objective is to create effective learning condition so that students are interested in the learning process and hence understand what is taught. Even for the online classes nowadays,

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teachers try their best to involve students so that lessons are not just mere lectures.

Involving students in the learning process requires lengthy and careful planning on the teachers’ part. Online conferencing application such as Zoom, Google Meet or Microsoft Teams allow the class to breakout into rooms, so that students can do online discussions and get the most out of the class. Teachers try to ensure that students cooperate, communicate and perform tasks that requires critical and creative thinking in the classrooms.

In fact, no matter what the platform is, whether in a real or virtual classroom, learning environment is still important. For teaching and learning to be more effective and more meaningful, teachers as ‘classroom managers’ are responsible to provide suitable and sound learning environments. Henceforth, this study is an extension of previous researches in that it is looking into the constructivist elements in the teaching and learning of mathematics especially for the topic on Algebra among Form One students in Malaysian school. However, the study started in 2017 and the data were collected just a few months before the pandemic outbreak, with the focus concentration only on the real or physical classroom learning environment.

1.2 BACKGROUND OF THE STUDY

In 2017, the Kurrikulum Standard Sekolah Menengah (KSSM) (Standard-Based Curriculum) for Secondary Schools was introduced to imbue a balanced set of knowledge and skills, as an initiative of Malaysian Education Blueprint 2013-2025. In fact, the properly designed Malaysia Education Blueprint 2013-2025 was to prepare the students for the future’s economy and the globalized world (Ministry of Education Malaysia, 2013) which include the employment of constructivist approaches to teaching and learning mathematics. Learning mathematics requires construction and

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active reception of knowledge, while knowing mathematics, requires constructive work with mathematical objects in a mathematics community (Davis, Maher &

Noddings, 1990). These views surfaced mainly in relation to constructivist approaches in a classroom. In general, constructivist learning environment (CLE) is an environment where learners work together, support each other and using various approaches to construct knowledge and to solve a real-life problem (Gu, Zhang, & Gu, 2020).

Learning environment is a signiﬁcant determinant of student learning and the research on learning environment has been carried out for more than four decades (Aldridge & Galos, 2018; Chipangura & Aldridge, 2017). Research related to the field of learning environments has included a wide range of contributions to the field of education (Chipangura & Aldridge, 2017; Lim & Fraser, 2018). In a quality mathematics learning environment, a student should be encouraged to think beyond what is presented, to explore further the concepts and to work collaboratively with the teacher and her/his classmates aligned with the constructivism principle (Zain, Rasidi,

& Abidin, 2012).

Basically, the main characteristics of constructivist approaches are that learners create knowledge on their own through participating, applying their previous knowledge or experiences. It is believed that learners as active agent of informants have the capability to adjust their mental model to incorporate new experiences and make sense of them (Zain, Rasidi, & Abidin, 2012). This conception of learning is influenced by Jean Piaget’s idea. Piaget (1972) suggests that concepts are formed by learners through a reconstruction of reality, not through an imitation of it. Another prominent scholar of this area of study, Bruner (1960) emphasizes that knowing is a

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process, not a product. Bruner stresses that learning is an active process in which learners construct new ideas or concepts based upon their current/past knowledge.

In learning mathematics specifically, many researchers believe that students’

learning can be improved through teaching using methods that align with the framework of constructivism (Ilyas et al., 2013). This is because constructivism focuses on preparing the learner to problem solving situations, negotiate and generate solutions through sharing and exchange of ideas, which is relevant to the mathematics classroom. Furthermore, in mathematics, the constructivist view of learning has been receiving a great deal of attention in the last three decades because of the proven positive impacts (Alsharif, 2014).

It was also proven that the constructivist approach increased positive attitudes towards lessons (Khalil & Aldridge, 2019; Toraman & Demir, 2016) and is significantly correlated with students’ attitude (Lim & Fraser, 2018). In fact, students’

attitudes toward mathematics have been the subject of research across several decades (Ma & Kishor, 1997). Attitudes is important in education and to prepare students for the future. In mathematics education literature, attitude has a long history and very high popularity (Tabuk, 2018). Some researchers argue that students’ attitude towards mathematics is the most important factor in determining students’ achievement (Ma &

Kishor, 1997; Tabuk, 2018). Hence, it is necessary to extend a study on students’

attitudes toward mathematics. This is because it has a potential to stimulate students’

learning in mathematics. Understanding student attitudes toward mathematics and its relationship with constructivist learning environment is crucial.

Multiple studies have found that constructivists approach is more effective and offers better learning prospects compared to traditional teaching method. Traditional method focuses on lecture-based instruction or teacher-centred teaching. It is the

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method where the teacher explains the theoretical knowledge and the students passively accept the knowledge by listening and taking notes (Bi et al., 2019). Thus, when Ilyas et al., (2013) conducted a study to examine the effect of teaching algebra through constructivist approach, he found that the treatment group taught through constructivist approach excelled in achieving learning outcomes compared to the control group taught through traditional one-way teaching.

Meanwhile, Baird, Andrich, Hopfenbeck, and Stobart (2017) suggested that evaluation of students’ performance is crucial in education to examine whether the students have learned according to the learning objectives or not. Since algebra plays an important role in students’ performance, it is crucial to understand the factors that influence the algebra performance (Simsek et al., 2019). Algebra is part of mathematics which deals with letters and symbols, (Khalid, Yakop and Ibrahim, 2020), hence, competency in interpreting and manipulating letters is crucial for the students to pursue higher level mathematics (Fey & Smith, 2017). Algebra proficiency is also known as the key to success in learning the next mathematical elements and must be mastered by students (Star, et al., 2015). Therefore, it is essential to learn algebra in order to understand mathematics (Egodawatte & Stoilescu, 2015).

Mathematics is important for many careers in modern technological societies, and it is also a fundamental skill for personal fulfilment, active citizenship and social inclusion (Stankous & Buibas, 2015). Mutai (2016) also emphasizes the importance of mathematics in this era of technology by stating that before an individual can function well in the society, he or she must possess relatively good knowledge of mathematics.

In Malaysia, mathematics is compulsory in accomplishing the upper secondary level of education in Malaysian Education System (Davrajoo, Tarmizi, Nawawi, & Hassan, 2010). Hence, a lack of sufficient mathematical skill and understanding might affect

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the ability to make critical important educational, life, and career decisions (Sherman, Richardson & Yard, 2015).

Algebra is used in our daily lives for a very long time to represent numbers and quantities in mathematical formulae and equations (Saleh & Rahman, 2016).

However, despite its importance, algebra can be very challenging to students. Studies have found that some students in many countries have an extremely shallow and imperfect knowledge of algebra (Witzel, 2016; Khalid, Yakop & Ibrahim, 2020). This is probably because algebra introduces more abstract representations, more complex relationships between quantities and also it can magnify the misconceptions that have their roots in earlier instruction (Egodawatte & Stoilescu, 2015). Thus, for some middle school students, algebra has become a nightmare subject (Alsaeed, 2017).

Therefore, it is essential to help students to confront the learning difficulties and misconceptions that can hinder their performance and learning in the subject by finding new ways to motivate them and to engage their interest in learning algebra (Alsaeed, 2017; Booth et. al., 2014). Currently in Malaysian KSSM secondary school syllabus, algebra is being taught to Form One students from Chapter Five (5) onwards.

There are thirteen (13) chapters to be covered in the Mathematics Form One Textbook (Ministry of Education, 2016). In chapter 5, Form One students are expected to know what is unknown and how to create algebraic expression. This offers the students a strong algebraic foundation because the later chapters will be utilizing this knowledge.

Thus, it is important to strengthen the foundation knowledge on algebra first because being unable to understand the basic will affect the next stages of mathematics learning.

Besides, the element of Higher Order Thinking Skills (HOTS) has also been emphasized as one of the agendas in the Malaysian Education Blueprint 2013-2025

A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy in Education

THE RELATIONSHIPS BETWEEN CONSTRUCTIVIST LEARNING ENVIRONMENT, MATHEMATICS ATTITUDE, AND ALGEBRA PERFORMANCE FROM

( شار ) ايقلل س (Rasch Measurement

DECLARATION

COPYRIGHT PAGE

DEDICATION

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

CHAPTER TWO: LITERATURE REVIEW ... 28

CHAPTER THREE: METHODOLOGY ... 73

CHAPTER FOUR: RESULTS AND FINDINGS ... 131

CHAPTER FIVE: DISCUSSION AND CONCLUSION ... 235

REFERENCES ... 265

APPENDIX H: PERMISSION FROM IEA ... 304

LIST OF FIGURES

CHAPTER ONE INTRODUCTION

1.2 BACKGROUND OF THE STUDY