ONG PUAY HQON

A thesis submitted in full EullilEment of the requiF.enments for the degree of

Doctorate of Philosophy

Faculty of Cognitive Sciences and Human Develapment UNNERSm MALAYSIA SARAWAK

June 2001

ABSTRACT

Every human being, through nature and n m , exhibits a propensity to employ Werent brain subshates for different thoughts, adorn and reactions to the m * o m t . The left hemisphere is said to be bepomible for processing verbal stimuli and the right hemisphere for visual-spatial processing. H e m , pupils with different brain laterality patterns will demonstrate different capacities in learning a!

particular topic or subject. This study aimed to identify the brain laterality patterns of pupils and to investigate its associations with visual-spatid ability, mathemtics and fraction scores.

Paper-and-pencil tests of brairz, lateralityr, visual-spatial and f~actions wem coinductal among a random sample of 5 12 Primary 5 children. 28 Mgh achievers and 28 low achievers in the fraction diagnostic tests were randomly selected for clinicd interview sessions. An attempt to undmtand the nature and specEcity of the learning difficdties eqerienced by cEIdn=n in fractions was made by identifying the cerebral substrates used in M o n processing. This was done by comparing task-related ElEG changes between 20 low and 19 high abiIity pupils during fiactian tasks. These pupils, randomly selected from the clinicill interview group, were all right-handed and had no previous history of neurological dyshdons .

It was found that pupils of wholehained dominance word statistically highest in fraction canceptml know1dge &st. High fiactian achievers were also high visual- spratid scorers and vice v m a The resuIts suggested that there was an involvemnt of both whole brain and visual-spatial pacessing in the leaning of fractions. The significant differences in task-related I E G activation patterns and asymmetry indices between these high and low achievers were interpreted in tams of their &fferential cognitive andl information processing strategies. Whilst the low achievers presented a tonic bias for a more active right hemisphere during rcsting conditions, they exhibited a tendency to employ more of theiir Ieft hemispheres in working out fraction tasks. This contrasted with the high achievers who while, presenting a resting mnic bias for a more active left hemisphere, wae able to utilize more of their right hemispheres during fraction tasks. As the right hemisphere has been widely postulated to be ihvolved in visual-spatial strategies, the findings suggested that the low ac0ieVm employed less efficient neural substrates and hence, Iess efficient cognitive sg-ategies whilst the high achievers were mom likely tc4 use mare efficient visual-bas& strategies. This obmation was discussed in relation tu the learning difficulties md misconceptions of rZle low achievers identified through a verbal protwo1 of the cognitive task analysis.

A neurobiological model of fraction laming was develop& to discuss the invalvement of appropriate neural resources in the processing of fraction tasks. This model was then used to describe the breakdown in fraction learning through an apparent mismatch of newd substrates and task demands. The implicarion~% towards using ado-visual aids, non-symbolic manipuIatiom and visual-based strategies in the t e ~ h i n g and learning of fractions were also discussed. Recommendations were d e for instructional and cmicdar refonns towards tfie teaching and learning of fractions. This study also d k e d attention to a ~nultidiseiplinitq approach of cogIzitive newscience in educational research.

Setiap mnranusia, mars smulajadi dan pengdmm, munjukkan suam kecendenmgm muntruk menggunakm substrat-substrat otak yang bez'laim untuk peaikkan, tindakan dm ger&alas kepada pemekitaran. Hemifera otak kiti dkt&n batmggung jawah, wtuk pemprosesanr stimuli lisan m k d a t u d s f a a k.anarr pemprosesan visual-slpatid. Oleh itu, murid-murid dengan corak latediti otak rang berbeza akan mnunj* keupayaan rang berbeza mtuk belajar samtu mpik atau mata plajaran. Kajian ini bertujum mtuk mengenalpasti corak Iateraliti otak para murid sata menyelidiki perktlitamya dmgan kbolehan visual-spatial, skor-sh mtmatik dm paahan.

Ujian-ujian kmas-pew1 mengenai 1atp;rajEiti ot&, visual-spatial d m poca$iha ditadbirkm kepada satu sample raw& terdiri daripada 512 mwid Tahun 5. 28 orang murid berkebolehan, tinggi dan 28 orang murid ~ k e b a l e ~ m d a h ddam ujian- wjim diagnthstik pecahan dipilih swam mwak mNk sessi temubual k l i d d . Daxi kmpdan id, 20 o m g murid berkebolelzan tinggi dan 19 orang murid berkebolehan rendah dipilih untuk kajian EEG yang bertujuan unW mengenalpasti substrat- substrat atak yang d i g u n h senma pemprosesarm pecah,an. Ahli-ahli sub-sample ini bukan kidzll &an tidak mempunyai sebmmg mudah mmlagikal.

napatan menmjukh bahawa inwid-murid ymg bedominan otak rnenyehmh mencapai paling tin@ dalam ujian pengetahurn komq pecahan. Murid-murid yang k r k e b l e h n tinggi &lam pecahan jnga mencapai tifuggi dalarn ujian-ujian visual- spatial dan s&aliknya, Dapatan nnencadangkan penglibatan otak yang m n y e ~ ~ dm pmprosesF visual-spatial ddam peahelajaran pmahan. Pmbezaarn yang sigdfikan di anfara 'wrak-corak k&fan EXG semasa tugas-tugas paahan dm indeh asime&i di a n m pencapai ti?@ dan pencapai rendah diintepretasikm dad segi strakgi kognitif d m pempmsesy m & w t yang berbeza. Pencapai-pencapai rendah rrnnunjukkan hemisfera W ) y a n g lebih aktif senma keadaan rehat, tetapi pula beocenderung rnenggunakan hemisfera kiri s e w tugas-tugas pwalm. Ini adalah berbeza dengan pencapai-pencapai tin@ yang menunj hc-misfera kiri ymg lebih aBrtif senma rehat telapi mm&gunakan hemisfera h smasa tugas- tugas pecahan. Oleh k e r n hemisfera hnm dljarigka terlibat da lm strategi visual- spatial, dapatan ini mencadarum bahawa pencapai-pemapai rendah rnenggmabn substrat otak dan algh :hitug strategi kognitif yyag kurmg k k e s a n manakala pencapai-petscapai tinggi lebih cendamg mnggwnakan strategi berasaskan vlisual yang lebih berkesm. Pernerhatian ini dibincang dari se@ masalah pembelzljaran dan salah anggapan pencapai-pencapai rendala rang dEkenalpasti melalui pratokol lism.

Suatu model newcrbiobgtkal untuk pembelajaran faahan dikembanw untulk membincangknn penglibatan mas - ram at& dalam pempmsesm pwahan. Model hi kemudiamya digmakin mtuk meneran@n kesutitan dalam pembelajmm pecaban melalui s a w padanan substrat otak dengan kepecluan tugas. ImpOrirsi terhadap dat pmdang-dengar, nmipulasi b a - s h b o I d m stratcgi. bmasi~skan visual &am pengajaran-pembelajaran pecahan juga dibincang, Penymuaian stfategi pengajwan dm kdkulium bagi pengajaran-pembelajaan pecahan &cadan&n. Kajian ini juga mennrlk perhatian kep& pendekatan integasi neurosains kognitif dalam penye l idk pendidikan.

DECLARATION

No portion of the work referred to in this thesis has been submitted in support of an application for another degree of qualification of this or any other university or institution of higher learning.

PENGAKUA N

Tiada bahagian daripada tesis ini telah dikemukakan untuk menyokong sesuatu permohonan lain untuk mendapatkan ijazah di universiti ini atau institusi pengajian tinggi yang lain.

Signature:

Name: Ong Puay Hoon

Date: 27 June 2001

To Kum Loy,

Kher Lee,

Kher Chinq,

Khai M e q ,

and

in sweet and everlastinq memcrq o f our first-born, Nq Khai %it

Only a while yar smiled at u5, Only too brief we held ym close, Dut forever, we treasure those moments, 5weet child , , , , ,

till we meet aqain at heaven's door,

ACKNOWLEDGEMENTS

I would like to acknowledge the award of the scholarship and study leave from the Ministry of Education and Aminuddin Baki Institute, Malaysia to pursue a course that leads to the Degree of Doctorate of Philosophy at this university.

The pursuit of the study that culminates in this academic thesis will not be possible without the help of many people. It is with a grateful and humble heart that I acknowledge all the help extended to me.

My supervisors have been an unfailing source of valuable inputs, insights and guidance, both in the work and preparation of the manuscript. You have given me direction, shape the quest, guide the writing, and most of all, each of you have ungrudgingly given me your time to listen me out. Your constructive and sharp criticisms gave me confidence and encouragement. With admiration and humility, I thank you all, Prof. Dr. Syed Hassan bin Syed Ahmad Almashoor, Prof. Dr. Razali bin Arof, Prof. Madya Dr. Peter Songan and Dr. Wang Yin Chai, from the bottom of my heart. Only God can repay this deed and may He continues to shower his blessings on each one of you.

I thank Mr. Kamal D. E. Quadra, Director of Sabah Education Department, who as ex-principal of Maktab Perguruan Gaya, Kota Kinabalu, unceasingly encouraged and supported his staff to pursue further studies.

I acknowledged the insightful and tireless comments of Prof. Dr. Dietrich Lehmann, Director, The KEY Institute for Brain-Mind Research, University Hospital of Psychiatry, Zurich, Switzerland for analysis of the EEG data and to Dr. Abu Bakar Abd. Rahman, Consultant Neurologist, Pantai Medical Center, Melaka for his invaluable help in editing the EEG output.

The late Prof. Madya Dr. Jamali Ismail had helped to guide my first few attempts in shaping the research proposal. I owe him my grateful thanks and pray that his soul rest in peace among God's chosen people in heaven.

Despite her heavy work schedule, Madam Elaine Khoo Guat Lein (Faculty of Cognitive Sciences and Human Development, UNIMAS) still found time to read, edit, comment and critique my research proposal and the thesis draft. Thank you Elaine, I am truly indebted to you. I record my heartfelt thanks to Dr. Akhbar Ibrahim who as my Principal in Tun Abdul Razak Teacher-Training College found time among his tight schedule to read and offer constructive suggestions to the draft.

I take this opportunity to thank Cik Norazuna Norahim (Deputy Dean, Center of Language Studies, UNIMAS), Puan Muriatul Khusmah Musa (Center of Language Studies, UNIMAS), Puan Monica Lau (Tun Abduk Razak Teacher-Training College, Kota Samarahan) and Encik Md. Zin Mohd. Amin (Gaya Teacher-Training College, Kota Kinabalu) who helped to translate the research instruments.

My grateful thanks also to Mr. Hong Kian Sam (Faculty of Cognitive Sciences and Human Development, UNIMAS) whose observations, comments and suggestions to the statistical procedures and analyses allowed a greater sense of confidence in the reporting. I gratefully acknowledged the help and assistance of Dr. Spencer Empading (Faculty of Social Sciences, UNIMAS), Dr. Charlie Laman (Faculty of Resource Science and Technology); Mr. Nagarajah Lee (Universiti Teknologi MARA, Kota Samarahan) and Dr. Toh Wah Seng (Batu Lintang Teacher- Training College, Kuching) who have most willingly given up their time to sort out my statistical difficulties and loaned out useful references.

I am indebted and grateful to the heads of schools, senior assistants and teachers of the schools involved in the pilot and research studies who have pleasantly and whole-heartedly rendered their approval, assistance and support. To the pupils involved in Sekolah Rendah St. Anthony, Penampang, Sabah; Sekolah Rendah Bantuan St. Theresa, Kuching; Sekolah Rendah Bantuan St. Thomas, Kuching; Sekolah Rendah Kebangsaan Encik Buyong, Kuching and Sekolah Rendah Merprti Jepang, Kuching and their parentslguardians, I thank you with all of my heart.

I am indebted to the Dr.Liding Jungian, Director of the Medical Services, Sarawak General Hospital for his approval to use the EEG machine and technician expertise. Very special thanks and gratitude to Mr. Berinoe Rojey, the EEG technician of Sarawak General Hospital, who has given so much of his time and sharing of expertise to do the EEG recordings and analysis. I acknowledged the free use of the Medelec's Power Spectrum Analysis program from Mr. Choong K.K. of Delta Medisains Sdn. Bhd., Kuala Lumpur and the expert comments on EEG procedures of Mr. Thomas Yeoh, Medical Neurophysiology Technologist, Loh Guan Lai Specialist Hospital, Penang.

I thank Miss Angela Anthonysamy (Faculty of Cognitive Sciences and Human Development, UNIMAS) for her valuable support and help in sourcing out much- needed research articles. My grateful thanks to Mr. Lee Kim Chong, Maktab Perguruan Gaya, Kota Kinabalu for his help in opening my way to the pilot-study school and his most patient tutoring in AutoCAD.

I thank Prof. Dr. Ibrahim bin Mamat, Dean, Faculty of Cognitive Sciences and Human Development, UNIMAS for his support and help in facilitating my study. I acknowledge and thank Dr. Abang Ahmad Ridzuan bin Abang Awit (Faculty of Cognitive Sciences and Human Development, UNIMAS) for opening my way into the research schools. Grateful thanks to the lecturers, tutors, managers, administrative assistant and support staff of Faculty of Cognitive Sciences and Human Development and Center of Language Studies, UNIMAS for their help and words of encouragement. I thank the managers and support staff of Center of Graduate Studies and Research Support, Center of Academic Information Services, Chancellery and the telephone operators for their joyful countenance and support.

I acknowledge the help of Dr. Soe Soe New (resident doctor, UNIMAS), Madam Park Young Soon (Faculty of Information Technology, UNIMAS), Dr. Ang Whye Tong (Faculty of Information Technology, UNIMAS) and Mr. Tan Chong Eng (Faculty of Information Technology, UNIMAS) for their help in facilitating the EEG study. I thank the many graduate and undergraduate students of W A S who have never failed to provide a smile, a pat on the shoulder or help with the computer whenever there was a need. To Fletcher Tan, Phang Khar Yeow, Donny Chan, Ng Giap Weng, Dunstan Goh, Crispin, Nelson, Rojin - my grateful thanks.

My heartfelt gratitude to my best friends, Miss Fong Lye Choo (Maktab Perguruan Persekutuan Pulau Pinang, Penang) and Miss Ewe Mee King (Tuanku Bainun Teacher-Training College, Bukit Mertajam for their constant encouragement, moral, physical and spiritual support to me and my family.

My sisters, Liu and Tee, have been a constant source of inspiration. When the going gets tough, they are always there to lend a helping hand. Their encouraging words lent some sense of sanity when hopelessness prevailed. Their attention and care to my children during these trying period when the matriach seemed lost in her quest for a dream provided strength to the structure of the family framework. Thank you, girls, for the notebook, editing the draft and . . . . . ... everything!

Upon urgent summons, my mother has flown across the seas to come and look after what's left of my family. Her unfailing support and love tug at my heart-strings and constantly reminded me not to quit. To both my parents who have stood by me and my family, I salute and love you. To my siblings and in-laws, thank you all for the unconditional support.

Kum Loy, my beloved, is a husband beyond compare. I am because you are! Your steadfastness and unconditional support allow me to be what I am and can do. Your love surrounds and gives me strength. Thank you, darling. To my children - Kher Lee, Kher Ching and Khai Meng who were initially mystified at their mother having to study and leave them, I thank you all for struggling with me.

To all of you who have provided moral support and, whether directly or indirectly, lent a hand to the successful completion of this work, I thank you.

And most importantly, thank you God, for your unfailing presence and blessings to me and my family.

TABLE OF CONTENTS

DECLARATION

1.1 Introduction

iii iv

v- vii viii - miii

1

1

3 4 5 7 8

9 11

12

12

15 19

21

21

23

24

25

25

athematics Achieyement in 25

viii

Page

2.3 Children's Difficulties with Fractions 26

2.4 Newpsychdo@d Modch of Normal Brain Functioning 29 2.4.1 Cerebral laterality in humans: The Hemisphericity 29

Theoty 2.4.1.1 Development of Iaterality 33 2.4.1.2 Devetopment of Iaterality over time 35 2.4.1.3 The roles of genetic and environmental

faems in the establishment of asymmetry 36 2.4.1.4 Brain 1attr;tliZation and handedness 38 2.4.1.5 Hemispheric sharing of control of behavi 2.4.1.6 Role of cotpus eaflosum and the

commissurcs as interhemispheric integnttors 39

2.4.1.7 Theoretical models of brain asymmetry 39

2.4.2 >Hierarchical brain orgdat ion 40 2.4.3 Modutar brain orpnhtion (localis model) 41 2.4.4 Pwdlel distributed prwaxw ( c o d o n i s t model) 42

rain Imaging Techniques in the Study of Asymmetries 42 5.1 Elm&alography (EEG) 42 3.2 EAagncto-e+o~aphy (ME01 43 5.3 Posimn Emission Tomography (PET) 43

25.4 F d 0 m . I M&e Resonance Imaging (tMRI) 44 25.5 Trmsmmial Stimulation (TMS) 2.5.6 Single sell recording ~~

2.7 EEG Power Spectral Analysis

2.8 Neurological Bases in Mathomata b%~ning 2

2.9 Cerebtal Bases of Ocher Cognitive Processes 7

Page

2.10 Brain Dominance, Mathomatics Achievement and Gender 59 2.10.1 Brain dominance and matfiematics achievement 2.10.2 Brain dominance and gender 2.10.3 Mathematics actiievement and gender .

2.1 1 Visual-Spatial Ability 2.11.1 Categories of visual-spatial ability and tests 2.1 1.2 Visd-spatid ability and gender 65 2.1 1.3 Visual-spatial ability and brain dominance 66 2 1 1.4 Visual-spatial ability and mathematics achievement 67

2 Impiications of Brain Laterdity to Education 68

70

CHAPTER 3 RESEARCH METHODULOGY 71

7 1

71

4.0 Introduction

4.1 Demographic Char-ties 4.1-1 Demographic &a (M the subjects for EE

invtrstigatiom (sub-sample SG2)

4.2 Levels of Academic Ability 42.1 Levels of academic ability and gender

4.5.4 Visual-spatial scores and brain dominance 111 (a) Spatial perception test 111 (b) Spatial visualization test 112 (c) Two-dimensional mental rotation test 113 (d) Three-dimensional mental rotation test 114

4.6 Fraction Knowledge 4.6.1 Achievement in the fraction tests

(a) Types of fraction tests (b) Types of representational models for fractions

4.6.2 Fraction scores and gender 4.6.3 Fraction scores and levels of academic ability 4.6.4 Fraction scores and mathematics scores 4.6.5 Fraction scores and brain dominance patterns 4.6.6 Fraction scores and visual-spatial ability

(a) Fraction representational knowledge (b) Fraction conceptual knowledge (c) Fraction procedural knowledge (d) Taat &action scores

4.7 Power Spectrum Analysis Of Electroencephalographic Data 123 4.7.1 D m transformation 1 24 4.7.2 Power spectral parameters and gender 124 4.7.3 Power specrral parameters and levels of fraction

ability 125 4.7.4 Effects of experimental conditions on the locations of

cenm of electric gravity. 127 4.7.5 Asymmetry indices for inter-hemispheric activation 129

4.8 Discussion 133 4.8.1 Differences, relationships and associations between

variables studied 133 4.8.2 Elechmncephalographic data 137

CHAPTER 5 FINDINGS AND DISCUSSIONS 2: CONCEPTUAL AND PROCEDURAL UNDERSTANDING OF FRACTIONS AMONG LOW AND HIGH ABILITY PUPILS

5.0 Introduction

5.1 Conceptual Understanding Of Basic Fraction Concepts

(a) Levels of academic ability . (b) Mathematics achievement (c) Fraction achievement (d) Brain dominance p d l e s (e) Visual-spatial ability

6.2.2 Conceptions and misconceptions of high and low achievers on the basic concepts and skills of fractions 179 (a) Fraction high achievers 179 (b) Fraction low achievers 179

6.2.3 Power spectral analysis of EEG data (a) Between gender (b) Between levels of fraction ability

6.3 Conclusions 6.3.1 The cerebral bases of fraction learning 6.3.2 Gender differences in fraction learning 6.3.3 Neurobiological model of fraction learning

6.4 Recommendations (a) For instructional reforms (b) For curricular reforms (c) Directions for further and future research (d) A call for a multidisciplinary approach of cognitive

neuroscience in educational research

6 5 Summary

BIBLIOGRAPHY LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES

198 xv xvi

xviii

xiv

LIST OF TABLES

f the left and right

tests and conversion

ships or differences

o the lev& of

xv

Pi%=

32 73 79

81

82 86

87 88

88

89

90

93 95 99 101 101

111

112

120

163

167

ability. Figure 16 Box-and-whisker plot for the spatial perception scores. Figure 17 Spatial vistabation scores with respect to levels of academic

ability. Box-and-whisker plot for the spatial visualization scores. Twodmensionai mental rotation scores with respect to levtls of academic ability. Box-and-whisker plot for the two-dimensiod mental rotation scores. Threedimensional mental rotation scores with respect to levels of academic abiiity. Box-and-whisker plot for the two-dimensional mental rotation scores. Spatial visualization scores with respect to brain dominance patterns. Two-dimensional mental rotation scores with brain dominance patterns. Thrtc-dimnsionat mental rotation scores with brain dominance

(concept) and procedural (procedur) kriowledge tests. hart for the achievement of fractions in region, discrete

number line models. and standard deviations of fraction scores with respect to

ution of fraction scores with respect to levels of academic

n of fraction scores with respect to mathematics scores. n of conceprtlal knowledge scores witb respect to brain

es between homologous cortical sites of the two

esentations of the differential activation patterns during

naadel of fraction learning

n the two groups of pupils.

115

116

117

118 119

121

124

126

128

129

132 185

186

xvii

LIST OF APPENDICES

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E Appendix F Appendix G1 Appendix G2 Appendix G3 Appendix HI Appendix H2 Appendix I1 Appendix I2 Appendix I3

Appendix I4

Appendix J l Appendix J2 Appendix K AppendixL

ApptndixM Appendix N

Mathematics Education In The Integrated Curriculum For Primary Schbls The Content Deveiopment of the Topic "FRACTIONS AND ITS OPERATIONS" in the Primary Mathematics Syllabus Letter of approval for study from the Educational Planning and Research Division, Ministry of Education Malaysia Letter of approval for study from the Sarawak Education Department Letter soeking consent for EEG investigations from parents The Handedness Inventory Fraction Diagnostic Test A (Ujian Diagnostik Pecahan A) Fraction Diagnostic Test B (Ujian Diagnostik Pecahan B ) Fraction Diagnostic Test C (Ujian Diagnostik Pecahan C) 'Style of Leaning and Thinking' (SOLAT) Caya Belajar dun Berjikir (SOLAT-BM) Spatial Perception Test (Ujian Paras Air) Spatial Visualization Test (Ujian Perkembangan Pennukaan) Two-Dimensional Mental Rotation Test (Ujian Putaran Mental Two-Dimensi) Three-Dinsional Mental Rotation Test (Ujian Putaran Mental Tiga-Dimemi) Rangku Temubual Berstnrktur Strucaved Interview Schedule (English Translation) Fraction Tasks During EEG Investigations Questionnaire before EEG tests (Soul-Selidik Sebelum Mengambil Ujian EEG) Photographs of EEG tests Ditribution of fraction scores with respect to brain dominance patterns Distribution of fraction scores with respect to visual-spatial scores

All men by nnturg desireth to kmw.

hhthematics being pat of universal education appears in the cuniculm of all schools. It has long been recognized that the basis of success of any society is predicated ta a certain extent on its ability to use mhematicd concepts as mathematical udmtading f a c f t i ~ a person's participation and appreciation of maoy daily human activities. F d m , mathematics achievement, in part, determill~s future educatian and emplayment pmpats (Samnwns, 1995). However, mathemtics educators dl over the w i d are expressing c o w m about the degree of rnathenoatical miemtanding displayed by children in schools (Shohmh, 1993). The declining bvels of mathematics achievement in same countries have caused a serious deterioration in thew iwiushiaf strength and handicap their progress in mseamh aad dewlq- Needless tn say, this decline has seriously constrained the comtries'eefforts ta mxnpete internationally.

The levei aad extent of mathematical t~n-ng a d computational expeak by MaEaysian childm is aka a m t i d camem Local research has shwn that the level of a c K e v m and attainment in mathematics in prhaq schook, especially quaIitativety, has nat been favorable (Jabatan Pendidikan S W , 1990). Although pupils were able to pass mathamtfcs, the grade achievement in a gale of A to E of a significant majotity hovered around C. A later study by the Ministry of Education s h o d that the of mathematics at primary school level in rural m s of the corntry w (Star, 1991). A survey by the P a k Education Offioe showed that at least 10% of Printaty 2 d 9% of Primary 3 pupils were still unabl Waharb HasMm ODCl pupils in our pimar)r schools as slow I-, that is, exhibiting serious learning difficulties in the 3Rs (Reading, Wting and "Rithmercic) {Star, 24 August 1998). These SEW leamer~ were to be put through a new and iotensive, and of course very costly, ret.l.rctdial beginning 1999. Very mendy, we Iwat that 70 p e ~ of pupils in Is failed in the Mnwy Schoal Evaluation Test (Ujian Plrnilaian S&hft Redah, UPSR), which was donally-mnistesrxi for all Primary 6 pupils (Vim&, 2000). This ought to be viewed with deep concern as the core subjects beiig in this national exmination include Mathematics, Science, M.aIay Language and English Language.

The Primary School Evaluation Test is an evaluation of the achievement of pupils &a are in their final yeiv of priw srrrhool education. In the evatuation of 1994, the perrentage of passes in Mathematics for pupils in the three types a€ primary schools were as foltows:

I j National NatiatdType s ch l s [Sekolah Kehngsam clan

2) National-Type (Chinem) schools

3) National-Type (Indian) schools

SekoiatF Rendah Kehgsaan) I... 64.0%

(Sckolah! Redah Kcbangsm {Gina)) .... 86.36

.... 58.1% . . . . 69.596

(Saiouxce: M&y$ian i3xudWWn Board Yeuriy Reportf i~ the Year 1989 - 1994)

(,!$eblhh Re& K e h g s m (Tamil)) Average percentage of passes in Mathematics

,----\

Schools with diffkenl: medium of instruction displayed different perfomces . "be Chi- language-medim schools performed best with 86% pass, foliowed by the Malay l a n g u a g e d u r n schools at 64% and lastly, the Tamil language-medium 8chmls which fared the wrst at 58%. An extensive doctoral study suppmd these findings and showed that not only pupils h r n Chinese language-medim schools scored hi- in &@cs, they indicated mrs positive attitudes towards the subject than their c o m a r t s (?&+sir, 1997). This disparity of achievment in mathematics was also seen between the urban anni rural schmh and between pupils of differeat socio-economic groups an$ ethnicity. Pupils from urban schaols, &am upper secio-ecorr?omic groups aiMi of Chinese parentage p e r f a d better, &a& in matbmnatics scaes and positive attitudes-

B e s W UnsatisfaCtMy performanee aa$ achievement, mathemtics learning in Malaysia is also plagued by otha problem. Students, both in the p m and s e o e w school levels, denaonstrate a high level of mathemtical anxiety and phobia m, 1998). Teack-- in tnatfimtics aLw exhibit anxiety in learning mathmatics (Gan, 2000) and display diffiicultiea and misconceptions on many basic trrpics in mathmnatics (Kaur, 1990; Koe, 1992; OR&, M~~hmad, Walaipc~n, Wild 8t W a d i , 1992). Teachtx-prepard mtmials a d published text/wwkbook9 used in scchools display miscomptions in m y flLndmWnta1 cmcepts oE r n a t h d c s and damnstrate Wted flexibiliv and diversity in its, fonrrs of representations CKOh, 1991). Srudents' mlment in secondary matfiematics and science is falling (fabartan Pen&* Sarawak, 1996; LRa, G h i , Yoona;, Khadijah, Loo, Lim, Munltah C 7Jn. 1996) despite being earlier w d by Omar (1993) that MUT country is facing a shortage of expert human resome M ~n d manage the uatiod science and tecblogy eatmphe. Chong Kslh Eat, Minister OT Tourism, Environmen, Science and T a h ~ l ~ g y Development of Sabah, even onhorted that science subjects were the least liked in schools among Sabah stuhnts {Borneo Pst , ZW).

An exledve research by Lee et d. (1996) revealed that m y secondary schcwl stlldents in Malaysia shy away from Science because of their paor foundations in Mithematics (58.0% of Porn 4 and 59. oE Form 6 students). They have no confidence in Mathemtics (42.8% of Form 4 and 48.1% of Fom 6 students) and

2

they do not m t M take Additional Miitfiemdcs (34.5% of Earn 4 students). Pupils in National Primary schools [ S e b k k Kebaagsmn), when intervitwed, declared having mental I lfo~ks in Mathematics to expiain their pe&mnce. The researchers cxlnjectumi Ulzlt this couId probably be due &.I s e l f fulfilling prophecy,

They alsa provided data that there were 38 681 Science students for 22.316) and 134 fit23 Nora-Sciencc students (or 77.69%) in 15393 for ail Forms 4 and 5 classes in all s~ in MaIaysi;l, thus giving a ratio of Science to Non-Science students at 22: 78. In our strife ta increase the ratio of Science to Nan-Science students to 6(E: 40 aw; in Vision 2020 by the hatn of the century ( A h d Sarji, I9933, it is timely to calr far greater attention ta research of children's learning of primary mthemtics especially in the way the brain is involved, Brain research has become the basis far decisions m improve education, especially in United States and as teaching is cortsidemd bath an art and science, it bmmes imperative to seriously look into the applications of brain research in the classmms.

The af%.cial nathematks curricula differ fram cauntry to wuntry according to the cultural values p k e d on mathematical knawledg and the current meeds and danands of faciety. In Waysia, our earnest drive far technoleg-literate howledge w r k m \ria the sntant s c k t fm the xientific-tecbnoIIo&cal challenges of Vision 20201 c& far attention and emphasis as to why ow students d those abrolaa are still under-achieving in Mathematics.

tNhik much attention d financial mwes have been din:& at the level of w~~ and attaimmnt in. ~~ mthemties by the central p v e m n a t , for exa1431e. thmugh the ' G e d Gempus Sainr d m Mae&' [Science a i d tics esash h & r a m ) [Ikrnm- Pendidikaa Malaysia, 1996), primary mathematics has remained h p l y a concern for the individual state, district or schml authorities. Yet, primary ma&matics lay the founda~on to& farther m z ~ p ~ o w , ns and inclmtiom of the pupils, not only towards ma;thematil=s, & m e and ~ 1 0 g y especially computer t e c h l o g y in latex e d d a u but also in in tbiewartd sf WQ& in du1&aad and support and guidance to& tt3ei ~ f f - q d g s BS parents. The "multiplier effect' of parents who are weak m *W m t h d c s influencing thein children on vcrcatianal choices might be a majm factor in turnkg off mmy students fiom the science s&am in Forms 4,5 and 6. This view supported Shohtpku's (1993) belief that "the education of girls a d boys in mathematics aad scitmce at an early stage has a strong impact upon theis aptitude for and i n w t in Wi choice of engineering and tmhmlogyWtelat subjects at higher stages of education." (p. 302). As these oanceptions aMjl perceptions are of utnmst importance in the nation" strife towards being a global d t playerq pupifs' level af success in primary mathematics demands attention.

2.2.1 myfmf&~?

M o b d ' s (1988) findings that the effective learning of mathematics is hind& by three main factors, that is, no interest in mathematics, a misconception that mathematics is difficult and a p r basic knowledge of mathematics provided: support that the grounding of the basic' concepts and skills of mathematics in primary schools will influence its interest and achievement during the set;.undq schmxrEs.

T k Malaysian primary! mathem;ll;ics cmicdurn, Integrated Cursiculum for Piinmy Schools (Ku&bwn Bersepdg Sekofnh Rendah, KBSR), consists of eight topics, of which one of them is fractions (Curriculum Deveiopment Center, 1994) (Appendix A). Fractions is taught beginning E%mq 3 untrl Primary 6 (Appendix 3). There are a number of reasens why the topic fractions has been chosen b be the pr imq focus in lkis study.

Firstly, fractiorrs b a campulsrory mpic in all primary school math~matics curricula of South-East Asia" Ministers of Education Organization (SEA1MEU) countries and generally taught in the schools' mathematics curricula of other countries. H w x , pupils in these c o ~ ~ e - s might encounter similar learning prabiems in fractions.

Secondly, fractions appear to be an inherently difficult abstraction right from w J y times. Peaple in early civifizatiuns went to great length ta avoid &em. One common ~ ~ : t h a d userE by the early people to deal with fractional parts withiout hving to resort to fractions was itvl creata! sub-units of a masure. For e m p i e , the m~hmts of Rome would speak of 'W ( ' / la of the 'as"), "'sextam" [flEI2 af the 'as'), "semmW of the 'as'l] and so un, the 'as' being a pawd of copper (Smith, 1958).

Thitdly, it has bee0 mkd as one of the three most difficult topics to i m by pupilf and to teach by teachefs in Malaysian schools (Farm, Leang Bt Liau, 1989; Seth & M e m ~ 1990; Ong a al., 1992). Seth (1990i) identified decimals, fi-actions and division in word pmblems as the h e three m s t difficult topics to teach in a&r of ciecmsing diffide. Teachers perceived the difficulty was brought about by children's n&mmceptiom and andk of instiwtionaE maktids. There was a sligbt misaarch between teachers students; preferences in the teaching-leaning pmxa as students ranked fixtiom as the mst difficult topic to learn befare decimals. Elsewhere, Suydam (1978) after reviewing data from a variety of assessments, wmk, 'Vu: itvlpics with which difficulty (or weakness) were reported sari be ranked in this anfm of E i e q w y : ficst, fractions, ..." (p. 212).

Fourthly, fractions is intrdwed early in the cmicdum, that is, in Primq 3 (Appendix la) and extends until Fonn 5 in the secondary schools. The existence of my gaps in learnring or misconceptions in this early stage if not systemticdly diagnosed and cp:mediakd might prevail during the learner's entire schding experience.

Fifthly, knawledge and skills in common fractions are important bases far subsequent tapics in mathematics - decimal fractions, percentages, ratio, proportion, multiplication, division, mmsuremmt (e.8. length, weight, valurne, area, time), algebra and graphs. Not only wii1 any initial 'minor' difficulty in early fraction work gets exacerbated as the cumulative *ition sylIabus gets spirzrily wider, the difFiculty migtit 'over-spill' into these related areas in mathematics and even other subjects.

Sixthly, e x p i e d (Fuller, 1997) and prospective fKaur, 19W. Ong et a[,, 1992) mathematics teachers have been shown to exhibit significant areas of weaknesses in basic concepts and computati.etna1 procedures of fractions. Thew were teachers who view fractions as consisting of two separate numbers, each number denoting different quantities (Ong et nl., 19921.

Seventhly, fhctions has proved important for life outside school - at home, playground and at work The process af sharing or dividing famething between two and more people could involve fractions. Far example, dividing a p i m between the six members in; a family or distributing a pack of c a d s among four players. M a m m n t s are often express& as fractional parts of main units, for exampIe at the supernxtrbt, there are g d s sold by the half kilogram (sugar, salt, butter, margarine) and the half titer (milk, lemonade, cooking oil). Material might be bought to the nearest haif or the nearest quarter meter, for example 1% meters for a skirt, or hO M meters for curtains. Time is often expressed as quarter past, half past and quarter to the b u r .

Thene: are a. n u d e r of featma i n h a n t in the structure of fractions that lead to it being difficult to ieant, ?each and use. I"he following list describes these features:

1) There are diverse i n ~ o n s and possible sub-constructs for rational nu is a subset) - as part-whole relationship, a measure, quatiat, rdo, aperatmimapping number and probability (Keiren, 1976). Fwtiom, itself, has s e d diffmnt eoncepttlal meanings:

(a) a rurmber (either the same as rational n m k or a subset of Ehe rationid 1:

(b) a ntmmd (symbal, expression); (c) a part of a regional whole; (d) a portSon of a d i b set of objects; (el a measurement pint an a number Iine; (iF) one number divided; by another; or (& an o r d a d pair (dads not muet whether written as (a b) or a: b ar 'id.