99696 Excellent Thesis Met a Cognitive Scaffolding and Cooperative Learning [1]
Transcript of 99696 Excellent Thesis Met a Cognitive Scaffolding and Cooperative Learning [1]
THE EFFECTS OF
METACOGNITIVE SCAFFOLDING AND
COOPERATIVE LEARNING
ON
MATHEMATICS PERFORMANCE AND
MATHEMATICAL REASONING AMONG
FIFTH-GRADE STUDENTS IN JORDAN
by
Ibrahim Mohammad Ali Jbeili
September 2003
Thesis submitted in fulfillment of the requirements for the degree of
Doctor of Philosophy
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Table of Contents iList of Tables viiList of Figures ixList of Appendices xAcknowledgements xiAbstrak xiiiAbstract xvi
Chapter One INTRODUCTION 11.1 Background to the Statement of the Problem 11.2 Statement of the Problem 91.3 Research Questions 151.4 Hypotheses 161.5 The Theoretical Framework 171.6 Significance of the Study 191.7 Operational Definitions 21
Chapter Two LITERATURE REVIEW 242.1 Introduction 242.2 Objectivist Views Regarding the Learning/Teaching
of Mathematics
25
2.2.1 Behaviorism and the Learning / Teaching of
Mathematics
25
2.2.2 Gagne and the Learning / Teaching of
Mathematics
27
2.2.3 Landa and the Learning / Teaching of
Mathematics
29
2.2.4 Scandura and the Learning/Teaching of
Mathematics
30
2.3 Constructivist Views Regarding the
Learning/Teaching of Mathematics
34
2.3.1 Nature of the Learning Process and
Construction of Knowledge
34
2.3.2 Piaget and the Learning / Teaching of
Mathematics
38
2.3.3 Vygotsky and the Learning / Teaching of
Mathematics
40
2.3.4 Bruner and the Learning / Teaching of
Mathematics
41
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2.4 Metacognitive Strategies and the Construction of
Knowledge
43
2.4.1 Regulation of Cognition 452.4.2 Metacognitive Strategies and Age 472.4.3 Metacognitive Scaffolding 48
2.5 Cooperative Learning and Learning Mathematics
with Understanding
52
2.5.1 Theoretical Perspective on Cooperative
Learning
53
2.5.2 Elements of Cooperative Learning 562.5.3 Teacher’s Role in Cooperative Learning 57
2.6 Cooperative Learning with Metacognitive
Scaffolding and Learning Mathematics with
Understanding
60
2.6.1 Conceptual Understanding 632.6.2 Procedural Fluency 652.6.3 Strategic Competence 672.6.4 Adaptive Reasoning 702.6.5 Productive Disposition 72
2.7 Cooperative Learning with Metacognitive Scaffolding and Mathematical Reasoning
74
2.8 Cooperative Learning with Metacognitive
Scaffolding and Real-Life Problem Solving
77
2.9 Cooperative Learning with Metacognitive
Scaffolding and Motivation
80
Chapter Three METHODOLOGY 83
3.1 Introduction 833.2 Population and Sample 833.3 Experimental Conditions 843.4 Research Design 863.5 Instructional Materials and Instruments 88
3.5.1 Instructional materials 883.5.1.1 Adding and Subtracting Fractions
Unit
88
3.5.1.2 The Metacognitive Questions
Cards
89
3.5.2 Instruments 903.5.2.1 The Mathematics Achievement
Test
90
3.5.2.2 The Scoring of Mathematics
Achievement Test
92
3.5.2.3 The Metacognitive Knowledge 96
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Questionnaire3.5.3 Materials and Instruments Validity 963.5.4 Instruments Reliability 98
3.6 Procedures 983.6.1 The Pilot Study 993.6.2 The Formal Study 993.6.3 Groups’ Equivalence 1003.6.4 Teachers’ Training 1003.6.5 Implementation of the Study 102
3.6.5.1 The Cooperative Learning with
Metacognitive Scaffolding (CLMS)
Method
102
3.6.5.2 The Cooperative Learning (CL)
Method
106
3.6.5.3 The Traditional (T) Method 1073.6.5.4 Implementation Fidelity 108
3.7 Data Analysis Procedure and Method 1103.7.1 The pre-Experimental Study Findings
Analysis
110
3.7.2 The Experimental Study Findings Analysis 1103.7.3 Justifications for using two-way
MANCOVA / MANOVA
111
3.7.4 Pearson’s Correlation 1123.7.5 Assumptions for MANOVA / MANCOVA 113
Chapter Four RESULTS 116
4.1 Introduction 1164.2 The pre-Experimental Study Results 117
4.2.1 Statistical Data Analysis 1174.3 The Experimental Study Results 121
4.31 Testing of Hypothesis 1 1214.3.2 Summary of Testing Hypothesis 1 (CLMS > CL
> T)
126
4.3.3 Testing of Hypotheses 2 1274.3.4 Summary of Testing Hypothesis 2 (CLMSH
>CLH >TH)
133
4.3.5 Testing of Hypotheses 3 1334.3.6 Summary of Testing Hypothesis 3 (CLMSL >
CLL> TL)
139
4.3.7 Testing of Hypotheses 4 139
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4.3.8 Summary of Testing Hypotheses 4 (There are interaction effects between the instructional methods and the ability levels)
145
4.3.9 Summary of Findings to Research Questions
1 – 4
146
Chapter Five DISCUSSION AND CONCLUSIONS 1505.1 Introduction 1505.2 Effects of the Instructional Methods on
Mathematics Performance, Mathematical Reasoning, and Metacognitive Knowledge
152
5.2.1 Effects of the Instructional Methods on Mathematics Performance
152
5.2.2 Effects of the Instructional Methods on Mathematical Reasoning
155
5.2.3 Effects of the Instructional Methods on Metacognitive Knowledge
160
5.3 Effects of the Instructional Methods on Mathematics Performance, Mathematical Reasoning, and Metacognitive Knowledge Based on Ability Levels
163
5.3.1 Performance of High-Ability Students Taught Via CLMS
164
5.3.2 Performance of High-Ability Students Taught Via CL
167
5.3.3 Performance of Low-Ability Students Taught Via CLMS
168
5.3.4 Performance of Low-Ability Students Taught Via CL
170
5.3.5 Performance of Low-Ability Students Taught Via T
171
5.4 Interaction Effects 1735.5 Summary and Conclusions 1765.6 Implications for Educators 1785.7 Implications for Future Research 180
5.8 Limitations of the Study 182
5
References
183Appendices
199
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List of Tables
Table Page
3.1 Mechanisms for the Three Groups 85
3.2 Research Design 86
3.3 Pearson’s correlation among the three dependent variables (MP, MR, and MK)
113
3.4 Pearson’s correlation among the covariates (pre-MP and pre-MR) and the dependent variables (MP, MR, and MK)
115
4.1 Means and standard deviations on each dependent variable (pre-MP and pre-MR), by the groups
118
4.2 Summary of multivariate analysis of variance (MANOVA) pre-MP and pre-MR results and follow-up analysis of variance (ANOVA) results.
120
4.3 Means, standard deviations, adjusted means and standard errors for each dependent variable by the instructional method
122
4.4 Summary of multivariate analysis of covariance (MANCOVA) results by the instructional method and follow-up analysis of variance (ANOVA) results
124
4.5 Summary of post hoc pairwise comparisons 1254.6 Means, standard deviations, adjusted means and standard errors for
each dependent variable for high-ability students by the instructional method
128
4.7 Summary of multivariate analysis of covariance (MANCOVA) results by the instructional method and follow-up analysis of covariance (ANCOVA) results of comparing high-ability students across the three groups.
130
4.8 Summary of post hoc pairwise comparisons between high-ability students across the three groups
131
4.9 Means, standard deviations, adjusted means and standard errors for each dependent variable for low-ability students by the instructional method
134
4.10 Summary of multivariate analysis of covariance (MANCOVA) 136
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results by the instructional method and follow-up analysis of variance (ANOVA) results of comparing low-ability students across the three groups.
4.11 Summary of post hoc pairwise comparisons between low-ability students across the three groups
137
4.12 Means, standard deviations, adjusted means and standard errors for each dependent variable by the interaction between the instructional methods and the ability levels (high-ability and low-ability)
140
4.13 Summary of multivariate analysis of covariance (MANCOVA) results by the interaction effect and follow-up analysis of covariance (ANCOVA) results across the three groups.
142
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List of Figures
Figure Page
4.1 Interaction effect between the instructional method and the students’ ability levels on MP
143
4.2 Interaction effect between the instructional method and the students’ ability levels on MR
144
4.3 Interaction effect between the instructional method and the students’ ability levels on MK
145
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List of Appendices
Appendix Page
Appendix A Metacognitive Questions Cards 200
Appendix B Mathematics Achievement Test 206
Appendix C Distribution of Scores across the Test Items 212
Appendix D Scoring Rubric 213
Appendix E Metacognitive Knowledge Questionnaire and Scoring Key 214
Appendix F Permission of Conducting Research in the First Public Educational Directorate Schools under the Jordan Ministry of Education
218
Appendix G Permission of Conducting Research in Irbid Governorate Schools under the First Public Educational Directorate
220
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Acknowledgements
In the name of Allah, Most Gracious, Most Merciful
I am grateful for all the bounties that Allah has showered on me which enabled me
complete this doctoral thesis. I also thank Allah for providing me with a supportive
family and supportive colleagues and friends during my graduate studies.
I would like to express my appreciation to all the individuals without whom the
completion of this thesis would not be possible. First of all, my heartfelt thanks go to
my thesis major supervisor, Associate Professor Dr. Merza bin Abbas, for his warm
personality, continual and unwavering encouragements, support, tutelage, patience,
and perseverance in guiding me through the entire research and thesis-writing process.
My deepest thanks also go to my co-supervisor Associate Professor Dr. Wan Mohd
Fauzy for his invaluable assistance.
I would also like to express my particular thanks to the faculty and administrative staff
of the Center for Instructional Technology and Multimedia, University of Science
Malaysia, who provided facilities, and advice and support. My thanks also go to the
administrative staff of the Institute of Post-graduate Studies, IPS, USM, for their
assistance and support. I would also like to gratefully acknowledge the principals,
teachers, and students of the primary schools which served as research sites: Al-
Muthana bin Harethah School, Huthaifa bin Alyaman School, and Abd Arrahman
Alhalholi School. My profound gratitude goes to the Director of the Educational
Development and Research Department, Jordan Ministry of Education and the
Director of First Public Educational Directorate in Irbid Governorate for their
assistance.
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I am grateful to my colleagues, Dr. Sayed Anwar, Dr. Sharifa, Zainal, Oi, Husaini, and
Aree in Center for Instructional Technology and Multimedia for their friendship over
the past few years and the immense help during my research process. My gratitude
also goes to the administrative staff of the University of Science Malaysia’s Library
for their patience and assistance.
My acknowledgement is also extended to Dr. Jeremy Kilpatrick, University of
Georgia, Dr. Marjorie Montague, University of Miami, Dr. David Johnson and Dr.
Roger Johnson, University of Minnesota, and Dr. Khattab Abu Libdeh, Jordan
National Center for Educational Research and Development who provided valuable
documents, articles, and suggestions throughout my thesis-writing.
Last but not least, my affectionate thanks go to my family for their unfailing love,
continual understanding, sacrifice, prayers and confidence, and selfless support: My
parents, my brothers and sisters, and my wife and my mother in-law. I would like to
express my gratitude to my father and my brothers who financially supported me
during my graduate studies and patiently waited for me to finish my study. My mother,
may Allah reward you for your patience and prayers before and during my graduate
studies. I thank Allah for having a very understanding and loving wife. She has given
me tremendous support and continuous encouragement during my thesis-writing.
Words are inadequate to express my gratitude for their sacrifice, support, and patience
and I love them with all my heart. “May Allah reward and bless all of my family
members”.
Ibrahim Mohammad Ali Jbeili
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ABSTRAK
Kesan Perancahan Metakognisi dan Pembelajaran Kooperatif Terhadap Prestasi
Matematik dan Taakulan Matematik Di Kalangan Pelajar Tahun Lima di Jordan
Tujuan penyelidikan ini ialah mengkaji kesan pembelajaran kooperatif berserta
perancahan metakognisi terhadap (a) prestasi matematik (MP), (b) taakulan matematik
(MR), dan (c) pengetahuan metakognisi (MK) di kalangan pelajar tahun lima di
Jordan. menyelidiki Penyelidikan ini turut mengkaji kesan pembelajaran kooperatif
berserta perancahan metakognisi terhadap skor-skor MP, MR, dan MK di kalangan
pelajar berpencapaian tinggi dan rendah. Skor-skor MP, MR, and MK diukur melalui
ujian pencapaian matematik dan soalselidik pengetahuan metakognisi.
Reka bentuk eksperimen kuasi yang menggunakan reka bentuk factorial 3 x 2 telah
digunakan dalam kajian ini. Faktor pertama ialah tiga paras kaedah pengajaran, iaitu
(a) pembelajaran kooperatif berserta perancahan metakognisi (CLMS), (b)
pembelajaran kooperatif tanpa perancahan metakognisi (CL), dan (c) pengajaran
tradisional (T), iaitu kaedah pengajaran tanpa pembelajaran kooperatif atau
perancahan metakognisi. Faktor kedua ialah pencapaian pelajar, iaitu Pencapaian
Tinggi dan Pencapaian Rendah. Pembolehubah bersandar ialah skor-skor di dalam
MP, MR, dan MK. Tiga sekolah rendah lelaki telah dipilih secara rawak dari
sekumpulan empat puluh empat sekolah rendah yang mengajar matematik di dalam
kelas-kelas heterogenus di mana pelajar tidak dikumpulkan atau ditindik mengikut
keupayaan. 240 pelajar lelaki di dalam kelas tahun lima dari tiga buah sekolah rendah
telah dipilih secara rawak, iaitu dua kelas dari setiap sekolah.
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Satu ujian pra matematik telah ditadbirkan dan dua bulan sebelum kajian dimulakan
kelas-kelas yang terlibat telah diperkenalkan dengan kaedah-kaedah CLMS and CL
dan melaksanakan unit-unit latihan yang disediakan. Tumpuan kajian ini ialah pada
unit “Penambahan dan Penolakan Pecahan” yang diajar di semua kelas selama 14 sesi
pada penghujung semester pertama pada tahun akademik 2002 / 2003. Setiap
kumpulan kooperatif CLMS terdiri dari dua pelajar pencapaian tinggi dan dua pelajar
pencapaian rendah dan setelah mendengar pengenalan dari guru belajar secara
kooperatif dan menggunakan kad-kad soalan metakognisi untuk memandu kerja-kerja
serta latihan-latihan menyelesaikan masalah matematik yang disediakan. Dalam
kaedah kooperatif ini, pembelajaran pelajar dibantu oleh perancahan oleh guru, oleh
kad-kad soalan metakognisi dan oleh interaksi sesama pelajar. Ahli-ahli kumpulan CL
juga terdiri dari dua pelajar pencapaian tinggi dan dua pencapaian rendah dan belajar
secara kooperatif setelah mendengar pengenalan dari guru. Pelajar di dalam kumpulan
kaedah T diajar secara lazimnya dan menyelesaikan masalah secara individu.
Dapatan kajian ini menunjukkan bahawa pelajar kumpulan CLMS menunjukkan
prestasi lebih tinggi yang berbeza secara signifikan dari kumpulan CL yang seterusnya
menunjukkan prestasi lebih tinggi yang berbeza secara signifikan dari kumpulan T di
dalam semua skor, iaitu MP, MR dan MK . Juga pelajar pencapaian tinggi di dalam
kumpulan CLMS menunjukkan prestasi lebih tinggi yang berbeza secara signifikan
dari kumpulan T dalam skor-skor MP, MR dan MK, serta prestasi lebih tinggi yang
berbeza secara signifikan dari kumpulan CL dalam skor-skor MR dan MK. Pelajar
pencapaian tinggi di dalam kumpulan CL pula menunjukkan prestasi lebih tinggi yang
berbeza secara signifikan berbanding pelajar pencapaian tinggi kumpulan T dalam
skor-skor MP, MR dan MK. Dapatan kajian juga menunjukkan bahawa pelajar
pencapaian rendah di dalam kumpulan CLMS menunjukkan prestasi lebih tinggi yang
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berbeza secara signifikan berbanding pelajar pencapaian rendah di dalam kumpulan
CL dan kumpulan T dalam skor-skor MP, MR, dan MK. Pelajar pencapaian rendah di
dalam kumpulan CL juga menunjukkan prestasi lebih tinggi yang berbeza secara
signifikan berbanding pelajar pencapaian rendah kumpulan T dalam skor-skor MR dan
MK. Dapatan kajian juga menunjukkan kesan-kesan interaksi yang signifikan dalam
kumpulan CLMS di antara pencapaian pelajar dan skor-skor dalam MR dan MK
dengan pelajar pencapaian rendah mendapatkan manfaat yang lebih dari kaedah yang
digunakan.
ABSTRACT
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The Effects of Metacognitive Scaffolding and Cooperative Learning on
Mathematics Performance and Mathematical Reasoning among Fifth-Grade
Students in Jordan
The purpose of this study was to investigate the effects of cooperative learning with
metacognitive scaffolding on (a) mathematics performance (MP), (b) mathematical
reasoning (MR), and (c) metacognitive knowledge (MK) among fifth-grade students
in Jordan. The study further investigated the effects of cooperative learning with
metacognitive scaffolding and cooperative learning on high-ability and low-ability
students’ achievement in MP, MR, and MK. The MP, MR, and MK scores were
measured through a mathematics achievement test and a metacognitive knowledge
questionnaire.
A quasi-experimental study design that employed a 3 x 2 Factorial Design was applied
in the study. The first factor was three levels of instructional method, namely, (a)
cooperative learning with metacognitive scaffolding (CLMS), (b) cooperative learning
without any metacognitive scaffolding (CL), and (c) traditional instruction (T) with
neither cooperative learning nor metacognitive scaffolding. The second factor student
ability levels, namely, high-ability and low-ability. The dependent variables were
student achievement in MP, MR, and MK. Three male primary schools were randomly
selected from forty four primary schools where mathematics was taught in
heterogeneous classrooms with no grouping or ability tracking. 240 male students who
studied in six fifth-grade classrooms were randomly selected from the three primary
schools i.e., two classes from each school.
A pre-mathematics achievement test was administered first, and then the CLMS and
CL methods were introduced to the students with practice units two months before
conducting the study. For the study, the focus was on the “Adding and Subtracting
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Fractions” unit that was taught in all classrooms for 14 sessions at the end of the first
semester for the academic year 2002 / 2003. In the CLMS method, after listening to
their teacher’s introduction, students in small groups of two high-ability and two-low-
ability students worked cooperatively and used metacognitive questions cards to
execute their mathematics exercises and solve mathematics problems. In this method,
students’ learning was scaffolded by the teacher, the metacognitive questions, and the
students’ cooperation. In the CL method, after listening to their teacher’s explanation,
students worked cooperatively in small groups of two high-ability and two low-ability
students. In the T method, students were taught in the usual manner and solved the
mathematics problems individually.
The results showed that overall the students in the CLMS group significantly
outperformed the students in the CL group who, in turn, significantly outperformed
the students in the T group in all measures. Additionally, the high-ability students in
the CLMS group significantly outperformed their counterparts in the T group in MP,
MR and MK, and significantly outperformed their counterparts in the CL group in MR
and MK but not in MP. The high-ability students in the CL group in turn significantly
outperformed their counterparts in the T group in MP, MR and MK. Also, the results
showed that the low-ability students in the CLMS group significantly outperformed
their counterparts in the CL group and in the T group in MP, MR, and MK. The low-
ability students in the CL group in turn significantly outperformed their counterparts
in the T group in MR and MK but not in MP. Finally, the results showed significant
interaction effects between student ability and the instructional method for the MR
and MK scores with the low-ability students in group CLMS benefiting more than the
high-ability students.
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CHAPTER ONE
INTRODUCTION
This study investigated the teaching of mathematics based on constructivist principles.
The study focused primarily upon the investigation of the effects of cooperative
learning with metacognitive scaffolding and cooperative learning on Jordan fifth-
grade students’ mathematics performance, mathematical reasoning, and metacognitive
knowledge in learning and solving problems involving the addition and subtraction of
fractions. This first chapter of the study presents the background to the statement of
the problem, specifies the statement of the problem and the purpose of the study, and
describes its questions and hypotheses, and presents the study theoretical framework
and the significance of the study. Finally, the chapter presents the operational
definitions.
1.1 Background to the Statement of the Problem
Children today are growing up in a world permeated by mathematics. The
technologies used in homes, schools, and the workplaces are all built on mathematical
knowledge. Many educational opportunities and good jobs require high levels of
mathematical expertise. Mathematical topics arise in newspaper and magazine
articles, popular entertainment, and everyday conversation. Mathematics is a
universal, utilitarian subject, that is, so much a part of modern life that anyone who
wishes to be a fully participating member of society must know basic mathematics.
Mathematics also has a more specialized, esoteric, and esthetic side. It epitomizes the
beauty and power of deductive reasoning. Mathematics embodies the efforts
accumulated over thousands of years by every civilization to comprehend nature and
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bring order to human affairs. For students to participate fully in society, they must
learn mathematics with understanding, how to connect mathematical ideas, and how
to reason mathematically. Students who cannot reason mathematically are cut off from
whole realms of human endeavor. Students without mathematical understanding are
deprived not only from opportunity but also from competence in everyday tasks
(Kilpatrick et al., 2001). So mathematics instruction should emphasize such variables
that result learning with understanding in order to meet the changing demands of the
society.
Mathematics instruction has moved through a series of development phases. The
move from behaviorism through cognitivism to constructivism represents shifts in
emphasis away from an external view to an internal view of learning. To the
behaviorist, the internal processing is of no interest; to the cognitivist, the internal
processing is only of importance to the extent to which it explains how external reality
is understood. In contrast, the constructivist views the student as a builder of his
knowledge (Jonassen, 1991). This turning point of learning processes asks for
instruction that deals with students as builders not receivers of knowledge, students
who construct knowledge through interaction and connecting their experiences with
the current situations, and students who have learning strategies to help in building
their knowledge and understanding. Thus, successful and effective mathematics
instruction emphasizes the teaching of strategies that enable students to plan, monitor,
evaluate, and then construct their own knowledge and understanding.
Particularly, for Jordan educational system, Jordanian human resource based economy
was hard hit in the wake of the 80s’ slump in the regional oil economy, which had
during its boom given tangible spillover benefits to the country in the form of
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remittances from Jordanian skilled workforce working in Gulf Cooperation Council
Countries. The slump also caused the general education system and particularly
mathematics education to gradually lose its utility. The technological revolution and
growing use of modern technologies in the industries as well as in other employment
sectors had changed the mathematical knowledge and skills requirements of labor
markets (Ahlawat and Al-Dajeh, 1996).
It became necessary for Jordan to upgrade the quality of school graduates in order to
meet the changing demands of the domestic labor market and to maintain its skilled
workforce advantage in the region wide labor market. Under these circumstances,
Jordan in 1989 launched a comprehensive 10-year-long Education Reform Plan (ERP)
to overhaul the general education system. Mathematics education was one of the core
subjects that received a lot of attention. The overarching objective of the reform plan
was to enhance student achievement levels. The key reform elements were
reconstructing the curricula, designing new textbooks and instructional materials, and
conducting in-service teacher training in classroom applications of innovative
instructional methods for using new textbooks and materials (Ahlawat and Al-Dajeh,
1996).
To determine if the mathematics education reform has had the desired effects, in 1995,
the National Centre of Human Resources Development (NCHRD) conducted a study
to investigate the changes in mathematics achievement levels after five years of
reform (Ahlawat and Al-Dajeh, 1996). The findings showed that there was a
significant improvement in the whole field of mathematics achievement. The
improvement, however, related to the routine mathematics concepts, procedures, and
problem solving. The newly designed textbooks and in-service training did not cover
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high-level cognitive skills, analytical thinking, and reasoning that help students to
build their knowledge and develop understanding (Innabi, Hanan; Kaisee, and Hind,
1995). Ahlawat and AL-Dajeh (1996) indicate that the mathematics materials after
reform covered only three cognitive skills (conceptual knowledge, procedural
knowledge, and problem solving), and the improvement of conceptual understanding
was the weakest according to the post-reform achievement tests. Moreover, from the
analysis of individual items of students’ responses, there was significant deterioration
in performance particularly with topics that involve abstract theoretical concepts
(Ahlawat and Al-Dajeh, 1996). This indicates that mathematics teachers and materials
developers in Jordan concentrate on learning procedures exclusively and do not pay
attention to the teaching of strategies that help students to build and develop
conceptual understanding and reasoning.
In the last year of the Jordan Education Reform Plan, the Third International
Mathematics and Science Study–Repeat (1999) was conducted to compare
mathematics and science eighth-grade students’ achievements among 38 nations.
Jordan was among the 38 nations that participated in the study. TIMSS-R assessed
five mathematics content areas: fractions and number sense, measurement, data
representation (analysis, and probability), geometry, and algebra. The findings showed
that the average mathematics achievement across the all five mathematics areas of
Jordanian students was 428, which was lower than the international average which
was 487. The lowest average was in fractions and number sense with an average score
of 432, while the international average was 487. In terms of ranking, Jordan was
placed at number 32 out of 38. Singapore was ranked first with an average of 604
points. In terms of quality of the scores (students scored 616 or higher), 46 percent of
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the scores of the Singapore students were in the top ten percent compared to only 3
percent for Jordan.
For Jordanian eighth-grade students, for example, on an item testing for concept
knowledge of fractions that asked students to shade 8
3 of 24 cells correctly, only
30.9% students responded correctly, while the rest gave different incorrect responses
like for example, shading 3, 8, 11 cells, or giving unclear responses. Abu Libdeh
(2000) indicates that mastering this item falls in the third-grade with 50% accuracy as
criterion and increases to 80% in the fifth-grade. So this finding shows that 30.9% of
eighth-graders accuracy level to this item is below the desired level.
Many factors may affect and contribute to the students’ mathematical understanding
and achievement. The understanding of mathematics as an academic subject and its
perceived importance in school and life plays a very crucial role. For example,
Singapore values mathematics very highly and its primary schools systematically
teach aspects of mathematics normally reserved for middle schools or junior high
schools (Kaur and Pereira, 2000). Studies by Sternberg and Rifkin (1979) and
Thornton and Toohey (1985) have indicated that young children can benefit from
using sophisticated strategies in solving mathematics problems. However, the current
practice in most schools in Jordan has been to underestimate students’ real abilities to
learn mathematics.
The nature of mathematics is also called into question. It is usually classified as one of
the usual science subjects together with physics and chemistry but it is not taught as a
science subjects. It is conceived as consisting of numbers, rules, formulas, and
algorithms, and the teaching has focused on the acquisition of procedures (Gagne,
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1985) and algorithms and heuristics (Landa 1983) in solving routine and novel
problems. Also Romberg (1988) indicates that mathematics in many cases is divorced
from science and other disciplines and then separated into subjects such as arithmetic,
algebra, geometry, trigonometry, and so on. Within each subject, ideas are selected,
separated, and reformulated into a rational order. This is followed by subdividing each
subject into topics, each topic into studies, each study into lessons, and each lesson
into specific facts and skills. This fragmentation of mathematics has divorced the
subject from reality and from learning with understanding. Such essential
characteristics of mathematics as abstracting, inventing, reasoning, and applying are
also often lost from Jordanian textbooks and teaching methods (Abu Libdeh, 2000).
Thus, the learning of mathematics becomes the learning of isolated facts and skills
that according to Gipps and McGilchrist (1999) “quickly disappear from the memory
because they have no meaning and do not fit into the student’s conceptual map.
Knowledge learned in this way is of limited use because it is difficult for it to be
applied, generalized or retrieved” (p 47).
By scrutinizing Jordanian fifth-grade mathematics textbook and teacher’s guide, it is
clear that the Jordanian mathematics classrooms are dominated by seatwork,
homework, and review. For example, instructions of teaching fractions emphasize the
demonstration of mathematical facts, computation skills, and procedures used to solve
routine problems. These instructions lead to teaching strategies that require
mathematics teachers to concentrate on mastering procedures needed to solve routine
tasks and problems. In addition, these instructions necessitate mathematics teachers to
fragment mathematics materials and to include many topics, with a considerable
amount of repetition of content which is divorced from reality. Therefore, mathematics
instruction in Jordan often concentrates on teaching mathematical facts, skills, and
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procedures with much less concentration on developing analytical thinking like, for
instance, reasoning and metacognitive strategies. This kind of teaching promotes a
sequential mastery of knowledge, with the teacher as a giver and the student as a
receiver, but does not promote the view that students have potential abilities to build,
plan, monitor, reason, and evaluate their knowledge (Wilkins and Jesse, 2000). Also
this teaching does not promote values and other knowledge associated with
mathematical proficiency as to be discussed in this study.
The traditional sequence of teaching placing value, digit numbers, fractions, ratio
decimals, percentages, and geometry tends to fractionalize mathematical knowledge
instead of integrating and connecting it. This instruction influences students’ views of
mathematics and may make them unable to transfer what they have learned to new
situations in the real life. Moreover, this instruction does not encourage students to
invest their reasoning, connection, and metacognitive strategies in their learning. That
is, students often receive problem-solving procedures from the teacher without actual
participation in planning, monitoring, and evaluating their learning. In this
environment there is no opportunity for students to connect mathematical relationships
between what they have learned and the current situations in which these connections
enable them to recognize the importance of mathematics in all parts of life (Baroody,
1998), and learn mathematics with understanding (Carpenter and Lehere, 1999;
Kilpatrick et al., 2001).
Mathematics teachers in Jordan primary schools generally teach their students by
means of conventional instructional methods. They select a set of mathematical
problems, demonstrate the necessary steps leading to their current solutions and their
students then follow the same steps in finding solutions to similar problems. This
24
pedagogical approach may be effective for high-ability students, but it may not work
with low-ability students. The explanations for low mathematics achievement of low-
ability students could be that they are not taught the appropriate strategies, cannot self-
regulate the study strategies, and do not understand how to apply these strategies
(Simpson, 1984).
The TIMSS-R (1999) findings reveal that Jordanian students’ achievement is still very
low even after ten years of educational system reform. Moreover, students’
understanding of mathematics has also not improved. As Abu Libdeh (2000) found, a
great deal of errors students gave is related to undetermined errors (unspecific errors
that comprise unrelated responses, deleted responses, or unreadable responses). This
means that most Jordanian eighth-grade students are lacking in or do not have
mathematical understanding, strategic competence, and adequate reasoning skills for
mathematics; therefore they responded with unrelated, unreadable, or other
meaningless responses. While students’ difficulties in doing mathematics are partly
attributed to misconceptions or shallow conceptions of domain knowledge (Feltovich
et al., 1996), they are, to a greater extent, due to a lack of metacognitive strategies
(Brown, 1987). Comparisons of good and poor comprehenders have consistently
shown that poor comprehenders are deficient in the use of metacognitive strategies
(Golinkoff, 1976; Meichenbaum, 1976; Ryan, 1981). So Jordanian students need to be
taught mathematics through an effective instruction that enables them to acquire and
apply metacognitive strategies, reason mathematically, and thus learn mathematics
with understanding.
In other words, students have to be taught and supported to plan, formulate and
represent the mathematical problems, analyze and identify the mathematical variables,
25
connect the relationships among the mathematical variables, ask themselves questions
regarding mathematical situations, reason mathematically, evaluate their strategies and
outcomes (Kilpatrick et al., 2001; King, 1992), and to work cooperatively to learn
with understanding (Palincsar and Brown, 1984). Particularly, students need to learn
how to learn, that is, to be metacognitively trained. To date, insufficient attention has
been given to the important role the metacognitive strategies play in improving
mathematics performance and mathematical reasoning.
National Research Council (NRC, 1998) declares that effective mathematics
instruction must start from the earliest grades. Simply offering or even requiring all
students to take a standard first year course in eighth-grade is no assurance that they
will succeed. This method virtually guarantees failure for a large number of students.
Instead, as Kilpatrick et al. (2001) suggest that students must begin to acquire the
rudiments of learning with understanding in the earliest grades, as part of a
comprehensive method to developing their mathematical proficiency. Moreover, Abu
Libdeh (2000) found that a great deal of Jordanian eighth-graders’ mathematical errors
and misunderstanding refer to the topics they learned in the fifth-grade.
1.2 Statement of the Problem
New applications and new theories have given emphasis to instructional methods that
play an important role in developing the learning of mathematics. Documents such as
those produced by the National Council of Teachers of Mathematics (NCTM, 1989,
1991, and 1995) and the National Research Council (NRC, 1989) suggest that
traditional mathematics instruction has been challenged by the changing expectations
of the skills and knowledge of workers, and therefore, mathematics instruction should
shift from concentrating on the products to the learning processes that comprise
26
learning strategies, planning, monitoring, evaluation, and reasoning. In other words
effective mathematics instruction gives special attention to teach students how to learn
and how to reason and evaluate their learning and solution processes.
This view has sparked debate in the mathematics education community around the
nature of the effective mathematics instruction and the experiences students need to
learn mathematics with understanding. Some advocate a focus on conceptual,
procedural, and reasoning competences (Mathematics framework for California Public
Schools, 1999), while others argue for a focus on applications of mathematics and the
study of realistic mathematics (Apple, 1992). Others state that learning mathematics
with understanding brings together problem solving, reasoning, and criticalness.
Frankenstein (1990), for example, calls for a mathematics method that emphasizes
teaching mathematics through its applications with a goal of helping students become
critical of the uses of mathematics in society. Schoenfeld (1985) argues that effective
mathematics instruction must require students to understand mathematical concepts
and methods, recognize relationships and think logically, and apply the appropriate
mathematical concepts, methods, and relations to solve problems. Still others argue
that learning mathematics with understanding (Kilpatrick et al., 2001) requires
mastering and transferring the mathematical proficiency strands which are: conceptual
understanding, procedural fluency, strategic competence, adaptive reasoning, and
productive disposition in an integrated manner. Others (Mugney and Doise, 1978;
Rogoff, 1990; Vygotsky, 1978) concentrate on cooperative learning to learn
mathematics with understanding. Finally, Flavell et al. (1970) and Brown (1987) focus
on metacognitive strategies to be taught to enable students to learn mathematics with
understanding.
27
There is general agreement that learning mathematics with understanding involves
more than competency in basic skills. Much more than mastering arithmetic and
geometry, learning mathematics with understanding deals with conceptual
understanding, procedural fluency, and reasoning (Kilpatric et al., 2001). Learning
mathematics with understanding is more than learning the rules and operations that
students learn in school. It is about connections, seeing relationships, and knowledge
reconstruction in everything that students do (Brown et al., 1994). In summary,
learning mathematics with understanding is about acquiring and improving conceptual
understanding and procedural fluency (mathematics performance), mathematical
reasoning, and activation of metacognitive knowledge.
One way of supporting and improving students’ mathematics performance,
mathematical reasoning, and metacognitive knowledge is through the provision of
metacognitive strategies, which is an instructional method that concentrates on
monitoring one’s current level of understanding and decides when it is not adequate
(Bransford et al., 2000). It helps students to manage their thinking, recognize when
they do not understand something, and adjust their thinking accordingly (Schoenfeld,
1992). In other words, metacognitive strategies guide students to think before, during,
and after a problem solution. It begins by guiding students to plan for selecting the
appropriate strategy to accomplish the task, and then continues as they select the most
effective strategy and afterward evaluate their learning process and outcomes (Hacker,
1998).
Metacognitive strategies according to Piaget’s cognitive development stages (1970)
require abstract thinking that students become proficient in when they reach the
formal operation stage (12 years and above). Young students, for example, 11 year
28
olds need to be supported, guided, or pushed to be metacognitive thinkers. Vygotsky
(1978) explains the differences between students’ current abilities and their potential
development as the distance between the actual students’ independent level and their
potential level under guidance, support, or in collaboration with more capable peers.
Scaffolding provides an opportunity for students to develop knowledge and skills
beyond their independent current level, and this closes the distance between what is
and what is possible. That is, with scaffolding, students are supported to go beyond
their current thinking, so that they continually increase their capacities (Schofield,
1992).
Researchers (e.g., Palincsar and Brown, 1984; Wood et al., 1976) have investigated
the role of scaffolding to facilitate student comprehension, understanding, and
reflection on complex tasks. In the studies of Palincsar and Brown (1984), Palincsar
(1986), and Palincsar et al. (1987), scaffolding involved modeling and dialogue to
enhance comprehension monitoring and strategy use. Scardamalia et al. (1984)
provided coaching through question prompts, while King (1991a; 1992) modeled and
guided students to use self-generate questions. These scaffolding strategies were
shown to improve students’ cognition by activating their learning, enhancing
knowledge retrieval, comprehension, and metacognition by making their thinking
explicit and guiding them to monitor their understanding.
Among the strategies of improving students’ mathematics performance, mathematical
reasoning, and metacognitive knowledge is a recommendation for using cooperative
learning (National Council of Teachers of Mathematics [NCTM], 1989; Kramarski,
2001). According to Vygotsky (1978) learning with understanding occurs within a
social context. When students interact with each other, they typically will learn,
29
receive feedback, and be informed of something that contradicts with their beliefs or
current understanding. This conflict often causes students to recognize and reconstruct
their existing knowledge (Rogoff, 1990). Cooperative learning has been strongly
recommended to be used in improving students’ cognitive performance, social
relationships, and metacognitive knowledge (Dansereau, 1988; Paris and Winograd,
1990; Weinstein et al., 1994). The report of the National Governors’ Association
(Brown and Goren, 1993) indicated that within cooperative learning setting, mixed
ability students work together to solve problems and complete tasks. In this setting,
low-ability students have the opportunity to model the study skills and work habits of
more proficient students. In the process of explaining the material, high-ability
students often develop greater mastery themselves by developing a deeper
understanding of the task.
However, there still exists uncertainty as to the mechanism by which improving
students’ mathematics performance, mathematical reasoning, and metacognitive
knowledge occurs within various cooperative learning environments. Does
cooperative learning alone improve students’ mathematics performance, mathematical
reasoning, and metacognitive knowledge? Or their cooperation needs to be structured
and guided? If metacognitive strategies are provided to guide students’ cooperation,
are students able to apply metacognitive strategies on their own, or do they need
external scaffolding to do so? Do high-ability students benefit more than low-ability
students from metacognitive scaffolding method? Elawar (1992) observed that low-
ability students are often found to be confused when they confront a mathematical
problem and they are unable to explain the strategies they employ to find a correct
solution. Costa (1985), Sternberg (1986 b) and Elawar (1992) indicated that low-
ability students generally lack well-developed metacognitive skills.
30
Although numerous research studies have been conducted on the separate effects of
metacognitive strategies or cooperative learning on mathematics achievement,
attitudes, and self-efficacy, no study was found that addresses the effects of
cooperative learning with metacognitive scaffolding and cooperative learning on high-
ability and low-ability students’ mathematics performance, mathematical reasoning,
and metacognitive knowledge.
Thus, the purpose of this study was to find out the extent to which the cooperative
learning with metacognitive scaffolding and the cooperative learning methods could
play an important role in improving Jordanian fifth-grade students’ mathematics
performance, mathematical reasoning, and metacognitive knowledge. Particularly, the
study was conducted to investigate if there were any significant differences in
mathematics performance, mathematical reasoning, and metacognitive knowledge
levels between students taught via the cooperative learning with metacognitive
scaffolding instructional method (CLMS), students taught via the cooperative learning
alone instructional method (CL), and students taught via the traditional instructional
method (T). The study also examined the effects of the instructional methods on high-
ability and low-ability students’ mathematics performance, mathematical reasoning,
and metacognitive knowledge. As such, the study was focused on the following
questions:
31
1.3 Research Questions
1. Would students taught via CLMS instructional method perform higher than students
taught via CL instructional method who, in turn, would perform higher than students
taught via T instructional method in (a) mathematics performance (MP), (b)
mathematical reasoning (MR) and (c) metacognitive knowledge (MK)?
2. Would high-ability students taught via CLMS instructional method perform higher
than high-ability students taught via CL instructional method who, in turn, would
perform higher than high-ability students taught via T instructional method in (a)
mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge (MK)?
3. Would low-ability students taught via CLMS instructional method perform higher
than low-ability students taught via CL instructional method who, in turn, would
perform higher than low-ability students taught via T instructional method in (a)
mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge (MK)?
4. Are there interaction effects between the instructional methods and the ability levels
(high-ability and low-ability) in mathematics performance, mathematical reasoning,
and metacognitive knowledge?
32
1.4 Hypotheses
Based on the research questions the following hypotheses were formulated:
1. Students taught via CLMS instructional method will perform higher than students
taught via CL instructional method who, in turn, will perform higher than students
taught via T instructional method in (a) mathematics performance (MP), (b)
mathematical reasoning (MR) and (c) metacognitive knowledge (MK).
2. High-ability students taught via CLMS instructional method will perform higher
than high-ability students taught via CL instructional method who, in turn, will
perform higher than high-ability students taught via T instructional method in (a)
mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge (MK).
3. Low-ability students taught via CLMS instructional method will perform higher
than Low-ability students taught via CL instructional method who, in turn, will
perform higher than low-ability students taught via T instructional method in (a)
mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge (MK).
4. There are interaction effects between the instructional methods and the ability levels
(high-ability and low-ability) on mathematics performance, mathematical reasoning,
and metacognitive knowledge (MK).
33
1.5 The Theoretical Framework
The theoretical base for this study comes from Piaget (1970) and Vygotsky (1978).
According to the constructivist paradigm, students learn because they have taken prior
knowledge and have reworked the new information into their current schema. A
schema consists of the pieces of knowledge already present in the person. The
processes, in Piagetian terms, that rework new information and incorporate it to prior
knowledge are called assimilation and accommodation. When a new experience is
incorporated into prior knowledge it is assimilated. Accommodation occurs when the
new knowledge alters the knowledge, or schema (DeLay, 1996).
Piaget (1970) believes that individuals work with independence and equality on each
other’s ideas, so when the student is opposed new knowledge and interacts with others
he or she encounters something that contradicts his or her believes or current
understanding. This is what Piaget calls “cognitive conflicts” (Mugny and Doise,
1978). This conflict results a case of disequilibrium. Working cooperatively and
activating metacognitive strategies such as planning, monitoring, and evaluation are
likely to enhance students to assimilate or accommodate their knowledge and
therefore reequilibrate their thinking. When students employ their metacognitive
strategies, they are more than likely enhanced to revise, evaluate, and guide their ways
of thinking to provide a better fit with reality.
34
While there is a general consensus that metacognitive strategies are developed with
age, the developing mind does not develop in isolation, but within a social, cultural
and linguistic environment such as in the case of the interaction with peers and adults.
The need to explain and justify to others, makes reflection on ones own thought thus,
developing metacognitive strategies (Pellegrini et al., 1996). Becoming more
reflective and metacognitive enables students to provide for themselves the supportive
and scaffolding role originally assigned to the adult or peer (Brown, 1987).
Vygotsky (1978) believes that there is a hypothetical region where learning and
development best take place. He identifies this region as the zone of proximal
development. This zone is defined as the distance between what an individual can
accomplish during independent problem solving, versus what can be accomplished
with the help of an adult or a more capable member of a group. This is often a higher-
ability individual. With cooperation, direction, or help, the individual is better able to
solve more difficult tasks than he or she could independently.
Furthermore, Vygotsky (1978) suggests that an active student and an active social
environment cooperate to produce developmental change. The student actively
explores and tries alternatives with the assistance of a more skilled partner, as in an
instructor, or a more capable peer. The teacher and the partner guide and structure the
students’ activity, scaffolding their efforts to increase current skills and knowledge to
a higher competency level. Scaffolding is the support during a teaching session, where
a more skilled partner (adult or peer) adjusts the level of assistance given based on the
level of performance indicated by the student. A greater level of support is offered if
the task is new, and less is provided as competency grows (Berk and Winsler 1995).
The student is able to move forward and continues to develop new capabilities.
35
Therefore, guidance i.e., cooperatively and metacognitively, should be provided to
support both cognition and metacognition. Cognition refers to domain-specific
knowledge and strategies for information and problem manipulation (Salomon et al.,
1989 and Schraw, 1998), and metacognition includes knowledge of cognition and
regulation of cognition (Jacobs and Paris, 1987), such as planning, monitoring, and
evaluation. The two constructs are interrelated. Although metacognitive knowledge
may be able to compensate for absence of relevant domain knowledge, its
development may also depend on having some relevant knowledge of the domain
(Garner and Alexander, 1989).
Thus, this study investigated the effects of cooperative learning with metacognitive
scaffolding and cooperative learning methods on mathematics performance,
mathematical reasoning, and metacognitive knowledge. That is, on learning with
understanding where students assimilate or accommodate their mathematical
knowledge.
1.6 Significance of the Study
The research on metacognition, scaffolding, and cooperative learning strategies to
enhance mathematical reasoning and understanding is based on meaningful learning.
The information age has challenged educators to reexamine the role of the student and
of instruction from the constructivist perspective. As the student’s role changes from a
passive knowledge recipient to an active meaning constructor, reasoning, planning,
monitoring, and evaluation have a significant value in instruction and play a
significant role in understanding particularly in subjects based on proof like
mathematics. Since learning mathematics with understanding requires skills more than
36
conceptual understanding and procedural fluency (Kilpatrick et al., 2001), cooperative
learning with metacognitive scaffolding focuses on helping students to ask
metacognitive questions that guide them to plan, understand, monitor, and evaluate
and reason their learning, not just guides them to master mathematical procedures. In
other words cooperative learning with metacognitive scaffolding focuses on helping
students to be metacognitively prepared to solve mathematical problems with
understanding and to plan, monitor, evaluate, and reason their solutions.
It is hoped that the findings of this study will contribute to further understanding of
the role of cooperative learning with metacognitive scaffolding and cooperative
learning strategies in improving mathematics performance, mathematical reasoning,
and metacognitive knowledge. If the use of cooperative learning with metacognitive
scaffolding and cooperative learning methods proves its effectiveness in improving
mathematics performance, mathematical reasoning and metacognitive knowledge,
teachers in Jordan will have additional instructional methods that can be used to
support students’ learning with understanding. Moreover, this will help educators in
Jordan in their search for an effective and efficient pedagogical strategy or model for
improving learning with understanding.
37
1.7 Operational Definitions
Metacognition:
The processes of considering and regulating student’s own learning that, include
planning, monitoring, and evaluation of the student’s current and previous knowledge.
These processes are activated before, during, and after the problem solution.
Scaffolding:
A technique in which the teacher provides instructional support as students learn to do
the task, and then gradually shifts responsibility to the students. In this manner, the
teacher enables students to accomplish as much of a task as possible without peer
assistance.
Cooperative Learning with Metacognitive scaffolding Method (CLMS):
An instructional method in which, the high-ability and low-ability students work
together in groups of four members (two high-ability and two low-ability students) to
solve a problem or complete an assignment. In this method, the teacher, the
metacognitive questions card, and the students’ interaction provide metacognitive
scaffolding to students in the form of planning, monitoring, end evaluation in
performing a given task. The teacher’s metacognitive scaffolding is gradually reduced
as the students are able to accomplish the task.
38
Cooperative learning Method (CL):
An instructional method in which, high-ability and low-ability students work together
in groups of four members (two high-ability and two low-ability students) to solve a
problem or complete an assignment. The teacher is allowed to assist the groups but the
groups and the teacher are not provided with any metacognitive questions card.
Traditional Instructional Method (T):
An instructional method in which, the teacher explains and manipulates the
mathematics concepts and procedures to the whole class. In this method, the teacher’s
teaching time is about 35 minute out of 45 minute of the session’s time. The teacher’s
concentration in this method is on mastering the task and developing specific skills.
High-ability students:
Students whose average scores on mathematics performance and mathematical
reasoning measured by the pre-test are above the median.
Low-ability students:
Students whose average scores on mathematics performance and mathematical
reasoning measured by the pre-test are below the median.
Mathematical reasoning (MR):
The student’s ability to make a decision about how to approach the mathematics
problem, select or generate the appropriate tactic to solve the problem, support the
solutions with evidence, and to generalize his solution processes to different
situations.
Conceptual understanding (CU):
39
The student’s ability to connect new mathematics ideas with ideas he has been known,
represent the mathematical situation in different ways, and to determine
similarities/differences between these representations.
Procedural fluency (PF):
The student’s ability to use mathematics procedures appropriately, flexibly, and
accurately.
Mathematics performance (MP):
The students’ score in conceptual understanding and procedural fluency items on the
mathematics achievement test.
Mathematics achievement test:
A mathematics test which assesses students’ conceptual understanding, procedural
fluency, and mathematical reasoning simultaneously.
Metacognitive Knowledge Questionnaire:
A questionnaire consists of 15 items that assesses students’ planning, monitoring, and
evaluation simultaneously , 5 items were designed to assess each skill.
Primary government schools:
Schools established by the Jordan Ministry of Education where students study from
the first to the tenth-grade. These schools are not coeducational, so there are male
schools and female schools.
40
CHAPTER TWO
LITERATURE REVIEW
2.1 Introduction
To understand how someone learns mathematics is an important matter.
Understanding this serves educators to determine what and how they should teach.
While understanding how someone learns mathematics is a difficult task, the study of
psychology offers many contributions and deep understanding of how students learn
mathematics. So reviewing the theories according to various psychological
perspectives contributes to the understanding of how mathematics learning occurs on
one hand, and serves in the understanding of how the teaching of mathematics should
be conducted on the other hand.
Mathematics is generally accepted as a very important school subject and thus the
teaching and learning of mathematics have been intensively studied and researched
over the past six decades. The study of the teaching of mathematics is always based on
the conception of learning held by the researcher as well as the mental tasks believed
to be necessary for performing mathematical tasks. This chapter reviews the
paradigms, theories, and models of learning based on the literature currently available
41
to identify the theory, model, and variables most promising for use in improving the
teaching of mathematics. The chapter also discusses the review of related literature on
metacognitive scaffolding as well as cooperative learning. Then the chapter describes
the mathematical proficiency model and discusses the role of metacognitive
scaffolding and cooperative learning plays in improving the teaching of mathematics.
The chapter continues with a discussion on mathematical reasoning and describes the
role of cooperative learning and metacognitive scaffolding plays in improving
students’ mathematics performance, mathematical reasoning, real-life problem
solving, and motivation.
2.2 Objectivist Views Regarding the Learning / Teaching of Mathematics
The objectivist theories postulate that knowledge exists independently of the student,
and then becomes internalized as it is transferred from its external reality to an internal
reality of the student that corresponds directly with outside phenomenon with the
mind acting as a processor of input from reality (Driscoll, 1994). Meaning is derived
from the structure of reality, with the mind processing symbolic representations of
reality (Jonassen, 1991). This belief is very popular among educators and researchers
and has produced numerous theories and models as represented by behaviourism, and
cognitivism as represented by the theories of Gagne, Landa, and Scandura.
2.2.1 Behaviorism and the Learning / Teaching of Mathematics
The learning paradigm of behaviorism represents the original Stimulus-Response (S-
R) framework of behavioral psychology: Learning is the result of associations forming
between stimuli and responses. Such associations become strengthened or weakened
42
by the nature and frequency of the S-R pairings. The paradigm for S-R theory is trial
and error learning in which certain responses come to dominate others due to rewards.
The hallmark of behaviorism is that learning could be adequately explained without
referring to any unobservable internal states.
The behaviorists’ earlier studies concentrated on animals before becoming interested
in human thinking. Thorndike (1932) states that in any given situation an animal has a
number of possible responses, and the action that would be performed depends on the
strength of the connection or bond between the situation and the specific action. The
bonds that go together should be taught together. In pedagogical terms, this yields a
drill and practice mode of instruction. At elementary school age for Thorndike, the
rules of arithmetic are said to have not been known. The purpose of instruction in
mathematics is thus seen to be one of drilling into the student the necessary rules and
connections until sufficient responses are obtained. Thorndike explains this in his law
of effect: “When a modifiable connection between a situation and a response is made
and is accompanied or followed by a satisfying state of affairs, that connection’s
strength is increased.” (p. 60).
Skinner (1968) further argues that an organism learns mainly by producing changes in
response to its environment. In other words, learning is characterized by changes in
behavior. This may seem to be a simple truism except for the fact that Skinner argues
that a change in behavior is the only characteristic of learning. He explicitly rejects
such concepts as purposes, deliberations, plans, decisions, theories, tensions, and
values as mentalism. Since these concepts are non-physical and therefore cannot be
measured, weighed, and counted, he refers to them as pre-scientific and says learning
need to move beyond such ideas and develop a true technology of behavior.
43
According to behaviorist principles, all learning processes are fully controlled by the
teacher. So the teacher has to understand all of the students’ behaviors and sub-
behaviors involved in the task, as well as the characteristics of the students. Also the
teacher has to create an instructional situation that requires students to practice the
appropriate behaviors, in proper sequence and with appropriate reinforcement,
gradually building more and more behaviors until the target behavior is achieved. This
process requires a great deal of time for complex, intricate tasks such as data
classification. The nature of mathematics as represented by behaviorism portrays
mathematics not as a product of human creation but, instead, as existing external to
the human mind. Tiene and Ingram (2001) assert that behaviorism is unable to
effectively address the critical issue like how students think, understand, reason, and
build knowledge. Students are more than just the sum total of the behaviors that they
engage in. Students make plans, remember things, forget things, solve problems,
hypothesize, and much more. These aspects of cognition could not be fully understood
just by looking at behavior. Moreover, the role of the student in this environment is
passive, namely, it is teacher-centered where the teacher selects, explains,
demonstrates, and evaluates the instructional activities.
2.2.2 Gagne and the Learning / Teaching of Mathematics
Gagne (1985) indicates that there are several different types of learning, and each type
requires different types of instruction. He classifies human learning into five domains:
intellectual skills, motor skills, cognitive strategies, verbal information, and attitudes.
Different internal conditions such as acquisition and storage of prior capabilities, and
external conditions such as the various ways that instructional events outside the
student function to activate and support the internal processes of learning are
44
necessary for each type of learning. For example, for cognitive strategies to be
learned, there must be an opportunity to practice developing new solutions to
problems; to learn attitudes, the student must be exposed to a credible role model or
persuasive arguments (Aronson et al., 1983).
Gagne suggests that learning tasks for intellectual skills can be organized in a
hierarchy according to complexity: discriminations, concept formation, rule
application, and problem solving (facts, concepts, principles, and problem solving)
(Aronson et al., 1983). In other words, the sequence in learning is from bottom up,
that is, from simple to complex (Gagne, 1985). Gagne asserts that the significance of
the hierarchy is to identify prerequisites that should be completed to facilitate learning
at each level. Doing a task analysis of a learning task identifies the prerequisites. He
determines two types of prerequisites: essential prerequisites which are the
subordinate skills that must be previously learned to enable the student to reach the
objective, and supporting prerequisites which are useful to facilitate learning but are
not essential for the learning to occur. He adds that learning hierarchies provide a
basis for the sequencing of instruction (Aronson et al., 1983).
In addition, Gagne outlines nine instructional events that provide the external
conditions of learning and corresponding cognitive processes: gaining attention
(reception), informing students of the objective (expectancy), stimulating recall of
prior learning (retrieval), presenting the stimulus (selective perception), providing
learning guidance (semantic encoding), eliciting performance (responding), providing
feedback (reinforcement), assessing performance (retrieval), and enhancing retention
and transfer (generalization) (Aronson et al., 1983).
45
For Gagne, mathematics is composed of a set of tasks to be learned and occurs
hierarchically. The learning task is analyzed to their subordinate elements, and
mastery of each subordinate element is essential to the attainment of the main task.
That is, learning of the task cannot occur without mastering their subordinate elements
and therefore mathematics instruction must start with the subordinate elements of the
task.
2.2.3 Landa and the Learning / Teaching of Mathematics
Landa’s Algo-Heuristic theory (1976) is a general theory of learning, but it is
illustrated primarily in the context of mathematics and foreign language instruction. It
is concerned with identifying mental processes that underlie expert learning, thinking
and performance in any area. His theory represents a system of techniques for getting
inside the mind of expert students and performers that enable one to uncover the
processes involved. Once uncovered, they are broken down into their relative
elementary components. Performing a task or solving a problem always requires a
certain system of elementary knowledge units and operations.
According to Landa, there are classes of problems for which it is necessary to execute
operations in a well-structured, predefined sequence (algorithmic problems). For such
problem classes, it is possible to formulate a set of precise unambiguous instructions
(algorithms) as to what one should do mentally and / or physically in order to
successfully solve any problem belonging to that class. There are also classes of
problems (creative or heuristic problems) for which precise and unambiguous sets of
instructions cannot be formulated. For such classes of problems, it is possible to
formulate instructions that contain a certain degree of uncertainty (heuristics).
46
For the content of learning, Landa maintains that students have to learn not only
knowledge but also the algorithms and heuristics of experts as well. Students also
have to learn how to discover algorithms and heuristics on their own. Landa
concentrates on the learning of cognitive operations, algorithms, and heuristics that
make up general methods of thinking (i.e., intelligence). According to this point of
view, Landa affirms that learning algo-heuristic processes is more important than
learning prescriptions (knowledge of processes).
For Landa, mathematics is also composed of a set of specific and general problems or
tasks that can be identified and taught sequentially. In this manner, Landa proposes the
“snowball” method of learning / teaching. This method entails the following sequence:
The first elementary operation in the chain is taught / learnt and practiced alone, then
the second elementary operation is taught / learnt alone, practiced alone then is
practiced together with the first, then the third is taught / learnt alone, practiced alone
and then practiced together with the first two, and so on, until all elementary
operations have been taught separately but practiced together (Landa, 1983).
2.2.4 Scandura and the Learning / Teaching of Mathematics
According to Scandura’s Structural theory (1980), learning occurs through learning
rules that consist of domain (internal cognitive structures of relevant environmental
elements of a learning situation), range (expected rule that corresponds to the
cognitive structure a student utilize to complete an objective), and procedures or
operations (sum of all decisions and actions to produce a specific range element).
Learning starts by using a very simple task as a prototype. Doing so requires
identifying the educational goals first and then identifying prototypic cognitive
processes (rules). There may be alternative rule sets for any given class of tasks.
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Scandura identifies two types of rules: higher order rules and lower order rules.
Higher order rules generate new rules, and problem solving may be facilitated when
higher order rules are used. Higher order rules account for creative behavior
(unanticipated outcomes) as well as the ability to solve complex problems by making
it possible to generate (learn) new rules. Lower order rules are simple rules that the
learning task starts with and are later reduced in number to derive higher order rules,
the redundant lower order rules from the rule set will be eliminated by the student
after practice and mastery that task.
The rules which are to be learned can be derived from educational goals through
structural analysis which is a methodology for identifying the rules to be learned for a
given topic or class of tasks and breaking them down into their atomic components.
The major steps in structural analysis are: (1) select a representative sample of
problems, (2) identify a solution rule for each problem, (3) convert each solution rule
into a higher order problem whose solutions is that rule, (4) identify a higher order
solution rule for solving the new problems, (5) eliminate redundant solution rules
from the rule set, and (6) continue the process iteratively with each newly-identified
set of solution rules. The result of repeatedly identifying higher order rules, and
eliminating redundant rules, is a succession of rule sets, each consisting of rules which
are simpler individually but collectively more powerful than the ones before.
Structural theory suggests that instruction has to start with the simplest solution path
for a problem and then move to the more complex paths until the student masters the
entire rule. The theory proposes that higher-order rules should be taught through
elaboration and replacement of lower order rules. The theory also suggests a strategy
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for individualizing instruction by analyzing which rules a student has / has not
mastered and teach only the rules, or portions thereof, that have not been mastered.
Structural theory has been applied extensively to mathematics. Here is an example in
the context of subtraction provided by Scandura (1977):
1. The first step involves selecting a representative sample of problems such as 9-5,
248-13, or 801-302.
2. The second step is identifying the rules for solving each of the selected problems.
This step can be achieved by determining the minimal capabilities of the students
(e.g., can recognize the digits 0-9, minus sign, column and rows). Then the detailed
operations involved in solving each of the representative problems must be worked
out in terms of the minimum capabilities of the students. For example, one subtraction
rule students might learn is the borrowing procedure that specifies if the top number is
less than the bottom number in a column, the top number in the column to the left
must be made smaller by 1.
3. The next step is identifying any higher order rules and eliminating any lower order
rules they subsume. In the case of subtraction, a number of partial rules can be
replaced with a single rule for borrowing that covers all cases.
4. The last step is to testing and refining the resulting rules by applying to new
problems and extending the rule set if necessary so that it accounts for all problems in
the domain. In the case of subtraction, problems with varying combinations of
columns may be used.
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Scandura also sees mathematics as a set of existing rules or procedures and the goal of
teaching is to develop expertise by deriving general rules from specific rules.
Therefore, teaching process starts with the simplest solution path first and then moves
to the more complex paths or rule sets, that is, the appropriate sequence in teaching is
from bottom up. The students’ prior capabilities have to be taken into account before
teaching rules. So rules must be composed of the minimum capabilities possessed by
the students.
The objectivist theories and models are based on the view that knowledge of the world
is fixed and can be quantified. These theories and models call for information or
knowledge to be taught to be divided into parts that are slowly assembled into whole
concepts. Mathematics according to these theories and models is seen as a set of
preexisting facts and procedures, free of context and value. Mathematics knowledge is
passed along from those who know to those who do not through authoritative means,
including memorization and practice (Borasi, 1996). So teachers serve as pipelines or
assemblers of knowledge and seek to transfer their thoughts and meanings to the
students. Lessons derived from these theories and models are teacher-centered and
depend heavily on textbooks for the structure of the course. The students are generally
passive or compliant, and there is little room for student-initiated questions,
independent thought or interaction between them. The role of the students is to
regurgitate the accepted explanation or methodology expostulated by the teacher
(Hanley, 1994). Also the assessment of performance is end-centered, that is, the
concentration is on mastering the task. Being content-based, these theories and models
produce lessons that are presented below the student’s true cognitive ability.
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These objectivist theories and models of teaching do not meet the needs of learning
mathematics with understanding where the student is actively doing mathematics
through the process of inquiry and investigation (von Glasersfeld, 1995). These
theories and models also do not promote mathematical reasoning, adaptive expertise
and an ability to deal with change or solve ill-structured problems characteristic of
today’s complex society (NCTM, 2000).
2.3 Constructivist Views Regarding the Learning / Teaching of Mathematics
2.3.1 Nature of the Learning Process and Construction of Knowledge
The central idea of objectivist is that learning performance could be defined solely in
terms of observable behavior, and the teacher’s job is just to give orders and monitor
student responses. New theories soon emerged to challenge the behaviorists, the
earliest being Gestaltism (Schoenfeld, 1987). Gestalt theory is one of the early
learning theories which emphasize the role of understanding. It is also one of the first
to deal with issues of problem solving and creativity. Wertheimer (1959) is one of the
principal proponents of Gestalt theory that emphasizes higher-order cognitive
processes. The focus of Gestalt theory is the idea of grouping, i.e. characteristics of
stimuli cause us to structure or interpret a visual field or problem in a certain way. The
primary factors that determine grouping were: (1) proximity - elements tend to be
grouped together according to their nearness, (2) similarity - items similar in some
respect tend to be grouped together, (3) closure - items are grouped together if they
tend to complete some entity, and (4) simplicity - items will be organized into simple
figures according to symmetry, regularity, and smoothness. These factors are called
the laws of organization and are explained in the context of perception and problem
solving.
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For learning mathematics, the essence of successful problem-solving behavior
according to Wertheimer (1959) is being able to see the overall structure of the
problem:
"A certain region in the field becomes crucial, is focused; but it does not become isolated. A new, deeper structural view of the situation develops, involving changes in functional meaning, the grouping, etc. of the items. Directed by what is required by the structure of a situation for a crucial region, one is led to a reasonable prediction, which like the other parts of the structure, calls for verification, direct or indirect. Two directions are involved: getting a whole consistent picture, and seeing what the structure of the whole requires for the parts." (P 212).
How humans learn has intrigued and troubled educators throughout history.
Constructivists believe that learning involves the generation of knowledge and
learning strategies. According to this view, learning in schools has to emphasize the
use of intentional processes that students can use to construct meaning from
information, experiences, and their own thoughts and beliefs. Mayer (1996) asserts
that successful students are active, goal-directed, self-regulating, and assume personal
responsibility for contributing to their own learning. So the learning of complex
subject matter is most effective when it is an intentional process of constructing
meaning from information and experience. von Glasersfeld (1995) argues that: “From
the constructivist perspective, learning is not a stimulus-response phenomenon. It
requires self-regulation and the building of conceptual structures through reflection
and abstraction” (p.14). Fosnot (1996) mentions that “Rather than behaviors or skills
as the goal of instruction, concept development and deep understanding are the foci”
(p.10). This view of learning sharply contrasts with the one in which learning is the
passive transmission of information from one individual to another.
The psychological theoretical base for constructivism comes from Piaget. He uses the
term schemata to describe mental or cognitive structures that allow one to think about,
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organize and make sense of experiences (Borich and Tombari, 1997). The individual
continuously constructs his or her schemata. Cognitive development, or learning, is
the lifelong process by which the student constructs and modifies his or hers own
personal schemata. This occurs in two ways: a) existing schemata are organized into
higher-order, more complex structures, and b) the individual adapts to new
information and experience through assimilating it or accommodating it.
Consequently, the constructivist point of view defines meaning as an act of
interpretation i.e., meaning does not exist independently of the student (Mugny and
Doise, 1978).
For mathematics, White (1998) maintains that “Educational research has shown that
students tend to comprehend complex concepts much better and to retain them as part
of their body of knowledge much longer when they become actively involved in their
learning process” (p.1). Mathematics can be actively learned by involving students in
their leaning process. Ahmed (1987) asserts that “Mathematics can be effectively
learned only by involving students in experimenting, questioning, reflecting,
discovering, inventing and discussing. Mathematics should be a kind of learning
which requires a minimum of factual knowledge and a great deal of experience in
dealing with situations using particular kinds of thinking skills” (p.24). Carpenter and
Lehrer (1999) indicate that the critical learning of mathematics by students occurs as a
consequence of building on prior knowledge via purposeful engagement in activities
and by discourse with other students and teachers in classrooms. So students must
engage in activities that encourage their mathematical understanding.
This view of learning mathematics leads to the characteristics of learning mathematics
with understanding. Hiebert and Carpenter (1992) assert that learning mathematics
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with understanding implies students not only must learn the concepts and procedures
of mathematics (its design features), but they must learn to use such ideas to solve
non-routine problems, and learn to mathematize in a variety of situations (its social
functions). Therefore the concentration should shift from judging student learning in
terms of mastery of concepts and procedures to making judgments about students’
deep understanding of the concepts and procedures and their ability to apply them to
mathematics problem situations.
The construction of relationships is one of the important forms of mental activity
where mathematical understanding emerges. For students to learn mathematics with
understanding, new ideas take on meaning by the ways they are related to other ideas.
Students construct meaning for a new idea or process by relating it to ideas or
processes that they have already understood. Although learning with understanding
entails forging connections between what the students already know and the
knowledge they are learning, it is not sufficient to think of developing understanding
simply as appending new concepts and processes to existing knowledge. Over the
long run, developing understanding involves more than simply connecting new
knowledge to prior knowledge; it also involves the creation of rich, integrated
knowledge structure (Carpenter and Lehrer, 1999).
Therefore, the role of the teacher and the role of students should be appropriate to this
learning environment. The teacher’s role should be shifted from being an orator to a
learning manager and facilitator who manages, directs, and encourages students’
creation or from sage on the stage to guides on the side where he provides students
with opportunities to test the adequacy of their current understandings. Doyle (1988)
argues that teachers should be especially attentive to the extent to which meaning is
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emphasized and the extent to which students are explicitly expected to demonstrate
understanding of the mathematics underlying the activities in which they are engaged.
Such an emphasis can be maintained if explicit connections between the mathematical
ideas and the activities in which students engage in are frequently drawn. Also,
Carpenter and Lehere (1999) assert that connections with what students already know
and understand what they are learning play an important role in engaging students in
high-level thought processes. The mathematical activities should therefore be selected
to encourage the students to link between the knowledge what they have already
learned and the new knowledge. The students’ role should also be changed from
obtaining knowledge from the teacher to assimilating or accommodating their own
knowledge by connecting the relationships between what they have known and what
they are learning. The students’ role should also be shifted to confront their
understanding in light of what they encounter in the new learning situation (Manion,
1995). If what the students encounter is inconsistent with their current understanding,
their understanding can change to accommodate new experience.
The teacher has to keep in mind that students come to the learning situations with
knowledge gained from previous experience, and that prior knowledge influences
what new or modified knowledge they will construct from the new learning
experiences. Bennett and Desforges (1988) affirm that a critical factor underlying
unsuccessful task implementation is a lack of alignment between tasks and students'
prior knowledge, interests, and motivation. Such mismatches may cause students to
fail to engage with the task in ways that will maintain a high level of cognitive
activity.
2.3.2 Piaget and the Learning / Teaching of Mathematics
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Cognitive structure is the central idea of Piaget’s (1970) theory. Cognitive structures
are patterns of physical or mental action that underlie specific acts of intelligence and
correspond to stages of child development. Piaget specifies four primary cognitive
structures (i.e., development stages): sensorimotor (0-2 years) where intelligence takes
the form of motor actions; preoperations period (3-7 years) where intelligence is
intuitive in nature; The concrete operational stage (8-11 years) where the cognitive
structure is logical but depends upon concrete referents; and formal operations (12 and
above) where thinking involves abstractions. While the stages of cognitive
development identified by Piaget are associated with characteristic age spans, they
vary from one individual to another.
Piaget indicates that cognitive structures are not stable and they change through the
processes of adaptation i.e., assimilation and accommodation. Assimilation involves
the interpretation of events in terms of existing cognitive structure whereas
accommodation refers to changing the cognitive structure to make sense of the
environment. Cognitive development consists of a constant effort to adapt to the
environment in terms of assimilation and accommodation. Piaget affirms that all
children construct, or create logic and number concepts from within rather than learn
them by internalization from the environment (Piaget 1971; Piaget and Szeminska
1965; Inhelder and Piaget 1964; and Kamii 2000).
Piaget explores the implications of his theory to all aspects of cognition, intelligence
and moral development. Many of Piaget’s experiments were focused on the
development of mathematical and logical concepts. Applying Piaget’s theory, results
in specific recommendations for a given stage of cognitive development. For example,
with students in the concrete operational stage, learning activities should involve
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problems of classification, ordering, location, and conservation using concrete objects.
So teachers should also try to provide a rich and stimulating environment with logical
matters that depend on concrete objects and try to prepare students to the next stage,
namely, formal operational that involves abstract thinking.
2.3.3 Vygotsky and the Learning / Teaching of Mathematics
A critical factor that relates to the learning process and construction of knowledge is
sociocultural development. Constructivists view sociocultural development as one of
the significant factors that contribute to the construction of knowledge. Vygotsky
(1978) states that cognitive development is dependent on social interaction, and that
cultural development has two levels: social and interpersonal. During social
interaction, the students recognize the new knowledge and then internalize it. So for
effective learning, students have to cooperate in an environment where social
interaction is taken into account (Bonk and Reynolds, 1997).
Vygotsky suggests that students can be guided by explanation, demonstration, and can
attain to higher levels of thinking if they are guided by more capable and competent
adults. This conception is better known as the Zone of Proximal Development (ZPD).
The Zone of Proximal Development is the gap between what is known and what is not
known, that is, generally higher levels of knowing. The ability to attain higher levels
of knowing is often facilitated and, in fact, depends upon, interaction with other more
advanced peers, who for Vygotsky are generally adults. Through increased interaction
and involvement, students are able to extend themselves to higher levels of cognition.
Vygotsky defines the Zone of Proximal Development as “the distance between the
actual developmental level as determined by independent problem solving and the
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level of potential development as determined through problem solving under the
guidance or in collaboration with more capable peers.” The ZPD is thus the difference
between what students can accomplish independently and what they can achieve in
conjunction or in cooperation with another, more competent person.
The sociocultural development enriches the active learning processes and contributes
in encouraging constructing knowledge. Moore and Kearsley (1996) have indicated
that sociocultural development is an area that is missing in most traditional or
objectivist learning environments. Interactions between the student and the content,
between the student and the instructor, and between the students themselves are
necessary for learning and for the shared social construction of knowledge (American
Psychological Association [APA], 1995; Moore and Kearsley, 1996).
2.3.4 Bruner and the Learning / Teaching of Mathematics
A major theme in the theoretical framework of Bruner (1960) is that learning is an
active process in which, students construct new ideas or concepts based upon their
current or past knowledge. The student selects and transforms information, constructs
hypotheses, and makes decisions, relying on a cognitive structure to do so. Cognitive
structure i.e., schema, mental models provides meaning and organization to
experiences and allows the individual to go beyond the information given.
Bruner believes that students can and have to discover knowledge by themselves. So
the teacher should encourage students to discover their knowledge. The teacher and
student should engage in an active dialog i.e., Socratic learning where the teacher’s
task is to translate information to be learned into a format appropriate to the students’
current state of understanding. To reach that learning environment, Bruner suggests
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that the curriculum should be organized in a spiral manner so that the students
continually build knowledge upon what they have already learned. He maintains that
learning starts from the top down, that is, it begins with problem solving that makes
students learn the fundamentals because they need them. Instruction for him is a roller
coaster ride of successive disequilibrium and equilibrium until the desired cognitive
state is reached or discovered (Shulman, 1973).
Bruner asserts that knowing is a process not a product. He describes three levels of
student’s representation: the enactive level where the student manipulates materials
directly. The second level is the ikonic level where the student deals with mental
images of objects but does not manipulate them directly. The final level is the
symbolic level where the student is strictly manipulating symbols and no longer
mental images of object. This sequence is an out growth of the developmental work of
Piaget (Shulman, 1973).
Transfer of learning for Bruner (1960) occurs when the student can identifying from
the structures of subject matters basic, fundamentally simple concepts or principles
which, if learned well, can be transferred both to other subject matters within that
discipline and to other disciplines as well. He gives an example of the concept of
balance. If the teacher teaches the balance of trade in economics in such a way that
when ecological balance is considered, students will see the parallel and this could be
extended to balance of power in political science, or to balancing equations.
In summary, learning according to Bruner is a process, and through the processes,
students discover and build their knowledge. Instruction should be concerned with the
experiences and contexts that make the student willing and able to learn (readiness), it
should be structured so that it can be easily grasped by the student (spiral
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organization), and instruction should be designed to facilitate extrapolation (going
beyond the information given) (Bruner, 1973).
2.4 Metacognitive Strategies and the Construction of Knowledge
The central point of constructivism is that learning involves more than just the transfer
of information from the teacher to a student; instead, each student plays an active role
in working with and integrating the information according to his or her own
background or experience. This integration involves applying personal study and
learning skills, and monitoring one’s own comprehension (Gordon, 1996). Therefore,
to construct or reconstruct his or her knowledge, the student needs to employ certain
techniques regarding managing his or her thinking like thinking about thinking,
planning, monitoring, and evaluation. In other words for students to reach the
equilibrium case (resolving the conflicts), and then assimilate or / and accommodate
their knowledge, they should employ various metacognitive strategies. Metacognitive
strategies are techniques that students use to plan, monitor and control, and evaluate
their own cognitive processes (Flavell, 1976). These strategies are seldom taught
directly and tend to develop naturally in only good students (Smith and Ragan, 1993).
However Jacobson (1998), Perkins and Grotzer (1997), and Halpern (1996) indicate
that metacognitive strategies can be systematically taught to most students. Also Paris
and Winograd (1990) argue that teachers can promote metacognitive strategies
directly by guiding students about effective problem solving strategies and discussing
cognitive and motivational characteristics of thinking.
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Metacognition has been defined as an awareness of one’s own cognitive processes
rather than the content of those processes together with the use of that self-awareness
in controlling and improving cognitive processes (Biggs and Moore, 1993). Other
researchers have referred to metacognition as “cognitive strategies” (Paris and
Winograd, 1990), “knowledge about executive control systems” (Brown et al., 1994),
“monitoring of cognitive processes” (Flavell, 1976), “resources and self regulating
learning” (Osman and Hannafin, 1994, Lawson, 1995) and “knowledge and regulation
of cognition, and evaluating cognitive states such as self-appraisal and self-
management” (Brown, 1987).
Flavell (1977) refers to metacognition as metamemory which he describes as
intelligent structuring and storage, intelligent search and retrieval, and intelligent
monitoring. His description suggests that metacognitive strategies are deliberate,
planful, intentional, goal-directed, and future-oriented mental behaviors that can be
used to accomplish cognitive tasks. Flavell mentions that metacognition is an
awareness of oneself as “an actor in his environment, that is, a heightened sense of the
ego as an active, deliberate storer and retriever of information” (p. 275). “It is the
development of memory as applied cognition” (p. 273), in which whatever
“intellectual weaponry the individual has so far developed” is applied to mnemonic
problems (p. 191).
What is basic to the concept of metacognition is the notion of thinking about one’s
own thoughts. These thoughts can be of what one knows (i.e., metacognitive
knowledge), what one is currently doing (i.e., metacognitive skill), or what one’s
current cognitive or affective state is (i.e., metacognitive experience). To differentiate
metacognitive thinking from other kinds of thinking, it is necessary to consider the
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source of metacognitive thoughts: Metacognitive thoughts do not spring from a
person’s immediate external reality; rather, their source is tied to the person’s own
internal mental representations of that reality, which can include what one knows
about that internal representation, how it works, and how one feels about it. Therefore,
metacognition sometimes has been defined simply as thinking about thinking,
cognition of cognition, or using Flavell’s (1979) words, “knowledge and cognition
about cognitive phenomena” (Hacker,1998, p. 906).
Although perspectives differ in emphasis, there is common agreement that
metacognitive strategies involve both the knowledge of and the regulation of
cognition. Pressley and McCormick (1987) identify two components of knowledge of
cognition, which are knowledge about and awareness of one’s own thinking and
knowledge of when and where to use acquired strategies.
Knowledge about one’s thinking includes information about one’s own capacities and
limitations and awareness of difficulties as they arise during learning so that remedial
action may be taken. Knowledge of when and where to use acquired strategies,
includes knowledge about the task and situations for which particular goal-specific
strategies are appropriate. In the absence of domain-specific knowledge or lack of
information in various content areas, students often need to apply general strategies,
which can be applied to the problems, regardless of their content. In the social science
study conducted by Voss et al. (1991), they found that experts were flexible in that
they take into account more factors than do novices in searching for information.
Additionally, experts used strategies of argumentation more often than novices did.
They concluded that argumentation may be an important strategy in problem solving
(Gick, 1986).
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2.4.1 Regulation of Cognition
Jacobs and Paris (1987) determine three components of regulation of cognition, which
are planning, monitoring, and evaluation. Planning (processes selected prior to any
task action) consists of setting goals, activating relevant resources, and selecting
appropriate strategies. Monitoring (processes selected to keep track of what has been
done, what is currently being done, and what still needs to be done for task solution)
involves checking one’s progress and selecting appropriate repair strategies when
originally-selected strategies are not working. Evaluation (processes selected to judge
the outcome of any action against criteria of effectiveness and efficiency, evaluation
refers to students’ ongoing assessments of their knowledge or understanding,
resources, tasks, and goals) involves determining one’s level of understanding. In
short, regulation of cognition is thinking before, during and after a learning task.
Jacobs and Paris (1987); and North Central Regional Educational Laboratory,
(NCREL, 1995) point out that successful students ask themselves metacognitive
questions before (through planning), during (through monitoring), and after (through
evaluation) the learning task. For example, at the planning stage the student asks him
or herself metacognitive questions such as: “What in my prior knowledge will help me
with this particular task? What should I do first? Do I know where I can go to get
some information on this topic? How much time will I need to learn this? What are
some strategies that I can use to learn this?” At the monitoring stage the successful
student asks him or herself metacognitive questions such as: “Did I understand what I
just heard, read or saw? Am I on the right track? How can I spot an error if I make
one? How should I revise my plan if it is not working? Am I keeping good notes or
records?” And at the evaluation stage the student asks him or herself metacognitive
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questions such as: “Did my particular strategy produce what I had expected? What
could I have done differently? How might I apply this line of thinking to other
problems?”
2.4.2 Metacognitive Strategies and Age
The idea of deliberate, planful, and goal-directed thinking applied to one’s thoughts to
accomplish cognitive tasks is deeply embedded in Piaget’s conceptualization of formal
operations in which higher-ordered levels of thought operate on lower-ordered levels.
During this stage of cognitive development, the abilities of the adolescent begin to
differentiate from those of the child (Hacker,1998). Flavell (1963) wrote:
“What is really achieved in the 7-11-year period is the organized cognition of concrete objects and events per se (i.e., putting them into classes, seriating them, setting them into correspondence, etc.). The adolescent performs these first-order operations, too, but he does something else besides, a necessary something which is precisely what renders his thought formal rather than concrete. He takes the results of these concrete operations, casts them in the form of propositions, and then proceeds to operate further upon them, i.e., make various kinds of logical connections between them (implications, conjunction, identity, disjunction, etc.). Formal operations, then, are really operations performed upon the results of prior (concrete) operations. Piaget has this propositions-about-propositions attribute in mind when he refers to formal operations as second-degree operations or operations to the second power” (p. 205-206).
Inhelder and Piaget (1958) provide further elaboration on second-degree operations:
“... this notion of second-degree operations also expresses the general characteristics
of formal thought. It goes beyond the framework of transformations bearing directly
on empirical reality (first degree operations) and subordinates it to a system of
hypothetico-deductive operations--i.e., operations which are possible” (p. 254). Thus,
first-degree operations, which are thoughts about an external empirical reality, can
become the object of higher-order thoughts in an attempt to discover not necessarily
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what is real but what is possible. “Formal thinking is both thinking about thought...
and a reversal of relations between what is real and what is possible” (p. 341-342).
Referring to Inhelder and Piaget’s work, Flavell (1977) wrote: “Another way to
conceptualize it is to say that formal operations constitute a kind of metathinking, i.e.,
thinking about thinking itself rather than about objects of thinking. Children certainly
are not wholly incapable of this and other forms of metacognition” (p. 107). So for
young students, 11 year olds for example, they can think metacognitively and apply
metacognitive strategies in their learning processes, particularly if they are taught and
supported deliberately how and when to use these strategies.
2.4.3 Metacognitive Scaffolding
Students in 7-11 years stage (concrete operations) have some abilities, and some
higher levels of thinking that enable them to work in the next stage (formal
operations), but they need a certain guidance and support from more capable and
competent adults to reach that stage. These children need to narrow their zone of
proximal development; they can be pushed to the next stage or can narrow their ZPD
by scaffolding and supporting them. Vygotsky (1978) believes that students cannot
independently narrow the zone of proximal development (Rosenshine and Meister,
1993). So the concept of scaffolding becomes a critical technique to bridge the gap
between what the students can accomplish independently and what they can achieve
with assistance or guidance of others. Therefore, scaffolding is a technique of teaching
where the learning is assisted by the teacher or / and other capable peers (Slavin,
1994; Rosenshine and Meister, 1993). When using scaffolding, students are provided
with “a great deal of support during the early stage of learning and then diminishing
support and having the students take on increasing responsibility as soon as they are
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able” (Slavin, 1994, p. 49). In this way, students are able to narrow the zone of
proximal development initially with support, and retain this level of achievement as
support is reduced. So awareness of a student’s ZPD helps a teacher gauge the tasks
student is ready for, the kind of performance to expect, and the kinds of tasks that will
help the student reaching his or her potential.
Fading of support during scaffolding should eventually result in self-regulated
learning, and thus more self-reliant students. Recent developments in pedagogy and
educational science also picture this more active, self-reliant role of students, self-
regulating their own learning process and actively creating new knowledge. For self-
directed learning, metacognition, “one’s awareness of one’s own cognition” (Alessi
and Trollip, 2001, p. 28), is needed which is so helpful for life long learning. As
students are being supported to work self-reliantly, they can learn how to learn, which
is critical for their professional futures where they will be required to keep themselves
up-to-date in their own professions.
Brown et al. (1991) describe scaffolding in reciprocal teaching which enhances
interactive learning. Interactive learning provides students with situations that push
the boundaries of their abilities and actively engage them in tasks. It also gives
students an opportunity to be students as they come to master a task and, once they
have achieved mastery, to be teachers of those who are still learning. Brown et al.
(1991) add that research indicates that problems which are too difficult at first for
students to handle on their own, later become problem types they can solve
independently when they have first received support and worked on them in a small
group setting. That is, the teacher scaffolds students and students scaffold themselves.
Therefore, scaffolding enables students to learn a body of coherent, usable, and
meaningful knowledge within their zone of proximal development and “to develop a
66
repertoire of strategies that will enable them to learn new content on their own” (p.
150).
King (1991a) affirms that matacognitive scaffolding, in the form of metacognitive
questions help students to clarify the problem and access their existing knowledge and
strategies when relevant. For example, to identify or redefine the problem, questions
such as, “What are you trying to do here?” can be asked which is expected to help
students determine the nature of the problem more precisely. Questions such as “What
information is given to you?” would presumably help students to access prior
knowledge, whereas the question “Is there another way to do this?” would foster
greater access to known strategies. A question to monitor problem solving may be
“Are you getting close to your goal?” Evaluation questions such as, “ Does the
solution make a sense?” enable students to reflect on their problem solving process,
for instance, to articulate the steps they have taken and decisions they have made,
facilitating their understanding of the reasons behind actions. In sum, metacognitive
scaffolding guide students’ attention to specific aspects of their learning process,
helping students to plan, monitor, and evaluate their own learning processes, and
therefore, helping students to learn with understanding (Lin et al., 1999).
King (1991a) found that many students lack the ability to engage in effective thinking
and problem solving on their own; therefore, scaffolding in the form of metacognitive
questions should be made to support students to plan, monitor, and evaluate their
learning, and therefore, learn with understanding. Moreover, this scaffolding is likely
to enable students to make judgments about what can be known and what cannot and
to justify the problem solution. Questions such as “What is your justification for that
67
solution?” would help students to construct cogent arguments for their point of view
(Jonassen, 1997).
Kramarskis et al. (2001) found that students with metacognitive scaffolding ask
themselves questions about the nature of the problem (what is the problem all about?),
about the appropriate strategies to solve the problem (what are the appropriate
strategies to solve the problem, and why?), and about the construction of relationships
between the previous and the new knowledge (What are the similarities / differences
between the problem at hand and the problems solved in the past?). So students who
are metacognitively scaffolded will more than likely be students who plan, monitor,
and evaluate their learning processes and outcomes. In other words, they will more
than likely be able to refer to the what, how, when, where and why of the learning
processes and solutions.
Wong (1985) affirms that teaching students to ask questions help them become
sensitive to important points in a text and thus monitor the state of their reading
comprehension. Palincsar and Brown (1984) indicate that in asking and answering
questions concerning the key points of a selection, students are likely to find that
problems of inadequate or incomplete comprehension can be identified and resolved.
Van Zee and Minstrell’s (1997) study described “a reflective toss” through a question-
answer cycle between the teacher and the students, which revealed the influence of a
teacher’s questions on a student’s reflective thinking process. It is evident that
metacognitive questions serve to facilitate metacognition in planning by activating
prior knowledge and attending to important information, in monitoring by actively
engaging students in their learning process, and in evaluation through reflective
thinking.
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Therefore, in the present study, the teacher in the cooperative learning with
metacognitive scaffolding group scaffolded students by asking metacognitive
questions and students were coached to ask themselves and their group members
metacognitive questions based on materials presented in the classroom. When students
in cooperative learning settings ask and answer the metacognitive questions, they are
more likely to understand the materials better, develop new perspectives, reason and
explain solutions, and recognize and fill in gaps in their understanding.
2.5 Cooperative Learning and Learning Mathematics with Understanding
A common response to the idea that students construct their knowledge is that students
should be encouraged to work with others. As Dewey (1961) and Vygotsky (1978)
suggest, people do not learn in isolation from others, they naturally learn and work
cooperatively throughout their lives (Petraglia, 1998).
Cooperative learning is defined differently by different researchers and theorists.
Vygotsky (1978), for example, views cooperative learning as part of a process leading
to the social construction of knowledge. Other scholars (Kohn, 1992; Sapon-Shevin
and Schniedewand, 1992) consider cooperative learning to be a form of critical
pedagogy that moves classrooms and societies closer toward the ideal of social justice.
Caplow and Kardash (1995) characterize cooperative learning as a process in which
“knowledge is not transferred from expert to student, but created and located in the
learning environment” (p.209). Others such as Burron et al. (1993) and Ossont (1993)
see cooperative learning as a strategy to help students improving intellectual and
social skills.
69
Although there seems to be some differences between the definitions of cooperative
learning, there is common agreement that cooperative learning is an instructional
method in which small groups of students work together to accomplish a shared goal
through changing or reconstructing their knowledge. The aim of this cooperation is
for students to maximize their own and each other’s learning, with members striving
for joint benefit.
2.5.1 Theoretical Perspective on Cooperative Learning
Piaget (1970) focuses on the individual as a starting point. Knowledge or information
is provided through cooperation for the individual to use when becoming aware of
differing perspectives and in resolving the differences between them. Cognitive
development from Piagetian view is the product of an individual, perhaps sparked by
having to account for differences in perspectives with others (Rogoff, 1990). The
process of knowing for Piaget comes about through the sequence of equilibrium,
disequilibrium, and reequilibrium. In these processes, existing schemes are altered to
accommodate new information or new information is being assimilated into existing
schemes, which are then strengthened. Piaget stresses that these processes can occur
either by way of cognitive conflicts, in which intraindividual discord during thinking /
problem solving leads to reequilibrium, or by way of sociocognitive conflicts, in
which intreindividual differences during thinking / problem solving are catalysts for
cognitive growth (Manion, 1995). Piaget believes that individuals work with
independence and equality on each other’s ideas, so when they interact they learn,
receive feedback, or are told of something that contradicts their believes or current
understanding. This is what Piaget calls “cognitive conflicts” (Mugny and Doise,
1978). This conflict initiates a process of cognitive or intellectual reconstruction in an
individual. Therefore, students’ interaction prompts the student to assimilate or
70
accommodate his or her knowledge. As Manion (1995) indicates, students revise their
ways of thinking to provide a better fit with reality when faced with discrepancies
between their own ways of viewing the world and new information.
Vygotsky is another psychologist who has done extensive work in social context. In
contrast to Piaget, Vygotsky (1978) focuses on the social basis of mind. He believes
that an individual makes use of the joint decision-making process itself to expand
understanding and skill. Cognitive development involves an individual’s appropriation
or internalization of the social process as it is carried out externally in joint problem
solving.
Vygotsky (1978) affirms that individual intellectual development cannot be
understood without reference to the social setting in which the student is embedded.
Students’ social interaction with more competent students is essential to cognitive
development. So students’ cognitive or learning is developed through interaction with
more skilled partners working in the zone of proximal development. This interaction
enables students to discuss and exchange their ideas and thoughts which in turn
emulate rational thinking processes such as the verification of ideas, the planning of
strategies in advance, the symbolic representation of intelligent acts, spontaneous
generation, and criticism. Student will then takes on and internalize these procedures
thus enhancing the development of his or her intellectual abilities such as his or her
problem solving capacities.
Although there seems to be some differences between Piaget and Vygotsky, Ismail
(1999) believes that they actually complement each other, and if they are combined,
they provide a profound conceptual base for cooperative learning.
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Cooperative learning is the instructional use of small groups so that students work
together to maximize their own and each other’s learning. Students perceive that they
can reach their learning goals if and only if the other students in the learning group
also reach their goals (Johnson, 1981). They are not only responsible for learning the
material that is presented, but also for ensuring everyone in the group knows the
material as well (Slavin, 1987). So students need not only to interact with materials
(i.e., textbooks, curriculum programs) or with the teacher, but also they need to
interact with each other to achieve their learning goals.
Johnson and Johnson (1999) identify three basic types of learning that goes on in any
classroom: competitive learning where students compete to see who is the best,
individualistic learning where students work individualistically toward a goal without
paying attention to other students, and cooperative learning where students work
cooperatively with a vested interest in each other’s learning as well as their own. Of
the three patterns, competition is presently the most dominant where the students view
the school as a competitive enterprise where one tries to do better than others.
Cooperation among students who celebrate each other’s successes, encourage each
other to do homework, and learn to work together regardless of ethnic backgrounds or
whether they are male or female, bright or struggling, disabled or not, is still rare.
Even though these three patterns are not equally effective in helping students learn, it
is important that students learn to interact effectively in each of these ways. Students
will face situations in which all three patterns are operating and they will need to be
able to be effective in each. They also should be able to select the appropriate pattern
suited to the situation. However, in the ideal classroom all three patterns are used. This
does not mean that they should be used equally, but the cooperative pattern should
72
dominate the classroom, being used 60 to 70 percent of the time. The individualistic
pattern may be used 20 percent of the time, and a competitive pattern may be used 10
to 20 percent of the time (Johnson and Johnson, 1999).
According to Johnson et al. (1986), about 600 experimental and over 100 correlational
studies have been conducted since 1898 which have compared these three learning
types or patterns. The majority of the studies show that cooperative learning has
advantages over the competitive and individualistic learning. Moreover, many
learning and instructional approaches that apply cooperative learning have resulted in
the students’ cognitive, intellectual, social, and affective growth (Johnson et al., 1991;
Slavin, 1996).
Johnson and Johnson (1999) clarify that there is a difference between simply having
students work in a group and structuring groups of students to work cooperatively. A
group of students sitting at the same table doing their own work, but free to talk with
each other as they work, is not structured to be a cooperative group, perhaps it could
be called individualistic learning with talking. For this to be a cooperative learning
situation, there needs to be an accepted common goal on which the group is rewarded
for its efforts. If a group of students has been assigned to do a report, but only one
student does all the work and the others go along for a free ride, it is not a cooperative
group. A cooperative group has a sense of individual accountability that means that all
students need to know the material well for the whole group to be successful. In other
words, a group of students can be a cooperative learning group if the elements of
cooperative learning are fulfilled.
2.5.2 Elements of Cooperative Learning
73
Johnson and Johnson (1987) have identified five basic elements of cooperative
learning. These include:
Promotive, Face to Face Oral Communication: Students are placed in heterogeneous
groups from 2 to 6 members. Team members are strategically seated in order to
encourage “eye-to-eye, knee-to-knee” interaction. Through team building activities,
promotive behavior is facilitated.
Positive interdependence: “All for one and one for all” “ Sink or swim together”. As
students work toward a common goal, team cooperation and fellow success becomes
imperative.
Individual accountability: What students can do together today, they can do alone
tomorrow (Vygotsky, 1978). Although students work together in a cooperative group,
each student is held accountable for individual learning. Individual student
performance is assessed and the outcome is reported and celebrated by the individual
as well as team members.
Interpersonal, Cooperative Social Skills: Students work together to reach a common
goal. In order for members to reach a common goal, students must utilize adequate
cooperative social skills to function successfully.
Evaluating and processing: Students are given time and encouraged to participate in
reflection about what was learned, how it was learned, and the skills used to process
and meet the goal.
2.5.3 Teacher’s Role in Cooperative Learning
74
Through the discussion above, the role of the teacher in cooperative learning can be
shown as a facilitator rather than prompter, a supervisor rather than instructor. The
teacher specifies the instructional objectives, monitors the learning groups, asks
questions, and intervenes when necessary. Also the teacher contributes in deciding the
group size and assigning group members and roles (e.g., recorder, summarizer,
encourager, checker, etc). Finally, the teacher contributes in refinement and evaluation
processes of learning outcomes. That is, teacher’s role is to guide and support students
to build or reconstruct their knowledge, to be a guide on the side rather than a sage on
the stage.
For learning mathematics, and according to the constructivist theories, information is
retained and understood through elaboration and construction of connections between
prior knowledge and new knowledge (Wittrock, 1986). Because providing
explanations is one of the best means for elaborating information and making
connections (Slavin, 1996) and students in cooperative settings often give
explanations to each other, the likelihood of constructing rich networks of knowledge
under these conditions increases. Also, when students work with peers who are in
various stages of mastering a task, mutual reasoning and conflict resolution are likely
to occur, which, in turn, facilitates learning (Mevarech and Light, 1992). Observing
other students solving a problem help students internalize either the cognitive
functions they are attempting to master or those that are within their zone of proximal
development (Vygotsky, 1978). As students interact cooperatively, they explain
strategies and mathematical ideas in their own words, thus helping one another to
process complex cognitive activity (Schoenfeld, 1985). Researchers have emphasized
the importance of mathematical communication to build students’ capacity for
mathematical thinking and reasoning (Stein et al.,1996).
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According to the National Council of Teachers of Mathematics (NCTM; 1991),
learning environments should be created in the way that promote active learning and
teaching; classroom discourse; and individual, small group, and whole-group learning.
Cooperative learning is one example of an instructional arrangement that can be used
to foster active student learning, which is an important dimension of mathematics
learning and highly endorsed by math educators and researchers. Students can be
given tasks to discuss, problem solve, and accomplish. Also Teachers can use
cooperative learning activities to help students make connections between the concrete
and abstract level of instruction through peer interactions and carefully designed
activities.
Finally, cooperative learning can be used to promote classroom discourse and oral
language development. Wiig and Semel (1984) describe mathematics as “conceptually
dense.” That is, students must understand the language and symbols of mathematics
because contextual clues, like those found in reading, are lacking in mathematics. For
example, math vocabulary (e.g., greater-than, denominator, equivalent) and
mathematical symbols (e.g., =, <, or >) must be understood to work problems as there
are no contextual clues to aid understanding. In a cooperative learning activity,
vocabulary and symbolic understanding can be facilitated with peer interactions and
modeling.
In almost all studies, the metacognitive strategies were employed in cooperative
learning settings where small groups of 4 – 6 students studied together (e.g.,
Schoenfeld, 1987; Mevarech and Kramarski, 1997; Hoek et al., 1999). The
cooperative learning with metacognitive scaffolding method is based on cognitive
theories of learning that emphasize the important role of elaboration in constructing
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new knowledge (Wittrock, 1986), and on a large body of research (e.g., Davidson,
1990; Qin et al., 1995; Stacey and Kay, 1992; and Webb, 1991, 1989a) showing that
cooperative learning has the potential to improve mathematics performance because it
provides a natural setting for students to supply explanations and elaborate their
reasoning. Since this potential has not always materialized, researchers suggested the
embedding of metacognitive strategies in cooperative setting in order to provide an
appropriate situation for students to elaborate their mathematical reasoning
(Schoenfeld, 1992; Mevarech and Kramarski, 1997; Mevarech, 1999; Kramarski,
2000; and Kramarski et al., 2001).
2.6 Cooperative Learning with Metacognitive Scaffolding and Learning Mathematics with Understanding
Most of people believe that mathematics in all is about computation. So most of them
are familiar with only the computational aspects of mathematics and are likely to
argue for its place in the school curriculum and for traditional methods of instructing
students in computation. For them, the broad goal of learning process is to master the
computational procedures regardless to what actually mathematics is about and
regardless to the learning process itself. That is, they have misconceptions about what
mathematics is about and they do not take how students learn, their experiences, their
metacognitive strategies, and their attitudes toward mathematics into account (Brown,
1987; national Council of Teachers of Mathematics, 1989; Thompson, 1992).
In most traditional mathematics classrooms, students are frequently expected to learn
facts, concepts, and skills divorced from any real context. They are drilled in
arithmetic without applying the skills to problems that mean anything to them. They
usually learn abstract formulas in mathematics out of realistic contexts. So their
77
learning are likely ineffective, and although they can acquire mathematical operations
they are usually unable to apply them in different situations as Clark (1995) has
concluded that when students interpret an activity as unrealistic and non-meaningful,
encoding, representation, and learning are likely to become over simplified and
narrowly school-focused.
Ertmer and Newby (1996) assert that “If schools are going to help all students become
expert students, the metacognitive strategies of students must be acknowledged,
cultivated, and exploited. A major function of all schooling must be to help create
students who know how to learn” (p.22). Therefore, effective mathematics instruction
should assist students to activate the metacognitive strategies in order to be able to
learn mathematics with understanding and reason mathematically.
Work in the area of mathematics problem solving suggests that the deployment of
cooperative learning with metacognitive scaffolding underlie successful performance.
Shoenfeld (1987) found that expert mathematicians engaged in decision-making and
management behaviors at critical junctures during the problem-solving process. In
contrast, novice problem solvers did not appear to use these metacognitive strategies
and often found themselves lost in the pursuits of “wild geese.” More recently, Artzt
and Armour-Thomas (in press), in their investigation of the analysis of problem
solving in small groups, found that a continuous interplay of cognitive and
metacognitive behaviors was necessary for successful problem solving. Other reviews
of studies of mathematical problem solving (e.g., Garofalo and Lester, 1985; Silver,
1987; King and Rosenshine, 1993) suggest that a fundamental source of weakness
underlying students’ performance may lie in students’ inabilities to actively monitor
78
and subsequently regulate and evaluate the cognitive processes used during problem
solution.
There is also some evidence about the role of cooperative learning with metacognitive
scaffolding in successful mathematics performance. For example, Peterson et al.
(1982, 1984) found that students’ metacognitive scaffolding for classroom learning
was significantly related to their performance. Using a stimulated-recall procedure, the
students were asked to recall their thoughts during mathematics instruction. They
reported that they were able to judge their own understanding, to diagnose and
monitor their understanding and specific cognitive processes such as reworking
problems, applying information at a specific level, and checking their answers. Other
researchers (e.g., Schonfeld, 1987; Xun, 2001; Kramarski et al., 2001) reported
positive effects of cooperative learning with metacognitive scaffolding on students’
mathematics performance and mathematical reasoning.
Kilpatrick et al. (2001) clarify that learning mathematics with understanding is not
about only computation or mathematical procedures. They believe that learning
mathematics with understanding has five interwoven and interdependent strands,
namely Conceptual Understanding, Procedural Fluency, Strategic Competence,
Adaptive Reasoning, and Productive Disposition. These five strands provide a
framework for discussing the knowledge, skills, abilities, and beliefs that constitute
mathematical proficiency. This framework has some similarities with the one used by
Donn and Taylor (1992a) that structures different facets of mathematical or
quantitative literacy (content knowledge, reasoning, appreciation of the societal
impact of mathematics, and disposition), with the California Public Schools
framework (1999) that includes three components (conceptsl, procedures, and
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reasoning) of mathematical competence, and also with the one used by the National
Assessment of Educational Progress, (NAEP, 2000), which features three
mathematical abilities (conceptual understanding, procedural knowledge, and problem
solving) and includes additional specifications for reasoning, connections, and
communication. The next section discusses the mathematical proficiency strands and
focus on the role of metacognitive scaffolding in teaching conceptual understanding,
procedural fluency (mathematics performance), and mathematical reasoning. Since the
mathematical proficiency strands are interrelated, the role of metacognitive
scaffolding in teaching each strand will be briefly discussed.
2.6.1 Conceptual Understanding
Conceptual understanding refers to an integrated and functional grasp of mathematical
ideas. Students with conceptual understanding know more than isolated facts and
methods. They understand why a mathematical idea is important and the kind of
contexts in which it is useful. They have organized their knowledge into a coherent
whole, which enables them to learn new ideas by connecting these ideas to what they
have already learned (Donovan et al., 1999). Students with conceptual understanding
are able to retrieve their knowledge and methods. That is, because they learned by
connecting facts and methods under their teacher’s guidance, it is easier for them to
remember and reconstruct the forgotten knowledge (Hiebert and Carpenter, 1992).
When students understand a method, they are unlikely to remember it incorrectly.
They monitor what they remember and try to figure out whether it makes sense. They
may attempt to explain the method to themselves and correct it if necessary. When
students are aware of their metacognitive thoughts, they describe their own thinking.
They can realistically assess what they are capable of learning and therefore they have
80
a good sense of what they know. Also they know what they are currently doing. They
have self-regulation, they are keeping track of what they are doing and knowing well
how to use their previous knowledge to guide the problem solving actions. So when
students are metacognitively trained, they are likely to learn with conceptual
understanding.
A significant indicator of conceptual understanding is being able to represent
mathematical situations in different ways and knowing how different representations
can be useful for different purposes. To find one’s way around the mathematical
terrain, it is important to see how the various representations connect with each other,
how they are similar, and how they are different. The degree of student’s conceptual
understanding is related to the richness and extent of the connections they have made
(Kilpatrick et al, 2001). Students who are working cooperatively and scaffolded
metacognitively ask themselves about the construction of relationships between the
previous and the new knowledge (What are the similarities / differences between the
problem at hand and the problems solved in the past?) (Kramarskis et al., 2001),
therefore they can identify the similarities and differences between the various
strategies used. So doing might help students to represent the problem in various
representations, and by connecting these representations with each other, they are
likely to gain the conceptual understanding and then to construct the correct solution.
Schonfeld (1987) and King and Rosenshine (1993) found that when students were
metacognitively trained they could make connections between mathematical concepts
in different areas.
Conceptual understanding helps students avoid many critical errors in solving
problems, particularly errors of magnitude. For example, “if they are multiplying 9.83
81
and 7.65 and get 7519.95 for the answer, they can immediately decide that it cannot be
right. They know that 10 x 8 = 80, so multiplying two numbers less than 10 and 8
must give a product less than 80. They might then suspect that the decimal point is
incorrectly placed and check that possibility” (Kilpatrick et al., 2001, p.6). Students
who work cooperatively and are scaffolded metacognitively are likely to understand
the concept of addition, connect the current problem with the previous one, represent
the problem in different ways, expect the product, and check their learning strategy
and the product, and therefore, they are unlikely to do critical errors.
2.6.2 Procedural Fluency
Procedural fluency refers to knowledge of procedures or algorithms, knowledge of
when and how to use them appropriately, and skill in performing them flexibly,
accurately, and efficiently (Kilpatrick et al, 2001). Flexibility requires the knowledge
of more than one approach to solving a particular kind of problem, such as two-digit
multiplication. Students need to be flexible in order to choose an appropriate strategy
for the problem at hand, and also to use one method to solve a problem and another
method to double-check the results. Accuracy depends on several aspects of the
problem-solving process, among them careful recording, knowledge of number facts
and other important number relationships, and double-checking results. Efficiency
implies that the student does not get bogged down in too many steps or lose track of
the logic of the strategy. An efficient strategy is one that the student can carry out
easily, keeping track of sub problems and making use of intermediate results to solve
the problem (Russell, 2000).
Students within cooperative learning with metacognitive scaffolding setting are likely
to represent the problem in different ways where they can select the appropriate
82
approach or procedure to solve the problem. That is, they ask themselves about the
appropriate approach to solve the problem (What are the appropriate approach /
strategy / procedures to solve the problem? (Kramarskis et al., 2001). They plan their
learning by understanding the whole problem before getting start, keep track of what
has been done, comparing the differences and similarities of the current problem and
the problems have been solved, and evaluate the outcome of any action. Therefore
these students are unlikely to make mistakes during procedures application. They
practice problems in a way that requires different types of procedures (Mathematics
Framework for California Public Schools, 1999). For example, they mix subtraction
and addition operations. This type of practice provides students with an opportunity to
understand better how different procedures work by making them think about which is
the most appropriate procedure for solving each problem. In other words, they
represent different procedures and evaluate the outcomes then select the appropriate
one and justify their selection.
Research indicates that long-term retention of mathematics procedures requires
frequent refreshers at different points in the students’ mathematical learning (Bahrick
et al., 1993). Students who work cooperatively with metacognitive scaffolding always
ask themselves questions like: what the whole problem is about, what are the
similarities and differences between the current problem and the problems already
were solved, and what is the appropriate procedure to solve the current problem.
Therefore, they are likely to refresh their mathematical learning and procedures.
Moreover, metacognitively trained students can modify or adapt procedures to make
them easier to use (Carpenter et al., 1998). For example, students who work
individually with limited metacognitive strategies in fractions addition, would
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ordinarily need paper and pencil to add 4
2and
2
1, while students within cooperative
learning with metacognitive scaffolding setting recognize that 4
2equals to
2
1 and
therefore they add 2
1 to
2
1 mentally to get 1 as the result.
2.6.3 Strategic Competence
Strategic competence refers to the ability to formulate mathematical problems,
represent them, and solve them. So for students to solve mathematical problems they
need to formulate the problem first then they can use mathematics to solve it. In other
words they need experience and practice in both problem formulating and problem
solving. Therwefore, they should know a variety of solution strategies as well as
determining which strategies might be useful for solving a specific problem. Students
with experience in solving mathematical problems and with limited or without
formulating experience usually encounter difficulties in figuring out exactly what the
problem is (Kilpatrick et al, 2001).
Students within the cooperative learning with metacognitive scaffolding setting are
encouraged to understand the whole problem first. They are encouraged to ask
themselves what the whole problem is about, represent the problem in different ways
and connect these representations, determine the similarities and differences between
84
the problem on hand and others they have solved, select the appropriate approach to
solve the problem, and evaluate the outcomes (Kramarskis et al., 2001). That is, they
are encouraged to plan (before the solution), monitor (during the solution), and to
evaluate (after the solution). So doing assisted them to formulate, represent, and solve
the problem. For example, if they encounter the following purchase problem:
“At Ahmad’s shop the price of a piece of cake is 4
3 Dinar. Ali’s shop price is
3
1 Dinar
less than Ahmad’s price. How much 3 pieces of cake cost at Ali’s shop?”
They may identify what the problem is about by studying the relationships among the
variables in the problem and determine what is known and what to be found. By doing
so, they are likely to conclude that subtraction and multiplication initially should be
used (problem formulation).
With a formulated problem in hand, students within the cooperative learning with
metacognitive scaffolding setting are more likely to represent it mathematically in
some fashion, whether numerically, symbolically, or graphically. They build a mental
image of the problem’s essential components. They avoid selecting numbers and
preparing to perform arithmetic operations on them directly. Rather they are likely to
construct a mental model of the variables and relations described in the problem. That
is, they generate a mathematical representation of the problem that captures the core
mathematical elements and ignore the irrelevant features (Kilpatrick et al, 2001). For
the purchase problem, students within the cooperative learning with metacognitive
scaffolding setting may draw a number line and locate each cost per piece of cake on
it to solve the problem. They may represent the problem by transforming the two
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fractions into equivalent fractions with a like denominator (12
9,12
4) to solve the
problem (problem presentation). Having many different representations students try
to connect the relationships among them by determining the common mathematical
structures. That is, they focus on structural relationships that provide the clues for how
the problem might be solved (Hagarty et al., 1995). They compare the current problem
with the previous one. For example, they recognize that this problem relates to
subtraction two fractions with unlike denominators and this is different from what
they solved previously with like denominator fractions. Therefore they are likely to
conclude that they cannot subtract the numerators directly and try to find equivalent
fractions with like denominators.
Students with strategic competence need to choose flexibly among the proposed
approaches to suit the demands presented by the problem and the situation in which it
was posed. Flexibility of approach can be seen when a method is adjusted or created
to fit the requirement of the problem (Kilpatrick et al., 2001). Students within the
cooperative learning with metacognitive scaffolding setting are flexible in their
approaches (Butler, 1995). They are likely to represent the problem in different forms
and therefore they have different solution approaches, and by comparing these
approaches they can select the appropriate one that will be evaluated to check its
appropriateness. For the purchase problem, they may select the approach of
transforming the two fractions into equivalent fractions with a like denominator as the
appropriate approach to solve the problem. They may offer 12
9and
12
4 as equivalent
fractions with a common denominator and formulate:
86
12
9 -
12
4 =
12
5 as the difference in price for a piece of cake. For 3 pieces of cake, the
formulation would be: 12
5 +
12
5 +
12
5 =
12
15
= 4
5
= 1 4
1
Because students within the cooperative learning with metacognitive scaffolding
setting are encouraged to ask evaluation questions, they are likely to check if the
solution makes sense by testing if 12
9equals to
4
3 and
12
4equals to
3
1. They may
draw a rectangle and divide it into 4 equal pieces and shade 3 pieces, and then they
divide the rectangle into 12 equal pieces where they find that the three shaded pieces
make nine pieces and so on for the other fractions (problem solving and evaluation).
Sternberg (1986b) refers to metacognitive strategies as Metacomponents.
Metacomponents are executive processes that control other cognitive components as
well as receive feedback from these components. According to Stemberg (1986b),
Metacomponents are responsible for “figuring out how to do a particular task or set of
tasks, and then making sure that the task or set of tasks are done correctly” (p. 24).
These executive processes involve planning, monitoring and evaluating problem-
solving activities. Research indicates that metacognitively scaffolded students are
more strategic and perform better than untrained students (Garner and Alexander,
1989). One explanation is that metacognitive scaffolding allows individuals to plan,
sequence, monitor, and evaluate their learning in a way that directly improves
performance (Schraw and Dennison, 1994).
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2.6.4 Adaptive Reasoning
Kilpatrick et al. (2001) refers to adaptive reasoning as the students’ ability to think
logically about the relationships among the mathematical concepts and situations. This
reasoning stems from careful consideration of alternatives, and includes knowledge of
how to justify the conclusions. Adaptive reasoning is the glue that holds everything
together, the lodestar that guides learning. Students use it to navigate through the
many facts, procedures, concepts, and solution methods and to see that they all fit
together in some way, that they make sense. Adaptive reasoning is much broader than
formal proof and other forms of deductive reasoning, it includes not only informal
explanation and justification, but also intuitive and inductive reasoning based in
pattern, analogy, and metaphor.
Since learning according to Piaget occurs by assimilation and/or accommodation
through resolving the cognitive conflicts (Mugny and Doise, 1978), in learning
mathematics, reasoning is used to settle disputes and disagreements and then
knowledge is changed or reconstructed. Mathematical answers are right because they
follow from some agreed upon assumptions through series of logical steps (Kilpatrick
et al., 2001).
Mathematics Framework for California Public Schools (1999) identifies the
mathematical reasoning steps as follows: a) making decisions about how to approach
problems through analyzing the problem by identifying relationships, distinguishing
relevant from irrelevant information, sequencing and prioritizing information, and
observing patterns, and determine when and how to break the problem into simpler
parts. b) using strategies, skills, and concepts in finding solutions through using
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estimation to verify the reasonableness of calculated results, applying strategies and
results from simpler problems to more complex problems, using a variety of methods,
such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to
explain mathematical reasoning, expressing the solution clearly and logically by using
the appropriate mathematical notation, terms and clear language; support solutions
with evidence in both verbal and symbolic work, indicating the relative advantages of
exact and approximate solutions to problems and give answers to specified degree of
accuracy, and making precise calculations and check the validity of the results from
the context of the problem. c) moving beyond a particular problem by generalizing to
other situations through evaluating the reasonableness of the solution in the context of
the original situation, realizing the methods of deriving the solution and demonstrate a
conceptual understanding of the derivation by solving similar problems, and
developing generalizations of the results obtained and applying them in other
situations. Pollack (1997) indicates that mathematical reasoning plays a significant
role in student’s ability to take an open-ended question and transform it into
unambiguous something to solve, that is, to formulate and summarize the real-life
problem.
In summary, mathematical reasoning refers to the students’ ability to identify the
similarities and differences among facts, concepts, procedures, and situations and then
think logically about the relationships among them. After the relationships were
identified, appropriate strategies for solving the problem are selected and reasonable
reasons about strategies selection and calculated results are provided. Finally, justified
strategies and results are applied in other situations (i.e., strategies generalization and
solving real-life problems).
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It is apparent from the steps of mathematical reasoning identified by the mathematics
framework for California public schools that mathematical reasoning comprises both
strategic competence and adaptive reasoning of Kilpatrick’s et al. (2001) model of
mathematical proficiency. Thus, the mathematical reasoning term in this study was
used to indicate strategic competence and adaptive reasoning simultaneously.
2.6.5 Productive Disposition
Resnick (1987) refers to the productive disposition as the tendency to see mathematics
as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own
efficacy. For students to learn mathematics with understanding (proficiently), they
have to believe that mathematics is understandable, not arbitrary, that, with diligent
effort, it can be learned and used, and that they are capable of figuring it out. It is
counterproductive for students to believe that there is some mysterious factor that
determines their success in mathematics (Kilpatrick et al, 2001). Therefore, learning
mathematics with understanding goes beyond being able to understand, compute, and
solve problems. It takes account of a disposition toward mathematics that is personal.
Students’ disposition toward mathematics may play a crucial role in their
understanding and success. For instance, Dweck (1986) indicates that students who
view their mathematical ability as fixed and test questions as measuring their ability
rather than providing opportunities to learn are likely to avoid challenging tasks and
be easily discouraged by failure, whereas students who view ability as expandable in
response to experience and training are more likely to seek out challenging situations
and learn from them.
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Kilpatrick et al. (2001) attribute the development of productive disposition to the
development of the other mathematical proficiency strands. For example, when
students build their own strategies to solve the mathematical task, their attitudes and
beliefs about themselves as mathematics students become more positive. In addition,
the more mathematical concepts they understand, the more sensible mathematics
becomes. In contrast, when students are seldom given challenging mathematical tasks
to do, they come to expect that memorizing rather than sense is the appropriate
approach to learn mathematics, and they begin to lose confidence in themselves as
students. Similarly, when students see themselves as capable to operate the
mathematical procedures and reason mathematically, their disposition is more likely to
be positive.
Since students within the cooperative learning with metacognitive scaffolding setting
are more likely to have conceptual understanding, procedural fluency, strategic
competence, and adaptive reasoning, they seem to have a positive attitudes and
beliefs. Also, when students apply the metacognitive strategies within the cooperative
learning environment, they discuss, share, and contrast their ideas and their teacher
ideas. This conflict environment is likely to enhance students to see themselves as
capable to learn with understanding (Cobb et al., 1995), which in turn, seems to help
them to have a positive attitudes and beliefs.
2.7 Cooperative Learning with Metacognitive Scaffolding and Mathematical Reasoning
Students within the cooperative learning with metacognitive scaffolding setting are
more than likely to reason mathematically in their learning situations. They are guided
about the knowledge of when, where, and why to use the strategies for problem
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solving (Pressley and McCormick, 1987). As metacognitive scaffolding comprises
planning, monitoring, and evaluation, metacognitively trained students are likely to
plan, monitor, and evaluate their learning strategies and solutions. Planning is
essential to formulate, identify, and define the problem and then building the
relationships among its concepts and procedures. To select the appropriate strategies,
the students need to regulate or monitor their problem performance by self-generating
feedback. Evaluation enhances students to reflect on their solutions or alternatives so
as to direct their future steps (Jacobs and Paris, 1987).
Since mathematical reasoning requires thinking about the relationships between
mathematical facts, concepts, and situations, students within the cooperative learning
with metacognitive scaffolding setting are enhanced to identify the similarities and
differences between the current problem and the ones they have already solved. Doing
so is likely to enable them to compare concepts, procedures, and strategies which in
turn, enable them to establish the relationships among them. For example, when the
students encounter the following task 4
3+
5
2 to solve, the teacher asks: what are the
differences / similarities between the current task and those you solved last class?
What in your prior knowledge will help you in this particular task? What you should
think about first? By answering these questions, students may possibly reach to the
conclusion that the current task is regarding adding two fractions with unlike
denominators. Studies conducted by Chi et al. (1994); Mevarech and Kramarski, (in
press); Slavin (1996); Cossey’s (1997); and Webb (1989) showed that metacognitive
scaffolding is one of the best means for making connections between mathematical
facts, concepts, and procedures.
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While mathematical reasoning requires selecting appropriate strategies for solving the
task and justifying both the strategies’ selection and the task’s solution, students within
the cooperative learning with metacognitive scaffolding setting seem to be able to
select and justify the appropriate strategies for solving the task. For the adding
fractions with unlike denominator example, the teacher asks metacognitive questions
that help students to plan, monitor, and evaluate their learning such as: Is the sum
would be less than or greater than 1? Thinking to answer this question is likely leading
students to answer “the sum will be greater than 1”, and then the teacher asks why?
By relating to their previous knowledge, students may justify that 5
2is greater than
4
1,
and 4
3+
4
1equals to 1, so the sum will be greater than 1. The teacher then asks what
are the appropriate strategies to find the sum of these two fractions? What should you
do first? Students with metacognitive scaffolding are likely to respond that they can
not add directly unless they make the two denominators equal. How you should do so?
The teacher asks. The students are likely to compare this task with the previous ones
and relate it to the equivalent fractions and offer 20
15and
20
8as equivalent fractions
with a common denominator. The teacher asks students to justify why they have
chosen 20 as the common denominator. By relating to the Least Common Multiple
(LCM), students seem to respond that 20 is the smallest number that both five and
four go into. How did you come up with that? the teacher asks. Again through
understanding and relating to the LCM, students are likely to answer “by multiplying
5 and 4. The teacher then asks students to write down the processes of solving the
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task. Within the cooperative learning with metacognitive scaffolding setting, students
are likely to write the solution as follows:
4
3+
5
2 =
54
53
××
+ 45
42
××
= 54
4253
××+×
= 20
815 +
= 20
23
The teacher asks, are you in the right track? How do you check if the solution makes
sense? How do you know that you have added the same fractions as in the original
task? How well did you do? The students may use different representations (graphs,
models, symbols, numbers line, etc) to prove that 4
3 equivalent to
20
15and
5
2
equivalent to20
8. Hoek et al. (1999) and Mevarech (1999) studies showed that
metacognitive scaffolding is effective for developing the selection of the appropriate
strategies for solving the problem.
Finally, mathematical reasoning requires applying strategies in other situations.
Students within the cooperative learning with metacognitive scaffolding setting are
more likely to generalize their learning strategies to other situations. Based on the
adding two fractions with unlike denominators example, the teacher asks: how might
you apply this line of thinking to other situations? Could you derive a rule that would
work for adding or subtracting any fractions with unlike denominators? The teacher
then provides different tasks, word-problems, and real-life problems regarding adding
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and subtracting fractions with unlike denominators and asks the students to solve them
through applying the same line of thinking they applied before.
2.8 Cooperative Learning with Metacognitive Scaffolding and Real-Life Problem Solving
Real-life problems are problems that people encounter in everyday life. They are
generally problems in which one or several aspects of the situation are not specified.
The general nature of these problems is that the goals are vaguely defined or unclear
(Voss and Post, 1988), their descriptions are not clear, and the information needed to
solve them is not entirely contained in the problem statements; consequently, it is not
obvious what actions to take in order to solve them (Chi and Glaser, 1985). Real-life
problems entail multiple solutions, solution paths, or no solutions at all (Kitchner,
1983). Since real-life problem solving may generate a large number of possible goals,
Sinnott (1989) insists that the solvers must have a mechanism or strategies for
selecting the best goal or solution.
Hong (1998) summarized the processes of real-life problems which their goals are
vaguely defined into three processes: (a) representation problem, (b) solution
processes, and (c) monitoring and evaluation. A representation problem is established
by constructing a problem space, including defining problems, searching and selecting
information, and developing justification for the selection. The solution process
involves generating and selecting solutions. Finally, the monitoring and evaluating
process requires assessing the solution by developing justifications for it. Since real-
life problems usually have no clear goals and require the consideration of alternative
solutions as well as competing goals, solving this kind of problems requires students
to regulate the selection and execution of a solution process. That is, when goals or
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action alternatives are unclearly defined, students have to organize and direct their
cognitive endeavors in different ways.
Because students need and use mathematics in their everyday lives, it is very critical
to learn solving real-life problems. For instance, students learn the concept of 2
1
effectively by solving a problem like if two ice cream cones cost 10 cents, how much
does one cone cost? That is, learning to solve real-life problems produces active
learning and easily retrieved knowledge as Brown et al. (1989) assert that when
learning includes real-life problems, students acquire content and skills through the
resolution of problems.
It is not that traditional teaching practices do not use examples, real-life problems and
other devices. It is that the overall approach is turned around the wrong way. Students
are taught the isolated basics and then are expected to apply them to artificial
problems (Tiene and Ingram, 2001). Lesh (1985) indicates that getting a collection of
isolated concepts in a student’s head (e.g., measurement, addition, multiplication,
decimals, proportional reasoning, fractions, negative numbers) does not guarantee that
these ideas will be organized and related to one another in some useful way; it does
not guarantee that situations will be recognized in which the ideas are useful or that
they will be retrievable when they are needed. Tiene and Ingram (2001) assert that the
best approach of teaching is to ground all learning as much as possible in tasks,
activities, and problems that are meaningful to the students. If it is important for
students to learn facts, they will learn them most effectively while engaged in
meaningful tasks.
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For learning mathematics, when situations are mathematized in the classroom such as
balancing a class budget, the students will be engaged in multiple mathematics
processes and they will learn how mathematical concepts are related to one another in
a useful and meaningful way. Such experiences also require students to talk and think
about mathematics with one another and with the teacher (Lesh, 1985).
Within the cooperative learning with metacognitive scaffolding setting, students are
enhanced to solve real-life problems because they are encouraged too plan, to monitor
problem-solving processes, to reflect on the goals and solution processes and to
construct cogent arguments for their proposed solutions. In a study of history experts,
Wineburg (1998) found that planning, monitoring, and evaluation helped students to
solve a real-life problem in the absence of domain knowledge. Lin and Lehman
(1999); Davis and Linn (2000); King (1991a, 1991b); Palincsar and Brown (1984,
1989; and Kramarski et al., 2002) found that planning, monitoring, and evaluation
enhanced metacognitive knowledge and reflective thinking which enhanced the
processes of solving real-life problems.
Students within the cooperative learning with metacognitive scaffolding setting are
likely to reason and defend their selections and solutions. As students select a good
solution from among the many viable solutions, they provide the most viable, the most
defensible and the most cogent argument to support their preferred solution, and
defend it against alternative solutions (Jonassen, 1997; Voss and Post, 1988). In
addition, students within the cooperative learning with metacognitive scaffolding
setting may also evaluate their selection by examining and comparing other
alternatives. Sinnott (1989) noted that during the process of solving a real-life
problem, successful students planed, monitored, and evaluated their own processes
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and movements from state to state, as well as select information, solutions, and
emotional reactions.
2.9 Cooperative Learning with Metacognitive Scaffolding and Motivation
Motivation is the reason why an individual behaves in a given manner in a given
situation. It exists as part of one’s goal structures, one’s beliefs about what is
important, and it determines whether or not one will engage in a given pursuit (Ames,
1992). Two distinct types of academic motivation interrelate in most academic
settings, intrinsic and extrinsic motivation. Academic intrinsic motivation is the drive
or desire of the student to engage in learning “for its own sake.” Students who are
intrinsically motivated engage in academic tasks because they enjoy them. They feel
that learning is important with respect to their self-images, and they seek out learning
activities for the sheer joy of learning (Middleton, 1992, 1993a). Their motivations
tend to focus on learning processes such as understanding and mastery of
mathematical concepts (Duda and Nicholls, 1992). When students engage in tasks in
which they are motivated intrinsically, they tend to exhibit a number of pedagogically
desirable behaviors including increased time on task, persistence in the face of failure,
more elaborative processing and monitoring of comprehension, selection of more
difficult tasks, greater creativity and risk taking, selection of deeper and more efficient
performance and learning strategies, and choice of an activity in the absence of an
extrinsic reward (Lepper, 1988).
On the other hand students who are extrinsically motivated engage in academic tasks
to obtain rewards (e.g., good grades, approval) or to avoid punishment (e.g., bad
grades, disapproval). These students’ motivations tend to focus on learning products as
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obtaining favorable judgments of their performance from teachers, parents, and peers
or avoiding negative judgments of their performance (Ames, 1992).
Teachers often complain that their students are not motivated and hence cannot or do
not learn well (Driscoll, 1998). The motivation that teachers wish their students to
have is intrinsic motivation. Intrinsic motivation is important because it contributes to
learning processes and achievement, but it is also important as an outcome (Ames,
1990). She adds “Effective teachers are those who develop goals, beliefs, and attitudes
in students that will sustain a long-term involvement and that will contribute to quality
involvement in learning” (p.413).
There are many ways to promote motivation. Garrison (1997) expounds that “to direct
and sustain motivation students must become active students” (p. 8). Task motivation
is integrally connected to task control and self-management. Driscoll (1998) declares
“Motivation appears to be enhanced when students’ expectancies are satisfied, when
they attribute their successes to their own efforts and effective learning strategies, and
when the social climate fosters interaction and cooperation among students” (p. 312).
According to constructivist point of view, learning is an active process in which the
students construct their own knowledge. So they encounter difficult problems that
they cannot solve by using only their current knowledge. In this case, students are
challenged by the task and they will be motivated more. Bruner (1973) maintain that
student may be motivated more quickly when given a problem they cannot solve, than
they are when given some little things to learn on the promise that if they learn these
well, three weeks later they will be able to solve an exciting problem (Shulman,
1973). Hein (1991) indicates that motivation is a key component in learning. Not only
is it the case that motivation helps learning, it is essential for learning. Motivation is
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broadly conceived to include an understanding of ways in which the knowledge can
be used. Unless the student knows the reasons why, he or she may not be very
involved in using the knowledge that may be instilled in him or her. Even by the most
severe and direct teaching.
Motivation is particularly important in learning mathematics with understanding.
When students are mathematically motivated, they will perceive mathematics as
useful and worthwhile and then they will see themselves as effective students of
mathematics (Resnick, 1987). Students who view their mathematical ability as
expandable in response to experience and training are more likely to seek out
challenging situations and learn from them. In contrast, students who view their
mathematical ability is fixed are likely to avoid challenging problems and be easily
discouraged by failure (Dweck, 1986).
Mathematics instruction plays an important role in encouraging students’ motivation
to learn mathematics. Students within cooperative learning with metacognitive
scaffolding setting, are likely to be motivated more to learn mathematics. These
students are encourager to negotiate among themselves the norms of conduct in the
class, and when those norms allow students to be comfortable in doing mathematics
and sharing their ideas with others, they see themselves as capable of understanding
and then doing mathematics effectively (Cobb and Yackel, 1995).
CHAPTER THREE
METHODOLOGY
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3.1 Introduction
The purpose of this study was to investigate the effects of cooperative learning with
metacognitive scaffolding (CLMS) and cooperative learning (CL) methods on
mathematics performance, mathematical reasoning, and metacognitive knowledge
among high and low ability fifth-grade students in Jordan. This chapter discusses the
methodology that was used in this study. It describes in detail the population and
sample, the experimental conditions, the research design, the instructional materials
and instruments, the procedures, and data analysis procedure and the method that was
used in the analysis of data.
It is important to note that everyday classroom instructions and all reading materials
(except for the English subject) used in the participating schools are in the Arabic
Language. Therefore, all the materials and instruments used in this study were
translated into Arabic.
3.2 Population and Sample
The population of this study comprised male fifth grade students enrolled in the first
public educational directorate in Irbid Governorate in the first semester for the
academic year 2002 / 2003. The first public educational directorate in Irbid
Governorate includes 44 male primary schools. Public schools in Jordan are not
coeducational.
In order to implement this study in a naturalistic school setting, existing intact classes
were used (O’deh and Malkawi, 1992). The sample consisted of 240 male students
who studied in six fifth-grade classrooms and were randomly (simple random sample)
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selected from three different male primary schools i.e., two classes from each school.
The three schools were also randomly selected from the primary schools where
mathematics was taught in heterogeneous classrooms with no grouping or ability
tracking. The size of the classes was approximately similar, and the mean age of the
students was 10.6 years. Students in the selected schools – as well as all Irbid
Government schools - were from approximately equivalent socioeconomic status as
defined by the Jordan Ministry of Education. Each of the three male teachers who
participated in this study taught two classrooms. All the teachers were men who had
similar levels of education (B.Ed. major in mathematics), had more than 7 years of
experience in teaching mathematics, and had taught in heterogeneous classrooms. The
teachers who taught the experimental groups were exposed to one week training on
the instructional methods. The participating students were informed that the purpose
of this study was to examine different learning strategies that may help in the
improvement of students’ mathematics performance, mathematical reasoning, and
metacognitive knowledge.
3.3 Experimental Conditions
The three schools were assigned randomly to one of the following conditions:
1. CLMS: students taught mathematics via the Cooperative Learning with
Metacognitive Scaffolding method (n = 80).
2. CL: Students taught mathematics via Cooperative Learning with no
Metacognitive Scaffolding (n = 79).
3. T (control group): students taught mathematics via the present classroom
practice (traditional method), that is, without Metacognitive Scaffolding or
Cooperative Learning methods (n = 81). (See table 3.1).
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Table 3.1: Mechanisms for the three groups
Group 1 (CLMS)
N = 80
Group 2 (CL)
N = 79
Group 3 (T)
N = 81
Cooperative Learning with metacognitive scaffolding
Cooperative Learning with no Metacognitive Scaffolding
The whole class with neither
cooperative learning nor Metacognitive
Scaffolding
Cooperative learningStudents worked, discussed, interacted in groups, and asked themselves and their members metacognitive questions
Cooperative LearningStudents worked, discussed, and interacted in groups with no MQ
Without
Cooperative
Learning
Metacognitive Scaffoldinga) The teacher asked metacognitive questions{MQ} and coached students to ask MQb) Students used metacognitive questions cards
Without teacher’s metacognitive scaffolding
Without metacognitive questions cards
Without teacher’s metacognitive
scaffolding
Without metacognitive questions cards
The three groups were different from one another in terms of the instructional method
and materials used. The CLMS group was asked metacognitive questions by the
teacher and students in this group used metacognitive questions cards in cooperative
learning setting. The CL group studied cooperatively with neither teacher’s
metacognitive questions nor using metacognitive questions cards, whereas the T group
studied in the usual manner with neither cooperative learning, teacher’s metacognitive
questions, nor metacognitive questions cards.
3.4 Research Design
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This quasi-experimental study was designed to investigate the effects of cooperative
learning with metacognitive scaffolding and cooperative learning methods on
mathematics performance, mathematical reasoning, and metacognitive knowledge.
The study employed Factorial Design 3x2. It was designed to investigate the effects of
the independent variable on the dependent variables at each of the two levels of the
moderator variable. The research design is illustrated in table 3.2.
Table 3.2: Research Design
Moderator Variable(Ability)
Independent Variable(Instructional Method)
CLMS CL THigh-ability (Y1) 1 2 3Low- ability (Y2) 4 5 6
O1 X1 Y1 O2 (1) X1: CLMS O3 X2 Y1 O4 (2) X2: CL O5 X0 Y1 O6 (3) X0: T
O7 X1 Y2 O8 (4) Y1: High-ability O9 X2 Y2 O10 (5) Y2: Low-ability O11 X0 Y2 O12 (6)
O1 = O3 = O5 = O7 = O9 = O11 = Pre-test.
O2 = O4 = O6 = O8 = O10 = O12 = Post-test.
The independent variable of this study was the instructional method with three
categories:
1. Cooperative learning with metacognitive scaffolding instructional method
(CLMS).
2. Cooperative learning instructional method (CL).
3. Traditional instructional method (T).
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The moderator variable was the ability level with two categories:
1. High-ability.
2. Low-ability.
The dependent variables were:
1. Mathematics performance (MP).
2. Mathematical reasoning (MR), and
3. Metacognitive knowledge (MK).
The design of the present study compares three instructional methods (a) cooperative
learning with metacognitive scaffolding instructional method, (b) cooperative learning
with no metacognitive scaffolding instructional method, and (c) traditional
instructional method with neither cooperative learning nor metacognitive scaffolding.
Slavin (1996) recommended the use of such research design because it enables
researchers to hold constant all factors other than the ones being studied.
3.5 Instructional Materials and Instruments
3.5.1 Instructional materials
In order to study the students’ mathematics performance, mathematical reasoning, and
metacognitive knowledge in a naturalistic setting of the classroom, the instructional
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materials used in this study were based on the fourth unit from the mathematics
textbook (Adding and Subtracting Fractions) designed by the Ministry of Education
for all fifth-grade students in Jordan, teacher’s lesson plans, and metacognitive
questions card.
3.5.1.1 Adding and Subtracting Fractions Unit
“Adding and Subtracting Fractions” was the unit chosen for this study. This particular
unit was chosen for two reasons: a) Jordan eighth-students’ performance regarding
fractions according to the TIMSS-R (1999) findings was very low and students
committed many undetermined errors. This low performance in fractions, particularly
addition and subtraction of fractions demands Jordanian educators to pay attention to
this particular topic. The TIMSS-R study was conducted on eighth-grade students but
according to the Jordanian curriculum the topic of fractions is taught in the fifth-grade.
Jordanian students start learning the basics of adding and subtracting fractions in the
fifth-grade; and b) This topic was scheduled by the schools to be covered by the
teacher in early December 2002 / 2003, which is also the same duration of time
planned for this study.
The “Adding and subtracting Fractions” unit consists of 12 lessons, which are,
Introduction to Fractions, Mixed Numbers, Equivalent Fractions, Simplifying
Fractions, Comparing and Ordering Fractions and Mixed Numbers, Adding Fractions,
Adding Mixed Numbers, Adding Fractions Problem Solving, Subtracting two
Fractions which one Fraction’s Denominator is Multiple of the second Fraction’s
Denominator, Subtracting two Fractions which one Fraction’s Denominator is not
Multiple of the second Fraction’s Denominator, Subtracting Mixed Numbers, and
Subtracting Fractions Problem Solving respectively. Within each school, the teacher
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conducted the class according to his assigned teaching method for 15 sessions. Each
lesson started with the new topic explanation and followed by mathematical exercises
and problem solving. One session (45 minutes) was conducted to teach the first ten
lessons and two sessions to teach the last two lessons as planned from the Ministry of
Education in teacher’s guide. All sessions were presented from written lesson plans to
ensure that all participating students in the three groups received the same quantity of
knowledge.
3.5.1.2 The Metacognitive Questions Cards
A set of metacognitive questions cards (see Appendix A) was developed by the
researcher based on the metacognition components (planning, monitoring, and
evaluation) designed by Jacobs and Paris (1987); and North Central Regional
Educational Laboratory, (NCREL, 1995). The students taught via the CLMS method
used the questions cards to scaffold their learning processes when they engaged in
cooperative learning activities with their respective peers. These questions were
categorized into the following groups of metacognitive questions:
Planning: “What is the problem all about?” “What are the strategies we can use to
solve the problem and why?” (There were 8 questions in this category).
Monitoring: “Are we on the right track?” (There were 9 questions in this category).
Evaluation: “What explanations can we make and what evidence do we have to justify
that our solution is the most viable?” (There were 5 questions in this category).
These questions were closely paralleled to the Kilpatrick’s model of mathematical
proficiency (2001). They consisted of questions supporting mathematical proficiency
and mathematical reasoning skills, such as “what”, “how”, and “why” as well as
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questions that were found in King’s generic question stems (King, 1991b). These
questions were designed to facilitate students’ understanding of domain knowledge, to
develop metacognitive thinking, and to develop mathematical reasoning, such as
questions regarding making decisions about approaching the problem, selecting the
appropriate strategies to solve the problem, and regarding generalizing the solution
processes to other situations. The students taught via the CLMS method were instructed
and reminded frequently to think about the questions, and use the questions to facilitate
their problem solutions.
3.5.2 Instruments
In this study, two major instruments were used to assess students’ mathematics
performance, mathematical reasoning, and metacognitive knowledge. A mathematics
achievement test was used to assess students’ mathematics performance and
mathematical reasoning and a metacognitive questionnaire was used to assess
students’ metacognitive knowledge.
3.5.2.1 The Mathematics Achievement Test (The pre-Test and post-Test)
The mathematics achievement test administered by the three groups’ participants in
this study was adapted from the mathematical competency test developed by Jbeili
(1999). The test-retest reliability coefficient of that test was .93. The test-retest
approach of measuring reliability is considered the best approach that provides the
test’s consistency over time (Tuckman, 1999). The mathematical competency test
consisted of 5 conceptual understanding items, 11 procedural fluency items, and one
problem solving regarding adding and subtracting fractions. Since there were no
mathematical reasoning items included in that test, the researcher constructed these
respective items (see appendix B) based on Kilpatrick’s model (2001), NCTM
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standards (2000), and Mathematics Framework for California Public Schools (1999).
With these new added items, the reliability coefficient was measured by applying the
test on a pilot sample.
The pre-test and post-test questions were similar in content but their order and
numbering were randomized. The mathematics achievement test questions consisted
of 24 mathematical items and sub items and a real-life problem. The mathematics
achievement test questions covered the following topics: equivalent fractions,
simplifying fractions, comparing and ordering fractions and mixed numbers, adding
and subtracting fractions, and adding and subtracting mixed numbers. Three
constructs, which tightly correspond to the mathematics performance and
mathematical reasoning, were identified as important for measuring mathematics
performance and mathematical reasoning for this study: (a) conceptual understanding,
(b) procedural fluency, and (c) mathematical reasoning. The mathematics achievement
test questions were composed of four kinds of items. One kind (10 items) was based
on multiple-choice items regarding conceptual understanding and procedural fluency.
The second kind (6 items) was based on open-ended tasks regarding conceptual
understanding and procedural fluency. The third kind (8 items) was specifically
designed to assess students’ mathematical reasoning. The mathematical reasoning
items were designed to require students to go beyond presenting facts to thinking
about those facts. The 8 items asked students to estimate the results, explain the
solution clearly, and justify and support the solutions with evidence. The fourth kind
was a real-life problem that involved conceptual understanding, procedural fluency,
and mathematical reasoning. The problem asked students to decide the better buy from
two different prices and quality of mixed fruit juice. The student had to calculate the
mixed fruit juice volume in each shop, compare the prices and quality, decide the
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better buy, and provide reasons for his decision. To gain a deeper understanding of
students’ mathematical reasoning, the items regarding mathematical reasoning and the
mathematical reasoning criteria in the real-life problem were separately analyzed
following the method used by Kramarski et al. (2001).
3.5.2.2 The Scoring of Mathematics Achievement Test
The total score of the test was 44. The distribution of the mathematics performance
and mathematical reasoning items and their scores across conceptual understanding
(CU), procedural fluency (PF), and mathematical reasoning (MR) is illustrated in
appendix C. The 24 mathematics items and sub items and the real-life problem scoring
were as follows:
Multiple-choice items: For each item, students received a score of either 1 (correct
answer) or 0 (incorrect answer), and a total score ranging from 0 to 10.
Open-ended task items: For each item, students received a score of either 1 (correct
answer) or 0 (incorrect answer), and a total score ranging from 0 to 6.
Mathematical reasoning items: The scoring procedure is adopted from
Kramarski et al. (2001) and has a repeated .90 interjudge reliability. For each item,
students received a score between 0 and 2, and a total score ranging from 0 to 16. For
example, “In the following item, 5
9 …
3
2, explain which sign >, <, or = that will
make the statement true.” A score of 0 indicates incorrect selection and explanations
or explanations that are irrelevant to the task (e.g., 5
9 <
3
2 because when the
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denominator is smaller the value is greater. Nothing is mentioned about numerator or
transferring it into a common denominator). A score of 1 indicates an explanation that
has some satisfactory elements but may has omitted a significant part of the task (e.g.,
5
9 >
3
2 because when transformed into a common denominator the numerator 27 is
bigger than the numerator 10. Nothing is mentioned about the denominators. A score
of 2 indicates a clear, unambiguous explanation of student’s mathematical reasoning
(e.g., 5
9 >
3
2, when transform into equivalent fractions with a like denominator
15
27
and15
10, the fraction with the larger numerator is the larger fraction if the denominators
are the same, since the denominators (15, 15) are same and the numerator 27 is bigger
than the numerator 10, 15
27 >
15
10.
The real-life problem: A scoring rubrics (see appendix D) was adapted from the
Kramarski et al. (2001) procedure with a repeated .86 interjudge reliability. Four
criteria, which tightly correspond to the conceptual understanding, procedural fluency,
and mathematical reasoning, were identified as important for measuring students’
ability to solve the real-life problem. Students’ answers were scored on these criteria,
each criterion ranges from 0 (no solution) to 3 (highest level solution), and a total
score ranging from 0 to 12. The criteria were:
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1. Referencing all data (referring to all data in each of the two offers: mixed fruit juice
volume, components, and prices. Identifying relationships, distinguishing relevant
from irrelevant information- Mathematical Reasoning).
2. Organizing information (summarizing the data in a table, diagram, or any other
representation for comparisons and identifying similarities/differences between the
representations- Conceptual Understanding).
3. Processing information (figuring the calculations correctly, writing the solution
processes, and provide an appropriate solution to the required task- Procedural
Fluency).
4. Making justifications for the suggested solution (giving reasons, providing
evidence, and justifying the suggestion- Mathematical Reasoning).
Example 1. If a student’s final response is “I suggest buying the mixed fruit juice from
Ali’s shop because both volumes are same and Ali’s price is cheaper than Ahmad’s
price”.
The student has given a very brief answer and the scoring will be as follows:
Reference to all data: The student refers to the prices and volumes but he does not
refer to the quality-Score 2.
Organizing information: The student does not use any representation to present his
calculation or conclusion-Score 0.
Processing information: The calculations are correct but the student does not write
explicitly the solution process-Score 2.
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Making justifications: The student explains his suggestion, but he does not justify his
reasoning (how Ali’s price is cheaper than Ahmad’s price)-Score 2.
The total score: 6
Example 2: If a students’ final response is “I suggest buying the mixed fruit juice from
Ahmad’s shop. Although the two volumes are same (each fraction in the first offer is
equivalent to the each fraction in the second offer), the components of Ahmad’s juice
are 100% fruit juice, but Ali’s juice contains 8
6 litters of water. So although Ali’s price
is cheaper by 4
1 (
2
1 -
4
1 =
4
1) dinar than Ahmad’s price; the better buy is Ahmad’s
juice because its quality is better than Ali’s juice.
The student has summarized all relevant and irrelevant data in table and given a
thorough explanation. The scoring will be as follows:
Reference to all data: The student refers to the prices, volumes, and each juice
components (quality)-Score 3.
Organizing information: The student summarizes all data in a table and provides
written explanations.-Score 3.
Processing information: The calculations are correct and the student writes explicitly
the solution process-Score 3.
Making justifications: The student explains his suggestion and justifies his reasoning
(how Ali’s price is cheaper than Ahmad’s price)-Score 3.
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The total score: 12
3.5.2.3 The Metacognitive Knowledge Questionnaire
The metacognitive knowledge questionnaire (see appendix E) was adapted from the
study of Montague and Bos (1990), assessed students’ metacognitive knowledge
regarding their problem-solving strategies, and from Xun (2001) self-report
questionnaire. Cronbach’s alpha reliability coefficient of those questionnaires were
.83, .86 respectively. The adapted metacognitive knowledge questionnaire consists of
15 items grouped into three categories. The first category (5 items) was focused on
strategies used before the solution process (planning) (e.g., “I tried to understand the
problem before I attempted to solve it”); the second (5 items) category was focused on
strategies used during the solution process (monitoring) (e.g., “I summarized what
were given and what were wanted in a table”); and the third (5 items) was focused on
strategies used at the end of the solution process (evaluation) (e.g., I tried to find
evidence to justify and support my solutions”).
Metacognitive questionnaire scoring: Each item was constructed on a 3-point, Likert-
type scale ranging from 1 (never) to 3 (always) and a total mean score ranging from 1
to 3.
3.5.3 Materials and Instruments Validity
Although the materials and instruments used in this study were derived from theories
principles and standards, after the translation to Arabic language, two experienced
mathematics teachers, two education mathematics supervisors, and two mathematics
education university lecturers in Jordan reviewed the lesson plans, the metacognitive
questions card, the scoring procedure of assessing mathematical reasoning items, and
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the scoring rubrics of assessing the real-life problem. The researcher met the
evaluators and discussed the questions regarding these materials and instruments
during the evaluation process. The evaluators’ suggestions, feedback, and comments
were taken into account until there were no discrepancies among the evaluators. Then
the evaluators reviewed the mathematics achievement test questions and the
metacognitive knowledge questionnaire items. Each looked at each question in the
test and at each item in the questionnaire and assessed which of the mathematical
proficiency strand (CU, PF, or MR) the question represented and which of the
metacognitive knowledge component (planning, monitoring, or evaluation) the item
represented, and rated their confidence in their response, using scale from 1 (weak) to
10 (strong). Only questions and items, which had received 7 or more scores from all
evaluators, were selected as test questions and questionnaire items following Chung
(2002).
Evaluators agreed that the questions 1, 2.1 ,2.2, 3.1, 3.2, 6, 7, 8, 9.1, 9.2, 10.1, and
10.2, were represented to CU, CU, MR, CU, MR, PF, PF, PF, CU, MR, CU, and MR
strands respectively, with all reporting confidence scores 10, questions 4.1(a, b, c, d),
4.2 (a, b, c, d), and 5 (a, b, c, d) were represented to CU, MR, and PF strands
respectively, with all 9 scales, and question 11 (real-life problem) (criterion 1 to 4)
were represented to MR, CU, PF, and MR strands respectively, with all reporting
confidence scores 8. Since evaluators were in disagreement about question 4.1 (e), 4.2
(e), 5 (e), and 11 (criterion 5), the questions were removed from the test. For the
questionnaire items validity, evaluators agreed that the first five items of each scale
(planning, monitoring, and evaluation) were represented, with all reporting confidence
scores 10. However, there were disagreement about the last three items of each scale
(9 items), therefore they were removed from the questionnaire. After an overall
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agreement was reached on the validity of the materials and instruments, they were
considered valid materials and instruments for conducting this study.
3.5.4 Instruments Reliability
Two major instruments were used in this study i.e., the mathematics achievement test
and the metacognitive knowledge questionnaire. Although the instruments used in this
study were adapted from reliable instruments, with the additional items and translation
to Arabic, a pilot test was carried out and the scores from the pilot study test and the
metacognitive questionnaire were collected and a set of reliability tests were
conducted to determine the Cronbach’s Alpha reliability coefficients. Cronbach’s
alpha reliability coefficient of the mathematics achievement test was .88, and it was
.84 for the metacognitive knowledge questionnaire. Cronbach’s Alpha reliability
coefficients for the metacognitive questionnaire categories were .64, .66, .60 for
planning, monitoring, and evaluation respectively. The Cronbach’s Alpha reliability
coefficients showed that the study instruments were satisfactory reliable.
3.6 Procedures
Prior to the implementation of the study, the researcher obtained permissions from a
number of different parties for conducting the pre-experimental study and the
experimental study. Permissions were sought from the educational development and
research department of the Jordan Ministry of Education (see appendix F), the First
Public Educational Directorate in Irbid Governorate (see appendix G) where the
participating schools are located, and from the participating schools’ principals.
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3.6.1 The Pilot Study
Prior to the formal study sessions, a pilot study was conducted to validate research
procedures. The researcher selected randomly 80 participants from a randomly
selected primary school, who were not going to participate in the formal study. There
were two purposes to the pilot study: first, to test the materials and instruments, in
terms of using the metacognitive questions cards, sessions duration, training of
teachers, and the test and the questionnaire durations; and secondly, to test the
instruments reliability i.e., the mathematics achievement test and the metacognitive
knowledge questionnaire. Two male teachers who had a similar level of education
(B.Ed. major in mathematics) and had more than 7 years of experience in teaching
mathematics were selected to teach the participants in the pilot study. The teachers
were exposed to one week training about teaching adding and subtracting fractions
with MSCL and CL methods. The participants were randomly assigned to the two
experimental conditions i.e., MSCL and CL groups. Within each condition, teachers
conducted classes according to their assigned teaching methods for 14 sessions. At
session 15, all participants were administered the mathematics achievement test and
immediately responded to the metacognitive questionnaire items.
3.6.2 The Formal Study
For the experiment, the researcher randomly selected three schools from the 44 male
primary schools in Irbid Governorate. Permission was sought from each school’s
principal. Three mathematics teachers with a similar level of education (B.Ed. major
in mathematics), had more than 7 years of experience of teaching mathematics in
heterogeneous classrooms were selected (one from each school). Each teacher taught
two classes in each school. Each teacher’s classes were randomly assigned into the
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three instructional methods described earlier. The researcher then discussed with each
teacher about his assigned instructional method and appointed one day with each one
to discuss about training.
3.6.3 Groups’ Equivalence
To test the assumption that the participants across the three groups were equivalent,
the pre-test was conducted two months before the beginning of the study. The pre-test
was focused on students’ conceptual understanding, procedural fluency, and
mathematical reasoning. The pre-test papers were scored by the researcher. To
determine if there were statistically significant differences between the groups’ mean
scores i.e., the high-ability students and the low-ability students, the scores by the
three groups were entered into the Statistical Package for Social Science (SPSS) for
Windows computer software (version 11.5).
3.6.4 Teachers’ Training
Prior to the beginning of this study, the teachers assigned to the experimental groups
participated in one week training sessions that focused on pedagogical issues
regarding teaching mathematics. The teachers were informed that they would be part
of an experiment in which new instructional methods were being tested. They worked
with the new methods and materials and learned how to use them with their students.
The materials included the mathematics textbooks, explicit lesson plans, and examples
of metacognitive questions. Within each school, the teachers continued conducting
classes according to their assigned teaching methods until the end of the first semester.
In the present study, the focus was on the “Adding and Subtracting Fractions” unit that
was taught in all classrooms for 14 sessions at the end of the first semester.
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The CLMS teacher was trained explicitly about using cooperative learning with
metacognitive scaffolding in the teaching of Adding and Subtracting Fractions. He
was exposed to some examples about the nature of the metacognitive questions and
how to use and train students to use them in a cooperative learning setting. He was
informed to use metacognitive questions in his explanations and coach his students to
use metacognitive questions when they solve the mathematical problems. The
procedures of selecting groups and assigning group members were explained to the
teacher. The researcher met the teacher for feedback and assessment regarding the
application of the teaching method. The CL teacher was trained about teaching
mathematics within cooperative learning setting, and about selecting groups and
assigning groups’ members. He was not exposed to any training about metacognitive
scaffolding method. The researcher met the teacher for feedback and assessment
regarding the application of the teaching method. Finally, the T teacher was not
exposed to the metacognitive scaffolding or to the cooperative learning training, he
was asked to teach as he used to teach in a usual manner. The researcher checked his
lesson plans and his methods of teaching to ensure that he followed the traditional
method.
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3.6.5 Implementation of the Study
3.6.5.1 The Cooperative Learning with Metacognitive Scaffolding (CLMS) Method
In this treatment, the pre-test was conducted first and then students were informed that
in the following weeks they would be exposed to an instructional method that would
help them become more effective managers of their own learning activities. The
teacher introduced the processes of cooperative learning with metacognitive
scaffolding method to the students. He discussed with them about the importance and
the role of this method in developing their mathematics performance, reasoning, and
metacognitive knowledge. The teacher spent some time on explicitly introducing the
concepts of how students can become metacognitive students within this learning
environment, why they would learn metacognitive strategies, and how they could
apply these strategies in solving real-life problems. After the discussion on
cooperative learning with metacognitive scaffolding method, students were assigned
into groups based on their ability. They were divided into high and low-abilities based
on their pre-test scores in mathematics performance and mathematical reasoning. The
median of the scores was the criterion of assigning students to the group. Because
students’ scores were interval variables, they were converted to nominal variables. The
scores were placed in numerical order and then the median score was located. Scores
above the median (16) were labeled as high-ability and below the median were labeled
as low-ability following Tuckman (1999). Each group was formed by randomly
choosing two high-ability students and two low-ability students. The remaining
groups were selected by repeating the same procedure with the reduced list.
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When the grouping was completed, students’ roles in their group were assigned.
Within each group, the teacher randomly assigned a metacognitive question asker,
summarizer, recorder, and presenter and then he described each role. The
metacognitive questions asker read the questions from the metacognitive questions
card and asked his group members. The summarizer summarized orally the main ideas
and the key points to solve the problem, and the recorder wrote down the solution
steps, the explanations, and the justifications of that solution. Finally, the presenter
presented, explained, and justified the solution to the whole class. These roles were
rotated among students after each session so that each group member played each one
several times.
The teacher applied the CLMS instructional method two months before the formal
experiment with practice units. For the formal experiment, just before the “Adding
and Subtracting Fractions” unit was taught, students were informed that at the end of
this unit, they would be asked to complete a mathematics achievement test and a
questionnaire. The formal experiment lasted 15 sessions (14 sessions for
implementing the method and 1 session for administrating the test and the
questionnaire). In the first session, the teacher introduced and explained the new topic
for about 30 minutes to the whole class by asking him-self metacognitive questions
regarding planning, monitoring, and evaluation. For example, before solving the
problem, instead of saying, First we..., next we..., then we..., the teacher said, “I need
to know what the whole task is about, is it about the whole numbers, fractions,
additions, or subtraction, etc? What is given and what is not given? What in my prior
knowledge will help me with this particular task? Last time I have learned about
adding fractions with the same denominators, but this task includes fractions with
different denominators, so what should I do? Do I know where I can go to get some
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information on this task? What are some strategies that I can use to learn this? I should
find a way to transform one of the denominators to be same as the other then I can
add, so how should I do this?”
The teacher then coached and encouraged students to ask these metacognitive
questions within the cooperative setting. Students were encouraged to talk about the
task, explain to each other, and represent it from different perspectives. During his
explanation process, the teacher also asked metacognitive questions regarding
monitoring. For example, did I understand what I have just decided to do? Am I on the
right track? How can I spot an error if I make one? How should I revise my plan if it is
not working? Am I keeping good notes or records? Again, students were encouraged
and trained to ask these questions. At the end of his explanation, the teacher asked and
trained students to ask metacognitive questions regarding evaluation such as: Did the
solution make a sense, and how can I decide that? Did my particular strategy produce
what I had expected? What could I have done differently? How might I apply this line
of thinking to other problems? Finally, the teacher summarized the learning processes
through metacognitive questions before (planning), during (monitoring), and after
(evaluation) of the learning task and encouraged students to apply them in learning
adding and subtracting fractions.
After the teacher’s explanation, the metacognitive questions cards were distributed to
the groups. The students were asked to do their exercises and solve the assigned
mathematical problems in groups for about 15 minutes. The teacher had explained to
the students about the reasons for doing each of the steps on the matacognitive
questions cards. This is important because according to Palincsar and Brown (1984),
providing reasons for doing a particular action (i.e., responding to the “why do we do
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this” question) during a learning strategy usage training will increase the likelihood
that the strategy will continue to be used by the participating students after the
training.
In this way, the metacognitive questions asker had read the problem and asked aloud,
the colleagues listened to the mathematics metacognitive question and tried to answer.
Whenever there was no consensus, the group members discussed the issue until the
disagreement was resolved. When the disagreement was resolved, the summarizer
orally summarized the solution, the explanation, and the justification and discussed
with his colleagues. With the solution, explanation, and justification were in hand, the
recorder has written them down and the presenter has presented to the whole class.
During these processes, the teacher monitored each learning group and intervened by
asking more metacognitive questions if necessary. At the end of the session, the
teacher collected the metacognitive questions cards and assessed and evaluated
students’ performance, discussed with the whole class to ensure that students carefully
process the effectiveness of their learning group, and had students celebrate the work
of group members.
For the next sessions, the teacher and students followed the same method and
procedures and the group members’ roles were rotated after each session. However,
the metacognitive scaffolding input by the teacher was gradually reduced, for
example, the teacher’s time in the first session was 30 minutes, in the second session it
was about 25 minutes, in the third session it was about 20 minute and so on until the
time became when the teacher taught for about 10 minutes regarding the new topic
and the students continued learning by their own using the metacognitive questions
cards. After one month of implementing the CLMS instructional method, namely in
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the last mathematics session of this experiment (session 15), the students were asked
to complete the mathematics achievement test. After completing the test, they were
immediately asked to complete the metacognitive questionnaire.
3.6.5.2 The Cooperative Learning (CL) Method
In this treatment, students taught via cooperative learning instructional method with
no metacognitive scaffolding. The pre-test was conducted first and then students were
informed that in the following weeks they would be exposed to an instructional
method that would help them to improve their learning activities. The teacher
introduced the cooperative learning method stages and discussed with the students
about the importance of using this method in mathematics classroom. Students were
assigned into heterogeneous small groups following the same procedures of assigning
students to the groups in the CLMS condition. Because there were students left over,
one group of three members was formed (one high-ability student and two low-ability
students). Within each group, the teacher randomly assigned reader, summarizer,
recorder, and presenter and then he has described each role.
The teacher applied the CL instructional method two months before the formal
experiment with practice units. For the formal experiment, just before the “Adding
and Subtracting Fractions” unit was taught, students were informed that at the end of
this unit, they would be asked to complete a mathematics achievement test and a
questionnaire. The formal experiment lasted 15 sessions (14 sessions for
implementing the method and 1 session for administrating the test and the
questionnaire). In the first session, the teacher introduced and explained the new topic
for 25 minutes to the whole class and then proceeded to teach in a usual manner. For
example, he used the board and explained the main ideas of today’s lesson. After the
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teacher’s explanation of the new topic to the whole class, students were asked to do
their exercises and solve the assigned mathematical problems in groups for 20
minutes. The reader read the problem aloud; the colleagues discussed about the
learning task and asked themselves different questions (but they were not trained to
ask metacognitive questions). The summarizer, the recorder, and the presenter played
the same roles of their counterparts in the CLMS group. At the end of the session, the
students ensured that all of them mastered the task. During the session, the teacher
intervened when needed to improve task work and teamwork, but he did not use
metacognitive scaffolding, namely, he asked questions regarding the task such as:
what are the procedures of adding two fractions with different denominators, and he
responded to students’ questions. Finally, the teacher assessed and evaluated students’
performance, ensured that students carefully process the effectiveness of their learning
group, and had students celebrate the work of group members. For the next sessions,
the teacher and students followed the same method and procedures and the group
members’ roles were rotated each session. After one month, namely in the last
mathematics session of this experiment (session 15), the students were asked to
complete the mathematics achievement test. After completing the test, they were
immediately asked to complete the metacognitive questionnaire.
3.6.5.3 The Traditional (T) Method
The control group served as a comparison group with no intervention. Therefore, the
teacher of this group continued teaching as he usually did, and the students were not
exposed to cooperative learning or metacognitive scaffolding. The pre-test was
conducted two months before teaching the “Adding and Subtracting Fractions” unit.
Just before the “Adding and Subtracting Fractions” unit was taught, students were
125
informed that at the end of this unit, they would be asked to complete a mathematics
achievement test and a questionnaire. In this condition, the “Adding and Subtracting
Fractions” teaching lasted 15 sessions (14 sessions for teaching and 1 session for
administrating the test and the questionnaire). In the whole 14 sessions of
implementing T method, the teacher introduced, explained, and manipulated the new
concepts and procedures of today’s lesson using the board and the textbook for 35
minutes to the whole class. After the teacher’s explanation, the students practiced the
mathematical items individually using their textbooks and teacher’s notes and
sometimes employed any method the teacher saw fit for 10 minutes. When the
students faced difficulties during solving the mathematical problems, and finally could
not find the solution, they asked for the teacher’s help. So the teacher intervened when
needed to help some students to solve their mathematical problems. Sometimes the
teacher explained and informed the students about the procedures of solving the
problem. At the end of each session, the teacher reviewed the day’s lesson with the
whole class. In session 15, the students were asked to complete the mathematics
achievement test. The metacognitive questionnaire was passed out to the students
immediately after they completed the mathematics achievement test.
3.6.5.4 Monitoring the Implementation of the Study
During the first two months of implementing this study, three mathematics education
supervisors, whose job was to regularly visit the three teachers in their classes, visited
the three teachers twice a month. Each mathematics education supervisor was
informed to observe his assigned teacher following the checklists prepared by the
researcher to ensure the fidelity to the implementation. The checklist of the CLMS
group contained questions such as: Did the teacher follow his lesson plans correctly?
Did the teacher ask metacognitive questions during his explanations? Did the teacher
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assign the groups correctly? Did the teacher gradually reduce his metacognitive
scaffolding input? Did the teacher distribute the metacognitive questions cards to the
all groups? Did each group member play different roles? The checklist of the CL
group contained questions such as: Did the teacher follow his lesson plans correctly?
Did the teacher assign the groups correctly? How long did the teacher's explanation
last? How long did the students work cooperatively? Did each group member play
different roles? The checklist of the T group contained questions such as: Did the
teacher follow his lesson plans correctly? How long did the teacher's explanations
last? How long did student spend to solve the mathematics problems individually?
During the last month of implementing this study, namely, during the teaching of
"Adding and Subtracting Fractions Unit", the three mathematics education supervisors
visited the three teachers twice a week and followed the same checklists to ensure the
implementation fidelity. Also the researcher met each teacher twice a week to ensure
fidelity to the treatment following the checklists used by the three mathematics
education supervisors. At the end of session 15, the researcher collected the
mathematics test and the metacognitive questionnaire papers from the three
participating groups. The mathematics test items and the metacognitive questionnaire
items were scored by the researcher using the scoring rubrics.
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3.7 Data Analysis Procedure and Method
3.7.1 The pre-Experimental Study Findings Analysis
Groups’ Equivalence
To test the assumption that the participants (high and low ability students) across the
three groups are equivalent, the participants’ average scores on the pre-test in
mathematics performance and mathematical reasoning(mathematics and Arabic) will
be were analyzed to determine if there are were statistically significant differences
between the groups’ mean scores. Since there are were two dependent variables i.e.,
mathematics performance (MP) and mathematical reasoning (MR), and a three groups
and one moderator variable with two levels (high-ability and low-ability), two-way
multivariate analysis of variance (two-way MANOVA) test willstatistical technique
was conducted. In addition, a reliability test was conducted for the mathematics
performance and mathematical reasoning test and for the metacognitive questionnaire
items to determine the Chronbach Alpha reliability values.
be used to compare the three mean scores, namely, two-way ANOVA will be used to compare the high-achievers across the three group and to compare the low-achievers across the groups.
3.7.2 Instruments Reliability
The test-retest reliability coefficient for the mathematical achievement test and the
metacognitive questionnaire will be measured through entering the findings into the SPSS
computer program. The correlation coefficient will be measured to ascertain the
instruments reliability.
3.7.2 The Experimental Study Findings Analysis
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At the end of this study, the two experimental groups and the control group
participants will completed the mathematics achievement performance and
mathematical reasoning test and filled out the metacognitive questionnaire. The test
and the questionnaire items were scored by the researcher and The results will be
analyzed to determine if there are were any statistically significantce differences
between the three groups on the dependent variables. While there arewas an
independent variable with (three levels, a moderator variable with two levels,) and
three dependent variables, and the pre-test as a covariate, two-way multivariate
analysis of variancecovariance (two-way MANCOVA) test will bewas conducted to
compare the three adjusted mean scores on mathematics achievementperformance
(MP), mathematical reasoning (MR), and metacognitive knowledge (MK).
MANCOVA will be was conducted first to compare MP, MR, and MK of the three
groups. Then MANOVA was conducted with splitting file technique to compare high-
ability students against high-ability students’ MP, MR, and MK across the three
groups. The same technique was used to compare low-ability low-achievers’ students
against low-ability students’ MP, MR, and MK across the three groups. Because the
overall two-way MANCOVA results were statistically significant, a series follow up
two-way analysis of covariance (two-way ANCOVAs) were used to identify where the
differences resided. Since the follow up ANCOVAs results were statistically
significant, the post hoc pair wise comparison technique using the /lmatrix command
was used to identify where the differences in adjusted means resided. Finally, by
conducting two-way MANCOVA without splitting files, the interaction effects
between the instructional method and the ability level (high-ability and low-ability)
was measured. All of the statistical analysis tests will bewere computed at 0.05 level
of significance.
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3.7.3 Justifications for using two-way MANCOVA / MANOVA
Two-way multivariate analysis of covariance is used to determine how each dependent
variable is influenced by two independent variables while controlling for a covariate
(Hair et al., 1998). MANCOVA is to reduce the size of the error term in the analyses
thereby increasing power (Stevens, 1986). Analysis of covariance adjusts the mean of
each dependent variable to what they would be if all groups started out equally on the
covariate. Analysis of covariance gives results preferable to those of a direct
comparison of gain scores i.e., post-test minus pre-test for the two groups, because
gains are limited in size by the difference between the test’s ceiling and the magnitude
of the pre-test score (Tuckman, 1999). In this study, pre-MP and pre-MR have been
shown to correlate with the dependent variables, thus they were considered as
appropriate covariates.
Two-way MANOVA is used to examine the effects of two or more independent
variables on a set of dependent variables (Stevens, 1986). A two-way MANOVA
enables us to (1) examine the joint effect of the independent variables on the
dependent variables, and (2) get more powerful tests by reducing error (within-cell)
variance (Stevens, 1986).
A moderate to strong correlation among the dependent variables is an additional
justification for using two-way MANOVA. If subsequent overall MANOVA results
are statistically significant, a one-way analysis (ANOVA) is conducted to further
examine or identify where the differences reside. If there is no correlation, or if the
correlation is weak among the dependent variables, MANOVA is not considered since
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a single outcome measure may be diluted in a joint test involving many variables that
display no effect. In such a situation, individual univariate tests are directly conducted.
3.7.4 Pearson’s Correlation
The scores of the mathematics performance and mathematical reasoning test and the
scores of the metacognitive questionnaire were analyzed by examining the
relationships among the multiple dependent variables by using Pearson's correlation
technique. The purpose was to determine if there were statistical justifications to use
multivariate analysis of variance (MANOVA) (Stevens, 1986). The results of the
Pearson's correlation (see Table 3.3) indicated an overall correlation among the three
dependent variables (mathematics performance MP, mathematical reasoning MR, and
metacognitive knowledge MK), significant at the .01 level.
Table 3.3Pearson’s correlation among the three dependent variables (MP, MR, and MK)
Variable MP MR MKParticipants (n = 240)
MP _
MR .746** _
MK .652** .757** _
Note. ** suggests that correlation is significant at the 0.01 level (2-tailed).
3.7.5 Assumptions for MANOVA / MANCOVA
A preliminary analysis was conducted to determine whether the prerequisite
assumptions of MANOVA / MANCOVA were met before proceeding the multivariate
analysis. Thus, the assumption of normality, equality of variance-covariance matrices,
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and the linear relationship between the covariates and the dependent variables were
examined.
The assumption of normality was supported by the data. The M-estimators had strong
agreement among 4 estimators. All Q-Q plots fall along the straight line showing that
the normality in all variables was reasonable.
Box’s M Test of Equality of Covariance Matrices tests the null hypothesis that the
observed covariance matrices of the dependent variables are equal across groups. The
Levene’s Test tests the null hypothesis that the error variance of the dependent
variable is statistically similar across groups.
The value for Box’s M of comparing the three groups regardless of the ability level =
68.217, F (30, 122811) = 2.200, p < .001 was significant, thus rejecting the
assumption of homogeneity of the variances. However, rejecting this assumption has
minimal impact if the groups are of approximately equal size i.e., if the largest group
size divided by the smallest group size is less than 1.5 (Hair et al., 1998). Therefore,
for this particular study, rejecting of this assumption has minimal impact since the
groups were of approximately equal sizes.
The value for Box’s M of comparing high-ability students across the three groups =
18.185, F (12, 65332) = 1.459, p > .001 was not significant. In addition, the value for
Box’s M of comparing low-ability students across the three groups = 29.420, F (12,
66085) = 2.360, p > .001 was not significant, thus accepting the assumption of
homogeneity of the variance.
The results from the Levene’s Test for homogeneity of variance of comparing the
three groups regardless of the ability level for each of the dependent variables
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indicated that homogeneity of variance has been met for all the three dependent
variables. For MP, F (5, 234) = 3.210, p > .001, for MR, F (5, 234) = 1.681, p > .001,
and for MK, F (5, 234) = 4.224, p >.001.
The results from the Levene’s Test for homogeneity of variance of comparing the
high-ability students across the three groups for each of the dependent variables
indicated that homogeneity of variance has been met for all the three dependent
variables. For MP, F (2, 117) = 2.801, p > .001, for MR, F (2, 117) = .427, p > .001,
and for MK, F (2, 117) = 1.955, p > .001. In addition, the results from the Levene’s
Test for homogeneity of variance of comparing the low-ability students across the
three groups for each of the dependent variables indicated that homogeneity of
variance has been met for all the three dependent variables. For MP, F (2, 117) = .339,
p > .001, for MR, F (2, 217) = .042, p > .001, and for MK, F (2, 117) = 4.378, p >
.001.
To examine the assumption that the covariates must have some relationship with the
dependent variables (Hair et al., 1998), the scores of the mathematics performance and
mathematical reasoning test and the scores of the metacognitive questionnaire were
analyzed by examining the relationships among the covariates and the dependent
variables by using Pearson’s correlation technique. The results of the Pearson’s
correlation (see Table 3.4) indicated an overall correlation among the two covariates
(pre-MP and pre-MR) and the three dependent variables (mathematics performance
MP, mathematical reasoning MR, and metacognitive knowledge MK), significant at
the .01 level.
Table 3.4Pearson’s correlation among the covariates (pre-MP and pre-MR) and the dependent variables (MP, MR, and MK)
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Covariate / Variable MP MR MKParticipants (n = 240)
Pre-MP .522** .513** .316**
Pre-MR .719** .658** .442**
Note. ** suggests that correlation is significant at the 0.01 level (2-tailed).
After determining that the assumptions were met, the multivariate statistical output
was examined. Then, providing the MANCOVA result was statistically significant, the
univariate results were examined for each dependent variable. For the significant
univariate results, the post hoc comparisons were performed to identify where the
differences resided. The pairwise comparisons statistic was used for the post hoc
results. The results of the multivariate tests, the univariate tests, the pairwise
comparisons among the three dependent variables, the interaction effect, as well as the
descriptive statistics for the dependent variables are reported in Chapter Four.
CHAPTER FOUR
RESULTS
4.1 Introduction
This chapter presents the results of the study from the data analyses of the pre-
experimental study as well as the experimental study. The analyses were carried out
through various statistical techniques such as the two-way multivariate analysis of
variance (MANOVA), the univariate analysis of variance (ANOVA), the two-way
multivariate analysis of covariance (two-way MANCOVA), the one-way multivariate
analysis of covariance (one-way MANCOVA), the two-way analysis of covariance
(two-way ANCOVA), and the post hoc pair wise comparison using the /lmatrix
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command analysis. The data were compiled and analyzed using the Statistical
Package for the Social Science (SPSS) for Windows computer software (version 11.5).
The results of the pre-experimental study, in response to the groups’ equivalence are
reported first. Hypotheses regarding the effects of the instructional methods on
students’ mathematics performance (MP), mathematical reasoning (MR), and
metacognitive knowledge (MK) are tested, and the findings of testing these
hypotheses are presented. Next the hypotheses regarding the effects of the
instructional methods on high-ability and low-ability students’ MP, MR, and MK are
tested, and the findings of testing these hypotheses are presented. Each hypotheses
tested is followed by a summary of testing that hypotheses. Finally, the summary of
findings to research questions 1 - 4 is presented.
4.2 The pre-Experimental Study Results
The purpose of the pre-experimental study was to test the assumption that the
participants across the three groups were equivalent in mathematics performance and
mathematical reasoning. To achieve this purpose, a pre-test that measures pre-
mathematics performance and pre-mathematical reasoning was conducted before the
beginning of the study. While there were three groups with moderator variable with
two levels i.e., high-ability and low-ability, and two dependent variables i.e., pre-
mathematics performance (pre-MP) and pre-mathematical reasoning (pre-MR), two-
way multivariate analysis of variance (MANOVA) with splitting file technique was
conducted to determine if there were statistically significant differences between the
groups’ mean scores i.e., high-ability against high-ability students and low-ability
against low-ability students across the three groups.
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4.2.1 Statistical Data Analysis
Table 4.1 summarizes the descriptive statistics for the dependent variables (pre-MP
and pre-MR) by the groups. Both dependent variables had the same points (22 points
for each). The scores of high-ability student on pre-MP across the three groups had
relatively similar means, 11.1750, 11.7368, and 11.0476 for CLMS, CL, and T
respectively. The scores of high-ability student on pre-MR had also relatively similar
means, 7.5000, 7.9474, and 7.9762 for CLMS, CL, and T respectively. For low-ability
students, the scores of the three groups on pre-MP were very close, (8.5500, 9.1220,
and 9.2564 for CLMS, CL, and T respectively). The scores of the three groups on pre-
MR were very close, (3.2750, 2.9756, and 3.4872 for CLMS, CL, and T respectively).
Table 4.1Means and standard deviations on each dependent variable (pre-MP and pre-MR), by the groups
Dependent Variables Pre-MP Pre- MR
Group
Ability High (H) Low (L)
CLMS H (n = 40) Mean 11.1750 7.5000SD 1.7525 .5991
L (n = 40) Mean 8.5500 3.2750SD 1.0857 1.5684
CL H (n = 38) Mean 11.7368 7.9474SD 2.0754 1.1377
L (n = 41) Mean 9.1220 2.9756SD 1.1289 1.4639
T H (n = 42) Mean 11.0476 7.9762SD 1.5134 1.8144
L (n = 39) Mean 9.2564 3.4872
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SD 1.5706 1.4346
Note. Total score on pre-MP = 22, and total score on pre-MR = 22
To examine if there were significant statistical differences between the high-ability
students on pre-MP and pre-MR across the three groups, and if there were significant
statistical differences between the low-ability students on pre-MP and pre-MR across
the three groups, two-way multivariate analysis of variance (MANOVA) was
conducted.
Table 4.2 presents the results of two-way multivariate analysis of variance, showing
overall differences between high-ability students and low-ability students across the
three groups on pre-MP and pre-MR. To evaluate the multivariate (MANOVA)
differences, Pillai’s Trace criterion was considered to have acceptable power and to be
the most robust statistic against violations of assumptions (Coakes and Steed, 2001).
The MANOVA results of comparing high-ability students against high-ability students
and low-ability students against low-ability students across the three groups were
statistically not significant (F = 1.773, p = .135), (F = 2.255, p = .064) respectively.
Further, the results of the univariate ANOVA tests, which are represented in Table 4.2,
indicated that there were no significant statistical differences between the high-ability
students in pre-MP and pre-MR, with an F ratio (2, 117) of 1.653 ( p = .196) and 1.700
( p =.187) respectively. Also the results indicated that there were no significant
statistical differences between the low-ability students in pre-MP and pre-MR, with an
F ratio (2, 117) of 2.803 ( p = .65) and 1.625 ( p = .201) respectively. This means that
there were no statistically significant differences between high-ability students and no
statistically significant differences between low-ability students across the three
groups in pre-MP and pre-MR. Therefore, the assumption that the high-ability
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participants across the three groups and the low-ability participants across the three
groups are equivalent in mathematics performance and mathematical reasoning was
met.
Table 4.2 Summary of multivariate analysis of variance (MANOVA) pre-MP and pre-MR results and follow-up analysis of variance (ANOVA) results.
MANOVA Effect and Dependent Variables
Multivariate F Univariate Fdf = 2, 117
High-ability
Group Effect
Pre-Mathematics Performance(pre-MP)
Pre-Mathematical Reasoning(pre-MR)
Pillai's Trace 1.773 ( p =.135)
1.653 ( p =.196)
1.700 ( p =.187)
Low-ability
Group Effect
Pre-Mathematics Performance(pre-MP)
Pre-Mathematical Reasoning(pre-MR)
Pillai's Trace 2.255 ( p =.064)
2.803 ( p = .065)
1.625 ( p = .201)
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4.3 The Experimental Study Results
The purpose of the experimental study was to examine the effects of the instructional
methods on mathematics performance (MP), mathematical reasoning (MR), and
metacognitive knowledge (MK), specifically on high-ability and low-ability students’
mathematics performance, mathematical reasoning, and metacognitive knowledge
while controlling students’ pre-MP and pre-MR on the pre-test. A two-way
multivariate analysis of covariance (MANCOVA) was conducted to analyze the
effects of the instructional method on the three dependent variables, as well as the
interaction between the instructional method and the ability levels effects on the three
dependent variables.
The statistical differences of the three groups were compared and analyzed according
to each of the three dependent variables. The research hypotheses were tested using
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the results from the two-way multivariate analysis of covariance (MANCOVA) and
univariate analysis of covariance (ANCOVA). The results of the analysis were used to
answer Research Questions 1-4.
4.3.1 Testing of Hypothesis 1
Students taught via cooperative learning with metacognitive scaffolding (CLMS)
instructional method will perform higher than students taught via cooperative
learning (CL) instructional method who, in turn, will perform higher than students
taught via traditional (T) instructional method in (a) mathematics performance (MP),
(b) mathematical reasoning (MR) and (c) metacognitive knowledge.
Table 4.3 presents overall means, standard deviations, adjusted means, and standard
errors of each dependent variable by the instructional method, CLMS, CL, and T.
Table 4.3Means, standard deviations, adjusted means and standard errors for each dependent variable by the instructional method
Dependent Variables The Instructional MethodCLMS CL TN= 80 N= 79 N= 81
Mathematics Performance (MP)
Mean 18.6500 17.5570 16.7654
SD 2.3390 2.7351 2.2928
Adj. mean 18.742a
17.611a
16.639a
Std. Error .156 .157 .155
Mathematical reasoning (MR)
Mean 16.1500 14.1646 12.7284
SD 2.2842 2.7336 2.3875
Adj. mean 16.289a
14.184a
12.576a
Std. Error .146 .147 .145
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Metacognitive Knowledge (MK)
Mean 2.2975 1.9485 1.7243
SD .2541 .3094 .2788
Adj. mean 2.299a
1.954a
1.718a
Std. Error .021 .021 .020
Note. a. Evaluated at covariates appeared in the model: pre-MP = 10.1417, pre-MR = 5.5250.
Total score on MP = 22, total score on MR = 22, and total score on MK = 05
To examine if there were statistically significant differences in mathematics
performance, mathematical reasoning, and metacognitive knowledge adjusted mean
scores between the CLMS, the CL, and the T groups, while controlling the pre-MP
and the pre-MR, multivariate analysis of covariance (MANCOVA) was conducted.
Table 4.4 presents the results of multivariate analysis of covariance (MANCOVA),
showing overall differences for the independent variable of instructional method effect
and the three dependent variables, while controlling pre-MP and pre-MR. The Pillai’s
Trace was used to evaluate the multivariate (MANCOVA) differences. The
MANCOVA results of comparing the three groups were statistically significant (F =
46.575, p = .000). The covariates pre-MP (F = 15.020, p = .000) and pre-MR (F =
16.553, p = .000) had significant effects. This means that there were some statistical
differences on at least one dependent variable.
Further, the results of the univariate ANCOVA tests, which are represented in table
4.4, indicated that there were statistically significant differences in the three dependent
variables (MP, MR, and MK). The F ratio of MP (2, 237) was 45.600 ( p = .000). This
means that the instructional method had a main effect on MP. This effect accounted
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for 28% of the variance of MP (Eta2 = .282). The F ratio of MR (2, 237) was 162.490
( p = .000). This means that the instructional method had a main effect on MR. This
effect accounted for 58% of the variance of MR (Eta2 = .583). The F ratio of MK (2,
237) was 202.729 ( p = .000). This means that the instructional method had a main
effect on MK. This effect accounted for 64% of the variance of MK (Eta2 = .636).
Table 4.4Summary of multivariate analysis of covariance (MANCOVA) results by the instructional method and follow-up analysis of variance (ANOVA) results.
MANCOVA Effect, Dependent Variables, and Covariate
Multivariate FPillai's Trace
Univariate Fdf = 2, 237
Group Effect
Mathematics Performance(MP)
Mathematical Reasoning(MR)
Metacognitive knowledge (MK)
Pre-MP
Pre-MR
46.575 ( p = .000)
15.020 ( p = .000)
16.553 ( p = .000)
45.600 ( p = .000)
162.490 ( p = .000)
202.729 ( p = .000)
142
The MANCOVA results of comparing the three groups on the three dependent
variables indicated that there were statistically significant differences between at least
two groups in the three dependent variables. Therefore, the researcher further
investigated the univariate statistics results (analysis of covariance ANCOVA) by
performing a post hoc pairwise comparison using the /lmatrix command for each
dependent variable in order to identify significantly where the differences in the
adjusted means resided. Table 4.5 is a summary of post hoc pairwise comparisons.
.
Table 4.5Summary of post hoc pairwise comparisons
Dependent Variable
Mathematics Performance
(MP)
Mathematical Reasoning
(MR)
Metacognitive Knowledge
(MK)
Comparison Group
Adj.Mean Difference
Sig Adj.Mean Difference
Sig Adj.Mean Difference
Sig
CLMSvs.CL
1.131 .000 2.105 .000 .345 .000
CLMSvs.T
2.103 .000 3.713 .000 .581 .000
CLvs.T
.972 .000 1.608 .000 .236 .000
Note.
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The adjusted mean differences shown in this table are the subtraction of the second
condition (on the lower line) from the first condition (on the upper line); for example, 1.131 (Adjusted Mean Difference for Mathematics Performance) = CLMS – CL.
Table 4.3 displays the means, standard deviations, adjusted means and standard errors
of different conditions by the dependent variables. Table 4.4 and table 4.5 show that
there are statistical adjusted mean differences among the three conditions in the three
dependent variables. The adjusted mean differences are presented below.
Mathematics performance. The cooperative learning with metacognitive scaffolding
(CLMS) group (Mean = 18.7, SD = 2.3, Adj.mean = 18.7, p = .000) significantly
outperformed the other two groups (CL and T), with an adjusted mean difference of
1.131 and 2.103 respectively. On other hand, the cooperative learning (CL) group
(Mean = 17.6, SD = 2.7, Adj.mean = 17.6, p = .000) significantly outperformed the
control group (T) (Mean = 16.8, SD = 2.3, Adj. mean = 16.6) with an adjusted mean
difference of .972. (Effect sizes on MP were .47 and .34 for comparing the CLMS and
CL, and CL and the T group, respectively).
Mathematical reasoning. The CLMS group (Mean = 16.6, SD = 2.3, Adj.mean = 16.3,
p = .000) significantly outperformed the CL and T group, with an adjusted mean
difference of 2.105 and 3.713 respectively. The CL group (Mean = 14.7, SD = 2.7,
Adj.mean = 14.9, p = .000) significantly outperformed the T group (Mean = 12.7, SD
= 2.4, Adj.mean = 12.6) with an adjusted mean difference of 1.608. (Effect sizes on
MR were .83 and .60 for comparing the CLMS and CL, and CL and the T group,
respectively).
Metacognitive knowledge. The CLMS group (Mean = 2.3, SD = .3, Adj.mean = 2.3, p
= .000) significantly outperformed the CL and T group, with an adjusted mean
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difference of .345 and .581 respectively. The CL group (Mean = 1.9, SD = .3,
Adj.mean = 2, p = .000) significantly outperformed the T group (Mean = 1.7, SD = .3,
Adj.mean = 1.7), with an adjusted mean difference of .236. (Effect sizes on MK were
1.25 and .80 for comparing the CLMS and CL, and CL and the T group, respectively).
4.3.2 Summary of Testing Hypothesis 1 (CLMS > CL > T)
The statistical results confirm the hypothesis, showing that students taught via
cooperative learning with metacognitive scaffolding instructional method performed
significantly higher than the students taught via cooperative learning instructional
method who, in turn, performed significantly higher than the students taught via the
traditional instructional method in (a) mathematics performance, (b) mathematical
reasoning, and (c) metacognitive knowledge.
4.3.3 Testing of Hypotheses 2
High-ability students taught via cooperative learning with metacognitive scaffolding
instructional method (CLMSH ) will perform higher than high-ability students taught
via cooperative learning instructional method ( CLH ) who, in turn, will perform
higher than high-ability students taught via the traditional instructional method (TH )
in (a) mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge.
Table 4.6 presents overall means, standard deviations, adjusted means, and standard
error of each dependent variable for high-ability students by the instructional method,
CLMS, CL, and T.
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Table 4.6Means, standard deviations, adjusted means and standard errors for each dependent variable for high-ability students by the instructional method
Dependent Variables The Instructional MethodCLMS CL TN= 40 N= 38 N= 42
Mathematics Performance (MP)
Mean 20.4500 20.000 18.4286
SD 1.6939 1.2945 1.6101
Adj. mean 20.528a
19.908a
18.438 a
Std. Error .244 .250 .236
Mathematical reasoning (MR) Mean 17.8000 16.5526 14.6190
SD 1.8701 1.4275 1.4808
Adj. mean 17.968 a 16.374 a 14.620 a
Std. Error .234 .241 .227
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Metacognitive Knowledge (MK)
Mean 2.4317 2.2123 1.9492
SD .2244 .1929 .1861
Adj. mean 2.419 a 2.218 a 1.956 a
Std. Error .031 .032 .030
Note. a. Evaluated at covariates appeared in the model: pre-MP = 11.3083, preMR = 7.8083.
Total score on MP = 22, total score on MR = 22, and total score on MK = 05
To examine if there were statistically significant differences in mathematics
performance, mathematical reasoning, and metacognitive knowledge adjusted mean
scores between the high-ability students in CLMS group, in CL, and in T group, while
controlling pre-MP and pre-MR, multivariate analysis of covariance (MANCOVA)
was conducted.
Table 4.7 presents the results of multivariate analysis of covariance (MANCOVA),
showing overall differences for the independent variable of instructional method effect
on high-ability students and the three dependent variables, while controlling pre-MP
and pre-MR. The Pillai’s Trace was used to evaluate the multivariate (MANCOVA)
differences. The MANCOVA results of comparing the high-ability students across the
three groups were statistically significant (F= 46.575, p = .000). The covariates pre-
MP (F = 15.020, p = .002) and pre-MR (F = 16.553, p = .000) had significant effects.
This means that there were some statistical differences between high-ability students
across the three groups on at least one dependent variable.
Further, the results of the univariate ANCOVA tests, which are represented in table
4.7, indicated that there were statistically significant differences between high-ability
students across the three groups in the three dependent variables (MP, MR, and MK).
147
The F ratio of MP (2, 117) was 45.600 ( p = .000). This means that the instructional
method had a main effect on high-ability students’ MP. This effect accounted for 26%
of the variance of the high-ability students’ MP (Eta2 = .258). The F ratio of MR (2,
117) was 162.490 ( p = .000). This means that the instructional method had a main
effect on high-ability students’ MR. This effect accounted for 48% of the variance of
the high-ability students’ MR (Eta2 = .477). The F ratio of MK (2, 117) was 202.729 (p
= .000). This means that the instructional method had a main effect on high-ability
students’ MK. This effect accounted for 49% of the variance of the high-ability
students’ MK (Eta2 = .494).
Table 4.7Summary of multivariate analysis of covariance (MANCOVA) results by the instructional method and follow-up analysis of covariance (ANCOVA) results of comparing high-ability students across the three groups.
MANCOVA Effect, Dependent Variables, and Covariate
Multivariate FPillai's Trace
Univariate Fdf = 2, 117
Group Effect
Mathematics Performance(MP)
Mathematical Reasoning(MR)
Metacognitive knowledge (MK)
Pre-MP
Pre-MR
46.575 ( p = .000)
15.020 ( p = .002)
16.553 ( p = .000)
45.600 ( p = .000)
162.490 ( p = .000)
202.729 ( p = .000)
148
The MANCOVA results of comparing high-ability students across the three groups on
the three dependent variables indicated that there were statistically significant
differences between high-ability students in at least two groups on the three dependent
variables. Therefore, the researcher further investigated the univariate statistics results
(analysis of covariance ANCOVA) by performing a post hoc pairwise comparison
using the /lmatrix command for each dependent variable in order to identify
significantly where the differences in the adjusted means resided. Table 4.8 is a
summary of post hoc pairwise comparisons between high-ability students across the
three groups.
Table 4.8Summary of post hoc pairwise comparisons between high-ability students across the three groups
Dependent Variable
Mathematics Performance
(MP)
Mathematical Reasoning
(MR)
Metacognitive Knowledge
(MK)
Comparison Group
Adj.Mean Difference
Sig Adj.Mean Difference
Sig Adj.Mean Difference
Sig
CLMSvs.
.620 .081 1.594 .000 .201 .000
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CL
CLMSvs.T
2.090 .000 3.348 .000 .463 .000
CLvs.T
1.471 .000 1.754 .000 .261 .000
Note.
The adjusted mean differences shown in this table are the subtraction of the second condition (on the lower line) from the first condition (on the upper line); for example, .620 (Adjusted Mean Difference for Mathematics Performance) = CLMS – CL.
Table 4.6 displays the means, standard deviations, adjusted means and standard errors
of high-ability students in the three groups by the dependent variables. Table 4.7 and
table 4.8 show that there are statistical adjusted mean differences among the high-
ability students in the three conditions on the three dependent variables unless no
statistical adjusted mean differences between high-ability students in CLMS and CL
groups in mathematics performance. The adjusted mean differences are presented
below.
Mathematics performance. The CLMS (Mean = 20.5, SD = 1.7, Adj.mean = 20.5)
high-ability students and the CL (Mean = 20.0, SD = 1.3, Adj.mean = 19.9) high-
ability students significantly outperformed the T high-ability students (Mean = 18.4,
SD = .1.6, Adj.mean = 18.4) (p = .000), with adjusted mean differences of 2.090 and
1.471 respectively. There were no statistically significant differences between high-
ability students in CLMS group and high-ability students in CL group (p = .081), with
an adjusted mean difference of .620. (Effect sizes on MP were .28 and .98 for
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comparing high-ability students in the CLMS and CL, and CL and the T group,
respectively).
Mathematical reasoning. The CLMS (Mean = 17.8, SD = 1.9, Adj.mean = 17.9) high-
ability students significantly outperformed the CL and T high-ability students, with an
adjusted mean difference of 1.594 (p = .000) and 3.348 (p = .000) respectively. The
CL (Mean = 16.6, SD = 1.4, Adj.mean = 16.4) high-ability students significantly
outperformed the T high-ability students (Mean = 14.6, SD = 1.5, Adj.mean = 14.6)
with an adjusted mean difference of 1.754 (p = .000). (Effect sizes on MR were .84
and 1.3 for comparing high-ability students in the CLMS and CL, and CL and the T
group, respectively).
Metacognitive knowledge. The CLMS (Mean = 2.4, SD = .2, Adj.mean = 2.4) high-
ability students significantly outperformed the CL and T high-ability students, with
adjusted mean differences of .201 (p = .000) and .463 (p = .000) respectively. The CL
(Mean = 2.2, SD = .2, Adj.mean = 2.2) high-ability students significantly
outperformed the T high-ability students (Mean = 1.9, SD = .2, Adj.mean = 1.9) with
an adjusted mean difference of .261 (p = .000). (Effect sizes on MK were 1.2 and 1.4
for comparing high-ability students in the CLMS and CL, and CL and the T group,
respectively).
4.3.4 Summary of Testing Hypothesis 2 (CLMSH > CLH > TH)
The statistical results partially support the hypothesis, that is, “CLMSH > CLH” is
confirmed in mathematical reasoning and metacognitive knowledge while in
mathematics performance is not. High-ability students taught via CLMS instructional
method performed significantly higher than high-ability students taught via CL
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instructional method in mathematical reasoning and metacognitive knowledge but
they did not perform significantly higher in mathematics performance. “CLMSH, CL H
> TH” is confirmed. High-ability students taught via CLMS and high-ability students
taught via CL instructional methods performed significantly higher than the high-
ability students taught via T instructional method in mathematics performance,
mathematical reasoning, and metacognitive knowledge.
4.3.5 Testing of Hypotheses 3
Low-ability students taught via CLMS instructional method will perform higher than
Low-ability students taught via CL instructional method who, in turn, will perform
higher than low-ability students taught via T instructional method in (a) mathematics
performance (MP), (b) mathematical reasoning (MR) and (c) metacognitive
knowledge.
Table 4.9 presents overall means, standard deviations, adjusted means, and standard
error of each dependent variable for low-ability students by the instructional method,
CLMS, CL, and T.
Table 4.9Means, standard deviations, adjusted means and standard errors for each dependent variable for low-ability students by the instructional method
Dependent Variables The Instructional MethodCLMS CL TN= 40 N= 41 N= 39
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Mathematics Performance (MP)
Mean 16.8500 15.2927 14.9744
SD 1.2517 1.4533 1.3858
Adj. mean 16.923 a
15.367 a
14.821a
Std. Error .198 .194 .199
Mathematical reasoning (MR)
Mean 14.5000 11.9512 10.6923
SD 1.2195 1.4992 1.1955
Adj. mean 14.662 a
11.967 a
10.509 a
Std. Error .177 .174 .179
Metacognitive Knowledge (MK)
Mean 2.1633 1.7041 1.4821
SD .2086 .1578 .1008
Adj. mean 2.170 a
1.706 a
1.472 a
Std. Error .026 .025 .026
Note. a. Evaluated at covariates appeared in the model: pre-MP = 8.9750, pre-MR = 3.2417.
Total score on MP = 22, total score on MR = 22, and total score on MK = 05
To examine if there were statistically significant differences in mathematics
performance, mathematical reasoning, and metacognitive knowledge adjusted mean
scores between the low-ability students in CLMS group, in CL group, and in T group,
while controlling pre-MP and pre-MR, multivariate analysis of covariance
(MANCOVA) was conducted.
Table 4.10 presents the results of multivariate analysis of covariance (MANCOVA),
showing overall differences for the independent variable of instructional method effect
on low-ability students and the three dependent variables, while controlling pre-MP
and pre-MR. The Pillai’s Trace was used to evaluate the multivariate (MANCOVA)
differences. The MANCOVA results of comparing the low-ability students across the
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three groups were statistically significant (F = 27.918, p = .000). The covariates pre-
MP (F = 12.202, p = .000) and pre-MR (F = 10.620, p = .000) had significant effects.
This means that there were some statistical differences between low-ability students
across the three groups on at least one dependent variable.
Further, the results of the univariate ANCOVA tests, which are represented in table
4.10, indicated that there were statistically significant differences between low-ability
students across the three groups in the three dependent variables (MP, MR, and MK).
The F ratio of MP (2, 117) was 29.823 ( p = .000). This means that the instructional
method had a main effect on low-ability students’ MP. This effect accounted for 34%
of the variance of the low-ability students’ MP (Eta2 = .342). The F ratio of MR
(2, 117) was 138.065 ( p = .000). This means that the instructional method had a main
effect on low-ability students’ MR. this effect accounted for 71% of the variance of
the low-ability students’ MR (Eta2 = .706). The F ratio of MK (2, 117) was 188.719 (p
= .000). This means that the instructional method had a main effect on low-ability
students’ MK. This effect accounted for 77% of the variance of the low-ability
students’ MK (Eta2 = .766).
Table 4.10Summary of multivariate analysis of covariance (MANCOVA) results by the instructional method and follow-up analysis of variance (ANOVA) results of comparing low-ability students across the three groups.
MANCOVA Effect, Dependent Variables, and Covariate
Multivariate FPillai's Trace
Univariate Fdf = 2, 117
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Group Effect
Mathematics Performance(MP)
Mathematical Reasoning(MR)
Metacognitive knowledge (MK)
Pre-MP
Pre-MR
27.918 ( p =.000)
12.202 ( p = .000)
10.620 ( p = .000)
29.823 ( p = .000)
138.065 ( p = .000)
188.719 ( p = .000)
The MANCOVA results of comparing low-ability students across the three groups on
the three dependent variables indicated that there were statistically significant
differences between low-ability students in at least two groups on the three dependent
variables. Therefore, the researcher further investigated the univariate statistics results
(analysis of covariance ANCOVA) by performing a post hoc pairwise comparison
using the /lmatrix command for each dependent variable in order to identify
significantly where the differences in the adjusted means resided. Table 4.11 is a
summary of post hoc pairwise comparisons between low-ability students across the
three groups.
Table 4.11Summary of post hoc pairwise comparisons between low-ability students across the three groups
Dependent Variable
Mathematics Performance
Mathematical Reasoning
Metacognitive Knowledge
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(MP) (MR) (MK)
Comparison Group
Adj.Mean Difference
Sig Adj.Mean Difference
Sig Adj.Mean Difference
Sig
CLMSvs.CL
1.555 .000 2.696 .000 .464 .000
CLMSvs.T
2.101 .000 4.153 .000 .698 .000
CLvs.T
.546 .053 1.458 .000 .234 .000
Note.
The adjusted mean difference shown in this table is the subtraction of the second condition (on the lower line) from the first condition (on the upper line); for example, 1.555 (Adjusted Mean Difference for Mathematics Performance) = CLMS – CL.
Table 4.9 displays the means, standard deviations, adjusted means and standard errors
of low-ability students in the three groups by the dependent variables. Table 4.10 and
table 4.11 show that there are statistical adjusted mean differences among the low-
ability students in the three conditions on the three dependent variables unless no
statistical adjusted mean differences between low-ability students in CL and T group
in mathematics performance. The adjusted mean differences are presented below.
Mathematics performance. The CLMS (Mean = 16.8, SD = 1.3, Adj.mean = 16.9)
low-ability students significantly outperformed the CL (Mean = 15. 3, SD = 1.5,
Adj.mean = 15.4) and the T (Mean = 14.9, SD = 1.4, Adj.mean = 14.8) low-ability
students with adjusted mean differences of 1.555 (p = .000) and 2.101 (p = .000)
respectively. However, there were no significant differences between low-ability
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students in CL group and low-ability students in T group (p = .053), with an adjusted
mean difference of .546. (Effect sizes on MP were 1.12 and .23 for comparing low-
ability students in the CLMS and CL, and CL and the T group, respectively).
Mathematical reasoning. The CLMS (Mean = 14.5, SD = 1.2, Adj.mean = 14.7) low-
ability students significantly outperformed the CL and T high-ability students, with
adjusted mean differences of 2.696 (p = .000) and 4.153 (p = .000) respectively. The
CL (Mean = 11.9, SD = 1.5, Adj.mean = 11.9) low-ability students significantly
outperformed the T low-ability students (Mean = 10.7, SD = 1.2, Adj.mean = 10.5)
with an adjusted mean difference of 1.458 (p = .000). (Effect sizes on MR were 2.13
and 1.05 for comparing low-ability students in the CLMS and CL, and CL and the T
group, respectively).
Metacognitive knowledge. The CLMS (Mean = 2.7, SD = .2, Adj.mean = 2.2) low-
ability students significantly outperformed the CL and T high-ability students, with
adjusted mean differences of .464 (p = .000) and .698 (p = .000) respectively. The CL
(Mean = 1.7, SD = .2, Adj.mean = 1.7) low-ability students significantly outperformed
the T low-ability students (Mean = 1.5, SD = .1, Adj.mean = 1.5) with an adjusted
mean difference of .234 (p = .000). (Effect sizes on MK were 4.6 and 2.2 for
comparing low-ability students in the CLMS and CL, and CL and the T group,
respectively).
4.3.6 Summary of Testing Hypothesis 3 (CLMSL > CLL > TL)
The statistical results partially support the hypothesis, that is, “CLMSL > CLL” and
“CLMSL > TL” are confirmed in mathematics performance, mathematical reasoning,
and metacognitive knowledge. “CLL > TL” is confirmed in mathematical reasoning
and metacognitive knowledge while in mathematics performance is not. Low-ability
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students taught via CLMS instructional method performed significantly higher than
low-ability students taught via CL instructional method and than low-ability students
taught via T instructional method in mathematics performance, mathematical
reasoning, and metacognitive knowledge. Low-ability students taught via CL
instructional method performed significantly higher than low-ability students taught
via T instructional method in mathematical reasoning, and metacognitive knowledge
but they did not perform significantly higher in mathematics performance.
4.3.7 Testing of Hypotheses 4
There are interaction effects between the instructional methods and
the ability levels (high-ability and low-ability) on mathematics
performance, mathematical reasoning, and metacognitive
knowledge.
Table 4.12 presents overall means, standard deviations, adjusted means, and standard
error of each dependent variable by the interaction between the instructional methods
and the ability levels (high-ability and low-ability).
Table 4.12Means, standard deviations, adjusted means and standard errors for each dependent variable by the interaction between the instructional methods and the ability levels (high-ability and low-ability)
Dependent VariablesMathematic
s Performance
MP
Mathematical Reasoning
MR
Metacognitive Knowledge
MK
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Instructional Method
Ability High (H) Low (L)
CLMS H (n = 40) Mean 20.4500 17.8000 2.4317SD 1.6939 1.8701 .2244Adj. mean
19.681 a
16.745 a
2.440 a
Std. Error .266 .249 .035
L (n = 40) Mean 16.8500 14.5000 2.1633SD 1.2517 1.2195 .2086Adj. mean
17.802 a
15.833 a
2.159 a
Std. Error .285 .267 .038
CL H (n = 38) Mean 20.000 16.5526 2.2123SD 1.2945 1.4275 .1929Adj. mean
18.997 a
15.155 a
2.219 a
Std. Error .296 .277 .039
L (n = 41) Mean 16.8500 11.9512 1.7041SD 1.2517 1.4992 .1578Adj. mean
16.226 a
13.213 a
1.690 a
Std. Error .287 .269 .038
T H (n = 42) Mean 18.4286 14.6190 1.9492SD 1.6101 1.4808 .1861Adj. mean
17.545 a
13.430 a
1.964 a
Std. Error .279 .262 .037
L (n = 39) Mean 14.9744 10.6923 1.4821SD 1.3858 1.1955 .1008Adj. mean
15.734 a
11.723 a
1.472 a
Std. Error .269 .252 .035
Note. a Evaluated at covariates appeared in the model: pre-MP = 10.1417, pre-MR = 5.5250.
Total score on MP = 22, total score on MR = 22, and total score on MK = 05
To examine if the effects of instructional method on mathematics performance,
mathematical reasoning, and metacognitive knowledge depend on the ability level in
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CLMS group, in CL, and in T group, while controlling pre-MP and pre-MR, two-way
multivariate analysis of covariance (MANCOVA) was conducted.
Table 4.13 presents the results of two-way multivariate analysis of covariance
(MANCOVA), showing overall differences for the interaction between instructional
method and ability level effect on the three dependent variables, while controlling pre-
MP and pre-MR. The Pillai’s Trace was used to evaluate the multivariate
(MANCOVA) differences. The MANCOVA results of the interaction effects on the
three dependent variables was statistically significant (F = 4.836, p = .000). The
covariates pre-MP (F = 15.020, p = .000) and pre-MR (F = 16.553, p = .000) had
significant effects. This means that there were some statistical interaction effects on at
least one dependent variable across the three groups.
Further, the results of the two-way univariate ANCOVA tests, which are represented in
table 4.13, indicated that there were statistically significant interaction effects across
the three groups in MR and MK. The F ratio of MR (2, 237) was 3.401 ( p =
.035). This means that the interaction effect was statistically significant on students’
MR. This interaction accounted for 3% of the variance of the students’ MR (Eta2 =
.028). The F ratio of MK (2, 237) was 10.557 (p = .000). This means that the
interaction effect was statistically significant on students’ MK. This interaction
accounted for 8% of the variance of the students’ MK (Eta2 = .083). However, there
were no statistically significant interaction effects across the three groups in MP. The
F ratio of MP (2, 237) was 2.917 ( p > .05).
Table 4.13Summary of multivariate analysis of covariance (MANCOVA) results by the interaction effect and follow-up analysis of covariance (ANCOVA) results across the three groups.
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MANCOVA Effect, Dependent Variables, and Covariates
Multivariate FPillai's Trace
Univariate Fdf = 2, 237
Group Effect
Mathematics Performance(MP)
Mathematical Reasoning(MR)
Metacognitive knowledge (MK)
Pre-MP
Pre-MR
4.836( p = .000)
15.020( p = .000)
16.553 ( p = .000)
2.917 ( p = .056)
3.401 ( p = .035)
10.557 ( p = .000)
The two-way MANCOVA results of the interaction effects on MR and MK indicated
that there were statistically significant interaction effects between the instructional
method and the students’ ability level in at least one group. Therefore, the researcher
further investigated the interaction effect results by plotting the interaction between
the instructional method and the students’ ability level on MR and MK to identify
significantly where the interactions resided. Also the interaction between the
instructional method and the students’ ability level on MP is plotted. Figure 4.1 shows
the interaction effect between the instructional method and the students’ ability level
across the three groups on MP.
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Instructional Method
TCLCLMS
MP
Adj
uste
d M
ean
Scor
es
20
19
18
17
16
15
Ability
High-ability
Low-ability
Figure 4.1Interaction effect between the instructional method and the students’ ability
levels on MP
Figure 4.1 shows that there is no interaction effect between the instructional method
and the students’ ability level on MP across the three groups. In other words, high-
ability and low-ability students taught via CLMS, CL, and T instructional methods
benefited equally in mathematics performance. Therefore, the effect of the
instructional methods on MP did not depend on the ability level.
Figure 4.2 shows the interaction between the instructional method and the students’
ability level across the three groups on mathematical reasoning (MR).
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Instructional Method
TCLCLMS
MR
Adj
uste
d M
ean
Scor
es
18
17
16
15
14
13
12
11
Ability
High-ability
Low-ability
Figure 4.2Interaction effect between the instructional method and the students’ ability
levels on MR
Figure 4.2 shows that the low-ability students taught via CLMS instructional method
benefited more than the high-ability students taught via the same instructional method
in mathematical reasoning. However, the figure shows that the high-ability and low-
ability students taught via CL and T instructional methods benefited equally in
mathematical reasoning.
Figure 4.3 shows the interaction between the instructional method and the students’
ability level across the three groups on metacognitive knowledge (MK).
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Instructional Method
TCLCLMS
MK
Adj
uste
d M
ean
Scor
es
2.6
2.4
2.2
2.0
1.8
1.6
1.4
Ability
High-ability
Low-ability
Figure 4.3Interaction effect between the instructional method and the students’ ability
levels on MK
Figure 4.3 shows that the low-ability students taught via CLMS instructional method
benefited more than the high-ability students taught via the same instructional method
in metacognitive knowledge. However, the figure shows that the high-ability and low-
ability students taught via CL and T instructional methods benefited equally in
metacognitive knowledge.
4.3.8 Summary of Testing Hypotheses 4 (There are interaction effects between the instructional methods and the ability levels)
The statistical interaction results and the interaction figures partially confirm the
hypotheses, showing that there were interaction effects between the CLMS
instructional method and the ability levels where low-ability students benefited more
than the high-ability students in MR and MK but benefited equally in MP. There were
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no interaction effects between the CL instructional method and the ability level. That
is, the performance of the CL instructional method did not depend on the ability level.
High-ability and low-ability students taught via the CL instructional method benefited
equally in MP, MR, and MK. Finally, there were no interaction effects between the T
instructional method and the ability levels. That is, the performance of the T
instructional method did not depend on the ability levels. High-ability and low-ability
students taught via the T instructional method benefited equally in MP, MR, and MK.
4.3.9 Summary of Findings to Research Questions 1 – 4
The findings to the four research questions are summarized below.
1. Would students taught via CLMS instructional method perform higher than students
taught via CL instructional method who, in turn, would perform higher than students
taught via T instructional method in (a) mathematics performance (MP), (b)
mathematical reasoning (MR) and (c) metacognitive knowledge (MK)?
Overall, CLMS instructional method has significant positive effects on students’ (a)
mathematics performance, (b) mathematical reasoning, and (c) metacognitive
knowledge. This is evidenced by the statistical results that the students taught via the
CLMS method significantly performed higher than the students taught via the CL and
the students taught via the T methods in (a) mathematics performance, (b)
mathematical reasoning, and (c) metacognitive knowledge. In addition, CL
instructional method has significant positive effects on students’ (a) mathematics
performance, (b) mathematical reasoning, and (c) metacognitive knowledge. This is
evidenced by the statistical results that the students taught via the CL method
significantly performed higher that the students taught via the T method in (a)
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mathematics performance, (b) mathematical reasoning, and (c) metacognitive
knowledge.
2. Would high-ability students taught via CLMS instructional method perform higher
than high-ability students taught via CL instructional method who, in turn, would
perform higher than high-ability students taught via T instructional method in (a)
mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge (MK)?
CLMS instructional method has positive effects on high-ability students’ (a)
mathematics performance, (b) mathematical reasoning, and (c) metacognitive
knowledge. The high-ability students taught via the CLMS method significantly
performed higher than the high-ability students taught via the T method in (a)
mathematics performance, (b) mathematical reasoning and (c) metacognitive
knowledge. In addition, except in (a) mathematics performance, the high-ability
students taught via the CLMS method significantly performed higher than the high-
ability students taught via the CL method. Also CL instructional method has
significant positive effects on high-ability students’ (a) mathematics performance, (b)
mathematical reasoning, and (c) metacognitive knowledge. The high-ability students
taught via the CL method significantly performed higher than the high-ability students
taught via the T method in (a) mathematics performance, (b) mathematical reasoning
and (c) metacognitive knowledge.
3. Would low-ability students taught via CLMS instructional method perform higher
than low-ability students taught via CL instructional method who, in turn, would
perform higher than low-ability students taught via T instructional method in (a)
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mathematics performance (MP), (b) mathematical reasoning (MR) and (c)
metacognitive knowledge (MK)?
CLMS instructional method has significant positive effects on low-ability students’ (a)
mathematics performance, (b) mathematical reasoning, and (c) metacognitive
knowledge. The low-ability students taught via the CLMS method significantly
performed higher than the low-ability students taught via the CL and the T methods in
(a) mathematics performance, (b) mathematical reasoning and (c) metacognitive
knowledge. In addition, CL instructional method has significant positive effects on
low-ability students’ (b) mathematical reasoning and (c) metacognitive knowledge.
The low-ability students taught via the CL method significantly performed higher than
the low-ability students taught via the T method in (b) mathematical reasoning and (c)
metacognitive knowledge, but they did not perform significantly higher in (a)
mathematics performance.
4. Are there interaction effects between the instructional methods
and the ability levels (high-ability and low-ability) on mathematics performance,
mathematical reasoning, and metacognitive knowledge?
There were interaction effects between the CLMS instructional method and the ability
levels with low-ability students benefited more than the high-ability students in
mathematical reasoning (MR) and metacognitive knowledge (MK) but benefited
equally in mathematics performance (MP). There were no interaction effects between
the CL instructional method and the ability levels i.e., high-ability and low-ability
students taught via the CL instructional method benefited equally in MP, MR, and
MK. Finally, there were no interaction effects between the T instructional method and
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the ability levels i.e., high-ability and low-ability students taught via the T
instructional method benefited equally in MP, MR, and MK.
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CHAPTER FIVE
DISCUSSION AND CONCLUSIONS
5.1 Introduction
The purpose of this study was to investigate the effects of cooperative learning with
metacognitive scaffolding (CLMS) and cooperative learning (CL) on (a) mathematics
performance, (b) mathematical reasoning, and (c) metacognitive knowledge among
fifth-grade students in Jordan. The study further investigated the effects of CLMS and
CL on high-ability and low-ability students’ (a) mathematics performance, (b)
mathematical reasoning, and (c) metacognitive knowledge. Students’ mathematics
performance, mathematical reasoning, and metacognitive knowledge were measured
through a mathematics achievement test and a metacognitive knowledge
questionnaire.
The sample consisted of 240 Jordanian male students who studied in six fifth-grade
classrooms and were randomly selected from three different male primary schools i.e.,
two classes from each school. They studied “Adding and Subtracting Fractions” unit.
The independent variable was the instructional method with three categories:
Cooperative learning with metacognitive scaffolding instructional method (CLMS),
Cooperative learning instructional method (CL), and Traditional instructional method
(T). The moderator variable was the ability level with two categories: High-ability and
Low-ability. The dependent variables were: Mathematics performance (MP),
Mathematical reasoning (MR), and Metacognitive knowledge (MK).
Data was collected during the first semester of the academic year 2002 / 2003. Two
months before the instructional treatment, the participating students were given the
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mathematics achievement test (pre-test). Students were randomly assigned to one of
the three conditions – CLMS method, CL method, or T method. Then students were
divided into high and low-abilities based on their pre-test scores in mathematics
performance and mathematical reasoning. In CLMS method, students worked
cooperatively and used metacognitive questions cards, while in CL method; students
worked cooperatively and did not use metacognitive questions cards. In T method,
students neither worked cooperatively nor used metacognitive questions cards.
Immediately, after the instructional treatment, the students were given the
mathematics achievement test (post-test) and the metacognitive knowledge
questionnaire. In this chapter, interpretations of the results are discussed.
The present chapter is organized in seven main sections. The first section focuses on
the general effects of the instructional methods on mathematics performance,
mathematical reasoning, and metacognitive knowledge. In the second section, the
effects of the instructional methods on mathematics performance, mathematical
reasoning, and metacognitive knowledge based on ability levels are discussed. The
third section focuses on the interaction effects. The fourth section presents the
summary and conclusions. The fifth section suggests implications for educators. The
sixth section proposes implications for future research. Finally, the seventh section
summarizes the limitations of the present study.
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5.2 Effects of the Instructional Methods on Mathematics Performance, Mathematical Reasoning, and Metacognitive Knowledge
CLMS and CL instructional methods had significant positive effects overall on
students’ (a) mathematics performance, (b) mathematical reasoning, and (c)
metacognitive knowledge.
Students taught via the CLMS method (working cooperatively and also using
metacognitive questions cards) significantly outperformed their counterparts taught
via the CL method who, in turn, significantly outperformed the students taught via the
T method in (a) mathematics performance, (b) mathematical reasoning, and (c)
metacognitive knowledge.
The findings on cooperative learning with metacognitive scaffolding (CLMS) support
the hypothesis that cooperative learning with metacognitive scaffolding not only
improves mathematics performance, as shown by the studies of Schoenfeld (1985);
Peterson et al.(1982); Peterson et al. (1984); and King (1991a), but also improves
mathematical reasoning and metacognitive knowledge.
5.2.1 Effects of the Instructional Methods on Mathematics Performance
The effectiveness of CLMS method on mathematics performance that consists of
conceptual understanding and procedural fluency support King and Rosenshine’s
(1993) study that found that guidance through questioning enhances problem
representation and improves conceptual understanding. The metacognitive questions
have provided the students with cues to important aspects of the problem and helped
them to identify the problem and identify relevant and important information. While
conceptual understanding is enhanced by constructing relationships between the
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previous and the new knowledge (Kilpatrick et al, 2001), the CLMS method
encouraged students to identify the similarities and differences between the problem at
hand and the problems solved in the past. The findings of this study are consistent
with studies by Schonfeld (1987) and Xun (2001) that questioning strategies enabled
students to connect what they learned with their current learning situation.
Metacognitive questions helped students to make connections between different
factors and constraints and link to the solutions. In this regard, metacognitive
questions assisted students to enhance their understanding of a given domain
knowledge.
Flexibility, accuracy, and efficiency are fundamental components of procedural
fluency (Kilpatrick et al, 2001). Students taught via the CLMS method were provided
with the opportunity to execute their mathematical procedures fluently. Working
cooperatively and using the metacognitive questions provided the students with more
than one approach to solve the problem. Metacognitive question such as “what is the
appropriate approach to …..?” helped the students to select the appropriate approach
from many approaches to solve the problem. Because students asked questions such as
“am I on the right track?, they were able to keep track of sub-problems and make use
of intermediate results to solve the problem and therefore to be more accurate and
more efficient learners. Thus, cooperative learning with metacognitive scaffolding
method enabled students to modify and adapt procedures to make them easier to use.
The high mathematics performance requires acquiring relevant conceptual
understanding and procedural knowledge. The performance of the problem solver acts
on these requisites. This probably accounts for the better performance of the students
taught via CLMS method over the students taught via CL method who, in turn,
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performed better than the students taught via T method. Cooperative learning with
metacognitive scaffolding method enabled students to acquire the appropriate
procedural problem solving techniques, and therefore, they were able to maneuver the
computations more accurately than the students in the other two groups. According to
Cross and Parts (1988), this is the self-management aspect of metacognitive strategies.
Students taught via the CL method worked cooperatively. The cooperative group
provides a more intimate setting that permits such direct and unmediated
communication (Shachar and Sharan, 1994). Such a context, proponents of
cooperative learning believe, is a key to students engaging in real discussion and
wrestling with ideas. Therefore, the CL method provided the students with the
opportunities to stretch and extend their thinking more than the students taught via the
T method who worked individually.
The low performance of the students taught via the T method in this study emerged
from the poor conceptual understanding and procedural techniques employed in
solving tasks and problems. In the last meeting with the control group’s teacher, the
teacher reported that students in this group worked individually and did not use
metacognitive questions, did not plan, monitor, or evaluate their solution procedures,
and mentioned that the students also immediately started the computations when the
questions were given to them. Also some of students were anxious as to the specific
demands of the questions. The control group’s teacher was satisfied with the
performance of his group. The teacher’s report shows that students in the traditional
group had insufficient conceptual understanding and procedural fluency and did not
sufficiently or elaborately engage in the planning, monitoring and evaluation phases in
solving their problems.
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5.2.2 Effects of the Instructional Methods on Mathematical Reasoning
The results of this study indicate that cooperative learning with metacognitive
scaffolding method enabled students to reflect on the similarities and differences
between previous and new tasks, as well as to comprehend each problem before
attempting a solution, and to consider the use of strategies that are appropriate for
solving the problem. The learning processes produced by the CLMS method enhanced
the students’ mathematical reasoning. The effectiveness of CLMS method on
mathematical reasoning support other findings by Chi et al. (1994); Mevarech and
Kramarski, (in press); Slavin (1996); and Webb (1989) that show that cooperative
learning with metacognitive scaffolding is one of the best means for elaborating
information and for making connections. By understanding why and how a certain
solution to a task and a problem has been reached, the students elaborated on the
information gained from the metacognitive questions and learned from it. Also
Kramarski et., al. (2001, 2002) found that working cooperatively and using
metacognitive questions facilitated metacognitive knowledge, which, in turn, affected
mathematical reasoning and students’ ability to transfer their knowledge to solve
mathematical authentic tasks.
According to constructivist theories, information is retained and understood through
elaboration and construction of connections between prior knowledge and new
knowledge (Wittrock, 1986). The ability of constructing networks of knowledge with
the CLMS method was greater than with the CL method which, in turn, was greater
than with the T method. These findings are similar to Cossey (1997) findings that
indicate that the more often seventh and eighth graders are exposed to metacognitive
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support such as pattern seeking, conjectures, and giving reasons for ideas, the greater
are their gains on mathematical reasoning.
The findings of this study support earlier findings (Hoek et al., 1999; Mevarech, 1999)
that cooperative learning with metacognitive scaffolding is effective for developing
problem-solving ability because it enables students to link quantitative knowledge and
situational knowledge. When the two types of knowledge are joined, a mental
representation is constructed that supports mathematical reasoning (Cecil and Roazzi,
1994).
The process of solving tasks at a high level of cognitive complexity (e.g.,
mathematical reasoning problems) depends on the activation of metacognitive
processes more than on solving tasks at a lower level of cognitive complexity (e.g.,
conceptual and procedural problems) because the former requires careful planning,
monitoring, regulation, and evaluation (Stein et al., 1996). The cooperative learning
with metacognitive scaffolding method forced students to activate such processes, so
they could reason mathematically better than the students taught via the CL method
that focused only on working cooperatively and the students’ interaction was not
structured. Specifically, the use of metacognitive questions guided students to analyze
the entire situation described in the task or in the problem and thereby did not only
enhance their understanding, but also enabled them to replace their earlier
inappropriate strategies with a new virtually errorless process which is an essential
element of mathematical reasoning.
Students taught via the CLMS method could reason mathematically because they were
guided about the knowledge of when, where, and why to use the strategies for the
problem-solving. The metacognitive questions comprise of planning, monitoring, and
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evaluation questions. Students taught via the CLMS method were required to plan,
monitor, and evaluate their learning strategies and solutions. Planning questions
enabled students to formulate, identify, and to define the task or the problem and then
build the relationships among its concepts and procedures. Monitoring questions
enabled students to regulate or monitor their problem performance by self-generating
feedback which enabled them to select the appropriate strategies. Evaluation questions
enabled students to reflect on their solutions or alternatives so as to direct their future
steps.
One of the most important components of mathematical reasoning is the appropriate
strategies selection and the justification of selecting these strategies. The students
taught via the CLMS method, were able to select and justify the appropriate strategies
for solving the problem because they were trained how to do so. They were trained to
ask metacognitive questions such as “what is the appropriate strategy to solve …? And
“how do we justify the appropriateness of our strategy?” Also mathematical reasoning
requires applying strategies in other situations. The students taught via the CLMS
method were supported to generalize their learning strategies to other situations.
Questions such as “how do we apply this line of thinking to other situations?” and
“can we derive a rule that would work for …?” enabled students to generalize their
strategies, and therefore enabled them to reason mathematically more than the other
two groups.
According to Piaget (1970), students work with independence and equality on each
other’s ideas. The students taught via the CLMS method encountered situations that
contradicted their believes or understanding. This is what Piaget calls cognitive
conflicts. This conflict created a case of disequilibrium for the students. Metacognitive
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questions that comprise planning, monitoring, and evaluation questions assisted
students to assimilate or accommodate their knowledge and therefore reequilibrate
their thinking. The students taught via the CLMS method were forced to revise,
evaluate, and guide their ways of thinking to provide a better fit with reality and
therefore their ability to reason mathematically was improved.
Since learning with understanding according to Piaget occurs by assimilation or
accommodation through resolving the cognitive conflicts, the students taught via the
CLMS method were actively able to adjust and construct their knowledge set and
strategies to settle disputes and disagreements and then their knowledge was
assimilated or accommodated. Also when the students discussed with each other,
different point of views emerged which pushed cognitive development by causing
disequilibrium, which directed students to rethink their ideas. This learning situation
created cognitive conflicts between the students and within every student which
helped then to reason mathematically to reequilibriate their thinking.
For Vygotsky (1978), students can be scaffolded by explanation, demonstration, and
can attain to higher levels of thinking. Vygotsky (1978) suggests the ZPD which is the
difference between what students can accomplish independently and what they can
achieve under support and guidance. The students taught via the CLMS method were
provided with the opportunity to be able to attain higher levels of knowing which were
facilitated by the interaction between the low-ability and the high-ability students.
Working cooperatively and using metacognitive questions provided students with the
opportunity to explain, modify, and justify their solutions which, in turn, enabled them
to extend themselves to higher levels of mathematical reasoning. In other words,
students taught via the CLMS method were scaffolded through the cooperation i.e.,
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high-ability and low-ability interaction, and through the use of metacognitive
questions which helped the students in narrowing their ZPD. The students taught via
the CL method were scaffolded through only the cooperation which enabled them to
reason mathematically better than the students taught via the T method whose learning
were not scaffolded. Therefore, when the metacognitive scaffolding was provided to
students for group cooperation, the benefits of cooperative learning in mathematical
reasoning were maximized.
While the findings of this study confirmed previous research (Lin et al., 1999;
Palincsar et al., 1987; Webb, 1982, 1989b; Brown and Palincsar, 1989; Kramarski et
al., 2001, 2002) on the effectiveness of cooperative learning in supporting students’
mathematical reasoning and cognitive and metacognitive development, they also
suggest that there were certain conditions in which the use of cooperative learning
fully worked to facilitate learning. Greene and Land (2000) found that cooperative
learning was useful in influencing the development of ideas only when group
members offered suggestions, when they were open to negotiation of ideas, and when
they shared prior experiences. There may be times when group members do not know
how to ask questions or how to elaborate thoughts, or there may be times when group
members are not willing to ask questions or respond to others’ questions, or there may
be times when group members do not see the need for cooperation. Webb’s (1989b)
model of cooperative learning further revealed that different conditions and patterns of
cooperation might lead to different learning outcomes. Webb (1989b) found that the
students who learned most were those who provided explanations to others in their
group. In this regard, metacognitive questions served to facilitate the cooperative
learning processes through eliciting responses from some students, and the responses
may invoke further questions from other students who may require elaboration,
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reasoning, or explanation from their peers. In this study the cooperative learning of the
students taught via the CLMS method was structured and guided by the metacognitive
questions cards and therefore these students were assisted to explain and reason their
solution processes.
5.2.3 Effects of the Instructional Methods on Metacognitive Knowledge
The effectiveness of the cooperative learning with metacognitive scaffolding method on
metacognitive knowledge confirms the results of previous studies (e.g., Lin and
Lehman, 1999; Davis and Linn, 2000; King, 1991a, 1991b; Palincsar and Brown,
1984, 1989), which were all consistent in concluding that cooperative learning and
questioning strategies enhanced metacognitive knowledge and reflective thinking. The
use of metacognitive questions directed students’ attention to plan, monitor, and
evaluate their learning processes, which helped them to obtain metacognitive
knowledge and transfer their understanding to novel problems and situations. Also
cooperative learning with metacognitive scaffolding directed students’ attention to
relevant information which made them aware of the important factors and aspects to
be considered, which in turn helped them to monitor their own understanding. The
findings of this study confirm that the cooperative learning with metacognitive
scaffolding method facilitated metacognitive thinking by directing the students’
attention.
The evaluation questions in the metacognitive questions card, such as “what are the
evidences to justify…?” helped the students to reflect upon and explain their own
actions and decisions. The findings of this study support Chi et al.’s (1989) and Lin
and Lehman (1999) findings that metacognitive questions and self-explanation
facilitated problem-solving processes and assisted students to make arguments for
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their solutions and decisions, and thus make thinking explicit. Metacognitive
questions also helped students to monitor their status of understanding in their
problem solving processes by constantly referring back to the goals of the problem.
Masui and De Corte (1999) findings show that students who used metacognitive
questions had more knowledge about orienting and self-judging themselves than did
students in the control groups.
The cooperative learning with metacognitive scaffolding method helped students to elicit
responses and explanations, which promote comprehension of the one who received
the explanation and the one who gave the explanation and feedback (Webb, 1989b).
The cooperative learning with metacognitive scaffolding method guided students to
develop solutions by building upon each other's ideas, questioning each other,
providing feedback, and checking the solution process. The students also were forced
to check each other’s ideas to test if the selected solution was feasible or not, which
required justification for a solution or suggestion, which facilitated the continuous
monitoring of the problem-solving process.
The students taught via the CLMS method were provided with multiple perspectives.
Multiple perspectives gave an opportunity for students to reflect upon and evaluate
their solution processes as the findings of Lin et al. (1999) showed. The choices of
perspectives direct students’ attention to the important aspects of the problem that they
might not have thought about, and as a result, students re-examine their thinking
process, elaborating or modifying their thoughts, recognizing limitations in their
solutions, or making justifications for their solutions or decisions. In this regard,
CLMS method facilitated students’ metacognitive knowledge in the problem-solving
process through planning, monitoring and evaluation. Metacognitive questions
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enhanced planning by activating prior knowledge and attending to important
information, monitoring by actively engaging students in their learning process, and
enhanced evaluation through reflective thinking.
A possible reason that the students taught via the CLMS method outperformed their
counterparts taught via the CL method who, in turn, outperformed the students taught
via the T method in metacognitive knowledge is that the CLMS method forced
students to ask more metacognitive questions than the CL method which, in turn,
forced students to ask more thinking and hinting questions than the T method.
Students taught via the CLMS method were constantly trained to produce
metacognitive questions and responses. The production of these questions, responses,
and feedback during the cooperative setting promoted higher level thinking and
understanding, and thus more metacognitive knowledge for participating students.
Previous research (Flavell, 1979; Palinscar and Brown, 1984) showed that learning
strategies that used cooperative learning and questioning activities function as a
testing mechanism that allows students to monitor their own comprehension. It also
helps students to realize what they know and more importantly, it helps the students to
know what they do not know (King, 1989). Therefore, the students taught via the
CLMS, who utilized this cooperative questioning strategy more extensively than the
CL and T students, reported higher metacognitive knowledge levels than the other two
groups. The students taught via the CL method worked cooperatively where multiple
responses were provided. This learning environment somewhat encouraged students to
produce high level thinking questions and provide evidence for their solutions more
than the T students.
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5.3 Effects of the Instructional Methods on Mathematics Performance,
Mathematical Reasoning, and Metacognitive Knowledge Based on Ability Levels
The results of this study showed that the high-ability students taught via CLMS
method (high-ability students working cooperatively and also using metacognitive
questions cards) significantly outperformed their counterparts taught via the CL
method in (b) mathematical reasoning and (c) metacognitive knowledge. The high-
ability students taught via the CLMS method and the high-ability students taught via
the CL method significantly outperformed their counterparts taught via the T method
in (a) mathematics performance, (b) mathematical reasoning, and (c) metacognitive
knowledge. However, there were no statistically significant differences between the
high-ability students taught via the CLMS method and the high-ability students taught
via the CL method in (a) mathematics performance.
Also the results showed that the low-ability students taught via the CLMS method
(low-ability students working cooperatively and also using metacognitive questions
cards) significantly outperformed their counterparts taught via the CL method and
taught via the T method in (a) mathematics performance, (b) mathematical reasoning,
and (c) metacognitive knowledge. The low-ability students taught via the CL method
significantly outperformed their counterparts taught via the T method in (b)
mathematical reasoning and (c) metacognitive knowledge. However, there were no
statistically significant differences between the low-ability students taught via the CL
method and the low-ability students taught via the T method in (a) mathematics
performance.
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5.3.1 Performance of High-Ability Students Taught Via CLMS
The high-ability students taught via the CLMS method outperformed the high-ability
students taught via the CL method in (b) mathematical reasoning and (c)
metacognitive knowledge, and outperformed the high-ability students taught via the T
method in (a) mathematics performance, (b) mathematical reasoning, and (c)
metacognitive knowledge.
Basically, high-ability students have acquired a great deal of content knowledge that is
organized in ways that reflect a deep understanding of their subject matter, and
therefore the high-ability students habitually use active learning strategies needed to
monitor understanding (Golinkoff, 1976; Meichenbaum, 1976; Ryan, 1981). In
addition to their habitual use of learning strategies, the high-ability students taught via
the CLMS method worked cooperatively and were provided with metacognitive
questions which assisted them to discuss, explain, and evaluate their and other
students’ learning processes. Also the CLMS method gave the opportunity to the high-
ability students to direct the low-ability students’ attention to the relevant features of
the problem they could not understand. Through directing and guiding the low-ability
students, the high-ability students’ reasoning, argumentation and justification were
supported. Working cooperatively and using metacognitive questions further gave the
opportunity to the high-ability students to actively engage in negotiation and meaning
sharing, they asked metacognitive questions that challenged one’s thinking and
required planning, monitoring, explanations, elaboration, and evaluation and
justifications. In such an environment, the CLMS method created a setting for the
high-ability students to construct arguments, reason, and make justifications. The
CLMS method forced the high-ability students to ask the low-ability students and
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themselves questions before, during, and after the solution processes. Asking and
receiving answers of these metacognitive questions assisted the high-ability students
to analyze the whole situation described in the problem, focus on the similarities and
differences between previous and new tasks, as well as on comprehending the problem
before attempting a solution and reflecting on the use of strategies that are appropriate
for solving the problem, and thus enhanced their understanding and enabled them to
evaluate, justify, and alter the inappropriate strategies with a new virtually errorless
process which are an essential elements of mathematical reasoning and metacognitive
knowledge.
Working cooperatively with the low-ability students and asking and answering
metacognitive questions provided the high-ability students with multiple perspectives
and guided them to see things they might have overlooked. Also formulating and
answering metacognitive questions forced the high-ability students to identify the
main ideas and the ways the ideas relate to each other and to the students’ prior
knowledge and experiences. Such a characteristic assisted the high-ability students to
reflect on their own thinking, actions, and decisions, and as a result, they modified
their thinking, planned remedial actions, evaluated their solutions, and monitored and
checked their and other students’ solution processes.
Additionally, the CLMS method assisted the high-ability students’ mathematics
performance that comprises conceptual understanding and procedural fluency.
Metacognitive questions within the cooperative learning setting provided the high-
ability students with prompts to important features of the task and helped them to
recognize the problem and recognize relevant and important information. Also the
CLMS method encouraged the high-ability students to perceive the similarities and
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differences between the current and the problems already solved previously. Working
cooperatively and asking and answering metacognitive questions assisted the high-
ability students to create connections between different aspects and constraints and
relate to the solutions. In this regard, the CLMS method improved the high-ability
students’ conceptual understanding and mathematical procedures.
However, the findings of this study showed that although the adjusted mean of the
high ability students taught via the CLMS method was higher, there were no
statistically significant differences between the high-ability students taught via the
CLMS method and the high-ability students taught via the CL method in (a)
mathematics performance. Thus, unlike mathematical reasoning and metacognitive
knowledge, metacognitive scaffolding did not assist the high-ability students to
outperform their counterparts taught via the CL method in mathematics performance.
This is due to the nature of the tasks and the problems that required conceptual
understanding and procedural fluency, the two major elements of mathematics
performance which were within their mastery.
The processes of solving these tasks and problems require mastering the procedures
and applying these procedures step by step more than the activation of metacognitive
strategies which, mathematical reasoning and metacognitive knowledge tasks and
problems require. Additionally, the high-ability students habitually often use active
learning strategies needed to monitor understanding (Golinkoff, 1976; Meichenbaum,
1976; Ryan, 1981). The high-ability students taught via the CLMS method worked
cooperatively with the low-ability students and asked and answered metacognitive
questions. In this situation, the CLMS method guided the high-ability students to
establish learning goals for tasks and problems, to assess the degree to which these
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goals are being met, and, if necessary, to modify the strategies being used to meet the
goals. Also the CLMS method forced the high-ability students to discuss with and ask
the low-ability students metacognitive questions before, during, and after the
processes of solving the mathematical tasks and problems. Therefore, the CLMS
method assisted and guided the high-ability students to activate their metacognitive
processes, and aided them to focus on formulating and understanding the problem
more than on mastering the procedures of solving the problem. Working cooperatively
and asking and answering metacognitive questions before, during, and after the
processes of solving the problem assisted the high-ability students to focus on the
processes of solving problems at a higher level of cognitive complexity, and thus, they
were more guided to execute the tasks and problems that required mathematical
reasoning and metacognitive knowledge than those that required conceptual
understanding and procedural fluency. The effect of this activity is evidenced by the
higher attainment in mathematical reasoning and metacognitive knowledge i.e., the
high-ability students taught via the CLMS method outperformed their counterparts
taught via the CL method in mathematical reasoning and metacognitive knowledge.
5.3.2 Performance of High-Ability Students Taught Via CL
The findings of this study showed that the high-ability students taught via the CL
method outperformed their counterparts taught via the T method in (a) mathematics
performance, (b) mathematical reasoning, and (c) metacognitive knowledge. Working
cooperatively with the low-ability students, the CL method gave an opportunity to the
high-ability students to discuss, clarify ideas, and evaluate each others’ ideas.
According to Vygotsky (1978), students are capable of performing at higher levels
when working cooperatively than when working individually. Group diversity in
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terms of knowledge and experience contributes positively to the learning process.
Within the cooperative learning environment, the high-ability students are confronted
with different interpretations of a given situation, and thus, the CL method created
cognitive conflicts among the students which then enhanced them to discuss, explain,
evaluate, and modify their opinions to reequilibriate their thinking to learn with
understanding. Also the CL method provided the high-ability students with
opportunities to learn from each other’s skills and experiences. Working cooperatively
helped the high-ability students to go beyond simple statements of opinion by giving
reasons for their judgments and reflecting upon the criteria employed in making these
judgments. Thus, each opinion was subject to careful scrutiny. The ability to admit
that one’s initial opinion may have been incorrect or partially flawed improved the
high-ability students’ mathematical reasoning and metacognitive knowledge.
5.3.3 Performance of Low-Ability Students Taught Via CLMS
The higher mathematics performance, mathematical reasoning, and metacognitive
knowledge scores of the low-ability students taught via the CLMS method is
explained by the fact that within cooperative setting, the metacognitive scaffolding
guided the low-ability students in the right direction through the metacognitive
questions and the questions generated by the high-ability students. Questions on
planning, monitoring, and evaluation guided the low-ability students to construct
sound arguments, evaluate solutions, and explain reasons for viable alternative
solutions. This indicates that the low-ability students during problem solving need
support and guidance in the problem solving process. The CLMS method forced each
student to be asker, summarizer, recorder, and presenter by rotation. Working
cooperatively with high-ability students and using the metacognitive questions
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assisted the low-ability students to generate more questions which served as a
guideline to help them start the problem solving task. The CLMS method gave the
opportunity to every student to ask questions before, during, and after the processes of
solving the problem, to generate more questions among the group members, and to
elaborate thoughts in responding to those questions.
The findings of this study overcome learning deficiencies involving low ability
students as found by Graesser and Person (1994) and others. Graesser and Person
(1994) found that low-ability students usually asked low frequency, short answer, and
shallow questions and argue that this phenomenon can be attributed to difficulties at
three different levels, one of them being the low-ability students’ difficulty identifying
their own knowledge deficits, unless students had high amounts of domain
knowledge. Gavelek and Raphael (1985) also pointed that low-ability students may
lack the background knowledge necessary to ask their own questions or even answer
the questions of others; and they may also lack the procedural knowledge for
discriminating what it is that they do know from that which they do not know. Xun
(2001) indicates that if the frequency of questions is low or if the questions asked are
superficial, there would not be many explanations elicited from other students or even
themselves. As found by Chi et al. (1989), self-explanation was an important
component to monitor one’s learning process.
In this study, the low-ability students taught via the CLMS method were provided with
metacognitive questions and worked cooperatively, which assisted them to ask
important and relevant questions before, during, and after the processes of solving the
problem which, in turn, helped to elicit their responses, elaborate their thinking and
articulate their reasoning and therefore, they solved the problems more correctly than
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the other two groups. Whether the response is verbally articulated or thoughtfully
considered, answering one’s own questions in the form of self-explanation can be an
effective strategy for enhancing reflection and metacognition (Chi et al., 1989).
Metacognitive scaffolding is especially important for low-ability students who tend to
jump immediately into computation aspects of problem solving when faced with the
task of solving complex problems (Lin et al., 1999). The low-ability students in this
study may lack the ability to engage in effective thinking and problem solving on their
own; thus the CLMS method enabled them to induce higher-order thinking and
provided them with tools that they did not already possess. Also the CLMS method
guided low-ability students’ attention to specific aspects of their learning process such
as planning, monitoring and evaluation of their own problem-solving processes
before, during and after the processes of solving the problems, which enhanced their
metacognitive knowledge.
5.3.4 Performance of Low-Ability Students Taught Via CL
The low-ability students taught via the CL method worked with the high-ability
students together to solve problems and complete tasks. In this setting, the low-ability
students had the opportunity to model the study skills and work habits of more
proficient students. Through cooperation, the low-ability students were provided with
different perspectives which helped them to evaluate and justify their solution
processes and therefore they outperformed their counterparts who taught via the T
method in (b) mathematical reasoning and (c) metacognitive knowledge. However, the
findings of this study showed the although the mean of the low ability students in the
CL group was higher there were the findings of this study showed that there were no
statistically significant differences between the low-ability students taught via the CL
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method and the low-ability students taught via the T method in (a) mathematics
performance. In this study, mathematics performance tasks required conceptual
understanding and procedural fluency. The process of solving tasks required
conceptual understanding and procedural fluency does not depend on the activation of
metacognitive processes as much as process of solving tasks required mathematical
reasoning. The low-ability students taught via the CL method worked cooperatively
with the high-ability students. This learning environment encouraged the low-ability
students to discuss with and ask the high-ability students questions regarding the
processes of solving the mathematical tasks and problems which, in turn enhanced the
high-ability students to provide the low-ability students with multiple perspectives,
guide them to activate their metacognitive processes, and assisted them to concentrate
on formulating and understanding the problem more than on the procedures of solving
the problem. The low-ability students taught via the CL method were assisted to focus
on solving problems at a higher level of cognitive complexity, and therefore, they
concentrated more on tasks and problems required mathematical reasoning and
metacognitive knowledge than those required conceptual understanding and
procedural fluency. This concentration is evidenced by outperforming their
counterparts taught via the T method in mathematical reasoning and metacognitive
knowledge.
5.3.5 Performance of Low-Ability Students Taught Via T
The explanations for the low mathematics achievement and metacognitive knowledge
of the low-ability students taught via the T method could be that they were not taught
the appropriate strategies, could not self-regulate the study strategies, and did not
understand how to apply these strategies. In the last meeting with the teacher who
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applied the T method, the teacher reported that most of the low-ability students were
confused when they encountered a mathematical problem and they were unable to
explain the strategies they employed to find the correct solution. The teacher’s report
confirmed that the low-ability students taught via the T method generally lack well-
developed mathematical reasoning and metacognitive knowledge. The low-ability
students taught via the T method were not scaffolded via cooperation and
metacognitive questions, and thus they might have been at a loss as to how to start to
solve the problems. They might not have known what questions to ask, they might not
generate many questions to ask; or even if they did ask questions, the questions might
not be focused or in-depth. Thus, they had limited abilities to solve problems require
mathematical reasoning and metacognitive strategies.
Within the traditional teaching method, the low-ability students often received less
teacher time, attention, and were asked a fewer number of process-oriented questions
(Leder, 1987). This may happened with the low-ability students taught via the T
method because the teacher reported that he himself determines the success of the
low-ability students, and thus may not gave these students more time and attention,
and may not encouraged them to participate in the whole class public interaction.
Also, the low-ability students taught via the T method were given much greater time
and emphasis to mathematical procedures.
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5.4 Interaction Effects
An interesting finding in this study is that the low-ability students taught via the
CLMS method benefited more than the high-ability students taught via the same
method in (b) mathematical reasoning and (c) metacognitive knowledge. This finding
is interpreted within the context of Piaget’s and Vygotsky’ theories.
Seen according to Piaget’s cognitive-development theory (Piaget, 1970), the CLMS
method played a critical role in enhancing cognitive development of low-ability
students. The CLMS method created cognitive discrepancies or cognitive conflicts and
therefore encouraged the students to resolve them. The causal sequence began with the
metacognitive questions that generated tension while creating the discrepancy, which
in turn caused disequilibrium, and the student then strived to resolve the discrepancies
via mental activity. In this case, CLMS method challenged low-ability students to
change their cognitive structure or schema to make sense of the environment, to think
about alternative solutions and consider various perspectives. The CLMS method
encouraged low-ability students’ mathematical reasoning and metacognitive
knowledge to settle disputes and disagreements and then the knowledge was
assimilated and accommodated. The metacognitive questions, particularly the “why”
questions activated the low-ability students’ prior knowledge related to the new
concepts. Therefore, the metacognitive questions helped to activate the low-ability
students’ schemata and thus enabled them to retrieve information, elaborate
knowledge, and represent understandings of the problem to be solved. It can be
concluded that the low-ability students who are habitually deficient in the use of
active learning strategies needed to monitor understanding (Golinkoff, 1976;
Meichenbaum, 1976; Ryan, 1981) were through MS enhanced to perform like the high
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ability students. The high ablity students were already habitual users of the processes
of metacognition (Golinkoff, 1976; Meichenbaum, 1976; Ryan, 1981) and thus despite
working cooperatively and asked and answered metacognitive questions did not
benefit as much from the CLMS method.
Low-ability students in the traditional teaching were rarely exposed to high-level
reasoning and mathematical discussions (Mevarech and Kramarski, 1997), and
frequently received less teacher time, attention, and were asked a fewer number of
process-oriented questions (Leder, 1987). Therefore, the low-ability students were in a
critical need of a learning method like the CLMS method that challenged them and
then forced them to attend to the instructions sufficiently. The CLMS method assisted
the low-ability students to organize the new material, integrate the information with
existing knowledge and guide the encoding of schema. Also, the low-ability students
taught via the CLMS method were supported to construct their mathematical
knowledge by their own and therefore learn mathematics with understanding.
Central to the notion of working cooperatively and using metacognitive questions is
Vygotsky’s (1978) zone of proximal development, that is, “the distance between the
actual development level as determined by independent problem solving and the level
of potential development as determined through problem solving under adult
guidance, or in collaboration with more capable peers” (p. 86). Through working
cooperatively with high-ability students and asking and answering metacognitive
questions, the low-ability students were provided with modeling of higher-level
thinking and more sophisticated ways of constructing arguments, understanding
textual materials, and solving problems, and thus the low-ability students reached
levels in mathematics achievement and metacognitive knowledge that they could not
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reach without cooperation and metacognitive scaffolding. Therefore, the CLMS
method helped the low-ability students to fully utilize their potential abilities and to
progress from what Vygotsky called their “actual developmental level” to their “level
of potential development” (1978, p. 86). The high ability students in all groups were
already independently functioning at ZPD levels which were higher than those of the
low ability students. The CLMS method had a positive effect on the ZPD levels of
high ability students but dramatically enhanced the ZPD levels the low ability
students.
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5.5 Summary and Conclusions
This study found that the use of metacognitive scaffolding helped the students to fully
benefit from cooperative learning. Overall the CLMS group outperformed the CL
group in all measures, showing that for fifth-grade mathematics cooperative learning
alone was not sufficient as a form of scaffolding.
The low-ability students taught via the CLMS method outperformed their counterparts
taught via the CL and T methods in mathematics performance, mathematical
reasoning, and metacognitive knowledge. The low-ability students taught via the CL
method in turn outperformed their counterparts taught via the T method in MR and
MK but not in MP. This study shows that the cooperative learning method, when
embedded with metacognitive scaffolding and implemented correctly in the
classrooms, is an effective method to achieve the goal of helping low-ability students
learn mathematics with understanding, reason mathematically, and obtain and apply
metacognitive strategies.
The high-ability students taught via the CLMS method outperformed their
counterparts taught via the CL method in MR and MK but not in MP, and
outperformed their counterparts in the T method in MP, MR and MK. The high-ability
students taught via the CL method in turn outperformed their counterparts taught via
the T method in MP, MR and MK.
The CLMS method was highly effective in the teaching of conceptual understanding
and procedural fluency (mathematics performance) for both high-ability and low-
ability students, but the interaction effects showed that the CLMS method is very
195
effective for enhancing mathematical reasoning and metacognitive knowledge among
low-ability students.
From these findings, it can be concluded that the use of metacognitive scaffolding
helped the students to fully benefit from cooperative learning. When students are
actively engaged in activities such as planning, monitoring, questioning, explaining,
elaborating, negotiating meanings, constructing arguments, and evaluation, they
benefit much from the cooperative learning process. Therefore, the cooperative
learning method is inadequate without metacognitive scaffolding or, cooperative
learning with metacognitive scaffolding method is superior to cooperative learning
method alone. It follows that the cooperative learning process should be scaffolded
appropriately, and modeling through metacognitive scaffolding. The metacognitive
scaffolding is especially effective in improving students’ mathematical reasoning and
metacognitive knowledge. The cooperative learning with metacognitive scaffolding
method is effective for younger students and for improving performance in all aspects
of mathematics. The cooperative learning with metacognitive scaffolding method
further is an effective method across abilities, but is especially beneficial for low
ability students.
.
196
5.6 Implications for Educators
From the discussion of the findings, it is evident in this study that cooperative learning
with metacognitive scaffolding method is effective in supporting students’
mathematics performance, mathematical reasoning, and metacognitive knowledge. A
close examination of the results revealed that cooperative learning alone is insufficient
as a form of scaffolding. Also it is inferred that the metacognitive scaffolding was
particularly effective in supporting low-ability students’ mathematics performance,
mathematical reasoning, and metacognitive knowledge. Therefore, metacognitive
scaffolding can be integrated in instructional design, curriculum design, computer
based design, or web-based design to develop mathematics performance,
mathematical reasoning, and metacognitive knowledge and facilitate self-regulated
learning (Brown and Palincsar, 1989).
The implementation of cooperative learning with metacognitive scaffolding method is
not costly. Therefore, the effectiveness, the high learn ability level and the cost
effectiveness of this method make this method a good candidate for inclusion in the
development of the pedagogical approach.
The cooperative learning with metacognitive scaffolding method should be included
in teacher education programs. There are several skills, such as grouping, drawing
metacognitive questions, and reflection, that pre-service and in-service teachers need
to be trained. Also the use of cooperative learning with metacognitive scaffolding
method in the classroom requires an approach to assessment and evaluation that is
different from the present system. A more authentic and performance-based
assessment criteria, that pre-service and in-service teachers need to be trained to
develop to accompany the implementation of this method in the classroom.
197
In the usual manner, low-ability students do not get the same attention and do not have
the knowledge and skills as high-ability students. In this study, the findings showed
that the cooperative learning with metacognitive scaffolding is particularly effective in
supporting low-ability students’ mathematics performance, mathematical reasoning,
and metacognitive knowledge. Therefore, teachers should give low-ability students
more attention and guide and assist them metacognitively. If low-ability students
receive more attention and assisted metacognitively, they can perform almost as high-
ability students as the findings of this study proved.
Finally, at present, many state proficiency tests and international examinations (e.g.,
TIMSS- 1999 or PISA- 2000 administered by OECD- 2000 countries) include
problems and tasks that ask students to explain their reasoning in writing. To acquaint
students with such tasks, teachers should use metacognitive questions cards as
guidelines and ask students to score one another’s reasoning by using the
metacognitive questions cards and activating metacognitive processes.
198
5.7 Implications for Future Research
The findings of this study raise several questions for further research:
First, no formal observations and / or interviews were conducted in this study.
Therefore, the quality of group interactions in CLMS and CL methods is not known. It
would be particularly interesting to examine how high-ability and low-ability students
interact with each other in the CLMS and the CL methods.
Second, Students’ motivation is an interesting area for future research. Webb and
Palincsar (1996) point out, “Groups are social systems. Students’ interaction with
others is not only guided by the learning task, it is also shaped by their emotions,
perceptions, and attitudes. Some social-emotional processes are beneficial for
learning, others are not." (p. 855). Therefore, the effect of cooperative learning with
metacognitive scaffolding and cooperative learning on students’ motivation is worth
further investigation.
Third, the findings of this study call for the design of additional learning environments
based on similar components. The extent to which the CLMS and CL methods used in
the present study are effective also for children at different grades, different gender,
different mathematical topics, or for different subjects is not known at present and
may be investigated in future research.
Finally, an interesting question raised in this study relates to the effects of providing
metacognitive scaffolding in cooperative learning setting versus providing
metacognitive scaffolding in an individual learning setting on mathematics
performance, mathematical reasoning, and metacognitive knowledge. To address the
issue, students who worked cooperatively and used metacognitive questions cards
199
should be compared with students work individually and use metacognitive questions
cards.
200
5.8 Limitations of the Study
This study sought to investigate the effects of cooperative learning with metacognitive
scaffolding and cooperative learning strategies on mathematics performance,
mathematical reasoning, and metacognitive knowledge among fifth-grade students of
two ability levels (high and low) in Jordan. This study was conducted in the natural
setting of the class. The following are some limitations may restrict the probability of
generalizing its findings:
First, Jordan Government schools are not coeducational, so this study samples limited
to the male fifth-grade students in the primary schools of Irbid directorate. The results
found in this study may not be generalizable to the female fifth-grade students in other
educational directorates in Jordan.
Second, this study limited to the “Adding and Subtracting fractions” unit in the fifth-
grade textbook, and this may restrict generalizing the study findings to the rest of
mathematics concepts and subjects.
The third limitation was associated with measuring students’ metacognitive
knowledge. Students’ metacognitive knowledge was assessed via a metacognitive
questionnaire; this assessment procedure may be insufficient as there are no verbal
report measures or/and direct observation of students’ interaction and strategy use and
development. However, the researcher justifies that the students’ age (11 years) may
restrict the implementation of these assessment procedures. Therefore, the researcher
relied primarily on the questionnaire data for the investigation of the students’
metacognitive knowledge.
201
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