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

1

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

2

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


Scaffolding and Real-Life Problem Solving

77


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

3

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

4

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

6

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

7

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

8

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

9

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

10

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.

11

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

12

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.

13

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

14

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

15

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

16

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.

17

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

18

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

19

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

20

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

21

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,

22

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

23

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

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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)



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


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



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



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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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:

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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


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.


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.


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


metacognitive scaffolding (CLMS) and cooperative learning (CL) methods on


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

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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)

133

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

134

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

137

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’


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

140

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

141

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

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.

143

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

144

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)



error of each dependent variable for high-ability students by the instructional method,

CLMS, CL, and T.

145

Table 4.6Means, standard deviations, adjusted means and standard errors for each dependent variable for high-ability students by the instructional method



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

146


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.




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.



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.


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.




Group Effect




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


(MP)


(MR)


(MK)

Comparison Group

Adj.Mean Difference



Sig

CLMSvs.

.620 .081 1.594 .000 .201 .000

149

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.


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

150

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

151

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.


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.


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


152


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


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.




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.



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.


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.




154

Group Effect




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




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(MP) (MR) (MK)

Comparison Group

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.


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-


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-


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.


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.


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


MR


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


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


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.


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



Group Effect




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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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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TCLCLMS

MR

Adj

uste

d M

ean

Scor

es

18

17

16

15

14

13

12

11

Ability

High-ability

Low-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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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


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


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

164

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


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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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




CLMS instructional method has positive effects on high-ability students’ (a)


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


166



CLMS instructional method has significant positive effects on low-ability students’ (a)


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

167

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


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

169

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)


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)


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

171

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.

173

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

174

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

175

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

176

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)


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


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

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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.

.

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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.

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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.

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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

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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’


201

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