Impact of Collaborative Teaching on 8th Gradeprr.hec.gov.pk/jspui/bitstream/123456789/9084/1... · Zafar Iqbal Distinguished National Professor Dr/2009-12 Dr. Munawar S. Mirza Division

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Impact of Collaborative Teaching on 8th Grade

Students’ Achievement in Mathematics in Punjab

MUHAMMAD ZAFAR IQBAL

Reg. No. 20961100002

DIVISION OF EDUCATION

UNIVERSITY OF EDUCATION

LAHORE

2014

2

PhD Thesis Certificate of Acceptance

3

Name of Student Student No.

Programme

Muhammad Zafar Iqbal 09-PhD-702 PhD

(Education)

TOPIC: IMPACT OF COLLABORATIVE TEACHING ON 8TH

GRADE

STUDENTS’ ACHIEVEMENT IN MATHEMATICS IN

PUNJAB

THESIS COMMITTEE

SR. # Name Position

Signature

1. Prof. Dr. Umer Ali Khan External

2. Prof. Dr. Munawar S. Mirza Supervisor

3. Dr. Ashiq Husain Dogar Controller of Examination

4

DECLARATION

It is certified that this Ph.D. dissertation titled “Impact of Collaborative

Teaching on 8th Grade Students’ Achievement in Mathematics in Punjab” is an

original research. Its contents have not been submitted as a whole or in parts for the

requirement of any other degree and are not currently being submitted for any other

degree or qualification. To the best of my knowledge, the thesis does not contain any

material published or written previously by any other author, except where due

references were made to the source in the text of the thesis.

It is further certified that the help received in developing the thesis and all

resources used for the purpose have been duly acknowledged at the appropriate

places.

________________________

________________________

Supervisor Muhammad

Zafar Iqbal

Distinguished National Professor Dr/2009-12

Dr. Munawar S. Mirza Division of

Education

University of

Education

Township Lahore,

Pakistan

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I Dedicate This Thesis

TO

My Father Muhammad Iqbal, Mother Parveen

Akhtar, and Wife Jahan Ara, My Son Jahanzeb

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ABSTRACT

The subject of mathematics plays a vital role in development of reasoning,

logical thinking, and problem solving skills of students. In Pakistan, predominantly

deductive method was used for teaching mathematics by a single teacher in the

classroom. In USA, UK, China, Australia, Canada and other developed countries

teachers were also using collaborative teaching (CT) in delivering instruction. It was a

teaching approach where two or more teachers plan, implement, and assess taught

lessons. This study aimed at designing a module and teaching of mathematics to 8th

grade students using CT. The study examined its impact on students’ achievement in

mathematics. The subsidiary objectives were to examine the impact of CT on

students’ achievement in different mathematical proficiencies (conceptual

understanding, procedural knowledge, and problem solving), different content strands

(algebra and geometry) of mathematics, and their beliefs about mathematics and

teaching of mathematics using CT. The nature of the study was mainly experimental

for which Solomon Four Group experimental research design was used. Semi

structured interviews were also conducted to investigate students’ beliefs about

mathematics and teaching of mathematics. One public school was selected

conveniently from Sargodha district of Punjab-Pakistan. The school was selected after

the due permission of the head teacher of the school. All students enrolled in 8th grade

i.e. 118 participated in the experiment. Two volunteer mathematics’ teachers, having

M.Sc. (mathematics) & B.Ed., of the sampled school were the part of this study. The

researcher having the same qualification also participated in the study as a co-teacher.

One mathematics teacher from the sampled school and the researcher himself

participated in CT; they developed the module to teach mathematics collaboratively,

in the content strands of algebra and geometry. Module was validated by two

mathematics education experts. Researcher divided subjects into four groups

randomly using Statistical Package for Social Sciences (SPSS). The researcher

adopted mathematics achievement test items from the item pool developed by the

National Educational Assessment System (NEAS). The findings of the study revealed

that CT is more effective than the traditional teaching of mathematics at 8th grade in

improving the overall academic achievement of students in the subject of

mathematics, in the mathematical proficiencies of conceptual understanding and

procedural knowledge both in the content strands of Algebra and Geometry. CT and

traditional teaching of mathematics at 8th grade have same effect on mathematical

ability of problem solving in subject of mathematics. The findings showed that

students’ beliefs were changed about mathematics as a subject and its teaching in

collaborative settings. Further research is suggested by including more factors such as

arithmetic, data analysis and probability, gender, school, and grades.

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ACKNOWLEDGEMENTS

First of all, I would like to express the deepest gratitude for almighty Allah,

the most gracious and the most merciful, who enabled me to complete this project.

Then, I would like to express cordial gratitude to my honorable supervisor, Prof.

Emeritus Dr. Munawar Sultana Mirza for her sincere cooperation, scholarly guidance,

useful observations and valuable suggestions. I can never forget the way she pruned

my immature ideas into ripe form. I must say that she is on the top of the people who

trained me in academic writing. She always made me capable of out of box thinking. I

thank her for everything she did in my favor for completion of this research.

I would like to express my deepest feelings of gratitude for the department of

Mathematics Education in Teachers College, Columbia University USA for providing

me a chance to learn more about research in the development of my dissertation. I

would like to thank from the core of my heart to Prof. Dr. Bruce Vogeli for providing

me the opportunity to visit Teachers College Columbia University. I must say that his

motivation, guidance, and cooperation were a source of inspiration for me. I would

like to express warm gratitude to Prof. Dr. Philip Smith for his support and guidance.

His guidance, encouragement, motivation, cooperation, and value able time put me in

a position to complete my research work on time. I will always remember him for

everything he did for my better learning. Especially the guidance related to literature

review and improvement of teaching module. I would also like to thank Prof. Dr Erica

Walker for her guidance to improve the final draft.

Much praise to my loving and affectionate parents Muhammad Iqbal and

Perveen Akhter, who supported me a lot in this journey. They always encouraged me

to achieve this goal, and always had an unceasing confidence in my abilities to

accomplish this task. My special thanks to my brothers Muhammad Mumtaz,

Muhammad Shahid Iqbal, and Muhammad Zahid Iqbal, and dear sister Haleema

Sadia who have always been a source of motivation in this journey,

I would like to express special gratitude to my beloved wife Jahan Ara to

whom I owe an immense debt for her constant support, time, and help. I always found

her as a source of motivation and encouragement in completing this task.

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Thanks to my all friends who were there to help me considering my project as

their own specially Dr. Bilal Cheema, Dr. Ayaz Khan, Dr. Shahzada Qaisar, Dr.

Azeem, and Dr. Nasir Mahmood.

Last, but not the least, I offer my thanks to the Higher Education Commission

Pakistan for financial support in accomplishing my mission. I would always cherish to

be an indigenous scholarship holder by HEC. The present boost in research in

Pakistan owes a great debt to HEC and it will always be a guiding star for promoting

research culture in Pakistan.

MZI

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TABLE OF CONTENTS

Chapter Topic Page

No.

1 Introduction 1

Rationale of the study 7

Statement of the Problem 9

Objectives 9

Hypotheses of the Study 10

Significance of the Study 12

Delimitations 13

Operational definitions of the terms 13

Development of the Dissertation 14

2 Literature Review 15

Collaborative Teaching 15

Difference between CT models: co-teaching and team

teaching

18

Origins and Development of Collaborative Teaching 20

Research in Collaborative Teaching 21

Collaborative Teaching Models 26

The Lead Teacher 26

Station teaching 26

Co-Teaching 27

Team Teaching 29

Consultation 30

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Collaborative Teaching of Mathematics: Approaches,

Methods, and Strategies

31

Characteristics of a Good Collaborative Teacher 32

Students Beliefs about Mathematics 32

Effective Co-teaching Pedagogy 35

Implementation of CT 37

Advantages, Disadvantages, Challenges of CT 39

3 Methodology 42

Research Design of the study 43

Population 44

Sampling 45

Difficulty in the Selection of Sample 46

Training of Volunteer Mathematics Teachers in this

Study

47

Collaboration of Co-teachers 48

Research Instruments 48

Mathematics Achievement Test (MAT) 49

Procedure of Finalizing MAT Items 52

Translation of MAT 55

Interview Protocol 56

Validity of Interview 57

Collaborative Mathematics Teaching Module (CMTM) 57

Profile of Participating Mathematics Teachers 59

Procedure of the Experiment 60

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Ethical Issues Related to the Experimental Design 62

Controlling Possible Confounding Factors 66

Data Analysis 69

4 Data Analysis 71

Quantitative Analysis of Data 72

Data Screening 73

Normality Test 75

Achievement Scores of Students in Mathematics

Taught through Traditional Teaching and Collaborative

Teaching

80

Analysis by Mathematical Proficiencies 82

Analysis by Mathematical Content Strands 86

Analysis of Mathematical Proficiencies and Content

Strands

89

Qualitative Analysis 97

Process of Axial Coding 98

5 Summary, Findings, Conclusions,

Discussion, and Recommendations

109

Summary 109

Findings 113

Conclusions 118

Discussion 120

Recommendations 124

Suggestions 125

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REFERENCES

127

APPENDICES 141

Appendix A: Mathematics Achievement Test 141

Appendix B: Collaborative Mathematics Teaching

Module (CMTM)

155

Appendix C: Selected Items for Mathematics

Achievement Test

236

Appendix D: Key of Achievement Test MCQs 237

Appendix E: Demographic Information of Experts 238

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LIST OF TABLES

Table No. Title Page No.

2.1 Differences and Similarities in Co-teaching and Team teaching 19

3.1 Sampling Detail of Students and Teachers 45

3.2 Qualifications and Experience of Practicing Teachers 46

3.3 Table of Specification of Mathematics Achievement Test (MAT) 50

3.4 Table of Specification of MAT Showing Number of Items with

Respect to Learning Outcomes

51

3.5 Detail of Judges’ Responses on 52 Items Selected on the Bases of

Statistical Values and Content Strands

53

3.6 Details of Groups 67

4.1 Number of Subjects in Control and Experimental Group 73

4.2 Mean Scores of Pre-tested and Not Pre-tested Subjects in the Control

and Experimental Groups

74

4.3 Normality of the Data of Control and Experimental Groups 76

4.4 Normality of the Data of Pre-tested and Not Pre-tested Subjects 78

4.5 Mean Scores of Experimental, Control, Pre-tested and Not Pre-tested

Groups

80

4.6 Difference Between Mean Achievement Scores of Control and

Experimental Groups

81

4.7 Comparison of the Conceptual Understanding Ability Mean

Achievement Scores of Students Taught through Collaborative

Teaching and those Taught through Traditional Teaching

83

4.8 Comparison of the Procedural Knowledge Mean Achievement Scores

of Students Taught through Collaborative Teaching and Traditional

Teaching

84

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4.9 Comparison of the Problem Solving Ability Mean Achievement

Scores of Students Taught through Collaborative Teaching and those

Taught through Traditional Teaching

85

4.10 Comparison of the Students’ Mean Achievement Scores in Algebra

Taught through Collaborative Teaching and those Taught through

Traditional Teaching

87

4.11 Comparison of Mean Achievement Scores in Geometry of Students

Taught through Collaborative Teaching and those Taught through


88

4.12 Comparison of the Conceptual Understanding Mean Achievement

Scores of Students Taught through Collaborative Teaching and

Traditional Teaching in Algebra

90


of Students in Algebra Taught through Collaborative Teaching and


91

4.14 Comparison of the Problem Solving Mean Achievement Scores of

Students in Algebra Taught through Collaborative Teaching and those

Taught through Traditional Teaching

92

4.15 Comparison of the Conceptual Understanding Mean Scores of Student

in Geometry Taught through Collaborative Teaching and those Taught

through Traditional Teaching

93


of Students in Geometry Taught through Collaborative Teaching and

those Taught through Traditional Teaching

94

4.17 Comparison of the Problem Solving Mean Achievement Scores of

Students in Geometry Taught through Collaborative Teaching and

those Taught through Traditional Teaching

95

4.18 Number of Students Interviewed before, during, and at the end of

Intervention

97

4.19 Change in Beliefs of Students about Mathematics and Mathematics

Teaching

107

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1 Detail of Sub-topics Contained in the CMTM 160

2 Selected Items for Mathematics Achievement Test 236

3 Key of Achievement Test MCQs 237

4 Demographic Information of Experts 238

LIST OF FIGURES

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ACRONYMS

Fig.

No.

Title Page No.

2.1 Levels of involvement of teachers while teaching

collaboratively

17

3.1 Groups formation in Solomon Four-Group research design 44

4.1 Box plot of control and experimental groups 74

4.2 Q-Q plot for experimental group 77

4.3 Q-Q plot for control group 77

4.4 Q-Q plot for not pre-tested subjects 79

4.5 Q-Q plot for pre-tested subjects 79

4.6 Mean scores of students of control and experimental

groups on conceptual understanding, procedural

knowledge, and problem solving mathematical

proficiencies

86

4.7 Mean scores of students of control and experimental

groups on the content strands of algebra and geometry

89

4.8 Mean scores of students of control and experimental groups

on mathematical content strands and proficiencies

96

4.9 Students’ beliefs about mathematics 108

4.10 Students’ beliefs about the teaching of mathematics 108

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NEAS National Educational Assessment System

CMTM Collaborative Mathematics Teaching Module

CT Collaborative Teaching

DSD Directorate of Staff Development

MAT Mathematics Achievement Test

UNESCO United Nations Educational, Scientific and Cultural

Organization

AERA American Educational Research Association

CHAPTER 1

INTRODUCTION

The rise and fall of societies is a direct outcome of education and educational

systems. The needs and demands of every new era are different from the previous

ones and consequently every new era needs a more developed educational system. If

the educational system for a particular society cannot meet the challenges of its time,

that society will vanish from the map of the world. Thus, it is very important to

understand those emerging demands and act accordingly. After the independence of

Pakistan, it was soon realized that Pakistan's education system was not based on

realistic objectives. The system, with its emphasis on liberal arts, was more geared to

serving colonial purposes. It served the objectives of a colonial empire, but major

modifications were urgently required to customize it according to the needs of the

new independent state. So, the education system needed overhauling and

restructuring, with a greater emphasis on mathematics, science and technology

(National Education Conference, 1947).

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According to the Longman Dictionary, mathematics includes algebra,

geometry and arithmetic, and it is the science of numbers and shapes. The Free

Dictionary explains that “the subject of mathematics is divided into Arithmetic, which

studies numbers; Geometry, which studies space; Algebra, which studies structures;

and calculus.” It is a sequential subject, and in mathematics it is difficult to follow the

topic when the topics related to it are not properly understood. One can’t follow

multiplication and division unless he knows addition and subtraction. Simple interest

discount and stock can’t be understood unless one knows percentage.

Mathematics is an important subject for development of reasoning faculties in

human beings. The famous educational theorists, Herbart, Froebel, and Montessori

recognized the importance of mathematics; in their opinion the intellectual and

cultural development of an individual were not likely to progress without studying

mathematics (Yasoda, 2009). Moreover, it plays an important role to develop

thinking, reasoning and problem solving abilities, and these abilities enable human

beings to become good citizens. Every businessperson, accountant, engineer,

mechanic, farmer, scientist, shopkeeper and even street hawker requires and uses a

knowledge of mathematics in everyday life. It is an important subject, for the study

of science subjects like Physics, Chemistry etc. that depend on the formulae and

equations of mathematics.

Mathematics has many characteristics, such as its peculiar language and

symbols that distinguish it from other subjects. Mathematics involves abstraction in

concepts. Due to its abstract nature, unique language and symbols, students face

difficulty in learning mathematics. In 2002, Western Great Blue Hills (WGBH)

Educational Foundation identified problems in mathematics learning such as errors

due to misread signs, difficulty in connecting the abstract aspects of math with the

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reality, difficulty in comprehending the relationships between quantities and the

numbers they embody, problems in visualizing mathematical concepts, difficulty in

remembering previously encountered patterns, difficulties in completing sequences of

multiple steps, difficulty in identifying the main features of a mathematical situation

(especially in word problems), difficulty with mathematics vocabulary, difficulty in

reading texts to direct students’ own learning, difficulties in interpreting and

manipulating geometric configurations.

Many problems which students face in learning mathematics are attributed to

teaching (Russell, 2006); the problem of poor or not clearly visible handwriting of the

Math teacher, the teacher’s answering the students’ questions at the same time when

busy in solving mathematical problems, students’ feeling of fatigue or exhaustion by

boring endeavor of copying for long periods of time, the difficulty in copying answers

correctly because of the writing speed of the teacher, the lack of teachers’ sufficient

interest in the problem to inspire the required mental efforts by the students, the fear

and anxiety of mathematics, lack of the understanding of mathematical concepts due

to a large number of students in a class (as a single teacher may not be able to focus

on individual student’s learning), and boredom due to one teaching style such

problems that most probably are caused by single teachers’ teaching. It is difficult for

a single teacher to cope with all of these problems effectively. A single teacher

usually has less interaction with all the students in a large class, less time available, as

well as less energy, knowledge, and techniques. Maeroff (1993) reported that

collaboration between teachers is an approach, contrasted with a traditional, “isolated”

one, to work in schools. Researchers Doebler, Smith, Hammer, & Giordano have

recommended that in contrast with single teacher teaching, team teaching, a model of

collaborative teaching, is valuable (Wadkins, Wozniak, & Miller, 2004). Robinson

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and Schaible (1995) claimed that “collaborative teaching helps us to overcome the

frequent sense of isolation felt by many teachers” (p. 59). Also in the context of

autonomy and control by a teacher, “Smylie et al. (1996) found that individual teacher

autonomy was negatively associated with student achievement. Conversely, team

control over resources and accountability for outcomes was positively associated with

student success.” (as cited in Marble & Green, 2011). Austin and Baldwin (1991)

suggested that “collaborators tend to be more creative and less averse to risk than

those who work alone" (p. 83).

In Pakistan, mathematics teaching, predominantly, is a process of transmission

of knowledge rather than a process of construction of concepts. Mathematics teachers

teach from text books without relating the concepts to everyday life. They start

lessons by dictating formulae and ask students to memorize those formulae in order to

solve questions. The mathematics teacher does not collaborate with colleagues to

discuss concepts or methodology of teaching mathematics content. As a result, the

quality of their teaching is not improving; thus, the problems like lack of

understanding of mathematical concepts and its applications and low achievement

persist. So, keeping in mind the problem of student learning due to limitations of a

single teacher and the difficulty in learning the concepts of mathematics, there is dire

need to change the teaching approach in mathematics in Pakistan.

Various mathematics teaching approaches are in use all over the world such

as direct instruction (teacher centered), collaborative teaching, shared mathematics, as

well as student centered, content focused, and classroom focused instruction (Kuhs &

Ball, 1986). Within these approaches, several teaching methods such as the lecture,

inductive, deductive, analytic, synthetic, project, problem solving, activity based,

simulations, heuristic (discovery), dogmatic, topical, laboratory, spiral, brain

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storming, and learning by doing are in use across the world. These methods are

complemented by various mathematics teaching techniques such as oral work, written

work, drill work, homework, self-study, group work, and supervised study (Papola,

2005; Sindu, 2010; Rohatgi, 2005).

The practice of using collaborative teaching in delivering instructions is found

in most of the countries of the world especially in USA, UK, China, Australia, and

Canada. There are many research studies from those developed countries which have

shown the positive effect of collaborative teaching on students’ academic

achievement (Mcduffe, Scruggs, & Mastropieri, 2007).

In order to understand the concept of collaborative teaching, it is necessary to

look at the origin of the word collaboration. It comes from co-labor which means

work together. Cook and Friend (1995) as cited in Glaeser (n. d.) defined

collaborative teaching as “a style of interaction between at least two co-equal parties

voluntarily engaged in shared decision making as they work toward a common goal”

(p. 3). Moreover, it is a process in which two or more teachers plan, present, and

assess classroom instruction. Their role varies based on the lesson activities and

teachers’ specific strengths in instruction. Collaborative teaching shows fruitful

results when it is carried out with vigilant planning and considered decision making.

Parrott (n. d.), an instructor at the University of Richmond, sees collaborative

teaching as proactive and reflective process. She was of the view that for

collaborative teaching, the focus should be on five key elements i.e. Collaborative:

presence, planning, presenting, problem solving, and processing. Researchers such as

Novicevic, Buckley, Harvey, & Keaton, stated several advantages of collaborative

teaching (Clarke & Kinuthia, 2009). First, different teaching styles enhance students’

academic achievement; they improve the capability of students to: critically evaluate

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problems, give reasons, and apply concepts effectively in different situations.

Secondly, faculty members develop mutual trust and respect for each other while

covering their specialty areas. They also get fruitful information concerning content

and teaching styles while collaborating with each other. Collaborative teaching

enhances the quality of teaching by transforming it into a more participative activity.

Thirdly, it is a means to achieve enhanced teaching outcomes because of its peer-

reviewed and monitored nature. Its strength lies in the combined forces applied to

address common goals or problems. Also, Murawski (2006) explored the notion that

teachers and students get benefits from working collaboratively through co-teaching.

Researchers like George and Davis-Wiley (2000), Jang (2006), Parker (2010),

Goddard, Goddard, and Moran (2007), Rigdon (2010), Almon and Feng (2012),

Witcher and Feng (2010), Wilson and Martin (1998), and Olverson and Ritchey

(2007) explored positive effect of teacher-teacher interaction on students’ academic

achievement.

So, collaborative teaching, having advantages and positive effects on

academic achievement of students, is a teaching approach which can cope with the

problems of learning and teaching mathematics. In the perspective of collaboration,

some research studies have been conducted in Pakistan but more focus was on

collaborative learning than teaching. Examples include research studies conducted by

Iqbal (2004), Khan (2008), Ahmad (2014), Qaisar (2011), Louise (1995), Akhtar,

Perveen, Kiran, Rashid, and Satti (2012). Less research has been done on

collaborative teaching in Pakistan e.g. Haider (2008), and Abbas & Lu (2013). Thus,

there is need to investigate the effectiveness of the phenomenon of collaborative

teaching in the context of Pakistan. It may prove its positive impact on student

academic achievement.

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RATIONALE OF THE STUDY

Many teaching approaches have been adopted by teachers across the world for

teaching mathematics. In Pakistan, the teaching of mathematics remains

predominantly a process of transmission of knowledge, rather than a process of

concept construction. The traditional method of teaching mathematics, mostly the

deductive method, is used by a single teacher in the Pakistani classroom. The focus

of instructors is to teach the textbook material without relating the concepts to

everyday life. Usually, they start their lessons by dictating formulae and requiring

students to memorize those formulae in order to solve exercise questions. They don’t

explain the background of the topics and generally do not explain the concept behind

those topics. In Pakistan, which is a developing country, the mathematics teachers do

not collaborate with their colleagues to discuss mathematical concepts, the

methodology of their teaching, assessment techniques, problems of students in

learning mathematics, or planning of lessons. As a result, the quality of their

teaching is not improving, and the problems existing, such as a lack of understanding

of mathematical concepts and performing inadequately on math tests, persist.

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In contrast to the deductive method of teaching used by single teachers,

collaborative teaching with different methods of teaching, such as inductive, analytic,

synthetic, project, and problem solving, is a recent approach that has proven its

effectiveness in USA, UK, China, and Canada. Some researchers suggest that team

teaching, a form of collaborative teaching, is better than one teacher teaching

(Wadkins et al., 2004). Collaborative teaching is a model in which two or more

teachers collaborate with each other in planning, presenting, and evaluating students.

When mathematics teachers plan, they share knowledge and information with each

other for enhancing the academic achievement of students.

Therefore, keeping in view the importance of the subject of mathematics as

well as collaborative teaching having several advantages such as the opportunity to

use different teaching styles to enhance student academic achievement, it improves

the capability of students to: critically evaluate problems, give reasons, and apply

concepts effectively in different situations. Secondly, faculty members develop

mutual trust and respect for each other, while covering their specialty areas. They

also receive fruitful information in their content areas and teaching styles while

collaborating with each other. Collaborative teaching enhances the quality of

teaching by transforming it into a participative activity. Lastly, it is a means to

achieve enhanced teaching outcomes because of its peer-reviewed and monitored

nature. Its strength lies in the combined forces applied to address common goals or

problems. So, in the Pakistani mathematics teaching context, collaboration between

mathematics teachers may produce a positive impact on student academic

achievements; it may also be helpful in reducing student learning problems in

mathematics which are common when a single teacher is teaching.

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STATEMENT OF THE PROBLEM

The subject of mathematics plays a vital role in student development in

reasoning, logical thinking, and problem solving skills. In Pakistan, predominantly a

deductive method is used for teaching mathematics by a single teacher in one

classroom. In the USA, UK, China, Australia, Canada and other developed countries,

teachers are also using collaborative teaching models in delivering instruction. Many

research studies from these countries have shown a positive effect of collaborative

teaching on student academic achievement. This study was aimed at designing a

module and teaching mathematics to 8th grade students using collaborative teaching.

The study examined its impact on student mathematics achievement in Punjab.

OBJECTIVES

The objectives of this study were to:

1. Design a module for teaching of selected content of grade 8th mathematics

using collaborative teaching.

2. Compare the impact of collaborative teaching and traditional teaching on

students’ overall achievement in mathematics.

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students’ achievement in the selected mathematical proficiencies i.e.

conceptual understanding, procedural knowledge, and problem solving.


students’ achievement in the two selected content strands of mathematics

i.e. algebra and geometry.

5. Examine student beliefs about mathematics and teaching of mathematics

in collaborative settings.

HYPOTHESES OF THE STUDY

The hypotheses of the study were as follows:

H01: There is no significant difference between the mean achievement scores of

students who are taught through collaborative teaching and traditional

teaching.

H02: There is no significant difference between the mean achievement scores in

conceptual understanding of students who are taught through collaborative

teaching and traditional teaching.


procedural knowledge of students who are taught through collaborative

teaching and traditional teaching.


problem solving of students who are taught through collaborative teaching and

traditional teaching.

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H05: There is no significant difference between the mean scores of students who are

taught through collaborative teaching and traditional teaching on achievement

test items of algebra.


taught through collaborative teaching and traditional teaching on achievement

test items of geometry.


taught through collaborative teaching and traditional teaching in conceptual

understanding in algebra.


taught through collaborative teaching and traditional teaching in procedural

knowledge in algebra.


taught through collaborative teaching and traditional teaching in problem

solving in algebra.


taught through collaborative teaching and traditional teaching in conceptual

understanding in geometry.


taught through collaborative teaching and traditional teaching in procedural

knowledge in geometry.


taught through collaborative teaching and traditional teaching in problem

solving in geometry.

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SIGNIFICANCE OF THE STUDY

The present study examined the impact of the use of collaborative teaching on

students’ achievement in mathematics. The study is significant as in Pakistan

mathematics teaching is being practiced by single teachers. Different research studies

show that it is not an effective approach to teach mathematics (Smylie et al., 1996 as

cited in Marble & Green, 2011). The alternative technique to single teacher teaching

is collaborative teaching which showed effectiveness in mathematics teaching in other

countries. This is also an emerging approach of mathematics teaching in Pakistan but

no teaching-learning material is available for teachers to follow. This study was an

experimental research study and the collaborative teaching intervention was done in

mathematics classes, after developing a module for teaching the selected contents. To

verify the findings of this study it may be replicated by using the same design. The

findings of this study may provide guidelines and directions for the policy makers and

curriculum developers to incorporate collaborative teaching approach in the

mathematics curriculum. The collaborative mathematics teachers’ module for algebra

and geometry developed in this study may be helpful for developing collaborative

mathematics modules for other topics and for different grade levels in Pakistan. It

may be helpful for the teacher training institutes to train mathematics teachers for

30

collaboration. It may be helpful for the mathematics teachers to modify their teaching

approaches, methods, and strategies to teach algebra and geometry. Likewise, it may

be helpful for mathematics students to know and understand collaborative teaching

and its effectiveness in learning mathematics.

DELIMITATTIONS

Keeping in view time and financial constraints the study was delimited to:

Public elementary school for boys

Following two content strands of Mathematics.

1. Algebra

2. Geometry

OPERATIONAL DEFINITIONS OF TERMS

Collaborative Teaching (CT)

For this particular research study a co-teaching model of collaborative

teaching was used. The co-teaching means two teachers teaching with collaboration

to the students. They collaboratively plan, organize, deliver lessons and assess

student achievement. Also they both remain present in the same class at the same

time. Both teachers with their agreement may use inductive, analytic, project,

synthetic, problem solving, learning by doing, and activity based teaching methods.

31

Traditional Method of Teaching

The traditional method is the deductive method of teaching mathematics by a

single teacher without any collaboration with other teachers of the subject; it is being

widely used in Pakistan. The teacher starts a lesson by dictating a formula on the

chalk board and asking the students to memorize it in order to solve exercise

questions.

Students’ Achievement

For this study, student achievement is measured by student mean scores

obtained in the mathematics achievement test.

Development of the Dissertation

The development of the thesis is as follows:

Chapter 2 is the literature review that presents detailed description of

collaborative teaching, its models, advantages, disadvantages, challenges, and the

ways to implement and make collaborative teaching effective. Chapter 3 is about

methodology: it presents the design of the study, details of population and sample,

mathematics achievement test, collaborative mathematics teaching module, process of

the experiment, ethical considerations, and details of confounding variables. Chapter

4 presents the data analysis and displays the findings of the study. Chapter 5 provides

a summary of the study, findings, conclusions and recommendations for policy and

further researches in the area of collaboration.

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CHAPTER 2

LITERATURE REVIEW

This chapter contains a review of relevant literature which provides a

theoretical basis for this study. In addition to printed literature sources, I have gone

through extensive digital database searches from Educational Resources Information

Center (ERIC), Pro-Quest, JSTOR, SAGE publications, and Google Scholar. The

purpose of this chapter is to discuss the literature regarding the concept of

collaborative teaching (CT); differences between models of collaboration i.e. co-

teaching and team teaching, and the origin, development, models, and research in CT.

It also treats the literature concerning approaches, methods, and strategies of teaching

mathematics as well as the characteristics of a good collaborative teacher, effective

co-teaching pedagogy, the implementation of CT, the advantages and disadvantages

of the practice, and challenges inherent in collaboration. Also, it considered the

research concerning student beliefs about mathematics and learning mathematics in a

collaborative setting.

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Collaborative Teaching

CT is a form of teaching which is based on collaborative settings.

Collaboration employs individuals who are somewhat equal but with different roles

working together to attain a unified purpose with conscious effort (Sperling, 1994).

Donaldson and Sanderson (1996) perceive collaboration as not only a state but as a

process. According to these researchers working collaboratively has two components:

one is a relationship among collaborative teachers, which is one of respect, and the

second is the process which helps them to do their work. Gray (1989) defined

collaboration as “a process through which parties who see different aspects of a

problem can constructively explore their differences and search for solutions that go

beyond their own limited vision of what is possible” (p. 5). Further, the concept of

collaboration is connected to the constructivist approach. Bartlett, in 1932, initiated

the idea of constructivism (Good & Brophy, 1990). The idea was that the individual

should construct his/her own reality (Jonassen, 1991). The most often cited

philosophical founder of this approach is John Dewey. In addition, Brunner’s (1996)

ideas on different styles of teaching supported the idea of constructivism.

Collaborative teaching is defined as an interactive style between two or more

co-equal individuals willingly involved to work on common goals and decision

making (Cook & Friend, 1995; Friend & Cook, 2003; Georgia Department of

Education (GDOE), 2006). According to Austin and Baldwin (1991) collaboration

(teacher-teacher) involves components such as coordinated effort, common goals, and

outcomes. Moreover, it is a process in which two or more teachers collaboratively

plan, present, and assess classroom instructions. Their role varies and is based on the

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lesson activities and teachers’ specific strengths. Wiedmeyer and Lehman (1991) also

defined CT as "a cooperative and interactive process between two teachers that allows

them to develop creative solutions to mutual problems" (p. 7). Parrott (n. d.), an

instructor in the University of Richmond, sees the collaborative teaching as a

proactive and reflective process. She was of the view that for collaborative teaching,

the focus should be on five key elements:

Collaborative Presence

Collaborative Planning

Collaborative Presenting

Collaborative Problem Solving

Collaborative Processing

The detail of collaborative teaching was shown in Figure 2.1.

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Figure 2.1. Level of involvement of teachers while teaching collaboratively

Collaborative Teaching and Co-teaching have been successfully used in an

extensive range of areas, including the programs of foreign languages (Greany, 2004),

subjects of science and mathematics (Jang, 2006), and interdisciplinary courses

(Davis, 1995; Shibley, 2006). According to Smith and Scott (1990) "Collaboration is

being increasingly recognized as not only a desirable but an essential characteristic of

an effective school." Additionally, co-teaching is using at tertiary level of education

(Davis, 1995; Shibley, 2006; Wilson & Martin, 1998), in Western and Asian countries

as well (Tajino & Tajino, 2000; Tajino, 2002; Jang, 2006).

Dieker (2001) found several types of teams engaging in relationships of co-

teaching: educators from general and special educations, two teachers from general

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education, an educator from special or general education and a community worker, a

volunteer and an educator from special or general education, teachers of music,

computers, art, etc. and an educator from special or general education. Teams of co-

teachers come together for a common purpose i.e. to deal learners with wide range

effectively. They may have a long-term agenda for working together (an entire

academic year) or short-term agendas such as completing a unit together or preparing

students for some specific skills (e.g., state testing, science project). Nevertheless, it is

relevant style of teaching geared toward changing reality of everyday life’s

(Wlodarczyk, 2000).

Difference between CT Models: Co-teaching and Team Teaching

Team teaching and co-teaching are well recognized forms of teachers’

collaboration (NCIPP, 2010). “Collaborative teaching” is often used interchangeably

with “co-teaching” (Morsink, Thomas, & Correa, 1991; Gerber & Popp, 1999;

Witcher & Feng, 2010) and “team teaching” – in which more than one teacher is

involved. The differences and similarities between team teaching and co-teaching are

given in Table 2.1.

Table 2.1

Differences and Similarities in Co-teaching and Team teaching

Team teaching Co-teaching

Student teacher ratio

30: 1 or

25: 1 Coffey (n. d.)

30: 2

http://www.learnnc.org/lp/people/1227

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Number of teachers Two or more than two Two

Responsibilities Distributed, for example one is

speaking while the other is

writing on the board

Distributed, for example one

is teaching while other is

assisting students.

Presence of teachers May or may not be in the same

classroom. Might be involved in

other activities outside the class

like planning etc. (Taylor &

Biddulph, 2001 )

In the same classroom

Teaching lesson Teachers teach same lesson at

the same time. For example one

writes on the chalk board other

shows related figure on chart.

May or may not teach the

same lesson together. It may

be one teaching and one

assisting or one teaching and

one observing

Lesson Lengthy and in-depth material

Requires additional hands or

assistance in the classroom

Leader There is a leader No leader

Origins and Development of Collaborative Teaching

The theoretical foundation of CT lies in the early 1960s. William M.

Alexander, known as the “father of the American middle school,” discussed the

structure of the junior high school at a conference at Cornell University and he

proposed the idea of team teaching, a model of CT, with three to five teachers

assigned to seventy-five to one hundred fifty students organized either in a single-

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grade or multi-grades. Team teaching is now used in all grade levels and across

disciplines (Coffey, 2008).

In the late 1970s the term co-teaching, a model of collaborative teaching, was

well studied and the term “team teaching,” two or more teachers teaching together,

has been synonymously used with co-teaching. Reinhiller (1996) noted that co-

teaching is also called collaborative teaching or co-operative teaching, but it was

known in the 1970s as team teaching. Walther-Thomas (1997), Jang (2006), and

Coffey (2008) contended that co-teaching, team teaching and cooperative teaching

refer to similar instructional delivery systems. In essence, these terms refer to two or

more teachers teaching to the same group of students through collaboration (Liu,

2008). Co-teaching practice initially emerged in the USA from general education

(Dieker & Murawski, 2003), to address issues related to the teaching of special

students in inclusive classrooms (Stanovich, 1996; Gately & Gately, 2001; Cook &

Friend, 1995; Vaughn, Schumm, & Arguelles, 1997; Keefe & Moore, 2004; Dieker,

2001; Tobin, 2005). In fact, in 1975 the US Congress passed public law 94-142, the

Education for All Handicapped Children Act, and in order to meet the requirement,

the co-teaching model of collaborative teaching was used to provide education to all

the children in inclusive classes.

Research in Collaborative Teaching

Most of the research studies in the context of collaboration are qualitative in

nature. Less work has been done using a quantitative approach. Moreover, research

studies focused on the nature of collaborative relationship; how to make CT effective;

training of co-teachers; perception and beliefs of parents, students, and teachers

towards CT; issues and problems in implementing CT; experiences of teachers in



http://en.wikipedia.org/wiki/Education_for_All_Handicapped_Children_Act

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collaborative settings; collaborative teachers’ efficacy; and challenges to CT. The

exploration of student’s achievement in mathematics using CT is becoming the center

of attention of the researchers and also recommended by Hill (2012). Research studies

of Magiera, Smith, Zigmond, and Gebauer (2005), Murawski (2006), Rea,

McLaughlin, and Walther-Thomas (2002), Scruggs, Mastropieri, and McDuffie

(2007), have found that it can be an effective method to fulfill the needs of students in

learning mathematics.

It is believed that CT can provide teachers with more opportunities to get

involved, to overcome teaching difficulties, to stimulate the growth of professional

knowledge and abilities, and to learn from each other (Moran, 2007; Trent et al.,

2003; Huffman & Kalnin, 2003; Rathgen, 2006). It is believed that CT has more

positive influences on learning than individual teaching does (Bullough, Young,

Birrell, Cecil, & Winston, 2003; Hoogveld, Paas, & Jochems, 2003; Vidmar, 2005).

In the perspective of collaboration a few research studies have been conducted

in Pakistan. For example, the PhD dissertations of Iqbal (2004), Khan (2008), and

Ahmad (2014) focused on examining the effectiveness of collaborative and

cooperative learning to gauge students’ achievement. Likewise, Ahmad (2010)

examined the effectiveness of cooperative learning on achievement and learning

experiences of prospective teachers, and Qaisar (2011) focused on examining the

effect of student work in groups through collaboration for concept development in the

subject of mathematics, Louise (1995) explored problems and possibilities in

implementing cooperative learning in classrooms, and Akhtar et al. (2012) investigated

student’s attitudes towards cooperative learning. However, Abbas and Lu (2013), and

Haider (2008) conducted researches in the context of collaborative teaching. Abbas

and Lu (2013) explored the attitudes of teachers towards collaborative teaching. On

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the other hand, Haider (2008) identified problems faced by collaborating teachers

during teaching practicum in pre-service teacher education.

In an international scenario, Jang (2006) examined the effect of team teaching

on students’ achievement. He used a quasi-experimental design on 8th grade students

of mathematics. Two certified math teachers and four classes participated in this

study. The study was a two-stage team teaching experiment, dividing the 12-week

period into two equal halves. The main research method was a combination of

quantitative and qualitative analysis. The collected data included student scores,

questionnaires, teachers’ self-reflection, video-taped records of teaching performances

and the researcher’s interviews with teachers. He found that there was significant

difference in team teaching and traditional teaching on students’ achievement in

mathematics. The mean scores on achievement test were 74.13 and 73.94 for

experimental groups which was higher than the mean scores 69.87 and 70.52 of

control groups. It seems like good research, but it would have been more interesting if

the researchers used true experimental design leading to more extensive

generalization of the results as in true experimental design a researcher needs to select

and assign the subjects randomly into control and experimental groups.

Olverson and Ritchey (2007) conducted action research and found that

collaboration between the teachers had significant impact in raising the students’

achievement. Murawski and Swanson (2001) carried out an analysis of all co-teaching

studies which indicates that co-teaching has a positive effect on student achievement

in mathematics. Since it was a meta-analysis and research studies were qualitative, it

is hard to strongly recommend that co-teaching is an effective teaching approach in

mathematics teaching. Also George & Davis-Wiley (2000) conducted a study with a

teaching team composed of one senior faculty member and one graduate student, each

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sharing the teaching responsibilities and it was concluded that team teaching showed

more fruitful results in students’ learning (Lester & Evans, 2009). Walther-Thomas

(1997) credited the students’ gain with better monitoring, which happened when one

teacher teaches and other monitors and assists the weaker students.

Parker (2010) investigated the impact of co-teaching on student achievement.

In this study, students who received instruction through co-teaching during their 10th

grade year in a mathematics class were compared with other 10th grade students

receiving instruction without co-teaching. An achievement test was used to determine

the impact of co-teaching on students’ efforts. The findings revealed that there was a

significant difference in the achievement of mathematics classes through co-teaching

and without co-teaching. He explored the positive effect of co-teaching on the

students’ achievement in mathematics at the 10th grade level. The study of Parker was

interesting and a good addition to the literature on co-teaching. It was important too

because there is less research conducted at this level in co-teaching context. In the

context of collaboration, Goddard et al. (2007) tested the relationship between teacher

collaboration in schools and students’ achievement in mathematics. They collected

data from elementary schools of the Midwestern United States and found that the

students obtained higher scores in the schools where the teachers taught through

collaboration.

Wilson and Martin (1998) started their CT at Muskingum College, Ohio.

They described their strategy: “co-teaching, you do this, and I’ll do that, strategy

which Bocchino and Bocchino (1997) called a tag team” (p. 6). They added strategies

such as “speak and chart,” “comment and perform,” and “add and speak.” They

found that collaborative effort resulted in improving students’ grades. They claimed

that “We each serve as the other’s sounding board, particularly in matters of student

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testing and grading. With two of us examining a situation, students are held to high

academic standards while being guaranteed that their work is evaluated fairly” (p. 9).

Witcher and Feng (2010) did research to compare co-teaching and solo

teaching on increasing the students’ achievement in mathematics at the 5th grade level.

They included two 5th grade classes in one school in the study. Achievement was

measured by using standardized and non-standardized tests. Independent samples, t-

tests, were used to compare math achievement scores of students. Findings of the

study indicated that co-teaching appears to have positive effect on students’

mathematics achievement at the elementary level. In this research the researchers had

limitations such as intact classes, no random selection of subjects, and small sample

size, and they suggested further research in this area.

In the context of students’ assessment of innovative teaching strategies in

enhancing mathematics achievement, Agommuoh and Ifeanacho (2012) did a survey

design research study. They used a purposive sampling technique to select a sample of

students. A self-developed questionnaire of the Likert type was used to identify the

teaching strategy which enhances achievement in mathematics. The reliability of the

instrument was 0.86. Data was analyzed by using mean and chi-square statistics.

Findings showed that team teaching was agreed to be an innovative teaching strategy

that could enhance students’ achievement in mathematics. They recommended that

mathematics teachers should use the team teaching strategy. It was a survey-based

study and teaching strategy was identified in the assessment of children, but we

cannot yet say that it is an effective strategy of teaching mathematics to enhance

students’ achievement.

Almon and Feng (2012) did research to compare the effects of solo-teaching

and co-teaching on students’ achievement in mathematics. It was an experimental

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study on fourth grade students. The achievement test was comprised of number sense,

multiplication, and division. Findings showed that solo teaching produced better

results than co-teaching in multiplication. On the other hand co-teaching was better in

the number sense section. They suggested that further experimental research is needed

to compare the effectiveness of co-teaching.

Likewise, the purpose of the research conducted by Rigdon (2010) was to

investigate the impact of co-teaching on regular education students’ achievement in

algebra at eighth grade. They used a mixed methods design, and data was collected

conveniently. They did an experiment and conducted interviews too. Students’

achievement in mathematics was measured through an Algebra assessment given

before and after the treatment of 12 weeks. ANOVA was used to assess the

differences. Study findings revealed that students taught through co-teaching obtained

better grades on algebra assessment as compared to those taught by single teachers.

Further interview data indicated that the teachers’ perception of student learning was

greater in the co-taught classroom.

Collaborative Teaching Models

There are many models consisting of collaborative settings. The details of

some popular models of CT are given below:

The Lead Teacher

Logsdon (2011) used Lead Teacher model of CT in classes with

mainstreaming special children. According to him in classrooms with a lead teacher,

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the regular classroom teacher, often delivers the instruction in the subject area. The

special education teacher is an observer who works with children after instruction to

provide designed instruction, to ensure understanding, and to provide adaptations and

modifications. A similar model was adopted by IOWA in 2004 from Marilyn Friend’s

co-teaching models.

Station Teaching

This model was also adopted by IOWA (2004) from Marilyn Friend’s co-

teaching models. In this model teachers are at stations, and students move from

station to station. The teachers divide the physical arrangement of the room into three

sections, two that support teacher-directed instruction and one for independent

seatwork. Course content and class work are also divided into three distinct ‘lessons’

that do not have to be completed in a particular order. One lesson is taught by each of

the two teachers, and the third lesson consists of a seatwork assignment that students

complete independently or with minimal supervision. The students in the class are

assigned to three separate groups, and each group rotates through each of the three

teaching stations. The composition of the groups can be homogeneous or

heterogeneous. A similar model was given by Logsdon (2011); he noted that each

teacher is responsible for instruction in a specific area of the room. Students are

assembled into groups that rotate through the centers for instruction.

Co-Teaching

Chapple (2009) mentioned the definition of co-teaching by Villa, Thousand &

Nevin as “two or more people sharing responsibility for teaching some or all of the

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students assigned to a classroom” (p. 5). For an effective co-teaching process, this

model features shared responsibility for the co-teachers in all aspects of teaching,

such as lesson planning, delivering of instruction, and evaluating student progress.

Also, the literature identified several critical elements of co-teaching such as

collaboration, planning, the roles of teachers in a co-teaching classroom, the nature of

co-teaching practices, and the process of implementation of co-teaching. Friend and

Cook (2000) described different ways to implement co-teaching, like one teaches and

one observes, one teaches and one assists, parallel teaching, station teaching and

alternative teaching.

In the one teaches one observes setup, one teacher is providing the instruction

while the other member is observing the students in the classroom. Teachers should

decide what behaviors for specific students need to be analyzed and what teaching

method they should use to record observations. According to Cuellar (2011), this

model should be used when teachers want to collect data about students’ behavior or

academic learning.

In the second model of co-teaching i.e. one teaches and one assists, one

teacher manages the class and he or she has the most important duty for planning and

teaching. At the same time second teacher moves within the classroom and helps

students (Friend & Cook, 2000). There are some advantages of the model, such as

timely and individual help for the students, second teacher monitoring of the students’

behavior which may not be seen by the single teacher, and closeness of the teacher to

help students to keep on task, as well as saved time in material distribution (GDOE,

2006). According to Wilson and Martin (1998) the advantage of collaboration is that

whenever students needed more attention, one teacher was always free to help them.

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The major disadvantage of this model is that movement of the second teacher may

distract some of the students.

In the station teaching approach both members are actively involved in

instruction. Students move from one station to the next station where instruction is

being provided. In this approach, it is helpful to have a third station where students

can work independently on an assignment or to complete independent seatwork. In

this model teachers impart pre-planned instructions while students move within

preplanned stations; it might be useful to use this design when styles of co-teachers

differ, or if teacher-student ratios are small (Cuellar, 2011). There are some

disadvantages of the model, like the importance of the pace of teaching in stations to

ensure the lectures end at the same time and increased noise; also, teaching materials

should be arranged before going to teach the students (GDOE, 2006).

Parallel teaching involves dividing a class into two separate groups and each

teacher presenting the lesson independently (Friend & Cook, 2000). Teachers being

able to work with smaller groups are the major advantage to this approach. This

model is considered good when teachers possess good subject knowledge and want to

meet students' diverse needs (Cuellar, 2011). A disadvantage is that in order to

equalize the students’ learning teachers should be competent in the area of subject

matter. The second problem with this setting is the availability of space for organizing

this kind of settings (GDOE, 2006).

In the alternative teaching approach, one member delivers instruction to the

larger group while the other member of the co-teaching team works with a small

group on something different than the rest of the class. The small group members can

be adjusted depending upon the purpose of the instruction: “For example, a teacher

could take an individual student out to catch them up on a missed assignment. A

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teacher could work, with an individual or small group for assessment purposes,

together for remedial/challenge work” (GDOE, 2006, p. 64). The major advantage of

alternative teaching is to meet the needs of the students.

Team Teaching

This involves both teachers simultaneously working together to teach students

in the classroom. In team teaching both co-teachers share teaching responsibilities and

are equally involved in leading instructional activities (Chapple, 2009; Vaughn et al.,

1997). According to Friend and Cook (2000, p.61), “team teaching requires the

greatest level of mutual trust and commitment and ability of teachers to mesh their

teaching styles.” Team teaching is characterized as taking turns in leading a

discussion or having the two teachers play roles in a demonstration. “Bauwens and

Hourcade (1997) describe team teaching as educators jointly planning and presenting

subject content to all students” (Flanagan, 2001). In team teaching, each teacher is

more likely to present the topics that truly interest him or her. When a teacher is

paired with another teacher who has more expertise on those topics, a more complete

and accurate picture can be presented to the students. Lester and Evans (2009)

believed that team teaching must extend beyond the idea of occasionally teaching

together to always teaching together. Cuellar (2011) suggested that this model is

appropriate when teachers “really hit it off; synergy and parity make or break this

approach.”(p.33). There are some advantages of the team teaching model such as,

“each teacher has an active role, students view both teachers as equals, both teachers

are actively involved in classroom organization and management, risk-taking is

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encouraged allowing teachers to try things in pairs they would not normally try

alone”( GDOE, 2006, p. 65).

Goetz (2000) stated that prior to implementation, the team members should

have sufficient professional development in the area of team teaching; they should

understand the philosophy behind team teaching. Team teaching partners need time to

foster a trusting and open relationship. For effective team planning, team members

should meet daily or weekly to make important decisions about: what will be

presented (e.g., the units, lesson objectives), what the order is to be, how the material

is to be presented, who is to present the information, how the students will be

assessed, how the class activities can be improved, what problems have arisen, and

how can these problems be solved?

Consultation

This model is adopted by IOWA (2004) from Marilyn Friend’s co-teaching

models. In this model a trainer, who knows the subject matter as well as

methodological expertise, may provide some instruction to students, but the majority

of service is indirect. The trainer teacher mostly provides guidance to the teacher on

how to modify instruction to meet the students’ needs.

Collaborative Teaching of Mathematics: Approaches, Methods, and

Strategies

Mathematics teaching plays a pivotal role in improving students’ learning of

mathematics. Mathematics teaching is dynamic as the subject of Mathematics consists

of various content strands such as algebra, geometry, arithmetic, probability, and

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information handling with different levels of difficulties in concepts. The teaching of

mathematics is unique as it has a unique language and symbols. Mathematics teachers

need to adopt different approaches, methods, and strategies to teach different

mathematical concepts.

Various mathematics teaching approaches are in use all over the world, such

as teacher centered, student centered (Banning, 2005), collaborative, content focused,

and classroom focused (Kuhs & Ball, 1986). Within these approaches, several

mathematics teaching methods such as lecture, inductive, deductive, analytic,

synthetic, project, problem solving, activity based, simulation, heuristic (discovery),

dogmatic, topical, laboratory, spiral, brain storming, and learning by doing methods

are in use across the world (Sindu, 2010). These methods are complemented by

various mathematics teaching techniques such as oral work, written work, drill work,

homework, self-study, group work (Papola, 2005; Sindu, 2010; Rohatgi, 2005),

asking questions verbally from the students during the lecture, and classroom tests.

Characteristics of a Good Collaborative Teacher

In order to be a successful and effective process of collaboration, the co-

teacher should have the characteristics like a strong psyche, strong personality

(Wilson & Martin, 1998), commitment, problem solving ability, strong interpersonal

skills, mutual trust, good communication, readiness for change (Griggs & Stewart,

1996), positive attitude, willingness (Cuellar, 2011), flexibility in philosophy of

learning and teaching (Wlodarczyk, 2000) and an excellent philosophy of teaching.

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According to Wilson and Martin (1998), it is imperative to CT that teachers should be

committed to work collaboratively to achieve shared goals. They further added that

“Both must be willing to solve problems as they arise rather than terminating the team

teaching experience” (p. 9). Robinson and Schaible (1995) noted that trust must be

developed between the teachers. They argued that disagreement between teachers in

front of students involves risks. According to Wilson and Martin, (1998) “If the

philosophies are shared, differences in the other elements-experience, backgrounds,

and approaches only enrich the team-teaching experience for both teachers and

students” (p. 8).

Students Beliefs about Mathematics

Kloosterman, Raymond, and Emenaker (1996), as cited in Ragland (2011),

defined beliefs as “the personal assumptions from which individuals make decisions

about the actions they will undertake.” Likewise, Pajares (1992) defined beliefs as

personal principles that are constructed from an individuals’ experience, often

unconsciously, and interprets new experiences and information and thus guides

action.

There are two terms that should be explained i.e. attitudes and values related

to beliefs. Attitude refers to the predisposition to respond positively or negatively

towards any certain idea, object, person, or situation, while the values are enduring

beliefs or ideals which are shared by the members of a society or culture about the

good, bad, desired and not desired. Defined in another way, value is a measure of the

worth which a person attaches to anything; our values are often reflected in our ways

of living our lives. For example, I value higher qualification, or I value my parents.

An attitude is the way people adopt to apply or express their beliefs and values using

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words and behavior. For example, “it makes me really upset to hear about the cruelty

to animals”, or “I hate playing games” (Anderson & Silva, 2009).

The researchers, Cooney (1985); Frank (1988); Spangler (1992); Thompson

(1984, 1985); Krosnick (2007); Taylor (2009); and Ragland (2011) investigated the

beliefs of students about mathematics. Keeping in view the importance of

mathematics in everyday life there is a need that students of mathematics should

achieve their learning objectives in mathematics. The majority of research in

mathematics seems to confirm that if someone has strong beliefs about one’s learning

then it has an impact on one’s achievement (Krosnick, 2007). Similarly, Ragland

(2011) found positive correlations between particular student beliefs and conceptual

learning gains. So one’s beliefs play an important role in his/ her achievement,

cognition, and behavior. Beliefs play a significant role in directing human’s

perceptions and behavior toward learning. In learning environments, students’ beliefs

might propagate the impulse to achieve as well as the smoothness of their learning.

Taylor (2009) found through empirical research that specific curricula and instruction

can change the beliefs of students about mathematics and the teaching of mathematics

and increase the performance of low-performing students, even in a short amount of

time. In the mathematics learning process, students’ belief about the nature of

mathematics and factors related to learning are two components that always concern

educators.

McLeod (1992) has suggested four categories of students’ beliefs. The first

category concerns beliefs about mathematics, such as mathematics is difficult or

follows some rules. The second category is beliefs about self which includes self-

confidence in learning of mathematics. The third category is beliefs about teaching

which includes beliefs about the desirable behavior a teacher should perform to help a

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student better learn mathematics. The fourth category is beliefs about social context. It

deals with the social influence of people such as parents and other members of society

on learning of mathematics. This category includes the beliefs about mathematics

learning as a competitive exercise.

According to Spangler (1992) there are some beliefs which the students

commonly hold. The beliefs are like thinking that mathematics is computation; the

solution of mathematical problems should not take more than five minutes and if it

takes, then there is any issue with the problem or the student who is solving it; the

sole purpose of mathematical problem solving is to get the correct answer; in the

process of teaching and learning, the teacher is active and the student is passive

(Frank, 1988). Generally, it is agreed that these beliefs are not “healthy” as they are

not conducive to the type of mathematics teaching and learning visualized in the

developed curriculum and evaluation standards for school mathematics (NCTM,

1989).

Ragland (2011) re-conceptualized the affective domain of McLeod and

divided the beliefs about social context, mathematics, mathematics teaching, and self.

In this study, belief of students was investigated in two aspects: mathematics as a

subject and teaching of mathematics. The students have mathematical beliefs such as:

mathematics is interesting subject, it is difficult subject, it is a boring subject, it is

useful in everyday life, it involves memorization to significant level etc. Most of the

mathematical beliefs of the students have strong relationship with their learning and

academic achievement in mathematics. Taylor (2009) suggested that specific curricula

and instruction can change the beliefs and increase the performance of low-

performing students, even in a short amount of time. There are many reasons to

change beliefs. For example, we want them to change or it happens in their lives.

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According to the English poet William Blake (as cited in Osment, n. d.), “The man

who never alters his opinion is like standing water, and breeds reptiles of the mind.”

Effective Co-teaching Pedagogy

Researchers Murawski and Dieker (2008) suggested in their research study on co-

teaching that co-teachers should follow the points for effective co-teaching before,

during, and after it. Such points include volunteer participation as a co-teacher,

communication of needs to the administrator, training of co-teacher, flexibility in

trying new things, informed consent, sharing strengths and weaknesses with other co-

teacher(s), demonstrating parity, sharing responsibilities (GDOE, 2006), co-planning,

and sharing the work load. Chapple (2009) stated that,

“There are many elements that must be discussed by co-teachers

prior to their implementation of this process. In preparing for co-

teaching, the teachers must discuss philosophy and beliefs of

teaching: what will the classroom routines look like? How will

discipline be handled? How and when will they find time to plan

appropriately? How will they deal with the increased noise level?

How can they provide each other feedback? How they will resolve

conflicts?” (p. 8).

Collaborative relationships, to be effective, may give rise to, contribute to, and

emerge from certain characteristics. Those characteristics include beliefs and thoughts

that support collaborations, increased mutual trust and respect, and the establishment

of a sense of community (GDOE, 2006). Additionally, for effective co-teaching, co-

teachers should contribute equally and participants have equal power in decision

making.

Moreover, effective CT takes place if implemented for an appropriate time.

Wlodarczyk (2000) identified that the amount of time for collaboration had a direct

54

influence on the effectiveness of collaborative endeavor. Different researchers took

different time periods for implementing CT. For example, Boeckel (2008) conducted

instructions of 45 minutes in one day, and he collected data in six weeks during the 4th

semester of the school year. Nevertheless, researchers and writers working on CT

have suggested appropriate times for collaboration. Austin and Baldwin (1991)

advised the collaborative teachers to spend considerable time to plan activities as a

team because the more time teachers spend on planning the more collaboration will

take place.

Furthermore, another crucial component of the collaborative process found by

Wlodarczyk (2000) is to identify and select teachers before initiating collaboration,

making sure that their styles of teaching are favorable to successful collaboration.

According to Friend and Cook (2000), “To keep co-teaching relationships and

instructional arrangements fresh and effective teachers should try several approaches

and varying their co-teaching methods regularly” (p.54).There are many mathematical

teaching methods such as inductive-deductive, problem solving, lecture, and the

activity method. Also there are different settings of co-teaching available such as

teaming, one teach and one assist, as well as parallel teaching. In order to obtain the

maximum value out of CT there is need to change teaching methods and CT settings.

In addition to this, there should be a small team of teachers to collaborate for

an effective collaborative process. Seaman (1981) identified that when he explained

that the group of teachers grows larger, it results in an increase in their problems. For

example, indirect and complex communication (as cited in Austin & Baldwin, 1991,

p. 72). Similarly, Robinson and Schaible (1995), advised, “Unless there are

compelling reasons for doing otherwise, restrict the teaching team to two. Good

collaborative teaching is too complex to do it well with more” (p. 57).

55

Implementation of CT

Before implementing CT, GDOE (2006) recommended that it’s better for co-

teachers to have a checklist covering the following aspects: a regular schedule to plan

together, lesson plans, collaborative settings, and management of teachers’ and

students’ time to ensure activities are available. Gray (1989) identified five

dimensions of the process of CT: interdependence of stakeholders; solutions emergent

through constructive dealing; collectively made decisions; collective responsibility

assumed by stakeholders; and an emergent collaborative process. The literature shows

that collaboration is not a random process. It occurs in stages which lead towards

collaboration. Austin and Baldwin (1991) suggested four steps for the CT process i.e.

selection of teachers, work division, guidelines formulation, and termination of CT.

The collaboration between teachers in the above four stages is likely to be the same as

those described by Tuchman (1996). He described the collaboration stages as

forming, storming, norming, performing, and reforming.

Friend and Cook (2000) have given some suggestions for effective

implementation of collaboration in co-teaching. There suggestions are given below:

Teachers must consider (a) willingness to let another colleague

observe a teaching content with which the teacher is not particularly

familiar, (b) willingness to consider and experiment with different

ways of teaching, (c) willingness to let someone else take

responsibility for tasks at which the teacher is particularly skilled,

(d) level of comfort about relying on someone else in a joint project,

(e) readiness to raise instructional issues with a co-teacher partner,

even if the situation is uncomfortable, and (f) preparedness to listen

and respond to instructional issues that a co-teacher partner may

raise with the teacher. (p. 63)

Additionally, GDOE (2006) suggested that when collaboration is getting started,

56

look at your instructional strategies, decide the best times for

meeting and planning, and consider and share with your partner

developed personality strengths and traits they should be aware of to

ensure successful team collaboration. (p. 65)

Moreover, Wlodarczyk (2000) claimed that two teachers have to deal

successfully in two areas, relationship issues and task issues, for CT to be effective.

He also claimed that enrichment of academic life can be achieved by managed

collaborative endeavor. When implementing CT, Robinson and Schaible (1995)

explained how the teachers give and receive constructive criticism. They can ask

certain questions of each other. For example, are we talking too much in class?

Furthermore, the format of lesson plans for co-teaching should be different from the

standard pattern of lesson plans. It should include a division of responsibilities and

definitely incorporate the collaborative teaching model in it (Cuellar, 2011).

Advantages, Disadvantages, Challenges of CT

For every kind of institute, teams achieve a variety of outcomes depending

upon the contribution of their members (Davis, 1995). Wlodarczyk (2000) noted the

personal benefits of collaborative teaching i.e. satisfaction and psychological well-

being. He argued that there is much evidence to support collaborative relationships

among teachers. Researchers Novicevic, Buckley, Harvey, & Keaton stated several

advantages of collaborative teaching (Clarke & Kinuthia, 2009). First, students have

the opportunity to see diverse teaching styles: it can lead to learners’ improved

57

capability to evaluate problems critically, to argue substantively, and to apply learned

concepts effectively to new situations or contexts. Moreover, Wlodarczyk (2000)

argued that “students improve in their ability to integrate and synthesize because they

are able to observe their instructors engaging in these processes.” (p.33)

Secondly, there are many benefits of CT for the teachers (Robinson &

Schaible, 1995). For example, teachers develop mutual trust and respect for each

other, cover their specialty areas, and learn valuable information from each other in

terms of content and teaching styles (Wlodarczyk, 2000). Also, CT enhances the

quality of teaching by transforming it into a participative activity that eliminates the

perception of being isolated (Fey, 1996). Robinson and Schaible (1995) reported that

CT helps teachers to create writing assignments. CT sparks teachers' commitment to

resolve their differences that strengthen their collaborative relationship. Thirdly, it’s a

means to achieve enhanced teaching outcomes because of its peer-reviewed and

monitored nature. Its strength lies in the combined forces applied to address common

goals or problems. Also Murawski (2006) explained that teachers and students get

benefits by working collaboratively through co-teaching. The CT model support

students as they grow emotionally, personally, socially, and culturally by being in a

regular education classroom and bonding with a variety of students instead of being

isolated in a traditional resource classroom (Boeckel, 2008). Moreover, it is the strong

perception of researchers that use of CT can improve the quality of education

(Wlodarczyk, 2000). Austin and Baldwin (1991) reported beliefs such as teachers’

collaboration motivate teachers, enhance productivity, and stimulate creativity.

According to Griggs and Stewart (1996, p. 1), “The interaction between individuals

not only lends mutual support but also provides greater vigor, energy, invention, and

enthusiasm. We learn best and accomplish more when we work together.”

58

In addition, collaborative endeavor improves classroom discipline (Wilson &

Martin, 1998). They further acknowledge that through collaboration “We become

more aware of opportunities for discipline integration” (p. 7). Robinson and Schaible

(1995) concluded that “Research on collaborative learning indicates that its benefits

for students include higher achievement, greater retention, improved interpersonal

skills, and an increase in regard for positive interdependence.” (p. 58). Nevertheless,

the fact of academic life is truly the collaboration (Austin & Baldwin, 1991).

Some of the researchers mentioned the disadvantages of CT; for example,

Davis (1995) reported dissatisfaction and frustration of collaborative teachers as there

was a lack of appreciation and support by administration of the institute and students

were confused when co-teachers explained the same concept differently (Gerber &

Popp, 1999). Furthermore, Austin and Baldwin (1991) noted that loss of autonomy of

collaborative teachers is an issue for the teachers; they might feel discomfort in the

presence of another teacher. There is less literature available to describe the

disadvantages of collaboration (AHED, 1996). Most of the literature on collaboration

focused on barriers to implementing collaboration. For example, personnel

organization, space and equipment, lack of specialized skills required in the related

fields, collaborative teaching schedule arrangement, extra burdens and lack of support

from the school administration (Lie & Xie, 2009).

Different researchers and writers identified some other challenges to CT, for

example, Conderman, Johnston-Rodriguez, and Hartman (2009) noted training of co-

teachers to develop effective communication skills is the biggest challenge for CT to

be successful. The key skill in collaboration is the ability to communicate properly,

which is a serious challenge to academic teachers (Griggs & Stewart, 1996).The

second challenge to CT is the students’ resistance to collaborative endeavors (Fey,

59

1996). Moreover, Austin and Baldwin (1991), Wilson and Martin (1998), and

Singleton (1998) wrote that amount of required planning time is the greatest barrier in

effective CT. Wilson and Martin (1998) mentioned that the tendency of students to

compare their teachers’ performance is a challenge to the collaborative endeavor.

It is concluded that CT is a kind of teaching approach in which two or more

teachers collaboratively plan, organize, present, and evaluate the students. It has many

settings such as one teach one assist; one teach one observe; teaming; parallel

teaching; alternative teaching; consultation etc. It has many advantages and has

proved its effectiveness in students’ achievement in mathematics. Most of the

research conducted on CT were qualitative in nature. This chapter provides the base

for this research study by detailed explanation about CT, how it can take place, what

measures should co-teachers take before its implementation, and the advantages and

disadvantages of using CT.

CHAPTER 3

METHODOLOGY

The purpose of this study was to examine the impact of collaborative teaching

(CT) on 8th grade students’ achievement in mathematics. CT is a teaching approach

in which two or more teachers collaboratively plan, organize, present, and evaluate

their teaching. It has different settings like one teacher teaches and one assisting,

teaming, and parallel teaching etc. Literature shows that the collaborative teaching

approach is more advantageous than other mathematics teaching approaches in terms

60

of students’ learning. The study aimed at examining the collaborative teaching

impact on two content strands of mathematics (Algebra and Geometry) and on three

mathematical proficiencies (conceptual understanding, procedural knowledge, and

problem solving). This chapter is all about the methodology and procedure used in

this study. In this chapter, I discussed the nature of the study, population on which to

generalize findings of this study, and brief explanation about the sample selection for

data collection as well as the difficulties faced by me in selecting the sample for

conducting experiment, using the Solomon Four Group research design. The details

of research design including the details of experimental design used, explanation

about finalization of mathematics Achievement Test (MAT) items used in this

research, data collection procedure, data analysis, controlling confounding variables,

and ethical consideration for this experimental study are also explained in this

chapter.

RESEARCH DESIGN OF THE STUDY

The nature of the study was mainly focused on quantitative aspects;

furthermore, it was an experimental research study: “The goal of the experimental

method is to establish a cause-and-effect relationship between two variables. One of

the variables is manipulated the second variable is observed for changes due to the

manipulation” (Gravetter & Walnau, 2005, p. 11). The hypotheses were developed

and literature suggested that experimental research design is best to use for testing

them. An experiment was conducted on 8th grade students of mathematics. The

quantitative data were collected through MAT. Austin and Baldwin (1991) reported

61

that most of the research studies focused on teachers’ collaboration were quantitative

in nature. This study also concerned the qualitative aspect. Qualitative data were

collected by conducting interviews from students of 8th grade. So both an experiment

and interviews were conducted in this study.

For this study, the Solomon Four Group experimental research design was

used. In such a design all the groups receive posttests (Creswell, 2002). Solomon

Four Group design is the combination of pretest-posttest control group design and the

posttest only control group design (Wiersma & Jurs, 2009; Best & Kahn, 2008). This

design consists of four groups (two experimental and two control groups), and the

assignment of subjects to groups is random. It is a true experimental design and

according to Fraenkel and Wallen (2006) random assignment of subjects to the

treatment groups is the essential aspect of true experimental design. This design best

control the threats to internal validity of the experiment i.e. subject characteristics,

mortality, instrument decay, testing, maturation, and regression (Gay, 2000; Creswell,

2002; Best & Kahn, 2008) as compared to other experimental designs. Best and Khan

(2008), and Creswell (2002) explained the sampling process, regarding Solomon

Four-Group, as shown in the figure 3.1.

R O1 X O2 Ex

R O3 - O4 C

R - X O5 Ex

R - - O6 C

Figure 3.1. Groups formation in Solomon Four-Group research design

62

In figure 3.1, O1 and O3 represents pre-test, O2, O4, O5, and O6 represent post-test,

R represents random assignment, and X represents treatment.

I, first of all conveniently selected a public school and took all the students at

8th grade as a sample of this study; furthermore, I randomly divided those students

into four equal groups. In this study the sample comprised 118 students.

Experimental groups were those who received intervention and control groups were

those who did not receive intervention. In this research the intervention was

collaborative teaching in the subject of mathematics. In the control groups traditional

method of teaching (i.e. the deductive method) was used by a single teacher in a

classroom.

POPULATION

All the 8th grade students in a public school of the Sargodha district of Punjab,

Pakistan constituted the population of this study.

SAMPLING

One public school was selected conveniently from the Sargodha district of

Punjab-Pakistan. The school was selected after obtaining permission from the head of

the school. All the available 8th grade students, who numbered 118, participated in

the experiment. Two volunteer mathematics teachers at the sampled school, whose

qualifications were M.Sc.in mathematics and B.Ed., were the part of this study. I,

having the same qualifications, also took part in the study as a co-teacher. One

63

mathematics teacher from the sampled school and I participated in CT. Both taught

mathematics to the experimental groups collaboratively. At the same time the second

mathematics teacher from the sample school was teaching mathematics to the control

group through traditional teaching method.

I divided the available (118) 8th grade students in the selected public school

into four groups randomly, by using Statistical Package for Social Sciences (SPSS).

The details of sampling are given in Tables 3.1 and 3.2.

Table 3.1

Sampling Detail of Students and Teachers

The demographic details of the mathematics teachers involved in this study

is given in the table 3.2.

Table 3.2

Qualifications and Experience of Practicing Teachers

Teaching

Experience

Qualifications

Category N

Public sector school 1

Mathematics teachers 3

Students 118

64

In Years

Teacher A 5 M.sc mathematics, B.Ed.

Teacher B 2 M.sc mathematics, B.Ed., M.Ed.

Teacher C 6 M.sc mathematics, B.Ed.

DIFFICULTY IN SAMPLE SELECTION

It was really a difficult experience for me to select a public school for

conducting the experiment specifically by using Solomon Four Group research

design. I visited many schools in cities that were easily accessible: Lahore, Sargodha,

and Faisalabad of Punjab province to get permission from the headmaster to conduct

the experiment. I faced hard experiences while communicating with the heads of

public sector schools. Most of the heads where I visited to get permission, did not

allow me to conduct experiments in their schools. Three heads agreed to the

conducting of a research experiment but when they came to know that I wanted to

randomly assign students to the four groups, they refused. Best and Kahn (2008)

stated that the major difficulty with this design is to assign the subjects randomly to

four equivalent groups. The second major problem was the availability of two

mathematics teachers with the qualification of M.Sc. B.Ed. In most of the public

sector schools the 8th grade mathematics teachers had B.Sc. B.Ed. degrees. Finally, I

found a secondary public school in Sargodha district of Punjab province where the

consent from the head was obtained and there were two mathematics teachers

available with the required qualifications. I also participated in the study as a co-

teacher in the experimental group.

65

TRAINING OF VOLUNTEER MATHEMATICS TEACHERS

I included two mathematics teachers, with their consent, in this study. I was

also part of the experiment as a teacher who collaborated with a mathematics teacher

in planning, organizing, teaching, and assessing mathematics. I held training sessions

for the two mathematics teachers of the sampled school for two days, and it took four

hours, two hours each day. On the first day I met with mathematics teachers,

introducing myself. I also explained the research objectives and details of the syllabus

to be covered to the two mathematics teachers. I shared a detailed schedule of periods

by topics and date. I requested that both the mathematics teachers be regular and

punctual during the experiment. On the second day I explained the CT separately to

one of the mathematics teachers of the sampled school; the following topics were

discussed: How will it take place? What things should be kept in mind before the

start of collaborative teaching?

COLLABORATION OF CO-TEACHERS

Both the collaborative teachers met on a regular schedule i.e. one hour every

day, during this study to: plan together, develop lesson plans, decide collaborative

settings, manage teachers’, and students’ time. They had informal meeting in the

school to discuss matters relating to school affairs management (classroom

availability, time table suitability, extra class schedules and other related facilities),

66

students, and teaching material. They made schedule of classes in advance with

details of topics and dates.

RESEARCH INSTRUMENTS

In this study two tools were used: achievement tests and semi-structured

interview protocol. The first instrument, the mathematics Achievement Test (MAT),

was used to measure the achievement of students. It was Multiple Choice Questions

(MCQs) type test. The items used in the test had been developed by the National

Educational Assessment System (NEAS). MAT was the appropriate tool to assess the

academic achievement of students at 8th grade level. I used the test as a tool to collect

the data in order to measure the achievement of the students.

Interviews are useful to collect the data in detail. They may be structured,

unstructured (informal), or semi-structured (Fraenkel & Wallen, 2006; Wellington,

2006). The second tool used in this study was the semi-structured interview protocol.

According to Bogdan and Biklen (2007), a researcher gets comparable data across

participants with semi-structured interviews. Wellington (2006, p.71) argued that

“Interviewing allows the researcher to investigate and prompt the things that we can’t

observe.” According to Creswell (2002) it is useful to collect data on instruments or

tests followed by in-depth exploratory interviews. Bogdan and Biklen (2007) also

discuss that interviews may be employed with other techniques of data collection. In

addition to that, researchers use more than one tool in research studies for

triangulation (Best & kahn, 2006).

MATHEMATICS ACHIEVEMENT TEST (MAT)

67

The National Mathematics Curriculum of 2000 includes four content strands:

arithmetic (43%), algebra (32%), geometry (20%, i.e. 10% measurement geometry

and 10% construction geometry), and data analysis & probability (5%) respectively.

Moreover, these content strands in the 8th grade mathematics textbook cover 12, 6, 5,

and 3 learning outcomes of National Mathematics Curriculum of 2000. Keeping in

view time and financial constraints, this study included the content strands of algebra

and measurement geometry which consists of 6 and 3 learning outcomes respectively.

Additionally, the content of 8th grade mathematics addresses three

mathematical proficiencies i.e. conceptual understanding, procedural knowledge, and

problem solving. The content percentage with respect to these proficiencies in the

National Curriculum mathematics of 2000 was found to be 30% for conceptual

understanding, 40% for procedural knowledge, and 30% for problem solving.

According to the New Jersey mathematics curriculum framework (1996), the

dominant mode of assessing the students’ achievement in mathematics is paper and

pencil tests. It includes items such as multiple choice, true/false, fill in the blanks, or

short answer questions. Multiple-Choice Questions (MCQs) are of higher quality

than short answer, true-false, or matching-type items and are recognized as the most

widely applicable and useful objective type test items for measuring achievement

(Miller, Linn, & Gronlund, 2009). They can measure a variety of learning outcomes

from simple to complex. Additionally, they are adaptable to most types of subject

matter content like mathematics, physics, social studies etc. It reduces guessing and

increases reliability (Miller, Linn, & Gronlund, 2009).

To ensure the validity of content strands and mathematical proficiencies of the

MAT a specification Table 3.3 was developed.

68

Table 3.3

Table of Specification of Mathematics Achievement Test (MAT)

S

r. n

o.

Conte

nt

Str

and

Conceptual

Understanding

(CU)

Procedural

Knowledge

(PK)

Problem Solving

(PS)

Total

No of

Items

%

weightage

in Test

No of

Items

%

weightage

in Test

No of

Items

%

weightage

in Test

No of

Items

%

weightage

in Test

1 Geometry 4 12.5 4 12.5 3 9.31 11 34.3

2 Algebra 6 18.7 9 28.12 6 18.7 21 65.6

Total 10 31.25 13 40.625 9 28.13 32 100

The MAT addressed three Mathematical proficiencies: conceptual

understanding (CU), procedural knowledge (PK), and problem solving (PS). The

number of items with respect to learning outcomes and Mathematical proficiencies is

given in Table 3.4.

Table 3.4

Table of Specification of MAT Showing Number of Items with Respect to

Learning Outcomes

Content

Strand

Learning

Outcomes

No of

items

CU

No of

items

PK

No of

items

PS

Total

Alg

eb

ra

Concept of algebraic expressions

1

3

-

4

69

Solve equations on the application

of basic algebraic expression

1 1 - 2

Know, derive, and apply the

formulae

2 1 - 3

Factorization 1 2 - 3

Solve linear equations 1 1 2

Solve problems based on linear

equations

- 1 6 7

G

eom

etry

Apply Pythagoras’ theorem

4

-

2

6

Find the area and volume of sphere

and cone

- 1 - 1

Find the area of rectangles - 3 1 4

Total 10 13 9 32

Table 3.4 shows that 10, 13, and 9 items address the Mathematical proficiency

of conceptual understanding (CU), procedural knowledge (PK), and problem solving

(PS) respectively with at least one item for each learning outcome.

PROCEDURE OF FINALIZING MAT ITEMS

In this research study, the Mathematics Achievement Test (MAT) was used to

gauge the achievement of students in mathematics in the 8th grade. The test items

were adopted from National Education Assessment System (NEAS) with due

permission from the National coordinator of the organization.

NEAS developed a pool of 280 mathematics items distributed into four booklets A,

B, C, and D; these booklets contain 70 items each. Out of 280 items, 65 items were from the

70

strand of Algebra and 30 items were from the strand of Measurement Geometry. NEAS

reported the values of Point-Biserial Correlation Co-efficient and Difficulty Level after pilot

testing the items. The formula for Difficulty Level is:

Difficulty Level = correct responses on an item / total responses

Firstly, I selected 52 items which included 35 items from the strand of Algebra and

17 items from the strand of Geometry. The initial selection was made on the values of Point-

Biserial Correlation Co-efficient i.e. greater than 0.15 is recommended (Kehoe, 1995), and the

value of Difficulty Level ranging from 0.20 to 0.80 is recommended for good items (Valentin

& Godfrey, 1996; Yuentrakulchai, Kamtet, & Dechsri, 2011; Mullen, Rieder, Gilk,

Luber, & Rosen (2004); koraneekij, 2008, Mcalpine, 2002).

Secondly, I took opinions of six Subject Matter Experts (SMEs) (Appendix E)

on mathematics and educational assessment to validate the measurement of the

selected 52 items addressing three mathematical proficiencies i.e. conceptual

understanding, procedural knowledge and problem solving. The detail is given in

Table 3.5.

Table 3.5

Detail of Judges’ Responses on 52 Items Selected on the Bases of Statistical

Values and Content Strands

Sr.

No

Co

nte

nt

Bo

ok

let

No

. It

em n

o

Judges labeled the items

Ju

dg

e A

Ju

dg

e B

Ju

dg

e C

Ju

dg

e D

Ju

dg

e E

Ju

dg

e F

Cu

mu

lati

ve

Decis

ion

Dif

ficu

lty

Ind

ex

Po

int-

Bis

eria

l

corr

ela

tio

n

coef

fici

ent

1

Al

ge

br

a A 53 CU CU CU CU CU CU CU 0.28 0.22

71

2 A 55 PK PK CU CU CU CU CU 0.37 0.47

3 A 56 PK PK CU CU CU CU CU 0.27 0.41

4 A 59 PK PK PS PK PK PK PK 0.29 0.33

5 A 61 PK PK PK PK PK CU PK 0.23 0.28

6 A 63 PS PK PS PK PK PK PK 0.24 0.24

7 A 64 PS PK PS PS PK CU UD 0.31 0.36

8 A 67 PS PK PK PS PS PS PS 0.20 0.23

9 C 53 PK PK PK PK PK PK PK 0.21 0.20

10 C 54 CU PK PK CU CU CU CU 0.47 0.37

11 C 55 PK PK PK PK PK PK PK 0.31 0.38

12 C 56 CU PK PK PK PK PK PK 0.26 0.29

13 C 68 PS PK PS PS PS PS PS 0.35 0.35

14 C 70 PK PK PS PK PK PS PK 0.33 0.34

15 D 54 PK PK PK PK PK CU PK 0.56 0.34

16 D 56 PK CU CU CU CU CU CU 0.25 0.27

17 D 58 PK PK PK PK PK CU PK 0.34 0.35

18 D 59 PS PK PK PK PK PK PK 0.46 0.30

19 D 60 PK PK PK PS PK PK PK 0.34 0.32

20 D 63 PK PK PK CU PK CU PK 0.22 0.39

21 D 64 PK PK PK CU PK PK PK 0.32 0.28

22 D 67 PS PK PK PK PK PK PK 0.20 0.23

23 D 68 PS PS PS PS PS PS PS 0.37 0.25

24 B 55 PK PK PS PK PK PK PK 0.25 0.22

25 B 56 CU CU CU CU PK CU CU 0.36 0.30

26 B 59 PK PK PS PK PK PK PK 0.22 0.20

27 B 60 CU PK PK PK PK PK PK 0.48 0.34

28 B 61 PS PK PS PS PK PK UD 0.15 0.16

29 B 62 CU PK PS PK PK PK PK 0.39 0.24

30 B 63 PS PK PK PK PK PK PK 0.34 0.20

31 B 65 PS PK PK PK PK PK PK 0.24 0.22

32 B 66 PS PK PS PK PS PS PS 0.20 0.17

33 B 68 PS PK PS PS PS PS PS 0.43 0.26

72

34 B 69 PS PK PS CU PK PK UD 0.24 0.19

35 B 70 PS PK PS PK PS PS PS 0.27 0.32

36

G

eom

etry

A

42

PS

PS

PS

PS

PK

PS

PS

0.25

0.21

37 A 31 CU PK PK CU CU CU CU 0.21 0.36

38 A 33 PK PK PS PS PS PS PS 0.22 0.21

39 A 35 CU CU CU CU CU CU CU 0.28 0.23

40 A 44 PK PK PK PK PK PK PK 0.24 0.22

41 C 36 CU CU CU CU CU CU CU 0.22 0.33

42 C 37 CU CU CU CU CU CU CU 0.31 0.23

43 C 41 PK PK PK PK PK PS PK 0.21 0.12

44 C 43 PK PS PS PS PS PS PS 0.20 0.12

45 D 35 CU CU CU CU CU CU CU 0.27 0.15

46 D 36 CU CU CU CU CU CU CU 0.68 0.32

47 D 37 PK CU CU CU PK CU CU 0.42 0.27

48 D 39 CU PK PK PK PS PK PK 0.26 0.20

49 B 31 PS PS PS PS PS PS PS 0.22 0.20

50 B 32 CU PK PK PK PK PK PK 0.34 0.25

51 B 34 CU PK PK PS PK PK PK 0.37 0.27

52 B 35 CU CU PK CU PK CU CU 0.31 0.27

Note: CU= Conceptual Understanding, PK= Procedural Knowledge, PS= Problem Solving,

UD=Undecided

Table 3.5 reflects the opinions of SMEs on 52 items about the mathematical

proficiencies. Based on the majority opinion, at least four of the six experts,

(Colomer, 2008), I selected 49 of the 52 items. Those items were selected on which

four out of six experts had the same opinion. All the rejected items were from the

strand of algebra. I finalized the MAT (Appendix A) by selecting 32 items from the

remaining 49 items having Point-Biserial Correlation Co-efficient greater than 0.15

73

(Kehoe, 1995) and Difficulty level ranging from 0.20 to 0.80 recommended for good items by

Rosen (2004) and Koraneekij (2008).

TRANSLATION OF MAT

The MAT items were originally developed in Urdu. Additionally, translated

versions of those items in the English language were also available. That translation

was done by two mathematics Subject Matter Experts (SMEs). I got the translated

items and in order to deal with the language translation threat to the validity of the test

items in his local context, furthermore, validated it through the collaboration of

mathematics SME with qualification M.sc in mathematics and English language

expert with qualification of MPhil English.

The mathematics and English language SMEs made changes to nineteen items

of the MAT (Appendix A) i.e. A(33), A(44), A(59), A(67), B(70), B(66), B(60),

B(62), B(56), B(35), B(34), B(31), B(32), C(70), C(68), D(39), D(58), D(59), D(68).

Most of the translation mistakes were related to typing and English grammar.

INTERVIEW PROTOCOL

Semi-structured interviews were conducted with the experimental group

students three times i.e. before, during and after the intervention. Bryman (2004)

affirmed that use of semi-structured interviews to explore any idea is useful. Green,

Camilli, and Elmore (2006, p.358) stated that “Ethnographers have been seen

interviewing as just one source of information about the multitudinous aspects of life

in society including behavior, attitudes, belief, and material culture.” In this study I

74

explored students’ views and beliefs towards mathematics and teaching of

mathematics. These interviews were conducted face to face with subjects individually

because subjects sometimes do not give information due to uncertainty of

confidentiality (Cohen, Manion, and Morrison, 2007). It was made clear to students

that the information provided by them will be kept confidential and will not be shared

with their mathematics teachers, parents and headmaster. Following the

recommendation by Creswell (2002) I video-taped the interviews. Fraenkel & Wallen

(2006) argued that a researcher can’t learn what he wants from interview if he does

not capture what the interviewee actually says. The interviews were conducted in the

Urdu language, the national language of Pakistan.

An interview protocol was developed by myself. It consisted of seven

questions explaining students’ beliefs about mathematics and mathematics teaching in

the collaborative setting. Respondents, students, were required to say “yes or no”

first, and then say something else, and some questions were asked by me in the result

of subjects’ response as Bernard (2006) also suggested probing details of different

related aspects during the interview. The interview protocol was as follows:

Is mathematics a difficult subject?

Is it an interesting subject?

Is it useful in everyday life?

Are its topics related to each other?

Is it necessary to memorize the formulae first to solve mathematical problems?

Is the purpose of mathematics to find out the solution of mathematical

problems only?

Is co-teaching in mathematics class useful?

Does learning mathematics involve activities?

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VALIDITY OF INTERVIEW

The interview protocol was validated through expert opinion and piloting.

Three experts (one bilingual (Urdu and English) and two Mathematics SMEs) gave

their opinion on the beliefs included in the protocol and about the language used in

the protocol. The actual interviews were conducted in Urdu. Therefore, I first

translated the interview protocol into the Urdu language. The protocol was also

piloted with 15 students studying mathematics.

COLLABORATIVE MATHEMATICS TEACHIG MODULE

(CMTM)

Mathematics is a compulsory subject in 8th grade. The learning of mathematics

depends heavily on its teaching. Predominantly, the traditional teaching approach - the

deductive method of teaching used by a single teacher - is practiced in Pakistan.

Collaborative teaching (CT) has many advantages such as students have the opportunity

to see diverse teaching styles and faculty members develop mutual trust and respect for

each other while covering their specialty areas. Moreover, it helps the teachers to learn

valuable information from each other in terms of content and teaching styles. It also

enhances the quality of teaching by transforming it into a participative activity,

eliminating the perception of being isolated. In spite of many advantages, CT is not

being used for teaching of mathematics in Pakistan.

Some Mathematics Teaching Modules (MTMs) are presently available in

Pakistan for 8th grade. These were developed by the Directorate of Staff Development

(DSD) Lahore and UNESCO. These modules lack in some areas; for example, DSD

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has developed MTMs only for the content strands of arithmetic and geometry and

missed the algebra section. They use lecture, demonstration, drill and practice methods

for teaching geometry. This module does not include the problem solving and

assignment method, and activity-based method to teach geometry. Similarly, UNESCO

has developed MTM at 8th grade. Although this module includes the content strands of

algebra and geometry, the module addresses few topics of algebra and geometry. This

module includes the inductive method, drill and practice method to teach algebra. It

does not include problem solving, activity-based method and assignment method to

teach algebra. Both of the modules had been developed for single teacher classrooms.

Although CT is important in mathematics teaching, still no Collaborative Mathematics

Teaching Module (CMTM) exists in Pakistan to teach mathematics in the 8th grade.

Therefore, keeping in view the significance of CT in mathematics, a CMTM

was developed for the present study consisting of lesson plans (Samantha, n. d.). The

format of lesson plans for this module was different from the standard pattern of lesson

plans as it includes the division of responsibilities and definitely incorporates the

collaborative teaching model in it, as is suggested by Cuellar (2011). This module

covers the two content strands algebra and geometry. The sub-topics of algebra are

evaluating algebraic expressions, addition and subtraction of polynomials up to degree

4, multiplication of polynomials up to degree 4, establishing formulae, factorization,

linear equations, simultaneous linear equations, and application of Pythagoras’

theorem. Similarly the sub-topics for the content strands of geometry are surface area

of spheres, volume of spheres, surface area of cones, volume of cones, and area of

rectangles.

The CMTM includes the three collaborative settings, namely one-teach one-

assist, teaming, and parallel teaching. Moreover, many mathematics teaching methods

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are used with these settings like inductive, activity based, assignment, and problem

solving. The CMTM also includes assessment techniques such as asking the students

questions verbally during the lecture, classroom tests, homework worksheets, or by

asking students to solve questions on the worksheets during the lecture.

PROFILE OF PARTICIPATING MATHEMATICS TEACHERS

Two mathematics teachers participated in the collaborative teaching i.e. I and

a mathematics teacher from the sampled school. The profile of both the teachers

follows:

Teacher A

He held an M.Sc in mathematics with a professional qualification of B.Ed. He

had excellent verbal communication skill and was good at using the chalk board. He

was an expert in the content strand of algebra with five years’ experience of

mathematics teaching in 8th grade. He was 45years old at the time of the study.

Teacher B

The teacher B was myself. I held an M.Sc in mathematics with a professional

qualification of B.Ed. and M.Ed and I was doing PhD (Education). I was good in

assisting and guiding the students and have sound command over the concepts of

geometry. I can draw geometrical figures with ease. My experience of mathematics

teaching in 8th grade was two years. I was 28years old at the time of the study.

PROCEDURE OF THE EXPERIMENT

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The experimental procedure was as follows:

1. As the first step I assigned 118 available students in 8th grade into four groups (two

experimental and two control groups). I and one mathematics teacher participated

in collaborative teaching in an experimental group. Another volunteer mathematics

teacher from the sampled school was involved in this study to teach the two control

groups. The experiment was conducted in two class periods of mathematics. In the

first period the collaborative teaching was taking place in one experimental group,

whose students took pre-tests, and at the same time traditional teaching with single

teacher was taking place in one control group, whose students took pre-tests, in two

different classes. Similarly, in the second period the collaborative teaching was

taking place in one experimental group, whose students did not take pre-tests, and

at the same time traditional teaching with single teacher was taking place in one

control group, whose students did not take pre-tests, in two different classes. At

any given time, two groups (one experimental and one control group) were studying

while two other groups were engaged by the class teacher of 8th grade class.

2. Treatment was started in the first week of 01-01-2013 and ended on 07-02-2013.

This duration was appropriate because it was nearly equivalent to the time given on

the chart, which gives the detail of total time allocated in the academic year for the

subject of mathematics according to the content strand of Punjab schools (available

on the website of Punjab schools). The time for collaboration in this study was

supported by the research study of Boeckel (2008). He took 45 minutes for CT each

day over six weeks during the fourth semester of the school year for data collection.

3. On the first day, I told the students about their groups. I also gave the MAT as a

pre-test to one experimental group and one control group of students. Moreover, I

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conducted interviews of the students of the experimental groups. I also shared a

schedule of class periods with the mathematics teachers.

4. The content strands were taught as per order given in the curriculum and in the

textbook of mathematics for 8th grade that is first algebra and then geometry.

5. In this study the duration of each class period was 60 minutes. I combined two

consecutive periods of 30 minutes each in the school into one period of 60 minutes.

6. The teachers taught collaboratively according to the CMTM. They used many

activities in the class. The students practiced those activities with the help of

mathematics teachers. They also solved many worksheet problems provided by the

teachers.

7. At the middle of the experiment the interviews were conducted again.

8. At the end of the experiment the post-test was conducted. Moreover, I conducted

interviews on that day as well.

ETHICAL ISSUES RELATED TO THE EXPERIMENTAL

DESIGN

Ethics play an important part in conducting meaningful and effective research

studies on humans. Keeping in view the importance of ethics, I also followed the

ethics of experimental research. The Ethical Standards of the American Educational

Research Association (AERA) stated that the researchers should keep in their minds

the privacy, dignity, and sensitivities of their research study subjects. They also need

to respect the rights of their participants. Moreover, educational researchers should be

especially watchful in conducting research studies on children (Hemmings, 2006).

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The American Sociological Association also gave code of ethics for researchers. The

American Sociological Association's Code of Ethics include: professional

competence, professional and scientific responsibility, respect for people's rights,

dignity, and diversity, and social responsibility. Ethics followed in the course of this

study are as follows:

PERMISSION FOR USING INSTRUMENT

In this study the MAT was used to measure the academic achievement of

students in mathematics. The items were developed by the National Educational

Assessment System (NEAS), Pakistan. In order to meet ethical considerations for this

study, I obtained permission from the National coordinator of NEAS for using

developed MCQs items in this research.

ENSURING THE PROTECTION OF HUMAN PARTICIPANTS

FROM ANY HARM OR EXPECTED HAZARDS

Creswell (2002), Fraenkel & Wallen (2006), Taylor, Sinha, and Ghoshal

(2006), Cohan, Manion, Morison, and Morison (2007) described the ethical concerns

in experimental research studies, especially related to the protection of participants of

the study from any harm. I tried to protect the participant from harm. I also tried to

avoid punishment and always treated the students in a polite and respectful manner.

VOLUNTARY PARTICIPATION OF MATHEMATICS

TEACHERS IN CT

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Voluntary participation is an important ethical consideration while doing

experimental research. In this study I needed two mathematics teachers with M.Sc.

and B.Ed. degrees for conducting the experiment. Two volunteer mathematics

teachers out of three from the sampled school who had the necessary qualifications

participated in the study. I also participated as a third mathematics teacher in the

study. I participated in this study as a co-teacher with one of the mathematics teachers

from the sampled school.

CONFIDENTIALITY OR PRIVACY OF THE INFORMATION

GIVEN BY THE SUBJECTS

All the information received from the participants (teachers and students) was

kept confidential. In experimental research, major ethical concern is confidentiality

with regard to the information, a point raised by Van Dalen (1979), Lodico, Spoulding

and Voeglte (2006), Sleber (1998), Fraenkel & Wallen (2006), Bogdan & Biklen

(2007), Dowson (2007), Leeuw, Hox, and Dillman (2008), Cohen et al.(2007). For

example, sometimes students commented (negatively or positively) on their

mathematics teachers and their teaching; but, I never shared it with the mathematics

teachers. Many times the co-teacher of mathematics commented on mathematics

teachers and their teaching, and I kept all those words secret in order to avoid

conflicts among the mathematics teachers. The co-teacher also commented on the

school’s management and teachers of other subjects; I kept his words confidential.

Additionally, in order to ensure confidentiality of the data, I myself conducted the

pre-test, post-test and semi-structured interviews.

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ANONYMITY OF THE SUBJECTS

One of the ethical concerns is to ensure the anonymity of the subjects by

removing their names from the data and using code numbers for response

identification instead (Cohen et al., 2007; Van-Dalen 1979). Also, I should report the

results as group averages so that the response of the individual can’t be identified. In

this study the interview data were reported by using serial numbers for the list of

experimental group members. The results were reported as a mean or percentage of

the whole group. The individual responses were narrated without using any names

wherever those were needed to explain the situation.

INFORMED CONSENT

The ethical concern of informed consent from the subjects or from the

institution is emphasized by a number of authors including Van-Dalen (1979), Best &

Kahn (2008), Leeuw, Hox, and Dillman (2008), as well as Cohen et al. (2007). I

gained the permission of the headmaster of the school for conducting the experiment.

CONSIDER THE CAPACITY OF THE SUBJECTS

I developed and used such activities and exercises compatible with the minds

and capacity of 8th grade students of mathematics. Also, during lectures sometimes

co-teachers made jokes to ease the minds of the students.

PRESENT THE ACTUAL RESULTS, NOT DECEPTIVE ONES

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The most important ethical consideration in any research is to present the

actual results. The researchers Taylor, Sinha, and Ghoshal (2006), Dowson (2007),

Creswell (2002) and Cohan, Manion, Morison, and Morison (2007) described it as a

major ethical concern. Researchers should not be deceptive and should report the

findings as they find them. In this research, I presented the actual findings.

RESPECT FOR THE SUBJECTS

Taylor, Sinha, and Ghoshal (2006), as well as Leeuw, Hox, and Dillman

(2008) highlighted that giving respect to the participants of the study is of great

importance. I treated the entire participant population with great respect. I used soft

language while talking with students and teachers. Conversely, the students also gave

respect and honor to me during the stay at school. They praised my efforts and

teaching and mathematical concepts.

CONTROLLING POSSIBLE CONFOUNDING FACTORS

The detail of possible confounding factors and the way I controlled these

factors are as follows:

TEACHING EXPERIENCE AND QUALIFICATION OF THE

TEACHERS

I selected mathematics teachers who have the same qualifications and almost

the same experience of mathematics teaching. There were three mathematics teachers

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available in the sampled school. They all possessed M.Sc. and B.Ed. degrees. I

selected two of them who were willing to participate.

INTERACTION OF PRE-TEST EFFECT AND TREATMENT

Sometimes it happens that treatment given in the experiment interacted with

the pre-test given to the subjects in the start of the experiment. If it happens, it is

mistake on the part of the researcher and indicates that he tried to focus the treatment

on the pre-test. If it happens, it directly affects outcomes. It can easily be identified

though the statistical data analysis technique, Factorial ANOVA, and it can be

controlled through Solomon Four-Group experimental research designs. I used

Factorial ANOVA in this study to find out this interaction.

EFFECT OF PRE-TESTING

Most of the experimental researchers agree that pre-testing only affects the

outcome of an experiment when the duration of study is too short, such as one day or

one week. So, this effect was controlled through the long duration of the treatment,

thirty seven days. It was also controlled through the Solomon Four-Group

experimental design because in Solomon Four-Group research design I randomly

assigns subjects to four groups as is shown in Table 3.6. If I found the effect of pre-

test then I can analyze the two groups that are not being pre-tested.

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Table 3.6

Details of Groups

Pre-tested Not pre-tested

Experimental Group E1 E2

Control Group C1 C2

NOVELTY EFFECT

Since the collaborative teaching practice was new for the students, they may

have been more enthusiastic and motivated for mathematics studies. This effect was

controlled by the long duration of the experiment, 37days (20 days for teaching of

mathematics, 2 days for pre-posttest, 5 holidays, 3 days for conducting semi-

structured interviews, classes could not be held on seven intermittent days due to

various issues at school such as non-availability of classrooms and students on school

test days, absence of the co-teacher etc. .

INSTRUMENT’S VALIDITY AND RELIABILITY

The validity and reliability of the instrument used for data collection may

affect the outcome of the experiment. This effect was minimized through the use of

items from a pool of items developed by NEAS Pakistan. The items used in the MAT

are validated by NEAS.

ATTENDANCE OF THE STUDENTS

If some of the sampled students do not attend classes, it may affect the overall

results of the class. This effect was minimized in two steps; firstly, I marked the

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attendance of students of 8th grade mathematics on daily basis and secondly at the

time of data analysis I removed the data of those students who were found absent

from more than 25% of the total period of experiment.

DATA ANALYSIS

In this study both quantitative and qualitative data were collected. The

quantitative data were collected through MAT and the qualitative data were collected

through interviews. The first part of data analysis was quantitative. The data

collected through MAT was analyzed through descriptive and inferential statistics

such as mean, independent sample t-test, and 2×2 factor ANOVA.

Independent sample t-test was used to compare the groups (experimental and

control) on mathematical proficiencies (conceptual understanding, procedural

knowledge, and problem solving) and content strands of mathematics (algebra and

geometry). Independent sample t-test was appropriate because all the assumptions of

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this test were fulfilled. It is used when there is one independent variable with two

factors and one dependent variable. It shows whether means of factors are

significantly different or not. In this study the independent variables were

mathematical proficiencies i.e. conceptual understanding, procedural knowledge, and

problem solving as well as content strands, algebra and geometry. The dependent

variable was academic achievement of students in MAT at the 8th grade level.

There are two terms used in ANOVA language i.e. main effects and

interaction effects. Campbell and Stanley (1963) suggested estimating the treatment

effect X by using column means, the main effect of pre-testing by row means, and

interaction effect of the test with treatment X by cell means. In this study the

treatment X is CT.

http://www.fammed.ouhsc.edu/tutor/solomon.htm

Factorial ANOVA was appropriate because it has been suggested and used by

several researchers in the field of experimental research for the data obtained from

Solomon Four-Group research design (Campbell and Stanley, 1963; Spector, 1981;

Wilke, 2003). Factorial ANOVA was also appropriate because all the assumptions of

this test were fulfilled.


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In addition to the quantitative analysis of the study, the second part of the

analysis was qualitative. This data was collected through interviews from the students

in 8th grade. The interviews were conducted before the start, middle, and at the end of

the experiment. The audio-video recorded interviews were first transcribed and then

translated from Urdu, the national language, to English. The qualitative data, after

transcription, were analyzed through open coding technique. Descriptive statistics

such as percentages of the categories were also calculated.

CHAPTER 4

DATA ANALYSIS

The purpose of this study was to investigate the impact of collaborative

teaching on the academic achievement of students in mathematics at the 8th grade

level in Pakistan. For this investigation, an experimental research design, the

Solomon Four Group method was used considering it as appropriate for this study. It

best controls the internal validity threats. The data were collected from 8th grade

students. In order to validate the results of the experiment, triangulation of data from

different sources was done i.e. the data collected through semi-structured interviews

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with the students of 8th grade. At the first stage I selected items from the Mathematics

Achievement Test (MAT) items developed by the National Education Assessment

System (NEAS) and he developed a Collaborative Mathematics Teaching Module

(CMTM), consisting of collaborative lesson plans of Mathematics for 8th grade

students. An interview protocol was also developed for students. Interviews were

conducted before, during, and after the intervention. I used descriptive and inferential

statistics such as Mean, Percentages, Factorial ANOVA, and t-test to analyze the data.

Factorial ANOVA was appropriate for this research design as it has been suggested

and used by several experimental researchers applying Solomon Four Group research

design (Campbell and Stanley, 1963; Spector, 1981; Wilke, 2003). The t-test was

appropriate to determine the significance of difference in the students’ achievement of

experimental and control groups as a whole by selected mathematical proficiencies

(conceptual understanding, procedural knowledge, problem solving), and by content

strands (algebra and geometry) of mathematics. The data collected from the MAT

fulfilled all the assumptions of parametric statistics. The audio-video recorded data

from interviews were first transcribed from Urdu, national language of Pakistan, to

English. The qualitative data were analyzed through open coding technique i.e. code

or label words and phrases found in the text.

The detail of both analyses, quantitative and qualitative, is given in this

chapter. The quantitative analysis is further divided into subcategories as analysis by

mathematical proficiencies (conceptual understanding, procedural knowledge, and

problem solving), and Mathematical Content Strands (algebra and geometry).

QUANTITATIVE ANALYSIS OF DATA

90

The first part of this chapter presents the analysis of quantitative data, collected

through the MAT, in the following steps:

Screening of the data

Box plot

Normality tests

Shapiro-Wilk

Q-Q-plot

Comparison of control and experimental groups’ achievement scores

Analysis by mathematical proficiencies (conceptual understanding, procedural

knowledge, problem solving)

Analysis by mathematical content strands (algebra and geometry)

Analysis by mathematical proficiencies and content strands

DATA SCREENING

The experiment was started with 118sampled students of8thgrade students

distributed into four groups (Table 4.1).

Table 4.1

Number of Subjects in Control and Experimental Groups

Group Pre-tested Not pre-tested Total

Experimental group E1

(n=30)

E2

(n=29)

59

Control group C1

(n=30)

C2

(n=29)

59

91

Total 60 58 118

Fifteen (twelve from experimental and 3 from control group) students’ post-

test scores were dropped from the analysis on the basis of their short attendance (less

than 75% of total classes). The attendance criterion was set prior to the experiment.

Scores of another five students identified as outliers were also excluded from the

analysis. An outlier represents an extreme value of data. The outliers often confound

the results of data. These values were identified by drawing a box plot (figure 4.1).

Three out of the five students obtained extreme high marks and remaining two

students scored extreme low marks on MAT.

Figure 4.1. Box plot of control and experimental groups

There were five outliers, three in the control group and two in the

experimental group leaving 98 subjects’ scores for analysis. The descriptive statistics

of 98 subjects is shown in Table 4.2.

Table 4.2

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Mean Scores of Pre-Tested and Not Pre-Tested Subjects in the Control and

Experimental Groups

Group Pre-tested Not pre-tested Total

Experimental group E1

(n=24)

= 12.29

E2

(n=21)

= 13.38

(n=45)

= 12.80

Control group C1

(n=24)

= 10.38

C2

(n=29)

= 9.83

(n=53)

= 10.08

Total (n=48)

= 11.33

(n=50)

= 11.32

n= 98

Table 4.2 showed that the mean scores of experimental groups with and

without pre-test were 12.29 and 13.38, respectively. Similarly, the mean scores of

control groups with and without pre-test were 10.38 and 9.83, respectively. Also, the

overall mean scores of experimental and control groups were 12.80 and10.08,

respectively. It means that the experimental groups performed better than the control

groups on the Mathematics Achievement Test. The table also showed that the overall

mean scores of pre-tested and not pre-tested groups were almost equal.

NORMALITY TEST

In order to check the normality of the data, the Shapiro-Wilk test of normality

was applied. This test is useful with more than 50 cases. In this study the number of

subjects was 98, so it was appropriate to use the Shapiro-Wilk test to check the

normality of the data. Its results are obtained by calculating the correlation between

the data and the corresponding normal scores (Ghasemi and Zahediasl, 2012). It is a

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better choice to check the normality of the data than other normality tests:

Kolmogorov-Smirnov, Lilliefors, and Anderson-Darling.

Razali and Wah (2011) compared the power of four normality tests and found

Shapiro-Wilk to be the most powerful normality test. According to Ghasemi and

Zahediasl (2012), “researchers recommend the Shapiro-Wilk test as the best choice

for testing the normality of data.” The details are shown in Table 4.3.

Table 4.3

Normality of the Data of Control and Experimental Groups

Group Shapiro-Wilk

Statistic

df Sig.

Control .97 53 .22

Experimental .96 45 .12

Table 4.3 shows that p values (0.22 for control group and 0.12 for

experimental group) are greater than the 0.05. The null hypotheses for the Shapiro-

Wilk test were accepted. It means that the data of both the groups is significantly

normal data. The normality of the data was also done by plotting Q-Q-plot.

This plot is used to check the normality of the data in the form of figures.

According to Razali and Wah (2011) these plots are commonly used as effective tools

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to check the normality of the data. These plots display the observed values against

normally distributed data (represented by the line). If the data is along the line in the

Q-Q plot, then data is considered as normal. Figures 4.2 and 4.3 show that data

collected through the MAT is along the straight line; therefore, the data is normally

distributed. Figure 4.2 shows that the data of the experimental group is normal and

Figure 4.3 shows that the data of the control group is also normal.

Figure 4.2. Q-Q plot for experimental group

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Figure 4.3. Q-Q plot for control group

Table 4.4

Normality of the Data of Pre-Tested and Not Pre-Tested Subjects

Group Shapiro-Wilk

Statistic

df Sig.

not pre-tested .96 50 .06

pre-tested .98 48 .54

Table 4.4 shows that p values i.e. 0.06 for not pre-tested subjects and 0.54 for

pre-tested subject are greater than the 0.05. The null hypotheses for the Shapiro-Wilk

test were accepted. It means that the data of both the groups is significantly normally

distributed. The normality of the data was also done by plotting a Q-Q plot. Figures,

4.4 and 4.5 show that the data are along the straight line; therefore, the data is

normally distributed. Figure 4.4 shows that the data of the not pre-tested group is

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normally distributed and the figure 4.5 shows that the data of the pre-tested group is

normally distributed.

Figure 4.4. Q-Q plot for not pre-tested subjects

97

Figure 4.5. Q-Q plot for pre-tested subjects

Achievement Scores of Students in Mathematics Taught through

Traditional Teaching and Collaborative Teaching

The difference between the mean scores of experimental and control groups’

students was calculated and, following that, the null hypothesis was tested using 2×2

ANOVA. The summary is presented in Table 4.6.

There are two independent variable groups (experimental and control) and

conditions (pre-tested and not pre-tested) and the dependent variable is students’

achievement scores in mathematics. Factorial ANOVA measures two effects i.e. the

main effect and the interaction effect. The main effect is the mean difference between

the levels of the particular factor. In this study there were two groups, one

experimental and one control, and condition was the pre-test (pre-tested and not pre-

tested). The interaction effect is the difference among cell means. The means of

factors and cells are given in Table 4.5.

Table 4.5

Mean Scores of Experimental, Control, Pre-Tested and Not Pre-Tested Groups

Conditions

Groups

Pre-tested Not pre-tested Total

Experimental group 12.29 13.38 12.80

Control group 10.38 9.83 10.08

Total 11.33 11.32

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students in mathematics taught through collaborative teaching and traditional

teaching.

Table 4.6

Difference between Mean Achievement Scores of Control and Experimental

Groups

Source Sum of Squares df Mean Square F P

Experimental and control groups 180.86 1 180.86 19.50 .000

Pre-tested and not pre-tested

groups

1.78 1 1.78 .19 .66

Groups *Conditions 16.19 1 16.19 1.75 .19

p=.05 (N=96)

Table 4.6 shows that there is significant difference in the mean scores of

control and experimental group as the p value i.e. .000 is less than 0.05. Thus, the

null hypothesis, H01, stating that there is no significant difference between the mean

achievement scores of students taught through collaborative teaching and traditional

teaching, was rejected. The mean achievement score of experimental group, 12.80,

was greater than the mean achievement score of the control group, 10.07 as shown in

table 4.5. It was concluded that experimental group performed better on the MAT

than the control group.

Table 4.6 also shows that there was no significant difference between the

achievement score of students who were pre-tested and those who were not pre-tested

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as the p value, 0.66, was greater than 0.05. No significant interaction was found

between the test and the treatment, as the p value, 0.19, was greater than 0.05.

ANALYSIS BY MATHEMATICAL PROFICIENCIES

The impact of collaborative teaching on students’ achievement was measured

on three mathematical proficiencies: conceptual understanding, procedural knowledge,

and problem solving. The comparison between the control and experimental groups

taught through traditional and collaborative methods was done by using independent

sample t-test. The data fulfilled the assumptions of the t-test. Therefore, it was

appropriate to compare mean scores on mathematical proficiencies by groups

(experimental and control). The detailed comparisons between groups on three

mathematical proficiencies are given below.

ACHIEVEMENT SCORES OF STUDENTS’ CONCEPTUAL

UNDERSTANDING

Mean score differences (between students taught through collaborative and

traditional teaching methods) on the conceptual understanding component of the test

was calculated. Ten items regarding conceptual understanding were included in the

MAT with one point for each correct item. The detail is given in Table 4.7. The

following null hypothesis was tested:

H02: There is no significant difference between the conceptual understanding ability

mean achievement scores of students taught through collaborative teaching and those

taught through traditional teaching.

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Table 4.7

Comparison of the Conceptual Understanding Ability Mean Achievement Scores

of Students Taught Through Collaborative Teaching and Those Taught Through


Group n SD df t Sig.

Control

53 3.69 1.66

96

-3.49

.001

Experimental 45 4.80 1.42

p═.05, (N═96)

Table 4.7 indicates that there was difference between the mean scores of

students’ achievement in conceptual understanding of mathematics. The value of t96 ═

-3.49, p ═ .001 was significant at 5% alpha. The null hypothesis, H02, stating no

significant difference between the mean scores of students’ achievement in conceptual

understanding of mathematics, was rejected. The mean achievement scores of students

taught through collaborative teaching and traditional teaching, 4.80 and 3.69, on the

conceptual understanding, showed that the students taught through collaborative

teaching performed better than the students taught by a single teacher. The graphic

representation of the mean achievement scores was given in figure 4.6.

101

ACHIEVEMENT SCORES OF STUDENTS’ PROCEDURAL

KNOWLEDGE

Mean score difference of students taught through collaborative and through

traditional teaching on the procedural knowledge component of the test was calculated.

Thirteen items regarding procedural knowledge were included in the MAT with one

point for each correct item. The detail is given in Table 4.8. The following null

hypothesis was tested.

H03: There is no significant difference between the procedural knowledge ability

mean achievement scores of students taught through collaborative teaching

and traditional teaching.

Table 4.8

Comparison of the Procedural Knowledge Mean Achievement Scores of Students

Taught Through Collaborative Teaching and Traditional Teaching

Group n SD d f t Sig.

Control

53 4.11 1.37

96

-3.71

.000


p═.05 (N═96)

Table 4.8 indicates that there was significant difference between the mean

scores of students’ achievement in procedural knowledge of mathematics. The value

of t96═ -3.71, p ═ .000 was significant at 5% alpha. The null hypothesis, H03, which

stated there is no significant difference between mean scores of students’ achievement

102

in procedural knowledge ability in mathematics, was rejected. The mean achievement

scores of students taught through collaborative teaching and traditional teaching, 5.4

and 4.11, on the procedural knowledge ability, showed that the students taught through

collaborative teaching performed better than the students taught by a single teacher. For

graphic representation of the mean achievement scores see figure 4.6.

ACHIEVEMENT SCORES OF STUDENTS’ PROBLEM SOLVING

Mean score difference of students taught through collaborative and through

traditional teaching of mathematics on the problem solving component of the test was

calculated. Nine items regarding procedural knowledge were included in the MAT with

one point for each correct item. The detail is given in Table 4.9. The following null

hypothesis was tested.

H04: There is no significant difference between the problem solving ability mean

achievement scores of students taught through collaborative teaching and those

taught through traditional teaching.

Table 4.9

Comparison of the Problem Solving Ability Mean Achievement Scores of

Students Taught Through Collaborative Teaching and Those Taught Through



Control 53 2.26 1.20

96

-1.47

.15


p═.05 (N═96)

103

There was no significant difference between the mean scores of experimental

and control groups students’ achievement in problem solving. The value of t96 ═ -1.47,

p ═ .15 was not significant at alpha 0.05. The null hypothesis, H04, stating that there

was no significant difference between the mean scores of the two groups’ achievement

in problem solving, was accepted. However, the mean achievement score (2.64) of

experimental group was higher than those of the control group students’ mean scores

(2.26). It means that CT had some positive effect on students’ achievement. The graphic

representation of the mean achievement scores was given in figure 4.6.

Figure 4.6. Mean scores of students of control and experimental groups on

conceptual understanding, procedural knowledge, and problem solving

mathematical proficiencies

ANALYSIS BY MATHEMATICAL CONTENT STRANDS

In this study I measured the impact of collaborative teaching on students’

achievement on two mathematical content strands, algebra and geometry. The

comparison between the experimental and control groups was done by using an

independent sample t-test. The data fulfilled the assumptions of the t-test. The detailed

comparisons between groups on mathematical content strands are given below.

0

1

2

3

4

5

6

Conceptual

Understanding

Procedural

Know ledge

Problem Solving

Control Group

Experimental Group

104

ACHIEVEMENT SCORES OF STUDENTS IN ALGEBRA

Mean score differences of students taught through collaborative and through

traditional teaching on the algebra component of the MAT was calculated. Twenty-one

items regarding algebra were included in the MAT with one point for each correct item.

The detail is given in Table 4.10. The following null hypothesis was tested.

H05: There is no significant difference between the students’ mean achievement

scores in algebra taught through collaborative teaching and those taught

through traditional teaching.

Table 4.10

Comparison of the Students’ Mean Achievement Scores in Algebra Taught

Through Collaborative Teaching and Those Taught Through Traditional

Teaching


Control

53 6.6 2.15

96

-4.13

.000


p═.05 (N═96)

Table 4.10 shows that there was significant difference between the mean

achievement scores of students’ achievement in algebra. The value of t96 ═ -4.13, p ═

.000 was less than 0.05. The null hypothesis, H05, stating no significant difference

between the mean achievement scores of students’ achievement in the content strand of

algebra, was rejected. The mean achievement scores of students taught through

105

collaborative teaching and those taught through traditional teaching in algebra are 8.6

and 6.6, respectively. It indicates that the students taught through collaborative

teaching performed better on the achievement test than the students taught by a single

teacher. The graphic presentation of the mean achievement scores is given in figure

4.7.

ACHIEVEMENT SCORES OF STUDENTS IN GEOMETRY

Mean score differences of students taught through collaborative and through

traditional teaching on the geometry component of the MAT was calculated. Eleven

items regarding geometry were included in the MAT with one point for each correct

item. The detail is given in Table 4.11. The following null hypothesis was tested:

H06: There is no significant difference between the mean geometry scores of students

taught through collaborative teaching and those taught through traditional teaching.

Table 4.11

Comparison of Mean Achievement Scores in Geometry of Students Taught

Through Collaborative Teaching and Those Taught Through Traditional

Teaching


Control 53 3.18 1.35

96

-2.68

.01


p═.05 (N═96)

Table 4.11 shows that there was significant difference between the mean scores

of students’ achievement in geometry. The value of t96 ═ -2.68, p ═ .01 was less than

0.05. The null hypothesis, H06, which stated there is no significant difference between

106

the mean scores of students’ achievement in geometry, was rejected. The mean

achievement geometry scores of students taught through collaborative teaching and

those who were taught through traditional teaching were 3.91 and 3.18 respectively. It

indicates that the students taught through collaborative teaching performed better on

the achievement test than the students taught by a single teacher. The graphic

representation of the mean achievement scores was given below in figure 4.7.

Figure 4.7. Mean scores of students of control and experimental groups on the

content strands of algebra and geometry

ANALYSIS OF MATHEMATICAL CONTENT STRANDS AND

MATHEMATICAL PROFICIENCIES

In this study I measured the impact of collaborative teaching on students’

achievement on three mathematical proficiencies i.e. conceptual understanding,

procedural knowledge, and problem solving as well as two content strands i.e. algebra

and geometry. The comparison between control and experimental groups was analyzed

0

1

2

3

4

5

6

7

8

9

10

Algebra Geometry

Experimental Group Control Group

107

by using independent sample t-test. The detailed comparisons between groups on three

mathematical proficiencies paired with two content strands are given below:

ACHIEVEMENT SCORES OF STUDENTS’ CONCEPTUAL

UNDERSTANDING IN ALGEBRA

Six items regarding conceptual understanding in algebra were included in the


following null hypothesis was tested.


students taught through collaborative teaching and those taught through

traditional teaching on conceptual understanding in algebra.

Table 4.12

Comparison of the Conceptual Understanding Mean Achievement Scores of

Students Taught Through Collaborative Teaching and Traditional Teaching in

Algebra


Control

53 2.16 1.28

96

-3.54

.001


p═.05 (N═96)


of students’ achievement on conceptual understanding in algebra. The value of

t96 ═ -3.54, p ═ .001 was less than 0.05. The null hypothesis, H07, which stated that

there is no significant difference between the mean scores of students’ achievement in

108

conceptual understanding ability in algebra, was rejected. The mean achievement

scores of students taught through collaborative teaching and those taught through

traditional teaching are 3.1 and 2.16, respectively on conceptual understanding in

algebra. It indicates that the students taught through collaborative teaching performed

better on the achievement test than the students taught by a single teacher.


KNOWLEDGE IN ALGEBRA

Nine items regarding procedural knowledge in Algebra were included in the


following null hypothesis was tested.


students’ procedural knowledge in algebra taught through collaborative

teaching and those taught through traditional teaching.

Table 4.13


in Algebra Taught Through Collaborative Teaching and Traditional Teaching


Control 53 2.73 1.19

96

-3.59

.001


p═.05 (N═96)


of students’ achievement on procedural knowledge items in algebra. The value of t96

109

═ -3.59, p ═ .001 was less than 0.05. The null hypothesis, H08, which stated there is no

significant difference between the mean scores of students’ achievement on items

related to procedural knowledge in algebra, was rejected. It also shows that the mean

achievement scores of students taught through collaborative teaching was higher than

those taught through traditional teaching.

ACHIEVEMENT SCORES OF STUDENTS’ PROBLEM SOLVING

IN ALGEBRA

Six items regarding problem solving in algebra were included in the MAT with

one point for each correct item. The following null hypothesis was tested:



traditional teaching on problem solving in algebra.

Table 4.14

Comparison of the Problem Solving Mean Achievement Scores of Students in

Algebra Taught Through Collaborative Teaching and Those Taught Through



Control 53 1.7 0.91

96

-.28

.78


p═.05 (N═96)

Table 4.14 shows that there was no significant difference between the mean

scores of students in problem solving ability in the content strand of algebra. The value

of t96 ═ -.28, p ═ .78 was greater than 0.05. The null hypothesis, H09, which stated there

is no significant difference between the mean scores of students in problem solving in

algebra, was accepted. The mean scores of students taught through collaborative and

110

traditional teaching were1.8 and 1.7, respectively on the problem solving in algebra

items were not significantly different.

ACHIEVEMENT SCORE OF STUDENTS’ CONCEPTUAL

UNDERSTANDING IN GEOMETRY

Difference in the mean scores of students on conceptual understanding items in

geometry taught through collaborative teaching and those taught through traditional

teaching was calculated. Four items regarding conceptual understanding in geometry

were included in the MAT with one point for each correct item. The detail is given in

Table 4.15. The following null hypothesis was tested:

Ho10: There is no significant difference between the mean achievement scores of


traditional teaching on conceptual understanding in geometry component of

MAT.

Table 4.15

Comparison of the Conceptual Understanding Mean Scores of Student in

Geometry Taught Through Collaborative Teaching and Those Taught Through



Control 53 1.52 0.75

96

-1.27

0.21


p═.05 (N═96)

There was no significant difference between the mean scores of students on the

conceptual understanding in geometry. The value of t96 ═ -1.27, p ═ .21 was greater

than 0.05. The null hypothesis, H010, which stated there is no significant difference

between the mean scores of students in conceptual understanding ability in geometry,

111

was accepted. However, the mean achievement scores (1.71) of experimental group

was higher than those of the control group students’ mean scores (1.52). It means that

CT had some positive effect on students’ achievement.


KNOWLEDGE IN GEOMETRY

Four items regarding conceptual understanding in geometry were included in

the MAT with one point for each correct item. The detail is given in Table 4.16. The

following null hypothesis was tested:

H011: There is no significant difference between the mean scores on procedural

knowledge in geometry by students taught through collaborative teaching and

those taught through traditional teaching.

Table 4.16


in Geometry Taught Through Collaborative Teaching and Those Taught

Through Traditional Teaching


Control

53 1.37 0.97

96

-1.10

0.27


p═.05 (N═96)

Table 4.16 shows that there was no significant difference between the mean

scores of students on the procedural knowledge portion in the content strand of

geometry of the MAT. The value of t96 ═ -1.10, p ═ .27 was greater than 0.05. The

null hypothesis, H011, which stated there is no significant difference between the mean

scores of students on procedural knowledge in geometry, was accepted. However, the

112

mean achievement scores (1.60) of the experimental group was higher than those of the

control group students’ mean scores (1.37). It means students’ taught through CT

performed slightly better than those taught through traditional method of teaching on

items measuring procedural knowledge ability in geometry.

COMPARISON OF ACHIEVEMENT SCORES OF STUDENTS’

PROBLEM SOLVING IN GEOMETRY

Three items regarding conceptual understanding in geometry were included in

the MAT with one point for each correct item. Mean scores difference of students

taught trough collaborative and traditional teaching on problem solving in geometry

was calculated. The detail is given in Table 4.17. The following null hypothesis was

tested:



traditional teaching on problem solving in geometry.

Table 4.17

Comparison of the Problem Solving Mean Achievement Scores of Students in

Geometry Taught Through Collaborative Teaching and Those Taught Through



Control 53 0.57 0.75

96

-2.06

0.04


p═.05 (N═96)


of students on the problem solving portion of the content strand of geometry. The value

of t96 ═ -2.06, p ═ 0.04 was less than 0.05. The null hypothesis, H012, which stated there

is no significant difference between the mean scores of students on problem solving in

the content strand of geometry, was rejected. The mean achievement score of students,

113

taught through collaborative teaching was higher than those taught through traditional

teaching.

Figure 4.8. Mean scores of students of control and experimental groups on

mathematical content strands and proficiencies

0

0.5

1

1.5

2

2.5

3

3.5

4

Algeb

ra C

U

Geo

metry

CU

Algeb

ra P

K

Geo

metry

PK

Algeb

ra P

S

Geo

metry

PS

Experimental Group

Control Group

114

QUALITATIVE ANALYSIS

In this study qualitative data were collected from students to find out their

beliefs about mathematics and teaching of mathematics in collaborative settings. Data

from students were collected through semi-structured interviews. Face-to-face

interviews were conducted with students individually because subjects sometime do

not express well in groups due to uncertainty of confidentiality (Cohen et al., 2007).

Interviews with the students were conducted three times in the study: before, during,

and at the end of the intervention. The total number of students interviewed was 45

from the experimental group. The detail of number of students interviewed is given

in Table 4.18.

Table 4.18

Number of Students Interviewed Before, During, and At the End of Intervention

Before the

intervention

Middle of

intervention

End of

intervention

Total

Number of

students

interviewed

45

45

45

135

The above table shows that the total number of interviews conducted was 135 at the

start, middle, and end of intervention. Due to students’ absences and time constraints,

only 45 students were interviewed at the start of the experiment. The problem of

students’ absences remained in interviewing students in the middle of the

intervention. I conducted interviews again with those 45 students in the middle of the

experiment in two sessions. In the first session 41 students were interviewed

115

individually and 4 students were interviewed in the second session individually. At

the end of intervention those 45 students were again interviewed in one session. In

order to ensure objectivity I did not rely on my memory and video-taped the students’

responses as suggested by Creswell (2002). The approximate average time for each

interview was five minutes.

PROCESS OF AXIAL CODING

The process of axial coding was as follows:

1. Transcription of collected data through interview

2. Translation of transcribed data in English

3. Review of translated document by language experts

4. Identification of main codes / categories from the translated data

The data collected through interviews were translated and critically reviewed by

two language experts. The reviewers were experts of the languages i.e. Urdu and

English. They highlighted some grammatical mistakes in the translated version and

did some rephrasing. Main codes were identified from the transcribed data that were

the same as the questions of the interview. The reason for this may be that the

students of a public sector school were somewhat shy and not confident enough to

discuss and answer the questions in detail.

Table 4.19 shows eight statements and each statement is based on a theme such as

1) explores belief about the difficulty level of mathematics. The preceding section

presents the description of change in students’ beliefs under each of the eight themes

in terms of narrations, qualitative data, and quantitative data by students.

116

1. DIFFICULTY LEVEL OF MATHEMATICS SUBJECT

Table 4.19 shows that before the start of the experiment, 58% of the students

believed that mathematics is a difficult subject for them. This percentage declined to

46% at the middle of the intervention and 23.5%at the end of intervention. At the

start of experiment, one of the students had the belief, “It seems to be a difficult

subject to me. Sometimes I do understand questions but most of the time I can’t.” In

the middle of experiment he added to his earlier comment by attributing the difficulty

of mathematics to the language problem. He claimed that “I do not understand the

content written in English. If the Mathematics subject is written and taught in Urdu

then it might prove an easy subject for me.” He further said that “I feel difficulty in

algebra, especially with word problems. I can’t understand the questions written in

English.”

Likewise, another student said, “I like to face the difficulties. It is a difficult

subject that is why I enjoy it.” He also ascribed difficulties in the subject of

mathematics with teaching of his mathematics teachers. He contended, “I feel it to be

a difficult subject because of the difficult long questions. Our mathematic teachers do

not teach us well. They mostly leave the long and difficult questions.” But at the end

of intervention there was change in that students’ belief about mathematics difficulty.

He stated, “I feel it a little bit easy now; because now I have started to understand the

concepts and I practice it a lot at home, also.”

Similarly, one student had belief that “Mathematics is a difficult subject for

me and I feel difficulty in algebra.” He further shared his experience in these words:

“The speed of Mathematics teacher’s writing on the chalk board is very fast. I cannot

copy the content sharply. The teacher speedily solves the questions and clears the

117

chalk board.” At the end of experiment, the same student expressed his belief in these

words, “I have no issue with learning of mathematics. I study it for 2hours daily at

home; it has become an easy subject for me.”

2. MATHEMATICS AS AN INTERESTING SUBJECT

It is evident from the table 4.19 that majority of the students had belief that

mathematics is an interesting subject. In the beginning of the intervention, 55% of the

students had the belief that it is “not a boring subject.” The percentage of this belief of

students rose to 58% at the middle of intervention; and reached to 80.4% at the end of

intervention and students were found to have belief that Mathematics is an interesting

subject.

Before the intervention, one of the students said, “It is boring subject for me

especially Algebra.” In the middle of experiment he said that “I feel ease with

Algebra section. It has become interesting for me due to performing activities to

solve algebraic problems. It was too boring for me before.” His belief changed at the

end of experiment and he stated that “It is an interesting subject for me and I use to

study it at least three hours, at home.”

Similarly, another student said, “It is boring subject for me because of its

difficulty.” At the end of experiment he shared his belief in these words “Now, I can

understand mathematics well which I could not understand previously. Now, it has

become an easy subject for me. I don’t get bored in the presence of two teachers in

mathematics classroom and spend good time there.”

118

3. USEFULNESS OF MATHEMATICS

The third belief investigated was about the usefulness of mathematics in

everyday life. At the start of the intervention, surprisingly, 51.6% of the students

believed that mathematics is useful and employed in everyday life. This belief was

speedily changed in the middle of the experiment with the percentage of 96%.

Almost all the students i.e., 98%, had change to this belief by the end of intervention.

Most of the students said that mathematics is only used in counting; they used the

Urdu word “Hisab Kitab.” Before intervention, one student believed, “There is no

other use of mathematics except counting.” In the middle of experiment he said that

“This room in which we are sitting is made using mathematical concepts e.g. the front

wall is a rectangular shape. I think it has limited use in the everyday life.” He defined

that, “Yes. Mathematics is used in many things. It is used in shops; it is used in

schools; it is used in houses; it is used in books; and it is used in banks and offices.”

4. RELATION AMONG MATHEMATICAL CONCEPTS

Fewer students (6.4%) were found with the belief that mathematical topics

were related to each other when asked at the start of the experiment. The number

changed to 40.6% in the middle of the experiment, and at the end of the experiment

the percentage of the students who believed that mathematical topics and concepts

were related to each other, was found to be 66.7%. Before the intervention, I noticed

that those students who thought that the topics are related with each other had no idea

about how the topics are related. The students appeared to be confused on the

connection among the mathematical topics. For example, one student said that “Yes,

119

I think the topics of mathematics are related with each other.” When I further

explored this by asking that how they are related with each other? He replied, “I

don’t know much about it.” But at the end on intervention that student said that

“operations of algebra are connected with each other such as plus and multiplication.”

5. MEMORIZING FORMULAE

Furthermore, students’ beliefs about the necessity of memorizing formulae to

solve mathematical questions were changed. At the end of the experiment the

percentage of those students who had this belief was 31.3%. This percentage was

96.7% in the beginning of the intervention. One student shared that “It will be

difficult for me to solve mathematical questions without memorizing the formulae.”

In the middle of the experiment the same student said that “In some topics it is

necessary to memorize the formulae first, but not always.” similarly, at the end of

intervention that student argued “if someone gets understanding of the mathematical

concept then there is no need to memorize the formulae.” In the beginning of the

intervention, another student had the belief that “I have always been cramming the

formulae before solving mathematical questions.” The belief of that student were

changed positively at the end of experiment and he stated that “Understanding is

more important than memorizing the formulae because sometimes when you forget

the formula even then you can solve the questions through understanding of the

concept involved.” Similarly one student said that “due to too many formulae of

mathematics it is difficult for me to memorize them.” Another student argued that an

“Individual needs a good memory for better mathematics learning.”

120

6. STUDYING MATHEMATICS TO SOLVE MATHEMATICAL

PROBLEMS

In the beginning, most of the students (80%), had the belief that the purpose of

studying mathematics is to find the solution to mathematical problems. This belief of

the students was found to be changed in the middle of the experiment and this

percentage was decreased to 62%. At the end of the experiment when students were

again asked about their belief, only 37.5% of the students were having the belief that

there was no purpose of mathematics other than finding the solutions to mathematical

problems.

One student at the start of the intervention had the belief that “Mathematics

has only one purpose i.e. to solve the questions. This is what I always did in my

whole academic carrier.” However, at the end of the intervention the same student had

the belief that “It has many purposes like preparing good problem solver and

enhancing intelligence.”

Another student believed that “The only purpose of mathematics is to pass

exams with good grades.” But his belief changed at the end of intervention and he

attributed the purpose of learning mathematics for development of the ability to deal

with everyday problems.”

7. USEFULNESS OF CT

CT also changed students’ earlier beliefs about the value of collaborative

teaching in mathematics instruction. In the start of the intervention only a small

number of students (13%) believed that CT in mathematics produces better results. In

the middle of the experiment an equal number of students (50%, 50%) were in the

favor of and against the CT. That figure went up to 100% in favor of two teachers in

the end of the experiment. Before the experiment almost all the students had a belief

121

that single teacher teaching was satisfactory and better than two teachers teaching in

mathematics class. They were apprehensive about the strict discipline, different

styles, differing accents, and methods of teaching used by two teachers. This belief of

students rapidly changed during the intervention. Before intervention, one student

had the belief that “I think one teacher teaching is better than two teachers. CT might

be fruitful if both the teachers cooperate with each other.” He also revealed that “I

studied from one teacher throughout the academic carrier. Therefore, I can’t say

exactly which teaching produces better results. ” the same student shared his views in

the middle of the experiment, “I like it. I am feeling better by learning from two

teachers in the same class. I have no experience of this sort of teaching before. I

found both teachers to be very polite dedicated and good in their ethical behavior” At

the end of intervention the student said that “Mathematics learning in collaborative

setting is really fruitful. I enjoyed a lot. It was wonderful experience.”

In the start of intervention, one of the students was of the view that “one

teacher teaches well due to one method of teaching but two teachers with different

styles and methods may confuse the students.” that students, in the middle of

intervention expressed his feeling that CT has enhanced his mathematical learning. In

this context, he said, “Two teachers teach well because one teacher solves question by

writing it on the chalk board, the other teacher manages the classroom discipline and

helps students in understanding and solving the questions. It really helps us to

enhance our knowledge about mathematics” at the end of experiment the same student

shared his experience with two teachers’ teaching in these words, “I learned better in

the presence of two teachers as compared to a single teacher’s teaching. I had the

opportunity to ask questions from the second teacher if could not understand from one

teacher.” Similarly, in the beginning one of the students was concerned about the

122

accent of the teachers and he said that “the different speaking accent of both the

teachers will be problematic.” that student in the middle of experiment shared his

belief in this way “both teachers come up with notes and helping teaching material,

charts, and work sheets. They treat the students with love and affection. It was

wonderful experience with two teachers present in the same class. I did not feel any

sort of worries. They taught us with hard work. I become more attentive in the

presence of two teachers in the same class. ” At the end of intervention, same student

described the method of teaching which he found his teachers to be following: “the

teachers teach us turn by turn. I mean some of the concepts and activities were

performed by one teacher and remaining by other one. CT is better than single

teacher’s teaching in mathematics. ”

Likewise, another student in the beginning of the experiment said that “if one

teacher teaches with honesty then there is no need of two teachers in the class.” The

same student stated in the middle of the experiment that “discipline is key factor in

learning mathematics in the big class size. I did not feel disturbance or noise in the

class in the presence of two teachers. Both teachers check the work in the notebook

properly.”

8. MATHEMATICS LEARNING THROUGH ACTIVITIES

At the outset of the experiment, students (16%) believed that learning of

mathematics should involve activities. This belief of students was dramatically

changed to 96% during the intervention and then rose up to 100% in the end of the

intervention.

123

Most of the students had the belief that activities could be used in mathematics

instruction, but they did not know any activity that could be employed for this

purpose. For example, before intervention, one student said that “I think we can learn

mathematical concepts through activities and models but I have not made any model

till today.” At the end of experiment the same student shared his experience in these

words “I did activities first time in mathematics class. I really enjoyed them a lot.”

Likewise, another student, in the start of the experiment, said that “No! It does

not involve any activity nor it involve any experiment. There is no need of any

activity in this as we can solve questions on our note books only.” During the

experiment his belief changed and he said that, “It was difficult subject for me but

now it is not, as two teachers teach us. They use lots of mathematical activities during

the class. Now, it seems an easy and interesting subject to me because of involving in

activities.”

Similarly, another student had the belief, “I have never performed any activity

in learning of mathematics, but I think there might be some activities about which I

have no idea” Same student in the middle of intervention explained that “I am

learning mathematics by activities. I am happy that I know some activities to learn

mathematical concepts. Thanks Allah.” At the end of experiment he had the belief

that “It is good to learn mathematics through activities. I really enjoyed a lot.

Mathematics teachers should use activities in the classrooms to teach mathematical

concepts.”

124

Table 4.19

Change in Beliefs of Students about Mathematics and Mathematics Teaching

Beliefs of Students

Beliefs before

experiment

Beliefs during

experiment

Beliefs after

experiment

Yes Yes Yes

In Percentage

1) Math is an easy subject 42 54 76.5

2) It is an interesting

subject

55

51.6

58 80.4

3) It is useful in everyday

life

96 98

4) Its concepts are related

to each other

6.4 40.6 66.7

5) One must to memorize

the formulae first to

solve math problems

96.7 76 31.3

6) Math is used to find the

solutions of

mathematical problems

only

80 62 37.5

7) CT in the Math class

produces better results

13 50 100

8) Math concepts can be

learnt through activities

16

96 100

Change in beliefs of 8th grade mathematics students is also shown in fig. 4.9 &

4.10 using line graph. The line graph shows some steeper lines which means rapid

positive change in the beliefs of students during the experiment e.g. usefulness of

mathematics in everyday life and relationship among mathematical concepts (see fig.

4.9). Figure 4.10 shows that all of the students’ beliefs were changed positively

125

about the teaching of mathematics in collaborative settings at the end of the

experiment.

Figure 4.9. Students’ beliefs about mathematics

Figure 4.10. Students’ beliefs about the teaching of mathematics

0

20

40

60

80

100

120

Before During End

Mathematics is an easy

subject

It is an interesting

subject

It is useful in daily life

Its concepts are related

to each other

It is must to memorize

the formulae first to solve

math problems

The purpose of Math is to

find the solutions of

mathematical problems

only

0

20

40

60

80

100

120

Before During End

CT in the same Math class

produces better results

Math concepts can be

learnt through activities

126

CHAPTER 5

SUMMARY, FINDINGS, CONCLUSIONS, DISCUSSION,

AND RECOMMENDATIONS

SUMMARY

Mathematics plays an important role to develop thinking, reasoning, and

problem solving abilities that enable humans to become good citizens. Mathematics

can be distinguished from other subjects due to its peculiar language, symbols, and

abstract concepts. Students face difficulties in learning mathematics, some of which

are attributed to teaching (Russell, 2006), especially with a single teacher teaching the

subject. A single teacher cannot cope with all problems effectively because of time,

energy, knowledge, methods, and lack of interaction with students individually. CT is

a teaching approach in which two or more teachers collaboratively plan, organize,

present, and evaluate their teaching. It has different settings like one teacher teaching

and one assisting, teaming, and parallel teaching. Literature shows that the CT

approach is more advantageous than other mathematics teaching approaches in terms

of students’ learning.

In the context of Pakistan, mathematics is taught predominantly by one

teacher. Moreover, mathematics teachers do not collaborate with colleagues to

discuss concepts or methodologies of teaching which results in low achievement of

students in this subject. Keeping in view the importance of CT, the objectives of this

study were to: examine the impact of CT on 8th grade students’ achievement in

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mathematics, examine CT’s impact on content strands of mathematics (algebra and

geometry), examine CT’s impact on mathematical abilities (conceptual understanding,

procedural knowledge, and problem solving), and explore the beliefs of students

about mathematics and teaching of mathematics in collaborative settings.

The nature of the study was mainly focused on quantitative aspects; using

experimental research. An experiment was conducted on 8th grade students using the

Solomon Four Group experimental research design. This design consists of four

groups (two experimental and two control groups) and randomly assigns the subjects

to the groups. The study was delimited to 8th grade in the subject of mathematics. All

the students of 8th grade studying in the public schools of Sargodha district of Punjab,

Pakistan was the population of this study. I faced difficulty in the selection of a

public school as a sample due to two reasons. The first reason was the lack of

willingness of the headmasters. Most of the heads of public schools refused to allow

the experiment because of random assignment of students into four groups. The

second reason was the lack of availability of two mathematics teachers, each with

M.sc (Mathematics) and a B.Ed. Finally, one public school was selected from the

Sargodha district. All the available students studying in the 8th grade, i.e. 118

participated in the experiment. I assigned 118 students to four groups randomly

through SPSS-16. Two volunteer mathematics teachers (each with M.Sc. in

Mathematics and a B.Ed.) from the sampled school participated in this study. I,

having the same qualifications, also took part in the study as a co-teacher. I held

training sessions for the two mathematics teachers over two days, with two hours each

day. I explained the research objectives to the two mathematics teachers, and shared

details of the syllabus to be covered and the schedule of periods by topics and dates.

I asked both the mathematics teachers to be regular and punctual during the

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experiment. Further, I discussed with the co-teacher of the study about CT separately.

I spoke about how it was to take place and what things should be kept in mind before

the start of collaborative teaching.

In this study two tools were used: an achievement test and a semi-structured

interview protocol. The first instrument, the Mathematics Achievement Test (MAT),

was used to measure the achievement of students as both pre-test and a post-test. It

was an MCQ type test. The items used in the test were selected from the pool of

items developed by the National Educational Assessment System (NEAS). These

items address three mathematical proficiencies: conceptual understanding (CU),

procedural knowledge (PK), and problem solving (PS) under two mathematical

content strands, algebra and geometry. There were 32 items in the MAT. The

number of items for the mathematical content strands of algebra and geometry were

21 and 11, respectively. Similarly, the number of items for mathematical

proficiencies of conceptual understanding, procedural knowledge, and problem

solving were 10, 13, and 9, respectively. The second tool used in this study was a

semi-structured interview protocol. According to Bogdan and Biklen (2007), a

researcher gets comparable data across participants with semi-structured interviews.

In this study, interviews were conducted before, during, and after the intervention.

The interview protocol consisted of questions based on various themes related to

perceptions and beliefs about mathematics i.e. Difficulty level of the subject, Math as

interesting subject, usefulness of mathematics in everyday life, usefulness of co-

teaching in mathematics classroom, relationship among mathematical concepts,

Mathematics learning using activities, memorizing math formulae first to solve

mathematical problems, and purpose of studying mathematics is only to find out the

solutions to mathematical problems. The interview protocol was validated through

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expert opinion and piloting. The three experts (one bilingual and two SMEs) gave

their opinion on the beliefs included in the protocol and about the language used in

the protocol. The protocol was also piloted with 15 students studying mathematics in

the 8th grade.

Moreover, a Collaborative Mathematics Teaching Module (CMTM),

consisting of collaborative lesson plans of mathematics for 8th grade students was

developed by us (me and a co-teacher of mathematics). This module covers two

mathematical content strands, algebra and geometry. The CMTM includes the three

collaborative settings: one-teach one-assist, teaming, and parallel teaching. Many

mathematics teaching methods were used with these settings like inductive, activity

based, the assignment method and the problem solving method. The CMTM includes

assessment techniques such as asking students questions during the lecture, giving

classroom tests, assigning homework worksheets, or asking students to solve

questions on worksheets during the lecture.

I used descriptive and inferential statistics such as Mean, Cluster Bar,

Factorial ANOVA, and an independent samples t-test in order to analyze the

quantitative data collected from the MAT. The Audio-Video recorded data from

interviews were first transcribed and then translated from Urdu, the national language

of Pakistan, to English. The qualitative data were analyzed through percentages, and

line graphs.

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FINDINGS

The findings of this study were as follows:

OVERALL STUDENT LEARNING ACHIEVEMENT

1. H01, stating no significant difference between the mean achievement scores of

students taught through collaborative teaching and traditional teaching, has not

been accepted. It was found that the mean score (12.80) of students on MAT

items taught through CT was significantly higher (F= 19.50, p < 0.01) than of

those taught through traditional method (10.08).


2. In case of conceptual understanding items, the mean score (4.80) of the students

taught through CT was significantly higher (t= -3.49, p < 0.05) than the mean

score (3.69) of the students taught through traditional methods of teaching.

Therefore, H02, stating no significant difference between the mean achievement

scores in conceptual understanding of students taught through collaborative

teaching and traditional teaching, has not been accepted.

3. Similarly, the mean score (5.4) of the students on procedural knowledge items

taught through CT was significantly higher (t = -3.71, p < 0.05) than the mean

score (4.11) of the students taught through traditional methods of teaching. So,

H03 i.e. there is no significant difference between the mean achievement scores in

procedural knowledge of students who are taught through collaborative teaching

and those who through traditional teaching, has not been accepted.

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4. H04 (there is no significant difference between the mean achievement scores in

problem solving of students taught through collaborative teaching and traditional

teaching), has been accepted and it was found that there is no significant

difference (t = -1.47, p > 0.0.5) in the mean score (2.64) of the students on

problem solving items taught through CT and that of traditional methods (2.26).

MATHEMATICAL CONTENT STRANDS

5. It was found that H05 i.e. there is no significant difference between the mean

scores of students who are taught through collaborative teaching and traditional

teaching on achievement test items of algebra has not been accepted. The mean

score (8.6) of the students on algebra items taught through CT was significantly

higher (t = -4.13, p < 0.05) than the mean score (6.6) of the students taught

through traditional methods.

6. On geometry items the mean score (3.91) of the students taught through CT was

significantly higher (t = -2.68, p < 0.05) than the mean score (3.18) taught through

traditional methods. So, H06 which states that there is no significant difference

between the mean scores of students who are taught through collaborative

teaching and traditional teaching on achievement test items of geometry has not

been accepted.

MATHEMATICAL PROFICIENCIES AND CONTENT STRANDS

7. The mean score (3.1) of the students on conceptual understanding in algebra items

taught through CT was significantly greater (t = -3.54, p < 0.05) than the mean

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score (2.16) of the students taught through traditional methods. H07: There is no

significant difference between the mean scores of students who are taught through

collaborative teaching and traditional teaching in conceptual understanding in

algebra has not been accepted.

8. Similarly, it was found that H08 i.e. there is no significant difference between the

mean scores of students who are taught through collaborative teaching and

traditional teaching in procedural knowledge in algebra has not been accepted.

The mean score (3.8) of the students on the procedural knowledge in algebra items

taught through CT was significantly greater (t = -3.59, p < 0.05) than the mean

score (2.73) of the students taught through traditional methods.

9. On problem solving in algebra items taught through CT the mean score (1.8) of

the students was not significantly higher (t = - 0.28, p > 0.05) than the mean score

(1.69) of the students taught through traditional methods. So, H09 (there is no

significant difference between the mean scores of students who are taught through

collaborative teaching and traditional teaching in problem solving in algebra.) has

been accepted.

10. It was found that the mean score (1.71) of the students on conceptual

understanding in geometry items taught through CT was not significantly higher (t

= -1.27, p > 0.05) than the mean score (1.52) of the students taught through

traditional methods. Hence, H010: there is no significant difference between the

mean scores of students who are taught through collaborative teaching and

traditional teaching in conceptual understanding in geometry, has been accepted.

11. It was found that H011, i.e. there was no significant difference (t = -1.10, p > 0.05)

in the mean score (1.60) of the students on the procedural knowledge in geometry

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items taught through CT and the mean score (1.37) of the students taught through

traditional methods has been accepted.

12. Similarly, the mean score (0.9) of the students on the problem solving in geometry

items taught through CT was significantly greater (t = -2.06, p < 0.05) than the

mean score (0.566) of the students taught through traditional methods. So, H012,

there is no significant difference between the mean scores of students who are

taught through collaborative teaching and traditional teaching in problem solving

in geometry, has not been accepted.

MATHEMATICAL BELIEFS

13. The percentage of the experimental group students who believed that mathematics

is an easy subject increased from 42 (before) to 76.5 (end of experiment).

14. Collaborative teaching was effective in creating interest for math among students.

The percentage of such students before the experiment was 55 which rose to 80.4

by the end.

15. Only 51.6% of the students considered math a useful subject prior to the

experiment. This percentage increased to 96 by the middle and to 98 by the end.

16. The students’ belief about co-teaching was changed positively at the end of

experiment. The percentages of students believing in the effectiveness of co-

teaching in mathematics class before, during and after intervention were 13, 50,

and 100, respectively.

17. There was positive change in the students’ belief about relation of mathematical

concepts. The percentage of students believing in the interrelation of

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mathematical concepts was changed considerably by the end of intervention i.e.

66.7 which was 6.4 at the start of intervention.

18. Students got involved themselves in activities during the experiment. The

students’ belief about learning of mathematical concepts by using activities was

changed positively and quickly with the passage of time. The percentages of

students having the belief that “mathematical concepts can be learned better by

doing activities” were 16, 96, and 100 before, in the middle, and at the end of

intervention respectively.

19. The percentages of students belief that “it is necessary to memorize the formulae

first to solve mathematical problems” before, during and after intervention were

96.7, 76, and 31.3, respectively.

20. The percentage of the students having the belief that the purpose of mathematics

is only to find out the solutions to mathematical problems was changed

considerably by the end of intervention i.e. 37.5 which was 80 at the start of

intervention.

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CONCLUSIONS

The conclusions drawn from the findings of the study were as follows:

COLLABORATIVE TEACHING

1. CT is more effective than the traditional teaching of mathematics at the 8th

grade level in improving academic achievement of the students.


2. CT is more effective than the traditional teaching of mathematics at the 8th

grade level in improving academic achievement of the students in the

mathematical proficiencies of conceptual understanding and procedural

knowledge.

3. CT did not improve students’ problem solving ability significantly more than

the traditional method of teaching mathematics.

MATHEMATICAL CONTENT STRANDS

4. CT is more effective than the traditional teaching at the 8th grade level in

improving academic achievement of students in the mathematical content

strands of algebra and geometry.

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MATHEMATICAL PROFICIENCIES WITH CONTENT

STRANDS

5. In the content strand of algebra the CT is better than the traditional method of

teaching in improving the academic achievement of students in conceptual

understanding and procedural knowledge. However, it did not improve their

problem solving ability significantly more than the traditional method of

teaching mathematics.

6. In geometry CT was more effective than the traditional method of teaching

mathematics in improving the academic achievement of students only in

problem solving but was not significantly more effective than the traditional

method of teaching in case of conceptual understanding and procedural

knowledge ability.

MATHEMATICAL BELIEFS

7. The students’ beliefs about mathematics and teaching of mathematics in the

collaborative setting can be changed positively using CT.

8. Through the use of CT in mathematics classroom at 8th grade, the beliefs of

students can be changed about the usefulness of mathematics in everyday life,

usefulness of co-teaching in mathematics class, and effective learning of

mathematical concepts through doing activities.

137

DISCUSSION

An experimental research design, Solomon Four Group, was applied to

examine the impact of collaborative teaching on students’ learning achievement in

mathematics at 8th grade. The experiment was conducted in a public school of

Sargodha district. There were four randomly assigned groups of mathematics

students two experimental and two control. The treatment of CT was given to the

experimental groups while the control groups were taught mathematics using

traditional methods. In the experimental groups two mathematics teachers taught

mathematics using collaborative teaching approaches. The teachers planned the

Collaborative Mathematics Teaching Modules (CMTM), and they collaboratively

taught mathematics to the students.

The study showed the effectiveness of CT as compared with single teacher

teaching of mathematics at the 8th grade level. This method of teaching has already

yielded positive effects on students’ learning achievement in developed countries like

USA, UK, China, Australia, Canada etc. (Mcduffe et al., 2007). The finding of the

present study is also in line with the findings of studies by Murawski and Swanson

(2001), Jang (2006), Parker (2010), and Goddard et al. (2007).

According to the National Educational Policy 2009, “English shall be used as

the medium of instruction for science and mathematics from class IV onwards.”

(p.20). Thus at the time of experiment the mathematics teachers were using English as

a medium of instruction in public schools. Earlier, Urdu was the medium of

instruction in public schools of Punjab province. I addressed three mathematical

proficiencies i.e. conceptual understanding, procedural knowledge and problem

138

solving in mathematics using CT. It was found that CT significantly affected the

students’ scores on conceptual understanding and procedural knowledge test items.

The research finding that CT did not significantly raise students’ scores in problem

solving might be due to using English as a medium of instruction (Khar, Lay,

Areepattamannil, Treagust, & Chandrasegaran, 2012) in the classes during the

experiment. Additionally, MAT was also administered in English. For problem

solving items one of the pre-requisite is good comprehension of problems stated in

English. The English language proficiency of the 8th grade students of public schools

is very low for this purpose. The cohort under study was at a special disadvantageous

position because they studied all subjects in Urdu up to grade 5 and were shifted to

English without any proper support and preparation. In the same vein, the general

practice of mathematics teachers is that they use English to promote mathematical

terms like square root, factorization, variable etc., but they deliver their lectures in

Urdu. Use of the English language in textbooks, teaching, and evaluation makes it

difficult for the students to understand word problems in mathematics. Another

reason of this finding might be mathematics students’ learning practices, as most of

the students in Pakistan learn mathematical concepts and procedures by memorizing

formulae and the drill method. Since students were exposed for the first time in their

academic carrier to a new teaching approach i.e. collaborative teaching, they might

have had problems in adjusting themselves to a new mode of teaching and learning.

It is evident from literature that specific teaching practices can change students’

beliefs about mathematics (Taylor, 2009). In this study, I conducted semi-structured

interviews of the students to explore their beliefs about mathematics and teaching of

mathematics in collaborative settings in the beginning, middle, and at the end of

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139

experiment. CT positively affected students’ academic achievement and changed

their beliefs positively about mathematics and its teaching.

The proficiency to solve mathematical problems is needed to develop and

improve: generic ability to solve real life problems, deep understanding of

mathematical concepts, imagination power, and ability to reason (NCTM, 1989).

Some of the tools used in teaching to develop such mathematical proficiency like

brainstorming, cause and effect diagrams, flowcharting, decision matrix, heuristics,

and algorithms are used all over the world. In Pakistan, mathematics students are

practicing formula driven rather than concept driven mathematics. Similarly,

mathematics teachers teach and ask students to memorize the formulae of

mathematics to solve mathematical problems i.e. deductive method. I also found that

even at the end of the experiment 31% of students held the belief that it is necessary

to memorize the mathematical formulae first to solve mathematical problems. Similar

were the findings by Kloosterman (2003) and Dossey et al. (1988).

We are living in the world of mathematics. Every economist, scientist,

businessperson, accountant, engineer, mechanic, farmer, shopkeeper and even street

hawker requires and need to use mathematical concepts in their fields. Therefore,

students of mathematics are needed to be aware of the use of mathematical concepts

in everyday life. In Pakistan students and teachers focus on getting good grades in

exams and do not relate the concepts with everyday life. I found that nearly half of

the students believed that there is limited use of mathematics in everyday life.

Kloosterman et al. (1996) also found that students’ believe in the limited use of

mathematics in everyday life.

A traditional method of mathematics teaching i.e. deductive method is usually

practiced by single teacher in Pakistan. CT in mathematics classes was new for 8th

140

grade mathematics students. Lin and Xie (2009), Krosnick (2007), and Ragland

(2011) found that CT changed the beliefs of students about the teaching of

mathematics in collaborative settings. Likewise, this study found that CT entirely

changed students’ belief about learning of mathematics in general and in collaborative

teaching settings.

This study opens new dimensions towards adaptation of new teaching

approaches in Pakistani schools. There are some suggestions for the school principals

in order to adapt this new teaching approach i.e. collaborative teaching in their

schools. First of all, they need to know the importance of CT in teaching of

mathematics. School Principals should also follow some steps to ensure that co-

teachers have a checklist covering the following aspects: a regular schedule to plan

together, lesson plans, collaborative settings, and management of teachers’ and

students’ time to ensure the availability of the activities (GDOE, 2006).

Moreover, it gives a move forward to the researchers and academicians in the field

of math education. The present study may be replicated in different grades. In

addition to that for the enhancement of critical thinking and better comprehensive

understanding of mathematics students in Pakistan, different mathematics teaching

methods like inductive method, question answer, analytical, drill and practice with

different collaborative settings may be used in class rooms. These methods may

change the students’ mathematics learning approach from memorization to

understanding and make them better learners.

141

RECOMMENDATIONS

In the light of findings of the study, the following recommendations were given:

1. In Pakistan single teacher teaching is being practiced in schools. Findings

show that CT is a better alternative to single teacher teaching of

mathematics. It is recommended that pre-service teachers’ training

institutions may include CT in the course of ‘Methods of Teaching’ with a

focus on teaching of mathematics. The topic may include the effectiveness

of CT, how it takes place, how to develop collaborative lesson plans and

how to practice CT in the classroom.

2. This research study found that CT changed the beliefs of the students

about mathematics and its teaching. Having strong beliefs about learning

does impact one’s academic achievement, both cognition and behavior.

Therefore, it is recommended that mathematics teachers may practice CT

in classrooms in order to enhance academic achievement of mathematics

students.

3. In order to incorporate CT in Pakistani public schools, Govt. may provide

two mathematics teachers and take measures to implement collaborative

teaching in large size high schools and allocate or reallocate teaching

resources such as stationary, models used for learning mathematical

concepts, AV-aids etc. to be used in collaborative settings.

4. In the beginning, 93.6 % students were having the belief that mathematics

concepts are not related to each other which were reduced to 34.3% by the

end of the intervention. So, mathematics teachers may put more effort to

142

relate various mathematical concepts for better and comprehensive

understanding.

5. 96.7% students believed that it was necessary to memorize the

mathematical formulae first in order to solve mathematical problems in the

start of intervention which were reduced to 31.3% by the end. Therefore,

it is recommended that mathematics teachers may use inductive method in

their teaching of mathematics to bring change in students’ mathematics

learning approach from memorization to understanding.

6. 80% students believed that the purpose of mathematics is to find the

solutions to mathematical problems only in the beginning of intervention

which were reduced to 37.5% by the end. It is recommended that

mathematics teachers may be educated during pre-service and in-service

training to teach the concepts of mathematics relating to everyday life.

SUGGESTIONS

Following are the suggestions of the study:

1. Educational policy makers may suggest or plan to design a module of

mathematics teaching for in-service mathematics teachers which may include

knowledge about CT: its usefulness in mathematics, its impact on academic

achievement of students, and procedure of practicing CT effectively in

mathematics classes. The module may include all content strands of

mathematics i.e. arithmetic, algebra, geometry (measurement and

construction), data analysis and probability.

143

2. The present study used one teach-one assist, parallel teaching, and team

teaching models of CT. It is suggested that further research in the context of

CT may use other collaborative settings such as alternative teaching, station

teaching etc. and it is also suggested that researchers may investigate the

effectiveness of various models of CT (i.e. one teach-one assist, parallel

teaching or team teaching, alternative teaching, station teaching models) and

the teaching methods (Inductive, analytic, synthetic, project) in enhancing

students learning outcomes.

3. This study was carried out on boy students of 8th grade public schools. The

findings of this study tell us that CT has positive impact on the students’

academic achievement in mathematics and has been effective in changing the

attitude of students towards mathematics. Further research may be conducted

on girl students of public schools, private schools, and on different grade

levels. It will help in comparing the impact of CT on students’ achievement

by gender, school, grade, and students of different ability levels. This may

lead to wider applicability of the method.

4. Additional factors like social learning of students and teachers, classroom

learning environment, students’ attitude towards mathematics teaching, and

classroom discipline may also be explored using collaborative teaching.

5. Collaborative teaching proved a better alternative to traditional teaching in

order to teach the two selected math strands algebra and geometry. Further

research may include other mathematical content strands, i.e. arithmetic, data

analysis and probability.

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158

APPENDICES

APPENDIX - A

Code

Mathematics Achievement Test

For Grade 8

Control Group Post – test Group

1 2 2

159

Mathematics Achievement Test for grade 8

Name: ----------------------------------------------Father’s Name: -------------------------------------

----

Name of the School: --------------------------------------------------------------------------------------

------------------------------------------------------------ Roll No: -----------------------------------------

Date: --------------------

Time: 90mins

This test covers the content strands of Algebra and Geometry and addresses three

mathematical proficiencies i.e. conceptual understanding, procedural knowledge, and problem

solving. In order to solve this test in a better way please read the instructions carefully.

Instructions for the students

The instructions are as follows:

i. Attempt all questions.

ii. Read each question carefully before answering.

iii. Encircle the correct option.

iv. Do not encircle more than one option.

v. Underline key words that help you to focus on what is expected.

vi. Check your calculations, even when your answer is one of the choices.

vii. Do not use mobile or calculator.

viii. Use the space in front of each question or the backside of the printed paper for rough

work.

Note: All questions carry equal marks.

160

If one angle of a triangle is 900, the triangle is called:

A. Acute Angle Triangle

B. Right Angle Triangle

C. Obtuse Angle Triangle

D. Equilateral Triangle

Simplify the expression a - (b – c).

A. b – c + a

B. b +c – a

C. a – b + c

D. a + b + c

Evaluate (7.7)2 - (2.3)2.

A. 50

B. 54

C. 56

D. 60

(903)2 is equal to:

A. ( 900 + 3 )2

B. ( 90 + 3 )2

C. ( 90 )2 + ( 3 )2

D. ( 900 )2 + ( 3 )2

161

Simplify the expression(x2- 5x + 6) / (x – 2)

assume x ≠ 2.

A. (x+ 3)

B. (x- 3)

C. (x+ 2)

D. (x- 2)

The age of Arif after 18 years will be three times

the present age of Arif. What is the present age

of Arif?

A. 7 years

B. 9 years

C. 11 years

D. 13 years

If the lengths of sides of a triangle are 3cm, 4cm, and

6cm, the triangle is called:

A. Equilateral Triangle

B. Scalene Triangle

C. Right Angle Triangle

D. Isosceles Triangle

162

Evaluate the expression 2x + 6y, when x + 3y = 4.

A. 4

B. 8

C. 10

D. 12

Which of the following expressions represents q + q + q + q?

A. q + 4

B. 4q

C. q4

D. 4( q + 1)

A rectangular carpet is 10 feet long and 7 feet

wide. What is the area of the carpet?

A. 17 square feet

B. 27 square feet

C. 34 square feet

D. 70 square feet

163

In a cricket match Waqar scored three times that of

Hanif and the sum of their scores is 80. How

many runs did Waqar score?

A. 20

B. 40

C. 60

D. 80

The degree of polynomial 3x4+2x2+x+1 is:

A. 1st degree

B. 2nd degree

C. 3rd degree

D. 4th degree

The present age of Ali is 40 years. If after two

years, his age becomes three times the present

age of his son, what is the present age of his son?

A. 14 years

B. 20 years

C. 26 years

D. 32 years

164

What is the area of a rectangle if its measurements

are 10cm and 5cm?

A. -25cm2

B. 15cm2

C. 30cm2

D. 50cm2

Which of the following expressions is equal

to 2x - 3y + 7x + 5y?

A. 5x + 2y

B. 5x + 8y

C. 9x + 2y

D. 9x + 8y

Evaluate the expression (2a + 3b) (2a - 3b)

when a = 6 and b = -3.

A. 441

B. 63

C. 24

D. 9

165

Sum of two consecutive even numbers is 82, the

first even number is equal to:

A. 40

B. 42

C. 78

D. 80

In the above figure a triangle ABC is given. What

is the measure of angle “C” in triangle ABC?

A. 300

B. 450

C. 600

D. 750

Simplify the expression (- 5x3) × (- 4x2).

A. 20x5

B. - 20x5

C. 20x6

D. - 9x6

166

Factor the expression 16a2 - 9b2.

A. ( 16a - 9b )( 16a + 9b )

B. ( 16a - 9b )( 16a - 9b )

C. ( 4a - 3b )( 4a - 3b )

D. ( 4a - 3b )( 4a + 3b )

Which formula is shown by the above

diagram?

A. ( a – b )2

B. ( a + b )2

C. a2- b2

D. a2 + b2

167

Which of the following expressions represents y3?

A. y × y × y

B. y + y + y

C. 3y

D. y2+y

If a number x is added to 4 times of itself, the

result is 45. What is the value of a number x?

A. 9

B. 36

C. 40

D. 41

The sides of a triangular garden are given in

the figure. What is the approximate area of the garden?

A. 9.0cm2

B. 11.6cm2

C. 14.7cm2

D. 33.0cm2

168

X 3 4 6 8

Y 0 1 3 5

Which of the following equations indicates the

relation between x and y in the given table?

A. y = x + 3

B. y = x - 3

C. 3y = x

D. x = 1 / 3y

A 5m long ladder is standing on a plain floor.

The distance between its lower end and the

wall is 3m. Its upper end is touching

the upper end of the wall. The height of the

wall is:

A. 3m

B. 4m

C. 5m

D. 6m

169

The diameter of inner base of a conical minaret is

6m and its height is 14m. Its volume is:

A. 140m3

B. 264m3

C. 176m3

D. 132m3

Z Y

W X

The given figure WXYZ is a rectangle. Which of

the following statements is not true?

A. Sides WX and ZY are Parallel

B. Sides XY and WZ are Parallel

C. Sides WX and YZ are Parallel

D. Sides WX and XY are Parallel

170

A 5m long ladder reaches a height of 4m in the wall.

How far will the foot of the ladder be from the wall?

A. 1m

B. 3m

C. 4m

D. 6m

In the given figure there is concrete path around a

rectangular pond. What is the area of the path?

A. 100m2

B. 161m2

C. 710m2

D. 1610m2

171

Price of one bat and two balls is Rs. 92 and price of

three bats and four balls is Rs. 234. What is the price

of a bat?

A. Rs. 60

B. Rs. 50

C. Rs. 40

D. Rs. 30

If double of a number x is 24, what is the value

of 1/3 of x?

A. 4

B. 6

C. 8

D. 12

172

APPENDIX B

Collaborative Mathematics

Teaching Module (CMTM)

For Grade 8

173

List of Contents

Sr. no. Topics Page no.

1 Introduction 156

2 Types of collaborative setting 158

3 Objectives of the Collaborative Mathematics Teaching

Module (CMTM)

159

4 Content Strands with subtopics 160

5 Collaborative Mathematics Teaching lesson plans 161

Introduction

Mathematics is an important subject; it develops the reasoning faculty of human

minds. Its importance has been recognized by eminent educationist. It is related with

other subjects like Physics, Chemistry, and Statistics etc. The subject of Mathematics

covers four content strands i.e. Arithmetic, Algebra, Geometry, and Probability at

elementary level in Pakistan.

Predominantly, the traditional teaching approach - deductive method of

teaching used by single teacher - is practicing in Pakistan. Collaborative teaching (CT)

which has many advantages such as students have the opportunity to see diverse

teaching styles and faculty members develop mutual trust and respect for each other

and also cover their specialty areas. Moreover, it helps teachers to learn valuable

174

information from each other in terms of content and teaching styles. Furthermore, it

enhances the quality of teaching by transforming it into a participative activity, and

eliminating the perception of being isolated. In-spite of many advantages, CT is not

being used for teaching mathematics in the country.

Moreover, some Mathematics Teaching Modules (MTMs) are presently

available in Pakistan for 8th grade. These were developed by the Directorate of Staff

Development (DSD) Lahore and UNESCO. These modules lack in some areas; for

example, DSD has developed MTMs only for the content strands of Arithmetic and

Geometry and missed the Algebra section. They used lecture, demonstration and drill

and practice methods for teaching geometry. This module does not include the problem

solving and assignment method, and activity based method to teach geometry.

Similarly, UNSECO has developed MTM at 8th grade. Although, this module includes

the content strands of algebra and geometry but the module addressed only few topics

of algebra and geometry. This module includes the inductive method and drill and

practice method to teach algebra. It did not include problem solving, activity base

method and assignment method to teach algebra. Both the available modules had

developed for single teacher teaching in the classroom. Although CT is important in

mathematics’ teaching but still no Collaborative Mathematics Teaching Module

(CMTM) exists in Pakistan to teach Mathematics at 8th grade.

Therefore, keeping in view the significance of CT in mathematics, CMTM was

developed for the present study. This module covered two content strands i.e. Algebra

and Geometry. CMTM includes the three collaborative settings such as one-teach one-

assist, teaming, and parallel teaching. Moreover, many mathematics teachings methods

were used with these settings like inductive, activity based, assignment method, and

problem solving method. Furthermore, CMTM includes assessment techniques such as

175

asking questions verbally from the students during the lecture, classroom tests,

homework worksheets, or by asking student to solve questions on the work sheets

during the lecture.

Types of Collaborative Settings

The detail of the collaborative settings in this CMTM is given below:

One Teach-One Assist

In this collaborative setting, one teacher delivers the instruction to the class

whereas the other teacher assists, helps, and guides the students in the classroom. He

assist the students in accomplishing their activities, help students to complete their

assignments and completing their work sheets in the classroom. He ensures that

students are doing their work properly.

Parallel Teaching

In the parallel teaching approach, the class will be divided into two equal

groups. Both teachers will assign mathematics’ worksheets, on specific topics, to their

group students and help them to complete those worksheets. They will help and guide

their group students independently the same tasks in the same class.

176

Team Teaching

It involves both teachers in delivering the instructions in the classroom. They

both simultaneously teach the concepts together. Their turns depends on the expertise

they have e.g. one is good in making drawing he should draw geometrical figures, one

may be good in writing on the chalk board, one may be a good in communicating and

explaining well verbally. So, both teachers will teach together the same concept.

Objectives of the Collaborative Mathematics Teaching Module

(CMTM)

The general objectives of CMTM are to:

Help and guide mathematics’ teachers to teach Algebra and Geometry

Enhance the knowledge of mathematics’ teachers in the subject of mathematics

Enhance the skills of mathematics teacher in teaching mathematics

Facilitate the mathematics teacher in various collaborative settings while

teaching mathematics

Enhance students learning in mathematics

177

Content Strands with Subtopics

CMTM consists of two content strands i.e. Algebra and Geometry of 8th grade

mathematics. The total number of collaborative lesson plans was 20. The time allocated

for each lesson was one hour. The detail of content strands with subtopics is given

below in Table 1.

Table 1

Detail of Sub-topics Contained in the CMTM

Content

strand

Sub-topic No. of

Collaborative

lesson plans

Allocated

Time

(minutes)

A

lgeb

ra

Evaluating algebraic expressions 1 60

Addition and subtraction of polynomials up

to degree 4

2 120

Multiplication of polynomials up to degree 4 1 60

Establishing formulae 4 240

Factorization 2 120

Solve Linear Equations 1 60

Solve Simultaneous Linear Equations 1 60

G

eom

etry

Apply Pythagoras theorem 2 120

Hero Formula 1 60

Concept of cone and finding slant height 1 60

Find the surface area of sphere 1 60

Find the volume of sphere 1 60

Find the surface area of cone 1 60

Find the volume of cone 1 60

178

Collaborative Mathematics Teaching Lesson Plans

Lesson plan no. 1

Content Strand: Algebra Grade 8

Topic: Evaluation of Algebraic Expressions

Specific

students’

learning

outcomes

After delivering this lesson the students will:

Demonstrate the evaluation of algebraic expressions.

Prerequisite

It is expected that students know the concept of constant, variable, and

algebraic expression.

Mate

rial

req

uir

ed

Charts of square boxes consisting of figures, work sheet

Collaborative

settings

Teaming Time

60mins

In

trod

uct

ion

At the start of the first lesson both teachers manage the

classroom sitting arrangement in a specific manner so that

teachers can access to every student during the lesson.

5mins Teacher A

Introduces the topic by

writing it on the chalk

board

Teacher B

Meanwhile, he maintains the

discipline of the class.

Pre

vio

us

kn

ow

led

ge

After the explanation of

teacher B, he writes x=1

and y=1,2, or 3 on the

chalk board and ask the

students to answer the

following questions

verbally:

Is x a constant or a

variable?

Is y a constant or a

variable?

(x is a constant and y is a

variable)

After this, he calls five students

of different height and ask the

following question to the class:

Do the students vary in height?

The students answer verbally

(yes). Then, he explains that

height is a variable.

5mins

179

Teacher A Teacher B Time

D

evel

op

men

t A

cti

vit

y

After checking the previous

knowledge, he tells the

students, variable is a

changing quantity and

constants are those

quantities or characteristics

which do not change.

Draws the following on the

chalk board.

Asks the students to write down

the expression for box (A).

10mins

Now, both teachers go to every student in the class, one by

one, and guide the students in writing the following

expression which represents the figure in box (A).

i.e. 3□ + 3 + 4▲

After ensuring that each student has written the expression,

both teachers distribute the worksheet among the students.

It includes:

180


Dev

elop

men

t A

ctiv

itie

s

Explains that we can

represent the properties

(price, weight) of the things

in box (A) with x, y, and z.

He further explains that x,

y, z at once might be the

prices of the things or it

may be the weights of the

things. Now, let us

consider that x, y, and z are

the prices of the things of

box (A) and we want to

purchase the things, present

in box (A), from the

market ‘S’ with prices x =

price of a bowl = 5Rs

y = price of a glass = 10Rs

z = price of a ball = 15Rs

the total expanse can be

calculated as

= 4(5)+2(10)+3(15)

= 20+20+45

= 85Rs

At the same time, he

ensures that all the students

have noted it down on their

note books.

After distribution of work sheet,

he asks the students to write

expressions for box (A) and box

(B). All the students make

expressions as follows:

For box (A)

2Bowls+2Glassess+3Balls

And for box (B)

2Cones+3Bats+2flowers

Ensures that all the students

have noted it down on their note

books. He also maintains the


Further elaborates the students

that we can calculate the

expanse for the same things

from the market ‘P’ with prices

as

x = price of a bowl= 10Rs

y = price of a glass= 20Rs

z = price of a ball = 30Rs

total expanse=

4(10)+2(20)+3(30)=170R

15mins

Both teachers ensure that students have written the

expressions for box (A) & (B)

181


Dev

elop

men

t A

ctiv

itie

s

Writes the following on the

chalk board and asks the

students to write it down on

their note books:

Let x, y, and z represent

the prices of the things as

x=price of a cone=50Rs

y=price of a bat= 100Rs

z=price of a flower=5Rs

and the students are

required to calculate the

total expanse of the things.

Ensures that students have

written the question on their

note books and also maintains


20mins

Both teachers go to every student, one by one, and if

needed they guide the students in calculating the total

expanse of the things in step by step.

Both teachers go to every student in the class and if they

needed teachers assist them in calculating the total

expanse step by step i.e.

Total expanse= 2(50) + 3(100) + 2(5)

= 100 + 300 + 10

= 410Rs

They assist them by telling the students how to solve

this question step by step

Both teachers ensure that students have written

the expressions for box (A) & box (B)

182

Hom

e w

ork

Use the value of x = 2, y = 3, and z = 1 to find the

value of 3 x + 4 y -5 z.

Find the value of x2 + y2 + z2 by using the values of x

= - 1, y = 2, and z = - 3.

Complete the following table

Make an algebraic expression consisting of three

variables; evaluate it by using any values of those

variables.

Time

5mins

Now teachers distribute worksheet among the

students. It includes:

183

Lesson plan no. 2


Topic: Adding polynomials up to degree 4

Specific

students’

learning

outcomes

After taking this lesson the students will:

Demonstrate the addition subtraction of polynomials up to

degree 4

Pre

req

uis

ite

It is expected that the students have the knowledge of algebraic

expressions, polynomials, term, degree of polynomial, addition and of

numbers, and about the rule – a ( b + c ) = - ab + ( – ac) = - ab – ac

Material

required

Charts (showing the objects presented in the boxes)

Collaborative

settings

One teach- One assist Time

60mins

Intr

od

uct

ion

At the start of lesson both teachers check the home work of

the students

5mins

Teacher A

After checking the home work,

he tells the students that today

we will study about addition of

polynomials and writes the topic

on the chalk board.

Teacher B

Maintains the discipline

of the class.

P

revio

us

kn

ow

led

ge

Writes down the following

expressions on the chalk board:

a) x4 + x2 b) x2 + x + 1

and asks the students to answer

the following questions:

What is the degree of given

polynomials? { a) four b) two}

How many terms are there in the

given polynomials?

{a) two b) three}

Writes down their answers on

their note books.

Ensures that all the

students have written the

polynomials on their note

books and check their

answers.

10mins

184


D

evel

op

men

t A

ctiv

ity

Show the following chart to the

students and all the students are

required to make expressions for

the boxes presented in the chart.

While, He go to every

student and if any student

needed assistance in

making expressions, He

provide them.

10mins

After checking the previous knowledge of the students,

teachers divide the class into groups of five students each.

185


D

evel

op

men

t A

ctiv

itie

s

Asks the students to combine

similar objects from both the

boxes.

Explains the students that in

addition of polynomials you

need to combine similar terms.

e.g.

(4x + 6y + 3z) + (2x + 4y + 4)

Writes down the above

expression on the chalk board

and explains that you need to add

4x into 2x and 6y into 4y.

finally, we get

6x + 10y + 3z + 4

Writes the following expression

on the chalk board and asks the

students to combine the similar

terms

i.e.

(3x2 + 4x + 1) + (2x2 + x + 7)

With the assistance of

teacher B, students

finally made the

expressions. i.e.

For box (A)

4 + 3 + 4 ■

For box (B)

3 + 1 + 2

Al the students combine

the similar objects.

i.e.

7 + 4 + 6 ■

Ensures that every

student has combined the

similar objects. If any

student needs assistance

he provides him.

Ensures that the students

have written the

expression on their note

books. He assists the

students if needed to

combine the similar

terms. Finally, the

students write the

following expression:

5x2 + 5x + 8

10mins

186


D

evel

op

men

t A

ctiv

itie

s

At the same time, he maintains

the discipline of the class.

Goes to every student

and assists the students if

they needed in

combining the similar

terms. He also ensures

that all the students are

working on the work

sheet.

20mins

Afterwards, both teachers distribute the work sheet to the


Work sheet

Simplify the following expressions:

1. (3bals+4pens)+(8balls+5pens)

2. (6apples+5mangoes)+(2apples+3mangoes)

3. (4x2+3x+2)+(5x2+9x+1)

4. (9xy+3xz+6yz)+(4xy+9yz)

5. (3x+5)+(2x+1)

Write two algebraic expressions on your own and

combine them.

187

H

om

e w

ork

Lastly, teacher A writes the following homework on the

chalk board:

Q no. 1

Simplify the following algebraic expressions:

(8x4+9x3+9x2+5) +(6x2+7)

(x4+2x3+2x2+1) +(6x2+2x+3)

(9x3+9x2) +(3x2+1)

(7x4+4x3+x2+3) +(4x4+9x3+6x2+7)

Q no. 2

Write two expressions on your own and add them.

Time

5mins

188

Lesson plan no. 3


Topic: Subtraction of polynomials up to degree 4

Specific

students’

learning

outcomes


Demonstrate the subtraction of polynomials up to degree 4

Pre

req

uis

ite

It is expected that the students have the knowledge of algebraic

expressions, polynomials, term, degree of polynomial, addition and of

numbers, and about the rule – a( b + c ) = - ab – ac, ( - x ) ( - x ) = x2,

( - x ) ( x ) = - x2

Material

required

Charts (showing the objects presented in the boxes)

Collaborative

settings


60mins

Intr

od

uct

ion

At the start of lesson both teachers check home work of

students

5mins

Teacher A

After checking the home work,

he tells students that today we

will study about subtraction of

polynomials and writes the topic

on the chalk board.

Teacher B


maintains discipline of

the class.

P

revio

us

K

now

led

ge

After introducing the topic, he

writes down the following

expressions on the chalk board:

a) 4x4 + 15x2 b)2x2 + x - 1

and asks the students to answer

the following questions:

What is the degree of given

polynomials? {a) four b) two}

How many terms are there in the

given polynomials?

{a) two b) three}

All the students write down their

answers on their note books.



polynomials on their note

books and check their

answers.

5mins

189


D

evel

op

men

t A

ctiv

ity

Afterward, he shows the

following chart to the students

and all the students are required

to make expressions for the

boxes presented in the chart.

While, he goes to every

student and if any student

needed assistance in

making expressions, he

provides them.

10mins


teachers divide class into groups of five students each.

190


D

evel

op

men

t A

ctiv

itie

s

Asks the students to subtract the

similar objects of box (A) from

box(B).

Afterward, he explains the

students that in subtraction of

polynomials you need to subtract

the similar terms. e.g.

(4x + 6y + 3z) - (2x + 4y - 4)

writes down the above

expression on the chalk board

and explain that you need to 2x

from 4x and 4y from 6y. finally,

we get

2x + 2y + 3z + 4

Next, he writes down the

following expression on the

chalk board and ask the students

to subtract the similar terms

i.e.

(3x2 + 4x + 1) - (2x2 + x - 7)

With the assistance of

teacher B, students

finally made the

expressions. i.e.

For box (A)

6 + 3 + 4 ■

For box (B)

4 + 2 + 3 ■

All the students subtract

the similar objects of box

(B) from box (A).

i.e. 2 + 1 + 1 ■

Ensures that every

student has subtracted the

similar objects.

Ensures that the students

have written the

expression on their note

books. He also assists the

students if needed to

subtract the similar

terms. Finally, the

students write the

following expression:

x2 +3x + 8

15mins

191


D

evel

op

men

t A

ctiv

itie

s

Mantains the discipline of the

class.


and assists the students if

they needed in

subtracting the similar

terms.


students are working on

the work sheet.

20mins

Afterwards, both teachers distribute the work sheet to the


Work sheet

Simplify the following expressions:

6. (3bals+4pens) - (2balls+1pen)

7. (6apples+5mangoes) - (2apples+3mangoes)

8. (4x2+3x+2) - (5x2+9x+1)

9. (9xy+3xz+6yz) - (4xy+9yz)

10. (3x+5) - (2x+1)

Write two algebraic expressions on your own and

subtract them.

192

H

om

e w

ork

Lastly, teacher A writes down the following homework on

the chalk board:

Q no. 1

Simplify the following algebraic expressions:

(18x4+7x3+9x2+4) - (6x2+7)

(x4+2x3-2x2+1) - (6x2+2x-3)

(9x3+5x2) - (3x2- 81)

(12x4+4x3+x2+3) -(4x4+9x3+6x2+7)

Q no. 2

Write two expressions on your own and subtract them.

Time

5mins

193

Lesson plan no.4


Topic: Multiplication of the polynomials

Specific

students’

learning

outcomes


Demonstrate the procedure of multiplication of polynomials.

Multiply of polynomials up to degree 4

Pre

req

uis

ite

It is expected that the students of 8th grade have the knowledge of

degree of a polynomial, multiplication of numbers, and the following

rules:

xa × xb = xa+b, (-x) ×(-x) = +x2, (-x) ×(+x) = - x2,

(+x) ×(+x) = +x2

Material

required

Work sheet

Collaborative

settings

Teaming Time

60mins

Intr

od

uct

ion

At the start of lesson both teachers check home work of

students.

5mins

Teacher A

Afterward, he writes down the

following topic on the chalk

board: Multiplication of the

polynomials

Teacher B


maintains the discipline

of class.

P

revio

us

k

now

led

ge

Writes down the following on

the chalk board and ask the

students to answer on their note

books.

x3 × x3= -------- ? (x6)

-1 x -1 = ------- ? (1)

-x (x) = -------- ? (-x2)

x (x2) = -------- ? (x3)

-2(-5) = -------- ? (10)



questions on their note

books.

10mins

Next, both teachers check the answers of the students by

checking their note books.

194


D

evel

op

men

t A

ctiv

ity

Asks the students to fill in the

blank cells of the table by

multiplying the elements of rows

and columns with each other.

Writes the expression on the

chalk board i.e. x2+x+2x+2

= x2+3x+2

Hence

(x+1) (x+2) = x2+3x+2

After checking the

previous knowledge of

the students, he makes

the table of two

polynomials x+ 1 & x+2

on the chalk board as

follows:

× x 1

x

2

10mins

Teachers go to every student in the class and if needed they

assist them in the following way:

First multiply the elements of first row i.e. x & 1 with the

first element of first column i.e. x. Next, multiply the

elements of first row i.e. x & 1 with the second element of

first column i.e. 2. Both teachers ensure that all the students

have completed the table i.e.

× x 1

x x2 x

2 2x 2

195


D

evel

op

men

t A

ctiv

itie

s

30mins

Work sheet

Students are supposed to write down the expression

for blank cells of each box.

a)

× x2 -1

x2

x

-1

b)

× x 3

x

-1

c)

× -x -1

x2

-2

Multiply

(x2 +2) and (-x-1) with the help of table.

Simplify

(x2 +2x+1) (x-1) by using table

Write two polynomials on your own and multiply

them by using table method.

Now teachers distribute work sheet among the students. It

includes:

Both teachers give the similar assistance to the students

as they give before in the lesson. They ensure that all

the students have answers all the questions of work

sheet on their note books.

196

Hom

e w

ork

Lastly, teacher A writes down the following home work on

the chalk board.

a)

Fill in the blanks of the following tables by multiplying rows

elements with column elements.

× X 1

x2

x

1

× -x -1

x2

-3

b)

Multiply the following with the help of tables:

(x+2)(x+1), (x+3)(x-1)

c)

Fill in the blanks

i) x2(x5) = ---------

ii) x(x2+1) = ----------

iii) (-x)(x4) = ----------

iv) (-2)(-x) = ---------

At the same time, teacher B ensures that all the students have

written the home work on their note books.

× x3 2x2 -2

x2

-x

1

Time

5mins

197

Lesson plan no. 5


Topic: Derivation of the formula

( x + a ) ( x + b ) = x2+ ( a + b )x + ab

Specific

students’

learning

outcomes


Derive the formula

( x + a )( x + b ) = x2 + ( a + b )x + ab

Simplify the polynomials multiplication by using this formula.

Pre

req

uis

ite

It is expected that the students of 8th grade have the knowledge of

multiplication of polynomials, multiplication of numbers, and the

following rules:

xa × xb = xa+b, (-x) ×(-x) = +x2, (-x) ×(+x) = - x2,

(+x) ×(+x) = +x2

Material

required

Plain sheet of hard paper, pair of scissors, work sheet, scale

Collaborative

settings

One teach- one assist Time

60mins

In

trod

uct

ion

Firstly, teachers check the home work of the students at the

start of the lesson.

3mins

Teacher A Tells the students that today we will

derive the following formula: (x + a)(x + b) = x2 + (a + b)x + ab Next, he writes it on the chalk board.

Teacher B At the same time, he


of the class.

P

revio

us

k

now

led

ge

Writes down the following on the

chalk board and ask the students to

answer on their note books.

x2 × (-x5)= -------- ? (-x7)

-1 × -x = ------- ? (x)

-3x (-2x) = -------- ? (6x2)

2x(x2) = -------- ? (2x3)


students have written

the questions on their

note books.

10mins

Next, both teachers will check the answers of the students

on their note books.

198


D

evel

op

men

t A

ctiv

ity

Start the activity and cut out the

rectangle of sides (x + a) & (x + b).

Suppose that x=5cm, a=2cm, and

b=3cm.

Draw the lines a=2cm and b= 3cm

on the rectangular piece as follows:

Assists the students

that take a scissor and

cut out the rectangle

by measuring the 8cm

length and 7cm width

with scale. He ensures

that all the students

have done the same as

the teacher A does.


students have drawn

the lines a=2cm and b=

3cm

10mins

Now teachers provide plain paper sheets to the students.

199


D

evel

op

men

t A

ctiv

itie

s Cuts out the rectangular piece of

sides (x + a) and (x + b) into four

parts as follows:

Explains to students the followings:

Area of the original rectangle of

sides (x + a) and (x + b) i.e.

(x + a)(x + b)= Area of square of

sides x+ Area of rectangle of sides

x and a+ Area of rectangle of side x

and b+ area of rectangle of sides a

and b

(x + a)(x + b) = x2+ ax + bx + ab

= x2+(a + b)x + ab

Asks the students to repeat the

activity for x =7cm,

a = 4cm, b = 3cm.


in the class and assists

them that first cut out

the square of sides x,

then cut out the two

rectangles of sides a, x

and x, b. the remaining

part is a rectangle of

sides a, b. He ensures

that all the students cut

out the rectangle of

sides

(x + a) and (x + b) into

four parts.

Assists the students in

similar way as he does

before in the lesson.


students have done it.

12mins

20mins

Next, teachers distribute the work sheet among the


Multiply the following polynomials

(d + 2) (d + 1)

(a + 8) (a + 6)

(d + 4) (d + 9)

(h + 3) (h + 5)

(m + 2) (m + 8)

200

D

evel

op

men

t A

ctiv

itie

s

All the students multiply the polynomials given in the work

sheet. Teachers go to every student in the class and assist

them in the following way:

Look at the formula (x + a) (x + b) = x2+ (a + b)x + ab, here

a and b are constants. Now, see the first question

i.e. (d + 2) (d + 1). Here in this question a = 2 and b = 1. You

can simplify it by using the formula as

(d + 2) (d + 1) = d2+ (2 + 1)d + 2 × 1

= d2+ 3d + 2

Both teachers ensure that all the students have simplified the

polynomials which are given in the work sheet.

Time

H

om

e w

ork

Lastly, teacher A writes down the following home work on

the chalk board:

a) By using the formula

(x + a) (x + b) = x2+ (a + b)x + ab, simplify the

followings:

(z + 2) (z + 4), (m + 3) (m + 12), (2x + 19) (2x + 7),

(4p + 11) (4p + 5)

b) Fill in the blanks

(x + 2) (x + 11) = x2+ ( )x + ( )

(3g + 1) (3g + 4) = ( ) + ( )g + ( )

(m + 6) (m + 9) = m2+ ( )s + ( )

(2s + 1) (2s + 9) = 4s2+ ( )s + 9

c) Write two polynomials on your own and multiply

them by using the formula

(x + a) (x + b) = x2+ (a + b)x + ab



5mins

201

Lesson plan no. 6


Topic: Derive and apply the formula

(a + b)2 = a2+ 2ab + b2

Specific

students’

learning

outcomes


Derive the formula

(a + b)2 = a2 + 2ab + b2

Use the formula to simplify the algebraic expressions.

Pre

req

uis

ite

It is expected that the students of 8th grade know the formula

(x + a) (x + b) = x2 + (a + b)x + ab, and have the knowledge of

multiplication of polynomials, multiplication of numbers, and the

following rules: xa × xb= xa+b, (-x) ×(-x) = +x2, (-x) ×(+x) = - x2,

(+x) ×(+x) = +x2

Material

required


Collaborative

settings


60mins

In

trod

uct

ion

Teachers check the previous home work of the students at

the start of the lesson.

5mins


derive the following formula: (a + b)2 = a2 + 2ab + b2 , and will

use it to simply the polynomials.

Writes the topic on the chalk board.

Teacher B Maintains the discipline of

the class.

P

revio

us

kn

ow

led

ge

Afterward, He write down the

following on the chalk board and

ask the students to write down the


(x + a) × (x + b) = ----- ?

(x2+ (a + b)x + ab

(x + 3) × (x + 4) = ------ ?

(x2+ 7x + 12)

(- xy) (- xy2) = -------- ? (x2y3)

(- 50)(10x2) = -------- ? (-500x2)




note books.

8mins

Both teachers check the answers of the students on their

note books.

202


D

evel

op

men

t A

ctiv

ity

Starts the activity and cuts out the

square of sides (a + b) and

suppose that a = 5cm, b = 2cm.

Draws the two lines

b = 2cm of length horizontally

and vertically on the square piece

as follows:

Assists the students that

take a scissor and cut

out the square by

measuring the 7cm

length with scale.


students have done the

same as the teacher A

does.


students have drawn the

lines b = 2cm

10mins

Now teachers provide plain paper sheets to the students.

203


D

evel

op

men

t A

ctiv

itie

s Cuts out the square piece of sides

(x + a) into four parts as follows:

Explains students the followings:

Area of the original square of

sides (a + b) i.e.

(a + b) (a + b) = Area of square of

sides a + 2(Area of rectangle of

sides a and b) + Area of square of

sides b

(a + b) (a + b) = a2+ ab + ab + b2

= x2+ 2ab + b2

Ask the students to repeat the

activity for a = 7cm,

b = 3cm.

Goes to every student in

the class and assists

them that first cut out

the square of sides a,

then cut out the two

rectangles of sides a, b.

The remaining part is a

square of side b.


students cut out the

original square of sides

(a + b) into four parts.



before in the lesson. He

also ensures that all the


12mins

20mins

Next, teachers distribute the work sheet among the


Use the formula to expand the following:

a) (3x + y)2

b) (3a2 + 4b2)2

c) (2a + 5b)2

Fill in the blanks

(2a + 3b)2 = 4a2 + ( ) + 9b2

(6a + 5b)2 = ( ) + ( ) + 25b2

(3a2 + 4b2)2 = 9a4 + ( ) + ( )

204

D

evel

op

men

t

A

ctiv

itie

s

All the students multiply the polynomials given in the work



Look at the formula (a + b)2= (a + b) (a + b) = a2+ 2ab + b2.

Now, see the first question i.e. (3x + y)2 Here in this question

a = 3x and b = y. You can simplify it by using the formula as

(3x + y)2= (3x)2+ 2(3x) (y) + (y)2

= 9x2+ 6xy + y2



Time

Hom

e w

ork

At the end of the lesson, teacher A writes down the following

home work on the chalk board:

d) By using the formula (a + b)2= (a + b) (a + b) = a2+

2ab + b2, simplify the followings:

(p + 2) (p + 2), (m + 12) (m + 12), (2x + 7) (2x + 7),

(5p + 11) (5p + 11), (5p + 4q)2

e) Fill in the blanks

(x + 11) (x + 11) = x2+ ( )x + ( )

(3g + 4) (3g + 4) = ( ) + ( )g + ( )

(t + 6) (t + 6) = t2+ ( )t + ( )

(2s + 1)(2s + 1) = 4s2+ ( )s + 9

f) Make a square of sides (a + b), where a = 6cm, b =

4cm and derive the formula

(a + b)2= (a + b) (a + b) = a2+ 2ab + b2



5mins

205

Lesson plan no. 7



(a - b)2 = a2- 2ab + b2

Specific

students’

learning

outcomes


Derive the formula

(a - b)2 = a2 - 2ab + b2


Pre

req

uis

ite

It is expected that the students have the knowledge of multiplication of

polynomials, multiplication of numbers, and the following rules:

xa × xb = xa+b, (-x) ×(-x) = +x2, (-x) ×(+x) = - x2,

(+x) ×(+x) = +x2

Material

required


Collaborative

settings


60mins

In

trod

uct

ion

At the start of the lesson, both teachers check the home work

of the students.

3mins


derive the following formula: (a - b)2 = a2 - 2ab + b2 , and will

use it to simply the polynomials.

Next, he writes the topic on the

chalk board.


the class.

P

revio

us

kn

ow

led

ge

Writes down the following on


students to write down the


(a + b)2 = ------ ? (a2+ 2ab + b2)

(- 3mn) (- 3) = -------- ? (9mn)

(- 12)(10g2) = -------- ? (-

120g2)



questions on their note

books.

10mins

Teachers check the answers of the students on their note

books.

206


D

evel

op

men

t A

ctiv

ity

Starts the activity and cuts out

the square of sides a Suppose

that a =10cm.

Draws the two lines

b = 3cm of length horizontally

and vertically on the square piece

of length a = 10cm as follows:

Assists the students that

cut out the square with a

scissor by measuring the

10cm length with scale.


students have done the

same as the teacher A

does.


students have drawn the

lines b = 3cm

10mins


teachers provide plain paper sheets to the students.

207


D

evel

op

men

t A

ctiv

itie

s Now, he cuts out the square

piece of sides (a - b) into four

parts as follows:

Explains students the followings:

Area of the original square of

sides (a - b) i.e.

(a - b) (a - b)= Area of original

square of sides a - 2(Area of

rectangle of sides a - b and b) -

area of square of sides b

(a - b)(a - b) = a2- ab - ab + b2

= x2- 2ab + b2

Asks the students to repeat the


b = 2cm.

Goes to every student in

the class and assists them

that first cut out the

square of sides

a - b, then cut out the two

rectangles of sides

a - b & b. The remaining

part is a square of side b.


students cut out the

original square of sides

‘a’ into four parts.



before in the lesson. He

also ensures that all the


12mins

20mins

Afterwards, teachers distribute the work sheet among the


Use the formula to expand the following:

d) (x - 8y)2

e) (2a2 - 9b2)

f) (5a - 6b)2

Fill in the blanks

(2a - 3b)2= 4a2 - ( ) + 9b2

(6a - 5b)2= ( ) - ( ) + 25b2

(3a2 - 4b2)2 = 9a4- ( ) + ( )

208

D

evel

op

men

t

A

ctiv

itie

s

All the students expand the given expressions in the work



Look at the formula (a - b)2= (a - b)(a - b) = a2- 2ab + b2.

Now, see the first question i.e. (x - 8y)2 Here in this question

a = x and b = 8y. You can simplify it by using the formula as

(x - 8y)2= (x)2- 2(x)(8y) + (8y)2

= x2+ 16xy + 64y2



Time

Hom

e w

ork

Finally, teacher A writes down the following home work on

the chalk board:

g) By using the formula (a - b)2= (a - b) (a - b) = a2- 2ab

+ b2, simplify the followings:

(m - 2) (m - 2), (m - 8) (m - 8), (2x - 17) (2x - 17),

(3p - 11) (3p - 11), (5p - 7q)2

h) Fill in the blanks

(y - 11) (y - 11) = y2- ( )y + ( )

(3g - 4 ) (3g - 4) = ( ) - ( )g + ( )

(m - 6) (m - 6) = m2- ( )m + ( )

(2s - 1) (2s - 1) = 4s2- ( )s + 9

i) Make a square of sides a, where a = 12cm, b = 5cm

and derive the formula (a - b)2= a2- 2ab + b2

At the same time, teacher B will ensures that all the students

have written the home work on their note books.

5mins

209

Lesson plan no. 8



a2- b2 = (a + b) (a - b)

Specific

students’

learning

outcomes


Derive the formula

a2 - b2 = (a + b)(a - b)


Pre

req

uis

ite

It is expected that the students have the knowledge of multiplication of

polynomials, multiplication of numbers, and the following rules:

xa × xb= xa+b, (-x) ×(-x) = +x2, (-x) ×(+x) = - x2,

(+x) ×(+x) = +x2

Material

required


Collaborative

settings


60mins

In

trod

uct

ion

At the start of the lesson, teachers check the home work of

the students.

5mins

Teacher A He tells the students that today we will

derive and use the following formula: (a - b)2 = a2 - 2ab + b2 , to simply the

polynomials. Then, he writes the topic

on the chalk board.

Teacher B Maintains the


P

revio

us

k

now

led

ge

Writes down the following on the

chalk board and ask the students to

write down the answers on their note

books.

(- 8) (- 4) = -------- ? (32)

(- 2)(12m2) = -------- ? (- 24m2)

- x (2x) = ---------? (- 2x2)

xa (xb) = -------? (xa+b)




note books.

10mins

Both teachers check the answers of the students on their

note books.

210


D

evel

op

men

t A

ctiv

ity

Starts the activity and cut out the

square of sides a =10cm.

Cuts out square of sides b = 2cm of

length.

Cuts the remaining part into two

pieces G1 and G 2as follows

Figure A


that cut out the

square with a scissor

by measuring the

10cm length with

scale.


students have done

the same as the

teacher A does.


students have cut out

the square of sides b

= 2cm.


students have cut out

the remaining piece

of original square of

sides ‘a’ into two

pieces G1 and G2.

10mins


teachers provide plain paper sheets to the students.

211


D

evel

op

men

t A

ctiv

itie

s Joins the two pieces to make them a

rectangle of sides a + b & a - b as

follows:

Figure B

Explains to students that from the

figure A and figure B:

Area of figure A= Area of figure B

a2- b2= (a + b) (a - b)

Assk the students to repeat the


b = 3cm.


in the class and

ensures that all the

students have made

the rectangle of

lengths a + b & a - b.


in similar way as he

does before in the

lesson to repeat the

activity for a = 8cm

and

b = 3cm. He also

ensures that all the


10mins

20mins

Afterwards, teachers distribute the work sheet among the


Use the formula a2 - b2 = (a + b) (a - b) to expand

the following:

(10x + 2) (10x - 2), (5a + 3b) (5a - 3b)

(x2y2 + 7) (x2y2 - 7)

Evaluate the following by using

a2 - b2 = (a + b) (a - b)

i) 46 × 54

ii) 197 × 203

212

D

evel

op

men

t

A

ctiv

itie

s

All the students answer the work sheet questions. Teachers

go to every student in the class and assist them in the

following way:

Look at the formula a2- b2= (a + b) (a - b). Now, see the first

question i.e. (10x + 2) (10x - 2) Here in this question

a = 10x and b = 2. You can simplify it by using the formula

as (10x + 2) (10x - 2) = (10x)2- (2)2

= 100x2- 4



Time

Hom

e w

ork

Afterward, teacher A writes down the following home work

on the chalk board:

Use the formula a2- b2= (a + b) (a - b) to expand the

following:

(6x + 7y) (6x - 7y), (2a + 3b) (2a - 3b)

(x2y2 + 5) (x2y2 - 5)

Evaluate the following by using

a2- b2= (a + b) (a - b)

i) 146 × 154

ii) 258 × 262

Make a square of sides a, where a = 10cm, b = 3cm

and derive the formula a2- b2= (a + b) (a - b)



5mins

213

Lesson plan no. 9


Topic: Factorization

Specific

students’

learning

outcomes


Demonstrate the procedure of factorization to factorize the

algebraic expressions.

Pre

req

uis

ite

It is expected that the students have the knowledge of expansion of

algebraic expressions, multiplication of polynomials, HCF, and the

following formulae:

(a + b)2, (a - b)2, a2- b2

Material

required

work sheet, scale

Collaborative

settings

Teaming Time

60mins

In

trod

uct

ion


the students.

8mins

Teacher A

After checking the home work, he

writes the topic on the chalk board

i.e. Factorization


the class.

Pre

vio

us

k

now

led

ge

Writes down the following pairs

of terms on the chalk board:

i) 4 and 12 ii) 24and 8

Then, he asks the students to tell

the HCF verbally. (4 & 3)

Asks the students the

following verbally.

(a + b) (a + b) =-------?

(a + b) (a - b) =-------?

10mins

214


D

evel

op

men

t A

ctiv

ity

After the explanation of teacher

B, he explains further as follows:

Expansion

3(2a + 3) = 6a + 9

and

Factorization

6a + 9 = 3 (2a + 3)

Conclude the concept of

factorization as “It is the process

of writing an algebraic

expression as a product of its

factors”.

After checking the

previous knowledge, he

draws two figures on the

chalk board as follows:

Explains the students that

factorization is the

reverse processes of

expansion.

.

12mins

5mins

Now, both teachers distribute the work sheet among the

students. It includes

Fill in the missing sides

Make the figure of the following expressions:

i) 3a2+6a

ii) 5x3+10x2

iii) 3ab2+6ab

Write an expression on your own and factorize it

Find the HCF of the following pairs:

9a & 36, 2a2 & 6a

215


D

evel

op

men

t A

ctiv

itie

s

All the students answer the work sheet questions. Teachers

go to every student in the class and assist him in the

following way:

Look at the question 3a2+ 6a, the teacher will ask the

students that what is the HCF in 3a2& 6a? (3a)

After this, tell them to take 3a common from both the terms

as 3a (a + 2). Now, teachers explain the student further that

given expression has two factors

i.e. 3a & (a + 2).

Teachers ensure that all the students have completed the

work assigned to them.

20mins

Hom

e w

ork

Afterward, teachers distribute the following home work sheet

among the students:

5mins

Resolve the following algebraic expressions into

factors:

i) 3x - 9y

ii) x2y2z2- xyz2

iii) 6ab - 14ac

iv) xy - xz

v) 15x2- 60xy

Write an expression on your own and factorize it.

216

Lesson plan no. 10


Topic: Factorization of algebraic expressions of the form x2+ ax + b

Specific

students’

learning

outcomes


Demonstrate the procedure to factorize the algebraic

expressions of the form x2 + ax + b.

Pre

req

uis

ite

It is expected that the students know the factorization of numbers and

the concept of factorization.

Material

required

work sheets

Collaborative

settings

Teaming Time

60mins

In

trod

uct

ion


the students.

8mins

Teacher A

After checking the home work, he

writes the topic on the chalk board.


the class.

Pre

vio

us

k

now

led

ge

Asks the following question to

the students:

What is factorization? The

students answer it verbally, that

it is the process of writing an

algebraic expression as a product

of its factors.

Writes the followings on the

chalk board:

Factorize it 9a+81, 12mn-48m.

Asks to students to write down

the factors on the note book. {9

(a + 9), 12m (n - 4)}



of the class and ensures

that all the students have

written the question on

their note books. He also

checks the answers of the

students.

10mins

217


D

evel

op

men

t A

ctiv

ity

Explain that you can remember

the factorization of x2 + ax + b

In steps as follows:

Firstly, multiply the first term

with the last term of the given

expression. Secondly, resolve the

term, which is the result of first

step, into its factors in such a

way that when you add or

subtract them, you should get the

middle term of the given

expression. Lastly, you take

common term.

e.g.

x2+ 7x + 12

at first step

12 × x2= 12x2

At the second step,

12x2 = 4x × 3x

Now, x2+ 4x + 3x +12

=x (x + 4) + 3 (x + 4)

= (x + 4) (x + 3)



steps to factorize

polynomial of degree 2

on their note books.

.

10mins

5mins

Now, both teachers distribute the work sheet among

students. It includes

Resolve the followings into their factors:

x2 + 7x + 12

m2 + 3m + 2

z2 + 6z + 8

x2 - 3x + 10

Write a polynomial of degree two on your own and

factorize it.

218


D

evel

op

men

t

A

ctiv

itie

s

All the students answer work sheet questions. Teachers go to

every student in the class and assist them to factorize the

polynomial of degree two.

Teachers ensure that all the students have completed the

work assigned to them in the work sheet.

22mins

H

om

e w

ork

At the end of the lesson, teachers distribute the following

home work sheet among the students:

5mins

Resolve the following polynomials of degree two

into factors:

x2 + 16x + 28

x2 + 3x + 2

x2 - 24x + 63

x2 + 9x + 14

x2 + 11x + 14

x2 - 4x + 3

Write an expression on your own and factorize it.

219

Lesson plan no. 11


Topic: Linear Equation

Specific

students’

learning

outcomes


Understand the concept of linear equations.

Solve the linear equations.

Solve the everyday life problems by using linear equations.

Prerequisite

It is expected that students knows about algebraic sentences, variables,

power of variables, algebraic expressions, degree of algebraic

expression.

Mate

rial

req

uir

ed

Work sheets, hand outs

Collaborative

settings

Teaming Time

60mins

intr

od

uct

ion

Both teachers check previous home work of all the

students.

5mins Teacher A

Introduces the topic by

writing it on the chalk board.

Teacher B

Maintains discipline of

class.

Pre

vio

us

kn

ow

led

ge

Writes down some

expressions and algebraic

sentences on the chalk board

and asks the students to

differentiate them.

4x+3, 3x – 4=0

x2+5x+6=0, 2x

The students answer verbally

on each algebraic expression.

Asks the students the

following questions: What

is the degree of

4x2+1, x+2, 2x

Students answer verbally

i.e. two, one, one

respectively.

5mins

220


Devel

op

men

t A

ctiv

ity

25mins

Definition of Linear Equation

It is an algebraic sentence with 1 as a maximum degree of

the variables involved.

e.g.

3x + 4 = 0

In the above equation degree of x is one. The general form

of linear equation is

ax + b = 0, where a ≠ 0

x is a variable with degree 1.

(1)

Find the value of x from the given figure

(2) 3x + 4 = 13

(3) Identify linear equations from the following:

(a) 3x + 4 (b) x2 + 4x + 4 = 0

(c) x = 0 (d) 2x + 5 = 0

(e) x + 1 = 0

After this both teachers distribute the following handouts

to the students.

All the students read the handout. After this both teachers

distribute work sheet to the students which includes:

Both teachers guide the students to solve the linear

equations.

Then again, they distribute work sheets in the students

which includes:

221

20mins

Hom

e w

ork

5mins

Work sheet

Solve word problems by using the concept of linear

Equation.

If fine books cast 100Rs then find the Price of one

book.

Ali is 5 years older than Waqas. The age of Waqas is

40 years. What is the age of A1i?

If 10 Pencils cost 50 Rs then what is the price of and

pencil?

All the students solve the tasks given in the work sheet

with the help of both teachers.

Now teachers provide home worksheet among the


Q. Identify linear equations

a) 2x + 4 = 0 b) 3x = 0

c) 2x2+ 5 = 0 d) 0x + 7 = 0

Q Solve for the value of x:

a) 2x+3=9 b) x – 7 = 6

c) x + 80 =100 d) 4x + 9 = 25

Q. If 5kg apples price 200 Rs then find the price of 1kg

apples.

Q. Make an equation on your own and find the value of

x from that equation.

Q.

Take any values and a variable in the above diagram and

show an equation then solve it for x.

222

Lesson plan no. 12


Topic: Simultaneous system of linear equations

Specific

students’

learning

outcomes


Demonstrate the concept of linear equations.

Solve the system of simultaneous linear equations

Solve the everyday life problem by using S.L. equations

Prerequisite It is expected that students know the concept and formulation of linear

equations in one or two variables.

Material

required

hand-outs, work sheets

Collaborative

settings

Teaming Time

60mins

Intr

od

uct

ion

Both teachers check the previous home work of the students.

5mins

Teacher A

Starts the lesson and writes down

the topic on the chalk board.

Teacher B

Meanwhile, he maintains

discipline in class.

Pre

vio

us

Kn

ow

led

ge

Asks the question verbally that is

linear equation contains equation

contains more than one variable?

Students will answer

verbally.(yes)

Then he explains further that

linear equations may contain

one, two, or more than two

variables but their degrees must

be 1. e.g.

2x+4y+z=0

or

x+y+3z+4w=0

After the teacher B explanation,

he asks the students to write

down five simultaneous linear

equations on their note books.

After the explanation of

teacher A, he writes the

following linear

equations and algebraic

sentences on the

chalkboard. He asks

students to identify linear

equations with reasons

verbally.

3x2+2=0 x+4=0

2x+7=9

x2+2x+4=0

Students answer verbally.

Tells the students that

simultaneous Linear

equation system consists

of two linear equations:

with two variables. e.g.

x +y=4

x – y=1

or 2x-y=7

x-3y=2

10mins

223


Dev

elop

men

t A

ctiv

ity

Explains to students the method

of elimination to solve the S.L.

equations on the chalk board e.g.

2x + y =5…….(1)

-x + y = 6…….(2)

Multiplying equation (2) by 2

and add it in equation (1)

Now put y = 6 in (1)

2x + 6 = 6

2x = 6 - 6

2x = 0

x = 0

40mins

Both teachers check and ensure that all the students have

written five S.L equations

Now both teachers provide work sheets to the students. It

will include:

Solve the S.L. Equations.

1) 5x + 2y = 10 2) 5x + y = 4

x – 2y = 2 3x - 4y = 30

(3) The sum of two numbers is 131. Twice of the one

number is 5 less than the other. Find the numbers.

All the students complete the tasks given in the worksheets

and both teachers help and guide the students in finding the

solutions.

224


H

om

e w

ork

5mins

After ensuring that all the students have completed the

tasks given in the work sheet, teacher A writes the home

work on the chalk board.

Solve the Simultaneous system of linear equations.

(a) 4x - 7y = 12 (b) 3x - 5y = 15

3x + y = 9 - 2x + y = 4

(c) The sum of two numbers is 215. And their difference is

53. Find the numbers.

225

Lesson plan no. 13

Content Strand: Geometry Grade 8

Topic: Pythagoras theorem

Specific

students’

learning

outcomes


Understand the concept of Pythagoras theorem

Derive the Pythagoras theorem

Use this theorem to solve everyday life problems

Prerequisite

It is expected that students already know the concept of triangle, and the

rule i.e. sum of angles of a triangle is 180o. They also know types of

triangles.

Material

required

Pair of Scissors, plain sheet of hard paper Charts (showing consisting of

triangles)

Collaborative

settings


60mins

Introduction

At the start of lesson both teachers check the home work of the

students

5mins Teacher B

Writes the topic on the chalk board. Teacher A

Control the

discipline of class

P

revio

us

kn

ow

led

ge

Proceed the lesson by drawing the

following geometrical figures on the

chalk board:

Then he asks the students to identify the

figures of triangles verbally. Afterwards

when students identify the figures of

triangles, he shows a card consisting of

the following triangles:

Asks students to identify right angle

triangle among them.

Control the

discipline of class

10mins

226

Teacher B Teacher A Time

D

evel

op

men

t A

ctiv

ity

After knowing that students have the

knowledge of triangle and its types, he

engages the students in the following

activity:

Draws a right angle triangle ABC with

sides a, b, and c where m<C=90o and a :

b : c = 3 : 4 : 5 on the plain sheet of hard

paper.

Draws squares of sides a, b, and c,

adjacent to the sides of right angle

ABC.

Next, He divides the lengths of sides of

squares a, b into 3 and 4 equal size strips

as follows

Provides the plain

sheet of papers to

the students and

helps them to

follow the teacher

B's work.

All the students

will follow the

same steps of

teacher B with the

assistance and

guidance of teacher

A.

All the students

perform the save

steps as the teacher

B does with the

support of teacher

A.

15mins

227


D

evel

op

men

t A

ctiv

itie

s Cuts the strips from a square side b with

pair of scissors and then place the square

of side ‘a’ in the middle and strips of

square with side b on the on the square

of side c.

Explains to the students that from the

figure it is obvious that the area of the

square of side c is equal to total area of

the square of side b and square with side

a Hence,

a2+b2 = c2

Note: The side opposite to the angle in

the eight angle triangle is called

hypotenuse and the side parallel to the

horizontal axis is base and the side

parallel to vertical axis is called altitude.

so

(Base) 2 + (Altitude) 2 = (hypotenuse)2

All the students

perform the save

steps as the teacher

B does with the

support of teacher

A.

Now, teacher A

distribute the

worksheets in the

students which

include:

15mins

Worksheet

Identify the base, altitude, and hypotenuse from the

following figures

228


D

evel

op

men

t A

ctiv

itie

s Both teachers check their work sheet

answers.

He solves the following problem on

chalk board:

As

c2 = a2+ b2

c = √(6)2 – (8)2

= √36 – 64

= √100

= 10cm

Writes questions on the chalk board and

ask the students to solve them.

Find the unknown side from the figures

Ensures that

students have noted

it down on their

note books

Guides and helps

the students in

finding the

unknown sides.

10mins

Now both teachers assist the students in finding unknown

side.

229

Both teachers distribute home work sheets among students. It

includes:

Time

5mins

Find the length of unknown sides of the following triangles

by using Pythagoras theorem

230

Lesson no. 14


Topic: Application of Pythagoras theorem

Specific

students’

learning

outcomes

After studying this lesson students will:

Apply the Pythagoras theorem in finding the diagonals of

squares and rectangles

Apply the Pythagoras theorem in everyday life problems to

solve them.

Pre

req

uis

ite

Students should know about the concept of right angle triangle and

know base, hypotenuse, and altitude of any right angle triangle. They

should remember Pythagoras formula i.e.

(Hypotenuse)2 = (Base)2 + (Altitude)2

Material

required

charts, handouts, and work sheets

Collaborative

settings

Parallel teaching

Time

60mins


P

revio

us

Kn

ow

led

ge

Teachers divide the students, in the class, into two equal

groups.

Both teachers give work sheets to their group students and

ask them to complete it. The work sheet includes:

Identify base, altitude, and hypotenuses of the following

right angle triangles:

10mins

231


Dev

elop

men

t A

ctiv

itie

s

After that both teachers provide handouts to their group

students which includes:

Figures A and B represent a rectangle and a square with

diagonal AC.

Both teachers provide worksheets to the students which

includes:

Teachers help their group students to find out the solutions of

the above questions.

45mins

Work Sheet

(i) Find the diagonal AC in the following figures.

(2) Find the length of ladder

232

Hom

e w

ork

Lastly, both teachers provide their group students home work

sheets. It includes:

Find the unknown side of the following figures by using

Pythagoras theorem:

Time

5mins

233

Lesson plan no. 15


Topic: Hero's Formula

Specific

students’

learning

outcomes

After taking this lesson students will:

Know the concept of Hero's formula.

Apply Hero's formula in finding the area of triangles and quadrilaterals.

Prerequisite Before the lesson should know the concept of triangle and hero formula i.e.

Material

required

Hard sheet of paper, charts, color pencils

Collaborative

settings

One teach one assist

Time

60mins

introduction

Both teachers check previous home work of students.

5mins

Teacher B Teacher A

Introduces the topic by writing it on

the chalk board

Maintains the class discipline.

Dev

elop

men

t A

ctiv

ity

Shows the following colored

triangles with inner sides are of

green color and boundaries with

yellow color. Then, He asks the

students the following questions:

Do colors (green/yellow) represent

the area of triangles?

He maintains class discipline.

5mins

234


D

evel

op

men

t A

ctiv

itie

s Asks the students how do we

calculate the area of a triangle? (the

answer will be 1/2(base) (Altitude)

Tell students that there was a Greek

Mathematician named "Hero" He

gave a formula for finding the area

of a triangle only when three sides

of a triangle are given i.e.

Where a, b, c are sides and

Work sheet

Complete the table below.

a B C S Area

of

5 6 9 - -

3 4 5 - -

7 9 11 - -

Performs an activity and all the

students follow that.

He takes a rectangular shaped paper

of length 4cm and width 3cm and

cut the rectangle from the opposite

ends as follows.

He maintains the discipline in

the class.

Students do the same as

teacher B does with the help

of teachers A.

30mins

Both teachers distribute work sheets to the students which

include:

Students complete the above table with the help of both teachers.

235


D

evel

op

men

t A

ctiv

itie

s

Shows the two pieces of the

rectangle ABCD.

By using Pythagoras theorem, he

finds side AC which will be 5cm.

Asks the students to find out the

Area of both triangles ADC and

ABC by using Hero's formula

= 4+ 3+5

2

= 6cm

= √ 6(6-3) (6-5) (6-4)

= √ 6(3) (1) (2)

= √ 36

= 6cm2

He helps the students in

Performing the same activity.


note it down on their note

books.

15mins

236

Hom

e W

ork

Teacher B will write down home work on the chalk board. It includes

Home work

Q No. 1 Complete the table

A B C S Area of

6 8 6 10 -

14 21 25 - -

4 9 11 - -

Q No. 2 Find the Areas of following quadrilaterals.

Time

5mins

237

Lesson plan no. 16


Topic: Concept of cone and finding slant height

Specific

students’

learning

outcomes

After studying this lesson students will be able to:

Understand the concept of cone

Know the parts of cone

Find out the slant height of cone

Pre

req

uis

ite

Students should know the diagram of cone, Pythagoras theorem.

Material

required

Glass, Sharpe pencil, Ice-cream, Carrot

Collaborative

settings

Teaming Time

60mins

Intr

od

uct

ion

Both teachers check the previous home work of the students

5mins

Teacher B

Maintains the discipline in the

classroom

Teacher A

Writes the topic on the

chalk board

Pre

vio

us

Kn

ow

led

ge

Puts the following objects in

front of students.

Asks the students to

identify sphere from these

objects.

5mins

238


Dev

elop

men

t A

ctiv

ity

Draws a cone and explains it

further that

Afterward he tells students

that they can find the third side

of triangle ABC if they know

two sides by using Pythagoras

theorem. i.e. b2 = a2+ c2

Draws a cone on the chalk

board and ask the student to

tell the name of the triangle

ABC. (Right angle triangle)

Explains the parts of the

cone by telling the students

that there are two parts of

cone. One is lateral part and

other is circular part. It is

shown in the figure below.


have noted it on their note

books.

20mins

239


Dev

elop

men

t A

ctiv

itie

s

25mins

Hom

e w

ork

At last teacher B write following questions for home work on

chalk board.

Home work

If the radius of circular cone is 3cm and its height is

4cm. Find slant side of the cone.

Find the height of cone when its radius of circular

base is 6cm with slant side 10cm.

Find out the missing values in the following table

a = radius C = height b = slant side

10 cm 9 cm -

15 cm - 25 cm

5mins

Both teachers distribute work sheets among students. It

includes:

Work Sheet


a = radius c = height b = slant side

4 cm 5 cm -

- 4 cm 7

21 cm 35 cm

Both teachers ensure that every student is working on the

work sheet questions and if anybody needs help teachers

help that student.

240

Lesson plan no. 17


Topic: Surface Area of Cone

Specific

students’

learning

outcomes


Make a cone

Find out surface Area of cone

Pre

req

uis

ite

Students should know about the radius of circle, area of circle, concept

of a sector, and concept of cone

Material

required

Charts, hard papers, pair of scissors.

Collaborative

settings


60mins

Intr

od

uct

ion


5mins

Teacher B

Starts the lesson by introducing

the topic by writing it on the

chalk board.

Teacher A

Maintain discipline in

class

Pre

vio

us

Kn

ow

led

ge

Shows a chart, of a come and ask

the students to identify the parts

of the come

He control the discipline

in the class

7mins

241


Dev

elopm

ent

Act

ivit

ies

After revising the parts of cone,

he makes a cone of a hard paper

with radius ‘r’. All the students

follow him and make a cone.

Takes a circular hard paper

with radius r.

Folds the paper into half as

follows

and then fold is into another

direction as

its new form becomes

After this, he cuts the sector AB.

It becomes.

and finally he make a cone

Provide all the students

the circular hard paper

with radius r.

Meanwhile, he helps the

students in following

Teacher B’s work.

10mins

242


Now, he cuts down the sector

ADB into small pieces (So small

that the circular edge of the

sectors can be considered as a

straight line).

Draws a line on a piece of paper

and place there pieces in such a

way that the vertex of one is on

the upper side and 2nd is on the

lower side and so one as follows

The above figure is in the shape

of rectangular region.

Area of the rectangular region

= l x (πr)

It is in fact the lateral = πrl

So total Area of come

= Lateral Area + base Area

= πrl + πr2

Afterwards, he writes down a

question on the board and

students are requiring finding out

area of cone.

Q. Find the area of come with

radius = 10cm and l = 5cm

He further, writes down a

question on black board as asks

students to find the area of cone.

Q Find the area of cone with

radius = 10cm and l = 5cm.

All the students follow

the work of teacher B

with the assistance of

teacher A


students write it down on

their

Helps students in finding

the area of cone.

35mins

243

Hom

e w

ork

Both teachers will provide home work sheets to the students.

It includes:

Make a come on the same steps

Complete the following table

r l Base

Area

Lateral

Area

Surface

Area of

Cone

10cm 5cm - - -

12cm 9cm - - -

24cm 15cm - - -

3mins

244

Lesson plan no. 18


Topic:Volume of Cone

Specific

students’

learning

outcomes

Students after studying this lesson will:

Determine the formula for calculating the volume of the cone

Understand concept of volume of cone.

Find the volume of cone

Pre

req

uis

ite

Before the lesson students should know about the radius of circle, area

of circle, concept of a sector, concept of cone, and volume of a cylinder

Material

required

Glass, ice cream cone, funnel, paper sheets

Collaborative

settings

Teaming Time

60mins

Intr

od

uct

ion


5mins

Writes down the topic on the

chalk board

Maintains the discipline

in the classroom

Pre

vio

us

Kn

ow

led

ge

Asks to identify the cones from

objects (funnel, ice cream cone,

glass)

After revising the

concept of cone, he tells

students that today we

study how to calculate

the volume of a cone.

5mins

245


Dev

elop

men

t A

ctiv

ity

Both teachers orange an

apparatus includes one hollow

cone of radius r and height h and

a cylinder of radius r and height

h. bases and heights of both cone

and cylinder should be same.

He fills up the sand into

the cone and pours it into the

cylinder.

He fills up the cone with sand

and pours it into the cylinder

three times. The cylinder gets

full with sand up to the top.

Meanwhile, he holds the

cylinder.

Conclude it as three

times volume(V) of a

cone with radius r and

height h = Volume of a

cylinder with radius r and

height h

i.e.

3V = πr2h

V = 1/3 πr2h

15mins

All the students perform the above activity in groups with

the help of both teachers

246


Dev

elop

men

t A

ctiv

itie

s

30mins

Hom

e w

ork

At last teacher B writes the following questions for home

work on chalk board.

Home work

Q.1 Find the volume of cone if

(a) r = 3cm, h = 4cm

(b) r = 7cm, h = 10cm

(c) r = 5cm, h = 7cm

Q. 2The volume of a right circular come is 2512cm3 and its

radius and height one in the ratio 5:12. Find its radius and

slant height.

5mins

After that both teachers provide students worksheets. It

includes:

Work Sheet

Complete the following table by finding the

missing values.

R H Volume of Cone

where π = 3.14

10cm 10cm -

5cm - 60cm3

- 11cm 90cm3

Then both teachers help students in completing the above

table.

247

Lesson plan no. 19


Topic: Surface Area of sphere

Specific

students’

learning

outcomes


Find out the area of sphere

Pre

req

uis

ite

Before the lesson students should know about the radius of circle, area

of circle, difference in sphere and circle

Material

required

Football, ring, bob, semi spherical bowl, rope, gum or glue, cylinder of

same radius as that of sphere and height equal to diameter of spherical

bowl, work sheet.

Collaborative

settings

Teaming Time

60mins

Intr

od

uct

ion


5mins

Teacher B

Maintains the discipline in the

classroom

Teacher A

Writes down the topic on

the chalk board

Pre

vio

us

Kn

ow

led

ge

Shows the following objects to

the students and asks them to

identify sphere.

After that he asks the

students about the

difference b/w circle and

sphere.

The answer will be

5mins

248


Dev

elop

men

t A

ctiv

ity

Performs an activity to derive

formula to calculate surface area

of sphere. For this He takes a

semi spherical bowl, a rope of

enough length, and a cylinder of

radius r same as bowl radius, and

height of cylinder is same as the

diameter of bowl.

Take a half piece of rope and

wind it on the bowl as shown

(Sticking it with gum)

Concludes that

Length of rope on semi spherical

bowl = length of rope on half

cylinder

So,

Curved surface area of sphere

with radius r = curved surface

area of cylinder with radius r and

height 2r.

= 2π r h

= 2π r (2r)

= 4π r 2

Meanwhile he divides the

piece of rope into two

equal halves.

Meanwhile, he winds the

rope on the cylinder as

shown (sticking it with

gum)

20mins

Both teachers cut the extra rope from bowl and cylinder.

249


Dev

elop

men

t A

ctiv

itie

s

25mins

Hom

e w

ork

At last teacher B writes down the following questions for

home work on chalk board.

Home work

Find the radius of a sphere if the area of its surface is

6.16m2

Find the area of surface of a sphere with radius 3cm.

5mins

After that both teachers provide students worksheets. It

includes:

Work Sheet


R = radius Area of sphere where

π = 3.14

3.5 m -

- 154m2

49cm -

- 616cm2

Then both teachers help students in completing the above

table.

250

Lesson plan no. 20


Topic: Volume of a sphere

Specific

students’

learning

outcomes


Derive a formula for calculating the volume of a sphere

Find out volume of sphere

Pre

req

uis

ite

Students should know about the radius of circle, area of circle, concept

of a sector, and concept of cone

Material

required

A solid sphere, A cylinder(radius r height 2r)

Collaborative

settings


60mins

Intr

od

uct

ion


5mins

Teacher B

Starts the lesson by introducing

the topic by writing it on the

chalk board.

Teacher A

P

revio

us

Kn

ow

led

ge

Makes the following figures on


students which one is sphere and

why?

Controls the discipline in

the class

5mins

251


Dev

elop

men

t A

ctiv

itie

s Divides the whole class into

groups and start activity.

Takes a sphere of radius r and a

cylinder of radius r with height

2r as shown

First of all, he fills out the

cylinder with water and

measures its quantity.

Place the sphere into that

cylinder the water drops out of

the cylinder. Now He takes out

the sphere from cylinder and

measures the quantity of water

again.

The quantity of water left in the

cylinder is one third of that

quantity measured in the start.

Volume(V) of sphere

= 2/3 (volume of cylinder)

= 2/3 (πr2) (2r)

= 4/3 πr3

All the groups follow the

same activity with the

helps and assistance of

teacher A

15mins

252


Dev

elop

men

t A

ctiv

itie

s Gives an assignment to the

groups to complete it

Assignment

Find the volume of

sphere of radius 4.2m

How much of water a

spherical tank can contain

whose radius is 5m?

Complete the table below

Radius Volume of

sphere

5.8cm -

- 616m3

15.2cm -

Helps students in

completing the

assignment.

30mins

Hom

e w

ork

Both teachers provide home work sheets to the students. It

includes:

home work sheet

A spherical tank is of radius 7.7m. how many liters of

water can it contain (1000cm3= 1 liter)

Complete the table

Surface Area of

sphere

Volume of sphere

5.544cm2 -

- 201.5cm3

5mins

253

APPENDIX C

Table 2

Selected Items for Mathematics Achievement Test Sr.

no.

Booklet

number

Item

number

Content

Strand

ability Difficulty

Index

Point Bi-

serial

1 D 36 Geometry CU 0.68 0.32

2 D 54 Algebra PK 0.56 0.34

3 B 60 Algebra PK 0.48 0.34

4 C 54 Algebra CU 0.47 0.37

5 D 59 Algebra PK 0.46 0.30

6 B 68 Algebra PS 0.43 0.26

7 D 37 Geometry CU 0.42 0.27

8 B 62 Algebra PK 0.39 0.24

9 A 55 Algebra CU 0.37 0.47

10 B 34 Geometry PK 0.37 0.27

11 D 68 Algebra PS 0.37 0.25

12 B 56 Algebra CU 0.36 0.30

13 C 68 Algebra PS 0.35 0.35

14 B 32 Geometry PK 0.34 0.25

15 D 58 Algebra PK 0.34 0.35

16 D 60 Algebra PK 0.34 0.32

17 C 70 Algebra PK 0.33 0.34

18 B 35 Geometry CU 0.31 0.27

19 C 55 Algebra PK 0.31 0.38

20 A 59 Algebra PK 0.29 0.33

21 A 53 Algebra CU 0.28 0.22

22 A 56 Algebra CU 0.27 0.41

23 B 70 Algebra PS 0.27 0.32

24 D 39 Geometry PK 0.26 0.20

25 D 56 Algebra CU 0.25 0.27

26 A 42 Geometry PS 0.25 0.21

27 A 44 Geometry PK 0.24 0.22

28 C 36 Geometry CU 0.22 0.33

29 A 33 Geometry PS 0.22 0.21

30 B 31 Geometry PS 0.22 0.20

31 A 67 Algebra PS 0.20 0.23

32 B 66 Algebra PS 0.20 0.17

254

APPENDIX D

Table 3

Key of Achievement Test MCQs

Item No. Answer Item No. Answer

1 B 17 A

2 C 18 C

3 B 19 A

4 A 20 D

5 B 21 B

6 B 22 A

7 D 23 A

8 B 24 B

9 B 25 B

10 D 26 B

11 C 27 C

12 D 28 D

13 A 29 B

14 D 30 C

15 C 31 B

16 B 32 A

255

APPENDIX E

Table 4

Demographic Information of Experts

Name Designation Institution Qualification Area

Dr. Nargis

Abbas

Assistant

Prof.

UOS

PhD from

France

Math Education

Dr. Shahzada

Qaisar

Assistant

Prof.

UE

PhD from UK

Math Education

M. Azeem

Subject

specialist of

mathematics

PEAS

M.Sc Math, PhD

(Education)

Scholar

Trainee in NEAS

test development

Zulfiqar

Shah

Subject

specialist of

mathematics

NEAS M.sc Math

B.Ed., M.Ed.

Trainee in NEAS

test development

Imran

Sarwar

Lecturer Punjab

Govt.

M.sc Math,

PhD(Math

Education)

Scholar

PhD Scholar at PU

Math Education

Asim Lecturer PU M.sc Math,

PhD(Math

Education)

Scholar

PhD Scholar at PU

Math Education

Impact of Collaborative Teaching on 8th Gradeprr.hec.gov.pk/jspui/bitstream/123456789/9084/1... · Zafar Iqbal Distinguished National Professor Dr/2009-12 Dr. Munawar S. Mirza Division

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