Higher Education Commission
THE EFFECTS OF MASTERY LEARNING STRATEGY ON STUDENTS’ ACHIEVEMENTS IN THE SUBJECT OF MATHEMATICS AT SECONDARY LEVEL
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Lubna Toheed
Ph.D Scholar
INSTITUTE OF EDUCATION AND RESEARCH
UNIVERSITY OF PESHAWAR, PAKISTAN
2016
THE EFFECTS OF MASTERY LEARNING STRATEGY ON STUDENTS’ ACHIEVEMENTS IN THE SUBJECT OF MATHEMATICS AT SECONDARY LEVEL
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Lubna Toheed
Ph.D Scholar
A thesis submitted to the Institute of Education and Research, University of Peshawar in partial fulfillment of the requirement for the award of the degree of
DOCTOR OF PHILOSOPHY IN EDUCATION
INSTITUTE OF EDUCATION AND RESEARCH
UNIVERSITY OF PESHAWAR, PAKISTAN
2016
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DEDICATION
I dedicate this research work with love and affections
To my mother
MRS. SAID QAMRESH
and to my sister
FARIDA JABEEN
for their continuous support and encouragement.
APPROVAL SHEET
This dissertation entitled “The Effects of Mastery Learning Strategy on Students’ Achievements in the Subject of Mathematics at Secondary Level” submitted by Miss Lubna Toheed in partial fulfillment of requirement for award of degree of Doctor of Philosophy in Education is hereby approved.
Internal Examiner/Supervisor _____________________________
Prof. Dr Arshad Ali
External Examiner ____________________________
Director ____________________________
Date________________________
DECLARATION
I, Lubna Toheed D/o Said Qamresh Roll No. 12, PhD scholar of The Institute of Education and Research (IER) University of Peshawar do hereby solemnly declare that the thesis entitled “The Effects of Mastery Learning Strategy on Students’ achievements in the Subject of Mathematics at Secondary Level”, is submitted in partial fulfillment of requirements for award of degree of Doctor of Philosophy in Education. It is my innovative work that has not been submitted or published before.
Date_________________ ___________________
Lubna Toheed
FORWARDING SHEET
The thesis entitled “The Effects of Mastery Learning Strategy on Students’ Achievements in the Subject of Mathematics at Secondary Level”, submitted by Miss Lubna Toheed in partial fulfillment for award of degree of Doctor of Philosophy in Education has been successfully accomplished under my guidance and supervision. I am fully satisfied with the quality of the work.
Dated: ______________ _________________ Supervisor
Prof. Dr. Arshad Ali
ACKNOWLEDGEMENTS
In the name of Almighty Allah, the most Merciful, the most Gracious, Who enabled me to complete this research work successfully.
The work presented in the current study was accomplished under the guidance and supervision of proficient and enlightened supervisor Prof. Dr. Arshad Ali, Institute of Education and Research, University of Peshawar. I could not find words to express the feelings of gratitude and appreciation for my respected supervisor for his valuable suggestions and motivational encouragement.
I am thankful to the Librarian, Institute of Education and Research, University of Peshawar for provision of books and research papers. I am also very thankful to all experts (Appendix-N) for their guidance, assistance and support in this study.
I acknowledge the help of Dr. Martha Tapia, associate professor of mathematics education at Berry College Georgia, who sent me Attitude Toward Mathematics Inventory (ATMI) via email and guided me in data analysis.
Special thanks to Dr. Asim, professor at the Department of Statistics, University of Peshawar and Mr. Waheed, Ph.D scholar for their practical assistance in data analysis.
I am very thankful to Dr. Farzana Bukhari, subject specialist at GGHSS, Shahdand Baba Mardan, for her encouragement and support to complete this work.
I am highly thankful to Mst. Saeeda Akhtar, Principal GGHS, No: 1 Mardan and Mst. Zainab Bibi, Principal GGHSS, Shehbaz Garhi, for their help and support in conducting the experiments. Special thanks to Mst. Janisar Akhtar , Secondary School Teacher (SST) and Miss Tayyba Taj, Subject Specialist (SS) in Mathematics, who voluntarily conducted experiments in their respective schools.
Finally, I am thankful to my all family members and friends who supported me in the accomplishment of this work.
Lubna Toheed
TABLE OF CONTENTS
Title Page No
Declaration -------------------------------------------------------------------------------- iii
Forwarding Sheet ------------------------------------------------------------------------- iv
Acknowledgements ---------------------------------------------------------------------- v
Table of Contents ------------------------------------------------------------------------ vi
List of Tables ----------------------------------------------------------------------------- ix
List of Figures ---------------------------------------------------------------------------- xi
Abstract ----------------------------------------------------------------------------------- xii
|CHAPTER-I |INTRODUCTION----------------------------------------------- |1-10 |
|1.1 |Background of the Study ---------------------------------------- |1 |
|1.2 |Statement of the Problem ---------------------------------------- |5 |
|1.3 |Objectives of the Study------------------------------------------- |6 |
|1.4 |Research Hypotheses -------------------------------------------- |7 |
|1.5 |Significance of the Study ---------------------------------------- |8 |
|1.6 |Delimitations of the Study-------------------------------------- |9 |
|1.7 |Abbreviations/Definitions of the Terms ----------------------- |9 |
|CHAPTER-2 |REVIEW LITERATURE ------------------------------------- |11-64 |
|2.1 |Concept and Nature of Mathematics --------------------------- |11 |
|2.2 |Importance of Mathematics ------------------------------------ |12 |
|2.3 |Importance of Mathematics for Secondary School Level --- |14 |
|2.4 |Factors Effecting Students’ Achievements in Mathematic-- |14 |
|2.5 |Instructional Strategies for Teaching of Mathematics ------- |17 |
|2.5.1 |Conventional Teaching Methods-------------------------------- |18 |
|2.5.2 |New Methods for Teaching Mathematics-------------------- |21 |
|2.6 |Role of Teacher in Teaching of Mathematics ---------------- |25 |
|2.7 |Mastery Learning Strategy -------------------------------------- |26 |
|2.7.1 |Background of Mastery Learning ------------------------------ |27 |
|2.7.2 |Contributions of Washburne (1922) --------------------------- |28 |
|1.7.3 |Contributions of Morrison (1926) ------------------------------ |29 |
|2.7.4 |Programmed Instructions (1950s) ------------------------------ |30 |
|2.7.5 |Carroll's Model of School Learning (1963) ------------------- |31 |
|2.7.6 |Contributions of Bloom (1968) --------------------------------- |33 |
|2.7.7 |Keller’s Personalized System of Instruction(1968) ---------- |36 |
|2.7.8 |Post Bloom Period (since 1971) -------------------------------- |37 |
|2.8 |Approaches to Mastery Learning Strategy -------------------- |38 |
|2.9 |Essential Elements of Mastery Learning ---------------------- |39 |
|2.10 |Mastery Learning Procedure ------------------------------------ |41 |
|2.11 |Variables of Mastery Learning --------------------------------- |43 |
|2.12 |Advantages of Mastery Learning Strategy -------------------- |47 |
|2.13 |Disadvantages of Mastery Learning Strategy ----------------- |47 |
|2.14 |Research Studies about Mastery Learning -------------------- |48 |
|2.15 |Why this Study ---------------------------------------------------- |63 |
|CHAPTER-3 |RESEARCH METHODOLOGY----------------------------- |65-80 |
|3.1 |Nature of the Study ----------------------------------------------- |65 |
|3.2 |Population of the Study ------------------------------------------ |65 |
|3.3 |Sample/participants of the Study ------------------------------- |65 |
|3.4 |Selection of Teachers -------------------------------------------- |67 |
|3.5 |Instructions for Teachers ---------------------------------------- |67 |
|3.6 |Development of Lesson Plans ---------------------------------- |68 |
|3.7 |Teaching Strategies----------------------------------------------- |69 |
|3.8 |Design for the Study---------------------------------------------- |70 |
|3.9 |Variables ----------------------------------------------------------- |71 |
|3.10 |Research Tools ---------------------------------------------------- |72 |
|3.11 |Research Tools’ Preparation and Validation ------------------ |72 |
|3.12 |Procedure of the Study ------------------------------------------- |77 |
|3.13 |Analysis of Data--------------------------------------------------- |79 |
|3.13.1 |Pre-test, Post-test and Retention Test Analysis --------------- |79 |
|3.13.2 |Attitude Scale Analysis ------------------------------------------ |79 |
|CHAPTER-4 |DATA ANALYSIS ---------------------------------------------- |81-104 |
| | | |
|CHAPTER-5 |SUMMARY, FINDINGS, DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS -------- | |
| | |105-117 |
|5.1 |Summary----------------------------------------------------------- |105 |
|5.2 |Findings ------------------------------------------------------------ |107 |
|5.3 |Discussion --------------------------------------------------------- |110 |
|5.4 |Conclusions ------------------------------------------------------- |114 |
|5.5 |Recommendations ----------------------------------------------- |116 |
|5.6 |Suggestions for Further Work ---------------------------------- |117 |
|REFERENCES ------------------------------------------------------------------------ |118-128 |
|APPENDIX-A |Pre-test-------------------------------------------------------------- |129 |
|APPENDIX-B |Post-test-1---------------------------------------------------------- |133 |
|APPENDIX-C |Post-test-2 --------------------------------------------------------- |137 |
|APPENDIX-D |Retention Test----------------------------------------------------- |140 |
|APPENDIX-E |Lesson Plan 1------------------------------------------------------ |144 |
|APPENDIX-F |Lesson Plan 2 ----------------------------------------------------- |150 |
|APPENDIX-G |Lesson Plan 3 ----------------------------------------------------- |155 |
|APPENDIX-H |Lesson Plan 4 ----------------------------------------------------- |160 |
|APPENDIX-I |Lesson Plan 5 ----------------------------------------------------- |165 |
|APPENDIX-J |Lesson Plan 6 ----------------------------------------------------- |167 |
|APPENDIX-K |Attitude Toward Mathematics Inventory (ATMI) ---------- |170 |
|APPENDIX-L |Urdu Translation of ATMI--------------------------------------- |172 |
|APPENDIX-M |Email from Marta Tapia ----------------------------------- |175 |
|APPENDIX-N |List of Experts ---------------------------------------------------- |176 |
|APPENDIX-O |No Objection Certificate-1 -------------------------------------- |177 |
|APPENDIX-P |No Objection Certificate-2 -------------------------------------- |178 |
|APPENDIX-Q |Experiment Completion Certificate-1-------------------------- |179 |
|APPENDIX-R |Experiment Completion Certificate-2-------------------------- |180 |
|APPENDIX-S |Test Scores of the Group 1(experimental) -------------------- |181 |
|APPENDIX-T |Test Scores of Group II (control)------------------------------- |183 |
|APPENDIX-U |Test Scores of the Group III(experimental) ------------------- |185 |
|APPENDIX-V |Test Scores of Group IV (control) ----------------------------- |187 |
LIST OF TABLES
|Table No |Titles |Page No |
|3.1 |Sample Size in GGHSS, Shehbaz Garhi Mardan and GGHS, NO:1 |66 |
| |Mardan-------------------------------------------------------------------------- | |
|3.2 |Relation of subscales of ATMI with Neal’s definition of mathematical attitude |77 |
| |--------------------------------------------------------- | |
|4.1 |Significant difference between the pre-test mean achievement scores of experimental and control groups|82 |
| |of rural area----------------------- | |
|4.2 |Significant difference between the pre-test mean achievement scores of experimental and control groups|83 |
| |of urban area --------------------- | |
|4.3 |Significant difference between the post-test-1 mean achievement scores of experimental and control |84 |
| |groups of rural area------------- | |
|4.4 |Significant difference between the post-test-2 mean achievement scores of experimental and control |85 |
| |groups of rural area ----------- | |
|4.5 |Significant difference between the post-test-1 mean achievement scores of experimental and control |86 |
| |groups of urban area------------- | |
|4.6 |Significant difference between the post-test-2 mean achievement scores of experimental and control |87 |
| |groups of urban area ----------- | |
|4.7 |Significant difference in short-term retention scores of experimental and control groups of rural |88 |
| |area-------------------------------------------- | |
|4.8 |Significant difference in long-term retention scores of experimental and control groups of rural area|89 |
| |------------------------------------------- | |
|4.9 |Significant difference in short-term retention scores of experimental and control groups of urban |90 |
| |area-------------------------------------------- | |
|4.10 |Significant difference in long-term retention scores of experimental and control groups of urban area|91 |
| |----------------------------------------- | |
|4.11 |Test of variability in the academic achievement scores of experimental and control groups of rural |92 |
| |area ---------------------------- | |
|4.12 |Test of variability in the academic achievement scores of experimental and control groups of urban |93 |
| |area --------------------------- | |
| | | |
|4.13 |Significant difference in self-confidence of rural students of experimental and control groups before |94 |
| |and after treatment--------- | |
|4.14 |Significant difference in the views of rural students about the value they give to the subject of |95 |
| |mathematics before and after treatment ---- | |
|4.15 |Significant difference in the views of rural students regarding enjoying the subject of mathematics |96 |
| |before and after treatment.-------- | |
|4.16 |Significant difference in students’ motivation for the subject of mathematics before and after |97 |
| |treatment in rural area -------------------- | |
|4.17 |Significant difference in self-confidence of urban students of experimental and control groups before |98 |
| |and after treatment ----------- | |
|4.18 |Significant difference in the views of urban students about the value they give to the subject of |99 |
| |mathematics before and after treatment ---- | |
|4.19 |Significant difference in the views of the urban students regarding enjoying the subject of |100 |
| |mathematics before and after treatment-------- | |
|4.20 |Significant difference in students’ motivation for the subject of mathematics before and after |101 |
| |treatment in urban area--------------- | |
|4.21 |Significant difference between the post-test-1 mean achievement scores of experimental group I (rural)|102 |
| |and Group III (urban) -------- | |
|4.22 |Significant difference between the post-test-2 mean achievement scores of group 1 (rural) and group |103 |
| |III (urban) -------------------------- | |
LIST OF FIGURES
|Figure No |Titles |Page No |
|1.1 |Process of Mastery Learning------------------------------------- |2 |
|1.2 |Variables of Mastery Learning ---------------------------------- |3 |
|2.1 |Carroll’s Model for School Learning (1963)------------------- |32 |
|3.1 |Pre-test Post-test Non-equivalent Group Design-------------- |71 |
|3.2 |Procedure of the Study for Experimental Group ------------- |78 |
ABSTRACT
The present study was designed to investigate the effects of Mastery Learning Strategy on students’ achievements in the subject of Mathematics at secondary school level. The objectives of the study were to examine the effects of Mastery Learning Strategy on students’ achievements, learning retention, achievement gapes, and attitude toward the subject of Mathematics. The research study was conducted in district Mardan of province Khyber Pakhtoonkhwa (KP), Pakistan.
The following null hypotheses were tested: 1) There is no significant difference in the achievement of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method. 2) There is no significant difference in the retention of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method. 3) There is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gap among the students. 4) There is no significant difference in the academic achievement of the students of urban and rural areas taught through Mastery Learning Strategy. 5) There is no significant difference in the attitude of students towards the subject of Mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
The study was experimental in nature. The design of the study was quasi-experimental. Pre-test post-test nonequivalent control group design was employed. Sample was selected through purposive and convenient sampling techniques. Two sections of grade 9th from GGHS NO.1 Mardan and two sections of grade 9th from GGHSS Shehbaz Garhi Mardan were the participants for this study. The sample size was 214, divided into four groups. Two groups, one from each school, were treated as experimental groups and two as control groups. Two teachers, one from each school, were selected to teach both groups in separate periods daily in their respective schools. The experimental groups were exposed to Mastery Learning Strategy and control groups were taught through Conventional Teaching Method. Teachers were trained for both approaches. Two chapters of algebra from grade 9th Mathematics were selected for teaching. Experimental period was two months and twenty days.
The tools used for data collection were pre-test, two post-tests, retention test and Attitude Toward Mathematics Inventory (ATMI). Pre-test were administered before experiment; post-test-1 was administered at the completion of first unit; post-test-2 was administered at the completion of experimental period, ATMI was administered before and after experimental period, and retention test was administered after two months of the experiment completion.
The collected data was analyzed through SPSS version 18. The independent samples t-test was performed on achievement scores of pre-test, post-tests, and retention test; and paired samples t-test was performed on the data received by ATMI. Degree of freedom, mean, standard deviation, t-value and p-value were obtained. Significant difference between the mean achievement scores of the experimental and control groups was tested at 0.05 level. The results showed better performance of the students of experimental groups than control groups. The results obtained from ATMI showed a positive change in the attitude of experimental group towards learning mathematics.
The study recommends that Mastery Learning Strategy may be used at higher secondary level for teaching of mathematics. This strategy may be applied in other subjects in different culture and may also be used at elementary level for concept development. Keeping Mastery Learning philosophy as base, curriculum and teacher training programmes may also be improved.
CHAPTER – 1
INTRODUCTION
The first chapter of the thesis provides background knowledge about current research study. There is discussion about the topic, focus of the study, significance, objectives, hypotheses and delimitations of the study.
1.1 BACKGROUND OF THE STUDY
Teaching is a complex activity. Teachers try to attain educational objectives through specific curriculum. Curriculum doesn’t mean only the academic subjects, but it also includes the educational objectives, teaching methodologies and assessment process. So the selection of appropriate teaching method is imperative for achieving the educational objectives. Moreover a single method is not sufficient to achieve the objectives, because inside a classroom there are students having different aptitudes, attitudes, interests and mental capacities. Individual differences are a vital force in our classes. These individual differences become more prominent on arriving at secondary school level. Differences in mental ability, the ability to reason and think reflectively, and the ability to solve problems, and differences in interest level results in the divergent level of attainment (Scopes, 1973). In addition, all students don’t learn with the same pace, some learn quickly while others need time to learn. So it cannot be expected from a teacher to enable all students of a class to achieve the learning objectives by using a single teaching method. If we wish for all students to achieve the learning objectives, we must have to employ such teaching strategy that could combine many methods and techniques; and that could relate the instructional process to the individual differences among students.
Mastery Learning is a teaching strategy that interrelate students’ interest and instructional strategy by taking into account the individual differences among students (Bloom, 1968). It works on the philosophy of “take your time”. Every student is provided the time needed for learning. Bloom (1976) states, that all students can achieve the same level of learning if the time and instructional strategies are made appropriate to their characteristics and needs.
In mastery learning method teaching activities are organized in small ordered steps. Before mastering one step students cannot move to the next. At the end of each step a formative assessment based on learning objectives is administered. Students who reach to the specified criterion in that assessment are given enrichment assignments. The purpose of these assignments is to extend and deepen their knowledge. Those who do not reach to the specified criterion are given feedback paired with corrective assignments, especially designed to correct their misunderstanding. The process continues till all the students reach a determined criterion or specified level of learning.
The process of Mastery Learning is given in the figure below:
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Figure 1.1: Process of Mastery Learning
Formative assessment is an essential part of the mastery learning process. Guskey (2005) says that for Bloom, formative assessment is very important, as it provides information about learning difficulties and thus assists teacher in the selection of appropriate corrective activities.
The idea of mastery learning received greater attention in the 20th century. Early attempts were made by Washburne and Morrison (Varughese, 2002). According to Block (1971) Washburne, in 1922, introduced a plan in which self-instruction practice materials were used. The role of teacher was that of a guide, s/he occasionally tutored the learners, either individually or in small groups. In 1926 Morrison introduced correctives, re-teaching and restructuring of learning activities (Block, 1971). His method was successful, but due to lack of technology, it was not brought in use then.
Carroll (1963) studied individual differences in learners and presented a theoretical model of mastery learning. Carroll argued that all students can learn, but they need different amount of time. Bloom (1968) converted Carroll’s theoretical model in to a practical model of classroom learning. The term “Mastery for Learning” was coined by him in 1968. He believed that if students are given multiple opportunities, proper time and quality instructions, more than 90% of them can achieve mastery learning level in a particular subject.
Keller (1968) improved the idea of Bloom and presented a model of personalized system of instructions. This model advocates 100% of performance, as it is individual based, while Bloom’s and Carroll’s model is group based which is more suitable for classroom situation.
Bloom believed that the basic indicators of learning outcomes are cognitive entry behavior (pre-learning for learning a unit), affective entry characteristics (motivation to learn a unit) and the quality of instructions (Kazu, Kazu & Ozdemir, 2005). These three dependent variables should be considered carefully for desirable learning outcomes. The quality of instructions means all the teaching activities such as clue, reinforcement, the students’ participation, feedback and corrections.
The variables of Mastery Learning are given in the figure below:
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Figure 1.2: Variables of Mastery Learning
In a hierarchically organized subject like mathematics, students face difficulties to learn a skill unless they have cognitive entry behavior for that. In other words, failure to learn prerequisite skills hinders further learning. In the conventional method, where time and type of instruction is constant, students move through the mathematics syllabus along with their deficiencies. This widens gap in their achievements year by year and finally, frustrated by the continuous failure in mathematics, many develop hate for the subject and many leave school. A National Statement on Mathematics for Australian Schools (1990) come straight to the point in stating that “there is considerable evidence to suggest that children come to school enthusiastic and eager to learn mathematics and that a great deal leave school with quite negative attitude”(p. 31).
It is often said that mathematics is too abstract for some students. Therefore, the presentation of the contents should be planned sequentially in such a way that all the students must acquire the required competencies. But unfortunately, the teaching of mathematics in the schools of KP is predominantly text-book based. Mostly, conventional methods are in practice in mathematics’ classrooms at secondary school level. The conventional method refers all students in a homogenous manner, where teacher is at the center of all teaching activities, teaching all students with the same method, without any consideration for individual differences (Bishara, 2015). Furthermore, Students passively receive information without any understanding. Conventional mathematics’ instructions consists of learning a concept by watching the teacher solving a problem at the writing board and students copy down each step during note taking (Fleener, Craven & Dupree, 1997). Students are then given several similar problems as homework that assess whether or not they can follow the procedures that they observed on writing board (Ferguson, 2010), thus only memorization of facts and the ability to follow procedure is the focus of instructions. The use of illustrative means in teaching is minimal, which makes it difficult for the students to attain prescribed competencies. It is also difficult to find any theoretical basis for a particular approach to follow in teaching of mathematics. If the selection of teaching approach is based on sound psychological principles of learning then it can be expected that the majority of the students will attain prescribed competencies in a particular grade.
Mastery learning takes into account individual differences, students’ interest and provides each student with appropriate time to learn.
It is based on the principle that all students can learn a set of reasonable objectives with proper instructions and sufficient time to learn. Mastery Learning puts the techniques of tutoring and individualized instructions in a group learning situation and brings the learning strategies of successful students to nearly all the students of a given group. In its full form it includes a philosophy, curriculum structure, instructional model, the alignment of student assessment, and a teaching approach. (Kazu et al., 2005, p.3)
Instead of accommodating learning difficulties, mastery learning finds out learning difficulties of students through frequent assessment and addressing these by providing the opportunities of corrective assignments and additional instructions. Bloom (1976, 1980) believed that by the application of mastery learning method the differences in learning rate will decrease and can approach to zero. Bloom felt that cognitive entry behavior will improve overtimes and a need for corrective instruction will be reduced gradually.
1.2 STATEMENT OF THE PROBLEM
It is desirable that all students should achieve learning objectives, but only few reach to the prescribed criterion. A recent example is the Secondary School Certificate two consecutive examinations conducted by Board of Intermediate and Secondary Education (BISE) Mardan (2013; 2014), in which only 4% of the total students achieved A-1 grade. Mostly, teachers are blamed for the poor performance of the students in examinations. It is said that the quality of instructions is directly related to the quality of students’ performance.
At secondary school level the poorest performance in annual examinations has been noted in the subject of mathematics (Aduda, 2003; Olunloye, 2010; Mbugua, Kibet. Muthaa & Nkonke, 2012; and Matthew &Kenneth, 2013). In Pakistan, the poor performance in the subject of mathematics has been declared by the gazette notification of BISE Rawalpindi (2002), BISE Islamabad (2002), and BISE Mardan (2013). Students do not pay attention to the learning of mathematics. They learn mathematics with the only purpose to pass their Secondary School Certificate examination (Mustafa, 2007, as cited in Nafees, 2011). Over the years, neither any change in the teaching methodology of the teachers nor any progress in the students’ performance in the subject of mathematics has been noted.
Keeping in view the unsatisfactory performance of the students in the subject of Mathematics at secondary school level the researcher designed this study to investigate the effects of Mastery Learning Strategy on students’ achievements, learning retention and achievement gaps in the subject of mathematics at secondary school level and its effects on students’ attitude towards the subject of Mathematics.
1.3 OBJECTIVES OF THE STUDY
The aim of the current study was to investigate the effects of Mastery Learning Strategy on students’ achievements in the subject of mathematics at secondary school level. To achieve the aim the following objectives were formulated.
1. To compare the effects of Mastery Learning Strategy with Conventional Teaching Method on students’ achievement in the subject of mathematics at secondary level.
2. To investigate the effects of Mastery Learning Strategy on students’ long term and short term learning retention.
3. To compare the effects of Mastery Learning Strategy with Conventional Teaching Method on achievement gaps of the students of the same grade.
4. To investigate the effects of Mastery Learning Strategy on students’ attitude towards the subject of mathematics at secondary level.
5. To compare the academic achievements of urban and rural students taught through Mastery Learning Strategy.
1.4 RESEARCH HYPOTHESES
To achieve the objectives, the following null hypotheses and sub null hypotheses were formulated.
H10: There is no significant difference in the achievement of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H1a0: There is no significant difference in the achievement of rural students taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H1b0: There is no significant difference in the achievement of urban students taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H20: There is no significant difference in the learning retention of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H2a0: There is no significant difference in the learning retention of rural students taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H2b0: There is no significant difference in the learning retention of urban students taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H30: There is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students.
H3a0: There is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students of rural area.
H3b0: There is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students of urban area.
H40: There is no significant difference in the attitude of students towards the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H4a0: There is no significant difference in the attitude of rural students towards the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H4b0: There is no significant difference in the attitude of urban students towards the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method.
H50: There is no significant difference in the academic achievement of students of urban and rural areas in the subject of mathematics taught through Mastery Learning Strategy.
1.5 SIGNIFICANCE OF THE STUDY
The present study provides a clear picture about the effectiveness of Mastery Learning Strategy. The study would be significant in the following ways:
1. In Pakistan the rate of students’ failure in the subject of mathematics at secondary school certificate examinations is very high. According to the Gazette notification of BISE Rawalpindi (2002), 80 percent of the total failed students were failed in the subject of mathematics; in BISE Islamabad (2002), 75 percent of failure was due to the failure in the subject of mathematics; and in BISE Mardan (2013), 80 percent of the unsuccessful students failed in the subject of mathematics in grade 9th annual examination. To cope with this situation, the application of Mastery Learning Strategy which involves “formative assessment”, “feedback” and “corrective” as its integral part will be helpful.
2. The present study provides a verified effective teaching strategy by comparing Mastery Learning Strategy with Conventional Teaching Method, thus it can be helpful for the school administrators and practitioners to adopt a better teaching strategy to achieve the desired learning outcomes.
3. This research study is an experimental trial of teaching the subject of mathematics, thus it will provide base for change in the existing teaching practices. It will be very helpful for teachers and trainers in the selection of appropriate learning strategies.
4. The present study will be helpful to students vicariously, as they will get multiple opportunities to master a concept.
5. The findings and suggestions of the study will be useful for the concerned authorities to revise textbooks and training manuals.
6. The study was conducted in district Mardan (Khyber Pakhtoonkhwa, Pakistan). It will be useful for further researches in other areas and other cultures as well.
1.6 DELIMITATIONS OF THE STUDY
From usability point of view and from research recommendation the scope of the study is large. But due to limited resources, it is not possible to conduct such a large scale study, so, the study was delimited to:
1. The Khyber Pakhtoonkhwa (KP) province.
2. The secondary school level.
3. The Government Girls’ schools.
4. The subject of mathematics and section algebra only.
1.7 ABBRIVATIONS/ DEFINITION OF THE TERMS
1. ATMI: Attitude Toward Mathematics Inventory
2. GGHS: Government Girls High School
3. GGHSS: Government Girls Higher Secondary School
4. KP: Khyber Pakhtoonkhwa (province)
5. MLS: Mastery Learning Strategy
6. PGC: Post Graduate College
7. PITE: Provincial Institute for Teacher Education.
8. RITE: Regional Institute for Teacher Education
9. SST: Secondary School Teacher
10. SS: Subject Specialist
11. SPSS: Statistics Package of Social Science
12. Achievement: to accomplish a task successfully with skill or efforts.
13. Achievement gaps: in education achievement gaps refer to the disparities/variations in academic performance of students of a group or groups.
14. Attitude: aggregate of an individual tendencies, emotions, preconceptions, ideas, fears and concrete beliefs about a thing, person, idea or situation.
15. Cognitive entry behavior: a form of pre-learning that is necessary in order to learn a concept or specific learning unit.
16. Effective entry characteristics: a combination of a students’ interest in, attitude towards and academic self-concept about learning a concept or specific learning unit.
17. Long term retention: preservation of information for relatively long period of time.
18. Short term retention: preservation of information for short period of time.
19. Secondary education: grade 9th – grade 10th
CHAPTER – 2
REVIEW LITERATURE
This chapter consists of three sections. The first section includes the concept and importance of learning mathematics, aims and objectives of teaching of mathematics and different methods in practice for teaching of mathematics; the second section includes the background and process of mastery learning; and the third section provides a review of the research studies on mastery learning.
2.1 CONCEPT AND NATURE OF MATHEMATICS
Mathematics is an exact science that deals with measurements, calculations, discovering relationships and dealing with the problems of space. As the widest science, mathematics is defined in a number of ways. According to the New English Dictionary, “Mathematics is the abstract science which investigates deductively the conclusions implicit in the elementary conception of spatial and numerical relations”.
“According to Lock, mathematics is a way to settle in the mind of children a habit of reasoning” (Sharan and Sharma, 2009, p. 2).
Ojeride (1999) defines mathematics as the communication system for the concepts of shapes, size, quantity and order that used to describe diverse phenomena both in physical and economic situation. The researcher further describes mathematics as a tool for science, technology and industries. In fact, the credit of all technical progress in this scientific world goes to mathematics. Ezeugo and Agwagah (2000) see mathematics as a scientific tool in realizing scientific and technological aspirations of a nation.
Otunu-Ogbisi (2009) defines teaching and learning of mathematics as an act of providing and acquiring knowledge, skills, abilities, aptitude, and so as to make an individual functional and industrious for effective all round attainment of a nation‘s developmental goals.
“Mathematics should be visualized as a vehicle to train a child to think, reason, analyze, and articulate logically. Apart from being a specific subject it should be treated as concomitant to any subject involving analysis and meaning” (National Education Policy 1986, as cited in Sharan & Sharma 2009, p. 1).
2.2 IMPORTANCE OF MATHEMATICS
Mathematics exists in all human activities either in the form of geometry, algebra, statistics, and trigonometry or as arithmetic. As the most important educational subject mathematics is taught to develop the thinking power of students and to systematize their experiences for the solution of the problems that surround men.
In today’s technology-based societies such individuals are required who think about and analyze complex issues arising in their lives, solve variety of problems, communicate effectively and can adapt themselves to the new circumstances. The study of mathematics prepares the learners to live successfully in such a society by giving them new information, skills and molding their habits in a desirable way. For more developed technological based society more skillful mathematical minds are required (National Curriculum for Mathematics 2006).
In Singapore Secondary Mathematics Syllabus (2006) it is stated through the learning and application of mathematics, students develop their reasoning, thinking, numeracy, and problem solving skills. These are appreciated not only in the field of science and technology, but also in everyday life. Scientifically and technologically skilled manpower requires a well-built basis in mathematics. An emphasis on mathematics education will make sure that we have an ever more competitive labor force to meet the challenges of the future.
Mathematical structures, operations and processes provide students with a framework and tools for reasoning, justifying conclusions and expressing ideas clearly. As students identify relationships between mathematical concepts and everyday situations and make connections between mathematics and other subjects, they develop the ability to use mathematics to extend and apply their knowledge in other fields. (National Curriculum for Mathematics 2006, p. 1)
There are uses of mathematics in all fields of knowledge, i.e., in all natural and social sciences such as physics, chemistry, biology, economics, sociology, and psychology; in all fields of engineering such as civil, mechanical, and industrial; in all technological fields such as computers, rockets, and communications; and even we can’t separate mathematics from arts, such as drawing, music and sculpture. New inventions, developments and innovations are the products of science and mathematics. The majority of daily used equipment, such as computer, vehicle, calculator, mobile, etc. has mathematics in their base. The application area of mathematics is wider than any other subject. Odumosu, Oluwayemi, and Olatunde (2012) described it as the artist’s pencil, the carpenter’s hammer, tailor’s tape, hair dresser’s comb, barber’s clipper, journalist’s pen, doctor’s stethoscope, broadcaster’s microphone, and lawyer’s wig.
Scopes (1973) described the importance of mathematics as follow:
Mathematics properly approached and organized can give moment of success to everyone, the satisfaction of mastering a skill or the pleasure of combining with others to present a discovery to a wider audience. It can help to build social values in projects corporation undertaken; it runs throughout the whole of our culture in art, music, language, logic, and science and it is virtually impossible to do without it; in intellectual pursuit it stands among the foremost accomplishment of the human mind; mathematical games and puzzles provide a lifelong avenue of recreation; and finally to some it provides a source of creative aspiration at least as valid as that provided by artistic pursuit to others (P.12).
Mathematics paves the ways to higher education. Mathematics is a life wire to study all disciplines (Usman, 2002) and an essential tool for the development of science and technology (Kurumeh, Jimin & Mohammed, (2012). Mathematics is indispensable for economic development and for technical progression (ACME, 2011b). Kiani, Malik and Ahmad (2012) considered mathematics crucial for the educated person as it held a key position in the school curriculum.
Mathematics plays vital role in individuals’ lives and in the society as a whole. It is much more than a subject to prepare learners mere for higher education or job qualification in the future. It is not mere a subject of calculations, statistics and algorithms, but it compels the human mind to discover problems, formulate theories and devise methods of solutions. It prepares individuals to face a variety of challenges in everyday life. Irrespective of status in life and position in job every human being uses mathematics in daily life, for example, counting, adding and subtracting, multiplying and dividing while doing shopping, paying bills, estimating income and expenditure, measuring distance, distributing time and so on.
2.3 IMPORTANCE OF MATHEMATICS FOR SECONDARY SCHOOL LEVEL
As the sample of this study consist of secondary school students and the subject selected for teaching is mathematics, so it is imperative to explain the importance of mathematics for secondary school level.
In the entire system of education, secondary education is the most important sub-sector. It provides middle level of workforce on one hand that is essential for the economy and provide feeder for higher education on the other hand. The quality of higher education depends largely upon the quality of secondary education. Higher education is expected to provide higher quality of professionals for economical (financial), political and social life of a state. Therefore, it is necessary to organize secondary education in such a way that it should prepare the young generation for the pursuit of higher level of education, as well as to prepare them for a productive practical life (Government of Pakistan, 2000).
2.4 FACTORS EFFECTING STUDENTS’ ACHIEVEMENT IN MATHEMATICS
Mathematics is the most important, most enjoyable, and most valuable subject. It gives moments of happiness and hope for success as learner proceeds through the course successfully. But unfortunately for majority of students, learning of mathematics is stressful and worthless. Students’ performance in the subject of mathematics declines as they move to upper grades. The achievement of the students has been reported poor in the secondary school examination by the research studies (Aduda, 2003; Obioma, 2005; Olunloye, 2010; Galadima & Okogbenin, 2012). There are many factors that effect the achievement of students in the subject of mathematics. Some of these factors are listed below:
2.4.1 Students’ attitude
Attitude is the tendency of a person to respond, either positively or negatively, towards some idea, something, someone or some situation. In the words of Eagly and Chaiken (1993) “attitude is a psychological predisposition that is expressed by assessing a particular individual or thing with some degree of favor or disfavor” (p. 1).
The student’s attitude towards a subject is crucial factor in learning and achievement in that subject. Eshun (2004) defines an attitude towards mathematics as “a disposition towards an aspect of mathematics that has been acquired by an individual through his or her beliefs and experiences but which could be changed” (p. 2).
Attitude toward mathematics is related to mathematics success in the classroom (McLeod, 1992, as cited in Schenkel, 2009). Papanastasiou (2002) and Schenkel (2009) found a positive relation between attitude and mathematics achievement. Students, having positive attitude towards the subject of mathematics, perceive mathematics as a useful and interesting subject (Callahan, 1971). Hence, Students with positive attitude towards mathematics achieve instructional objectives with little efforts and vice versa.
2.4.2 Home environment
Home environment and parents’ level of education play a vital role to develop positive attitudes towards mathematics. In the words of Ainley, Graetz, Long & Batten, (1995) parental education and their professional status also affects students’ achievement in mathematics. Kulm (1980) argues that parents’ desires and expectations are important factors for success or failure of students in the subject of mathematics.
2.4.3 Untrained and unqualified teachers
Teacher is the most important agent of bringing desirable changes in the behavior of students. Subject Specialists and trained teachers for teaching of mathematics are not available in our schools. Unqualified and untrained teachers of mathematics don’t know teaching tactics and techniques to elicit students’ interest to learn the subject of mathematics. Henderson and Rodrigues (2008) blame teachers for developing negative attitudes in learner towards the subject of mathematics.
2.4.4 Instructional strategies
Instructional strategies, adopted by the teacher, are of vital importance and have unavoidable effects on students’ learning.
Strategy refers to the overall approach towards a topic, subject or area of study. There are three aspects of it. The first concerns differences of approach that will be necessary because of the individual differences among children; the second concerns with recognition that there are different kinds of learning or stages within the learning process, each require a different approach; and the third arises from the fact that for any given topic there are frequently different approaches possible from a purely mathematical point of view (Scopes, 1973, p.72).
Instructional strategies, particularly in the teaching of mathematics, have significant effects on students’ achievements. In a mathematics classroom teacher should use different teaching methods and techniques according to the need and the level of the students.
2.4.5 Time for learning
Time is the most important factor for learning mathematics. All students cannot learn with same pace, some require more time to understand mathematical concepts. Unfortunately, in KP’s schools, teacher tries to complete course in a specific period of time, thus a teacher paced leaning process is followed that does not yield a desirable result in mathematics.
2.4.6 Gender differences
Mathematics is generally considered the domain of boys. It is assumed that only boys can do better in mathematics. A meta-analysis conducted by Etsey and Snetzler (1998) concluded that small amount of gender differences exist in student attitudes toward mathematics.
2.5 INSTRUCTIONAL STRATEGIES FOR TEACHING OF MATHEMATICS
Effective education depends upon effective teaching. Teaching cannot be made effective without the selection of proper teaching strategies. Nafees (2011) says that it is a procedure that helps to deliver the whole course, instructional segment, or an instructional module. The selection of suitable instructional strategies makes sure the achievement of the prescribed instructional objectives. Mizrachi (2010) described that instructional strategy is effective if it is helpful in the achievement of prescribed instructional objectives.
Learning of mathematics is a complex activity. It demands inclusive approaches of instructions, so as to make sure the achievement of prescribed instructional objectives. It lay down great responsibility on teachers to give proper attention to the knowledge of content and teaching method. It is the responsibility of teachers to provide the students with the update knowledge and lifelong learning skills such as precise reasoning, critical thinking, problem solving ability etc. In today’s dynamic society critical thinking and lifelong learning skills are very important for survival. Teachers need to reflect on their present practices and reorganize their teaching strategies according to the new challenges and to equip their students with all skills that are needed for a productive social life.
Research studies suggest that effective learning takes place through the presentation of daily life problems that in turn demand proper approach of instruction. Nafees (2011) says that new instructional strategies that emphasized learning, such as project method, inquiry-based learning and learning based on problem solving, are getting prominence in the classroom environment.
There are so many methods of teaching mathematics, but we can’t justify a specific method to be good for all contents and all students. No doubt that each method has some good but it is also a fact that no method has all goods. In teaching of mathematics, no single approach is accepted as best and suitable for all subject matter of mathematics and fit in all situations (Herrera, Kanold, Koss, Ryan & Speer, 2007). It depends on the mathematics teacher to select suitable methods for teaching different mathematical topics in different situations. Some of the conventional methods and new approaches of teaching mathematics are discussed below.
2.5.1 Conventional Teaching Methods
In Government schools of Pakistan conventional methods of teaching are still dominating the teaching learning process. Teachers think it economical in term of money and time; and appropriate for large classes. Agwagah (2004) stated that conventional teaching is still, as a custom in our schools, dominated in the mathematics classroom.
In conventional teaching method teacher is the center of all activities and s/he controls the whole instructional process. S/he is regarded as the source of learning. S/he delivers content/subject matter to the whole class and put emphasis on the factual information while students listen passively. In other words, the teacher gives lectures and the students listen to the lecture. Thus, the learning process tends to be passive and the learners play little role in the learning process (Orlich et al., 1998, as cited in Damodharan and Rengarajan, 2007). Students are treated as the ‘knowledge holes’ that need to be filled by teacher with information. The main focus of the teacher is on memorization, repetition and absorption of knowledge. S/he ignores the recommendations of modern scientific researches and professional organizations of mathematics about students’ learning (Perveen, 2009; Battista, 1999).
In conventional methods, particularly in the mathematics classroom, teacher solves exercise on writing board and students copy it in their notebooks and then practice it in the classroom or at home.
Farooq (1980) states that teaching method that is used traditionally in secondary schools in Pakistan is a mixture of lecture method and solving mathematical problems only on the blackboard.
Some common traditional methods are:
i. Rote learning.
Mathematical definitions, concepts, tables and formulas are learnt by repetition and memorization without understanding meaning or supported by mathematical reasoning.
ii. Imitation/copying.
Mathematical skills are acquired by imitating teacher’s demonstrations and textbook communications. Teacher solves examples from mathematics book and students copy it in their notebooks. And try to solve homework assignment on the same pattern. If there is a little change in the solution of a problem, it is left for the teacher to solve.
iii. Expository method.
Expository method is teacher centered approach. Teacher provides information verbally and students silently take notes. Students are never put into situation to get insight into the situation and move to understanding and logical reasoning. Expository method is a telling approach (Perveen, 2009) and one way channel of communication (Farooq, 1980).
iv. Drill/Exercises.
A large number of similar questions such as LCM, HCF or quadratic equations are given to student for solution to improve their mathematical skills.
Drill and verbal recitation, rote memorization, worksheet completion and teacher directed instructions failed to develop students’ basic skills and interest as students are still trying to comprehend abstract mathematical concepts since the approach lacked concrete experiences (Shannon, 2004). As a result, students’ achievements in the subject of mathematics have been reportedly poor, particularly in algebra at both internal and external examinations (Aduda, 2003; Obioma, 2005; Olunloye, 2010;Galadima & Okogbenin, 2012).
Some limitations of conventional teaching method are:
• There is one way flow of information, teacher use only “talk” and “chalk”.
• Most of the time teachers deliver lecture and talk for a long period without receiving any response and feedback from the learners.
• Presentation is based on lecture notes and textbook.
• This is “plug and play” method of teaching having no consideration for practical life.
• Teacher’s notes decide the fate of the subject.
• There is very little interaction between students and teacher in the classroom.
• Emphasis is put on theory without any practical and real life situations.
• Learning is done through memorization while understanding is ignored at all.
• Marks rather than result oriented.
( Damodharan & Rengarajan , 2007; Jusoh& Jusoff, 2009; and Usmani & Dawani, 2013)
2.5.2 New approaches for teaching Mathematics
A balanced mathematical program comprised of the concept learning, skills developing and maintaining, handling and tackling applications. These should be taught in such a way that students develop the ability to think mathematically. This ability develops effectively by applying mathematical concepts and skills in attractive and realistic contexts, that personally significant to them (Perveen, 2009). Instead of drill and practice if mathematic is learnt conceptually, it will develop students’ reasoning skills, problem-solving skills and logical-thinking.
Following are some new methods of teaching mathematics:
Inductive Method of Teaching Mathematics.
The inductive method is based on the process of induction. This is a teaching learning process that leads the learner from concrete to abstract, specific to general and from examples to formula. In this method first some particular examples are taken as a process and at the end formula is proven and generalized. This method is very helpful in the teaching of mathematics (Sharan & Sharma, 2009). Induction means to provide universal truth by showing that if it is true for a particular case, it will be true for all such cases. The inductive teaching process is psychological in nature and it sustained students’ interest till the end. Students’ attain knowledge of mathematical formulas and principals with the help of facts and experiments. Thus the knowledge attained by this process is solid and durable (Sharan & Sharma, 2009).
The following steps are followed by this method:
Teacher presents several similar examples and solves these examples with the help of students.
After the solution of examples, students observe and analyze these and try to reach to some conclusion.
After analysis and logical discussion, both teacher and students make a decision about a common formula, principle or law.
Testing and verification is done.
Deductive Method of Teaching Mathematics.
In this method, learning process proceed from general to specific and from abstract to concrete. Using this method in mathematics, a pre-constructed formula is taught to the learners and they are required to solve the relevant problems with the help of that formula. The formula is accepted by the students as a pre-established and well-established truth (Sidhu, 2003). This method is mostly used in algebra, trigonometry and geometry. Assumptions, postulates and axioms are supportive in this method (Sharan & Sharma, 2009).
Problem solving method of teaching mathematics.
This is a teaching method where students are trained to solve problems. Skinner (1984) defines ‘Problem Solving’ as the framework or pattern that involves creative thinking. It is a process of overcoming difficulties that appear in the process of attainment of a goal. Students apply already learned rules to the novel problem. The Teacher presents the problem and gives enough information to establish the background of the problem. Students realize the problem, collect and interpret information and try to arrive to one or more solutions.
The following steps are involved in problem solving method:
Recognizing the problem
Defining the problem
Collecting relevant information
Organizing information
Formulating a tentative solution
Arriving at the concrete solution
Verifying the result
Laboratory Method of Teaching Mathematics.
This method works on these principles: 1) learning by doing; 2) learning by observation; and 3) moving from concrete to abstract. This method is based on activity that leads the students to find out mathematical facts. Students are stimulated and encouraged to do activities and make discoveries (Kumar & Ratnalikar, 2004). Laboratory method makes mathematics more interesting and significant. In laboratory, students verify certain mathematical facts through experimentation and testing (Kumar & Ratnalikar, 2004). The success of this method depends on a well-resourced mathematics laboratory (Sarita & Sarivasta, 2005).
Project Method of Teaching Mathematics.
In project method students are assigned a task or problem and they are required to design and carry on their own plan. The students look for material and manage everything by themselves. The supporters of project based learning advocate that it enhances the learning retention of students, motivate them and develop their interest in learning. Project method gave opportunities of making connections to the outside world. This method develops in students the skills of creativity, critical thinking and problem solving. Students work collaboratively that also develops their skill of communication (Nastu, 2009). Project method is based on the principles of learning by doing and learning by living.
Generally the following steps are involved in project method;
The teacher provides tasks or problems.
Students are guided in choosing project related to their requirements.
The teacher provides a condition where students feel comfortable to work on projects.
Students discuss and plan the process of carrying out the project.
Responsibilities are distributed among the members of the group.
According to the planning made each student take a share in the execution of the project.
The project is then evaluated; the knowledge gained is reviewed and recorded.
Heuristic Method of Mathematics teaching.
This method involves self-education on the part of the students. The students try to explore and discover instead of merely telling by the teachers. This method is based upon the Herbert Spencer’s assumption that little should be told to the student and much should be discovered by themselves. The word heuristic has been derived from the Greek word “Heuresco”, which means “I find”. This method emphasizes efforts and experimentation. The teacher’s role is that of an observer or spectator and the learner proceeds independently without any help (Kumar & Ratnalikar, 2004). This method develops scientific and heuristic attitude in the learners. This method develops self-confidence, independencies, originality and thinking and judgment power of the learner. Objectives behind heuristic method are to develop problem solving skills, logical and imaginative thinking capabilities, and scientific attitude of students towards the solution of problems (Narayana, Krishma and Rao, 2004, as cited in Mustafa, 2011). This is an active method as it provides opportunities of the discoveries thus make the students self-reliant and independent.
Analytical Method of Teaching Mathematics.
In this method the unknown problem is break-up into simple parts and then these parts are recombined to find the solution. Students start with what is to be found out i.e. the result of the problem or theorem and then think of further steps or possibilities. Explanation of every step is carried out, then these possibilities and explanations are associated with the conditions of the problem or theorem and investigate the solution. This method is very helpful in understanding of the subject and developing the reasoning power of the learners. It is the process of unfolding the problem to understand its hidden aspects (Kumar & Ratnalikar, 2004) and to make a connection of the unknown to the known to achieve the desired results (Sharan & Sharma, 2009).
Synthetic Method of Mathematics Teaching.
The word “synthetic” is derived from the word “synthesis. Synthesis is the complement/opposite of analysis, which mean combine together. In this method we proceed from known to unknown. A number of facts are collected together; certain mathematical operations are performed and try to arrive at a solution. Start is taken from the known data and its relation is made with the unknown part. It is the process of putting together known bit of information to arrive at a point where unknown information become obvious and true (Kumar & Ratnalikar, 2004).
2.6 ROLE OF TEACHER IN TEACHING OF MATHEMATICS
Kumar and Ratnalikar (2004) in their book “Teaching of Mathematics” quotes Dr. S. Radhakrishnan as,
The teacher’s place in the society is of vital importance. S/he acts as a pivot for transmission of intellectual traditions and technical skill from generation to generation, and helps to keep the lamp of civilization burning. S/he not only guides the individuals, but also, so to say, the destiny of nations. Teachers have therefore to realize their special responsibility to the society. (P.151)
In today's modern world the role of teacher has been shifted from “dispensing knowledge” to an “investigator”, a “planner” and “manager”. The teacher has to investigate and plan tasks; to consolidate and systemize students’ informal knowledge; to manage a supportive learning environment, and to help students’ to understand different concepts.
Teaching in a mathematics classroom requires understanding students’ needs and their level of thinking, setting and analyzing tasks and their outcomes in order to understand how students construct meanings ((National Curriculum for Mathematics 2006).
Teacher should ensure not only that student build up the mathematical knowledge, but also that they value mathematics and feel good about their capability to deal with situation in which mathematics is used (A National Statement on Mathematics for Australian School, 1990).
Pirie and Kieren (1992) have accepted the Social Constructive view of mathematics and put forward four very important beliefs about mathematics teachers:
1. A teacher has the intention to move students toward particular mathematics learning objectives, but s/he will be well-aware of the fact that such progress may not be achieved by some of the students and may not be achieved as accepted by others.
2. In creating an environment for children for modification of their mathematical understanding, the teacher will act on the belief that there are different ways to similar understanding.
3. The teacher will be well-aware of the fact that different learners will hold different mathematical understandings.
4. The teacher will also be aware that for any topic, there are different levels of understandings, but that are never achieved “once and for all”.
The teacher’s role in a classroom is just like a gardener in a garden, who attend each plant, examine its leaves and flowers for diseases, water each plant according to its need and ensure that each plant take its nourishment. The teacher should take care of the students’ needs, capabilities, and differences in them while teaching in the classroom.
2.7 MASTERY LEARNING STRATEGY
Mastery learning works on the philosophy that all students can learn and attain instructional objectives if they are guided by appropriate teaching method and appropriate time for learning is given to them. Specifically, mastery learning is a teaching strategy whereby learners are required to progress a subsequent learning activity when they display proficiency with the existing one.
According to Bloom (1971) each teacher begins a new unit or course with a set of three expectations that about 1/3 of his students will learn adequately; 1/3 students will either fail or will just get by; and 1/3 will learn a good deal but still they can’t be regarded as good students. This set of expectations is the most destructive and wasteful aspect of the present system of education as it fuses the educational goals of teachers and students. It condenses the aspirations of both the teacher and the learner. Mastery learning, on the other hand, is a method of instructions that focus on the achievement of instructional objectives by all the learners. Mastery learning strategy was developed as a way to give students appropriate and high quality of instructions so that all learn well and truly master the subject (Guskey, 1997).
2.7.1 Background of Mastery Learning
The concept of "Mastery Learning" is not new; it has its roots in the scripts of Aristotle and other ancient Greek philosophers. The idea of mastery learning was stressed in different ways by Comenius in the 17th century, Pestalozzi in the 18th century, and Herbert Spencer in the 19th century. It was introduced into the western education system over 90 years ago. However, only a few schools of America used mastery learning during 1920s. Washburne (1922), Morrison (1926), Carroll (1963), and Bloom (1971) were the major proponents of this philosophy. According to Block (1971), in 1922 Washburn made an attempt to introduce mastery in students' learning. His plan was named as the Winnetka Plan, in which "primarily self-instructional practice materials were used, although the teacher occasionally tutored individuals or small groups"(p. 4). A further attempt was made by Morrison in 1926. Morrison used a variety of correctives i.e. “re-teaching, tutoring, restructuring the learning activities, and redirecting students’ study habits"(p. 4). Block (1971) stated that Morrison's method was successful, but was failed to be used due to lack of technology.
In the early 1950s the idea of mastery learning was invigorated in the form of “Programmed Instruction”. Initially this program was derived from work of B. F. Skinner that was further developed by other behaviorists. This programmed instruction was parallel to mastery learning in the sense that its focus was on the function of feedback in learning and on the individualized learning. And like the mastery learning model, programmed instruction allowed students to progress at their own learning speed, and to receive immediate feedback on their existing level of mastery.
The most eminent work on Mastery Learning is connected with John B.Carroll and Benjamin Samuel Bloom. Both are credited as the founder of idea of Mastery Learning, though many of its components were strongly influenced by Washburn (1922), Morrison (1926) and other behaviorists in the 1960s. The theoretical model of Mastery Learning is explained by Carroll (1963) and it is transformed into a practical strategy by Bloom (1968).
2.7.2 Contributions of Washburne (1922)
Washburne (1922) and his associates attempted to produce mastery in students’ learning through Winnetka plan. This plan was introduced in American schools in early 1920’s. The program was flourished during that decade, but gradually, due to lack of technology and other facilities, the interest of developers in that program was diminished. Winnetka plan was a self-paced learning program as it allowed each learner to move at his/her own pace by taking his/her own time to achieve the objectives of a unit. The particular features of this program were:
1. Defining objectives. Mastery was defined in terms of particular educational objectives and each student was required to achieve these objectives. The cognitive objectives were given prime importance.
2. Planning of instruction in well-defined learning units. Instructions were planned into well-defined learning units. Each unit consisted of a set of learning materials organized systematically to achieve the desired objectives for a unit
3. Mastery of the unit. Mastery of a unit was necessary for students before progressing to the next.
4. Diagnostic progress test. Administration of diagnostic progress test at the end of each unit with the purpose to provide feedback about learning difficulties.
5. Supplementary/additional materials. On the basis of diagnostic progress test, arrangement of supplementary/additional materials for further learning to be provided.
3. Contributions of Morrison (1926)
After Washburne (1922) the name of Henry, C. Morrison is prominent in the history of mastery learning. Henry, C. Morrison was a professor at the University of Chicago's laboratory school. According to Morrison’s teaching method, the product of teaching is not only memorization of facts, but mastery of content, that is attained only when planned knowledge is grasped fully. The major steps involved in Morrison’s teaching method were:
• Categorizing and defining instructional objectives, i.e. Cognitive, affective and psychomotor objectives.
• Splitting up of each subject into small units and to develop and arrange related facts in a unit sequentially.
• Each unit should present a particular concept and specific understanding in such a deliberate way that the majority of students achieve mastery. A unit is covered only when all or almost all students thoroughly understand the concept, its factual origins, its’ possible reliability, its generalization and its application.
• Administration of un-graded diagnostic test at the end of each unit to provide feedback on students’ progress.
• Based on the diagnostic test variety of corrective activities are to be used, such as re-teaching, re-structuring the learning activities, tutoring, and re-directing students’ study habits etc.
In Morrison's method of teaching each student was allowed as much time as needed for the mastery of a unit.
2.7.4 Programmed Instructions and Contribution of Skinner (1950s)
Due to lack of technology Winnetka plan and Morrison's method were difficult to implement properly, thus its developers and implementers lose their interest and gradually these plans vanished. Mastery Learning was invigorated in the form of programmed instruction in 1950's. The term Programmed Instruction (PI) was certainly derived from Skinner’s paper, “The Science of Learning and the Art of Teaching” that he presented at a conference on Current Trends in Psychology and the Behavioral Sciences in the University of Pittsburgh in 1954. PI was based on the idea that the learning of complex behavior rest upon the learning of a series of less-complex component behaviors (Skinner, 1954). So the most complex skills or behavior can be mastered if component behaviors are sequentially organized like a chain and mastery of each link is ensured (Varughese, 2002).
Skinner (1954) argued that each successive step should be made as small as possible, the frequency of reinforcement should be raised to the maximum and the negative consequences of being wrong should be reduced to minimum.
The basic features of Program Instruction have been sum-up by Hartley (1974) as follow:
• A clear idea about the terminal behavior should be given to the learner.
• The instructions are organized into less-complex small sequential steps. These steps were termed as ‘frames’.
• The learner is permitted to work at his/her own pace on each step.
• The learner is encouraged to respond actively at each step.
• The learner gets immediate result about his/her responses.
Immediate feedback, provide at the end of each “frame” reinforce the learner to move to the next “frame”. Skinner (1954) argued that reinforcement should be provided through a machine in the form of correct response as teachers can’t provide this immediately to every student.
Skinner calculated that during the first four years of school life a child need 50,000 reinforcements to achieve proficient mathematical behavior, but that in a conventional classroom a teacher could provide only a few thousands. This problem, according to Skinner (1954), could be solved through the introduction of instrumental aid in the classroom.
2.7.5 Carroll's Model of School Learning (1963)
Although mastery learning is rooted in the writings of the early Greeks, its current version has developed from Carroll's “Model for school learning”. John B. Carroll was a professor at Harvard University. He challenged the traditionally held notion about aptitude that view aptitude as a level up to which students could learn specific contents or a particular skill. Those who have low aptitude will learn only simple concepts and those having high aptitude will learn complex concepts. Thus, they divided students into two groups, i.e. good learners and poor learners.
Contrary to that idea Carroll’s viewed children as slow and fast learner. He argued that all children have the potential to learn but require different amount of time. In his 1963 article, Carroll defined all the variables in terms of time that influence the learning of children directly.
Carroll (1963) in his “Model for School Learning” assumed that children vary in the amount of time they required for learning and if they are provided with that time then the degree of learning can be measured as following:
Degree of learning = f Time actually spent
Time needed
If required time is not provided or if provided, but the learner does not spend it appropriately the degree of learning will be less than the specified level. Carroll model comprised of six variables; one input variable, 4 intermediate variables and one output variable. These variables are given in the figure 2.1:
Figure 2.1: Carroll’s Model for School Learning
1. Aptitude is the major descriptive variable which Carroll (1989) defined as the amount of time that children required to learn a given skill, task or content to a specified level or set criterion under suitable condition.
2. Opportunity to learn means the amount of time available for learning in the classroom as well as at home.
3. Ability to understand instruction means language comprehension and learning skills. If student have ability to understand they need less time to learn.
4. Quality of instruction includes instructional material, good instructional design and presentation. If quality of instruction is good, time needed to learn will decrease.
5. Perseverance is the amount of time that a student spends willingly on learning a given unit or skill.
6. Academic Achievement is the output variable that is measured by different tests.
Carroll’s article of 1963 was a worthwhile contribution to learning theories that opened the way for researchers into a new concept of aptitude. His concept of “learning rate” paves the way for “individualized instructional programs”.
2.7.6 Contributions of Bloom (1968)
Benjamin Samuel Bloom was an American psychologist and a professor at the University of Chicago. In the mid 1960's Benjamin S. Bloom started a series of investigations in order to improve teaching learning process. His focus was on individual differences and variation in learning outcomes.
Bloom (1964) recognized that many factors inside and outside the school have influences on students’ learning. He convinced about teacher’s potential influences on students’ learning. Bloom observed that in traditional classroom the time is kept constant and very little variation are introduced in the classroom instructions. He believed that these learning variables are suitable for some students, but not for all. Thus, those for whom time is sufficient and instructions are appropriate, learn quite well while others do not. According to Bloom (1964) if students were distributed normally, based on their aptitude for a subject, and uniform instructions were given to all (in terms of quality and time), then at the completion of subject their achievement would also be normally distributed. However, if instructions are differentiated and time allowed were made appropriate to individual students’ characteristics and needs than most of the students could be expected to attain specified level of learning.
In searching a strategy that would be helpful to achieve better results, Bloom worked on two sources of information. First, he studied perfect teaching-learning conditions where an excellent teacher is paired with an individual student and attempt to determine how the crucial elements of individualized instruction can be transformed to group based instructional process. Second, he gathered information regarding the strategies employed by academically successful students in grouped based learning condition that differentiated them from academically less or unsuccessful fellows. Regarding Individualized instructions, he studied the work of Washburne (1922) and Morrison (1926), especially Washburne’s Winnetka Plan and Morison’s experiments that he had been conducted in his University of Chicago Laboratory School. Regarding the second source of information he considered the work of Dollar and Miller (1950). He also studied Carroll’s idea of aptitude and was much impressed by that.
In Carroll’s perspective aptitude was predictive of time that a child need for learning. Therefore, according to Bloom (1968), it should be possible to fix a criterion for learning at some mastery level that we expect from our students. Then under the teacher’s supervision and control, by attending instructional variables, all or almost all could attain a specified level of learning. In other words, providing good quality of instruction and proper opportunities, all students could learn to the same level. Bloom (1968) transformed Carroll’s “Model for School Learning” and into a practical learning strategy. He named it “Learning for Mastery” (Bloom, 1968) and later on shortened it to “Mastery Learning” (Bloom, 1974). He explained that it is a process whereby students attain the same level of content mastery but at different time interval. One aspect of Mastery Learning, according to Bloom, is learning in sequence, where sequencing is explained as hierarchical. Thus, mastery of each step is essential before moving to the next step. Throughout the instructional period, students are given regular feedback about their learning progress. This feedback is helpful for both partners, i.e. student and teacher. Students identify areas that they have learned well and those that still need time to learn, and teacher become aware of the points where students feel difficulty to learn. Thus, on the basis of feedback teacher give more time and modify or change the teaching method if it is needed. Contrary to conventional methods “mastery” is held constant and time allowed for the “mastery” is varied.
Benjamin S. Bloom was not contended with teachers' traditional practice where curriculum contents are arranged into units and students' progress is checked at the end of units. He thought that this check, if used as a part of the teaching - learning process, would be very valuable.
The various stages of Bloom’s Mastery Learning Strategy are:
• The concepts and materials are first organized into small instructional units. A unit is composed of the similar or interconnected concepts presented in about a week or two.
• A suitable method is adopted for Initial instruction on the unit.
• An assessment is administered for giving information or feedback on learning unit. Bloom called it “formative assessment”. Formative assessment may be in the form of written test or a quiz.
• Based on formative assessment, remediation activities, termed correctives, are administered for those who didn’t master the concept.
• A parallel formative assessment is administered after completion of correctives.
Formative assessment is used as a part of the learning process as an informative tool, its basic purpose is to give feedback on students’ learning. Correctives are individualized, especially designed for those students who couldn’t master the concept during the initial instructional process. “Corrective” may point out additional or alternative sources of information on a particular topic, such as the page numbers in the course workbook and textbook where the topic is discussed or alternative learning resources such as alternative worksheets, different textbooks, peer-tutoring, or computer aided instructional lessons; or they may simply suggest additional sources of practice, such as learner centered activities, study guides, independent practice, guided practice activities etc. At the completion of “correctives”, a second formative assessment is administered to check the effectiveness of “correctives” in addressing learning difficulties. It serves as a motivational device as it provides a second chance of success. Bloom (1968) believed that through the process of formative assessment coupled with the organized correctives, all students could be provided with a more appropriate and quality instructions that would be helpful to attain mastery in specified task. The students who demonstrate mastery on the first formative assessment would be supplied with enrichment assignments.
2.7.7 Keller’s Personalized System of Instruction (1968)
While Bloom (1968) developed his Mastery Learning for classroom practice, Keller (1968) developed a system of instruction for individual student called as Personalized System of Instruction (PSI). PSI was suitable for higher education. Keller integrated the ideas of mastery learning with the of reinforcement learning theory of B.F. Skinner. The Plan is based on three fundamental observations.
1. If we expect all students to attain mastery, then it cannot be expected from all to do it at the same time.
2. The small quantity of substance is more easily digested than larger quantity. This directed to adopt "modularized" courses.
3. Students will learn well, if they are given frequent and instant rewards.
In the Keller Plan, these rewards are instant feedback on tests and credited marks for success in the test.
Keller’s Personalized System of Instruction (PSI) is individual based and student paced. It specifies instructional objectives, provides reinforcement for the successful achievement and gives more options and opportunities for personal interaction. Students are given multiple opportunities of retaking mastery test without any penalty. This system was first openly announced in 1968 in Keller’s article “Goodbye Teacher”. In this system the role of teacher is that of a guide. S/he provides assistance only when needed. Classroom presentation is viewed as vehicle of motivation rather than a source of information. Well organized “learning modules” are the essence of the PSI program (Kulik, Kulik, &Cohen, 1979).
There are five key elements in PSI as described by Keller in his 1968 paper:
1. The unit-perfection requirement in advance, which allow the learner to proceed to new material only after demonstrating mastery of the existing unit.
2. The go-at-your-own pace element, which permits a learner to advance through the course at a speed, appropriate with his/her abilities and other demands upon his/her time.
3. The stress upon the written word as a primary source of teacher-student communication.
4. The use of proctors for repeated testing, instant scoring, immediate feedback, and almost unavoidable tutoring.
5. The use of lectures and demonstrations as a source of motivation and providing occasions of getting together, rather than source of critical information.
Keller argued that the mastery means “perfect performance”. Even a single error should not be left incorrect. If done so, it will be unwise. According to Keller and Sherman (1974), the requirement of eighty-five percent will definitely raise the questions about eighty-three percent, as it is close enough. There may be requests for partial credit or bargaining about grades. Therefore Keller advocates 100% of performance.
2.7.8 Post Bloom Period (since 1971)
After Bloom many people start working on the Mastery Learning philosophy. Many of his colleagues and students devoted their attention to develop practices for the Bloom’s Theory of mastery learning. Extensive mastery learning researches have been conducted that helped in the development of successful learning strategies at all levels and in all subjects. At the initial stage the efforts of the researchers were concentrated to the improvement of classroom instructions and school wide practices. But soon the interest in mastery learning approach had spread far beyond the school level. According to Block (1979) the entire local, regional, and even national school systems wished to investigate the potential of the evolving mastery learning approach to their specific problems.
After Bloom’s group-based mastery learning approach, hundreds of studies have been conducted on the effects of Mastery Learning. These studies extended to all levels of learning and to all subjects ranging from science to languages. Dozens of reviews of those studies, including several meta-analyses were done by different researchers. The results of those reviews declared mastery learning an effective teaching strategy for all levels of learners.
Mastery Learning is operating in nations around the world at every level of education. The range of its application is from pre-school to graduate and professional schools. Moreover, evaluations of these practices demonstrate that students in mastery learning classes constantly learn better, achieve higher levels of learning and develop greater confidence as learners (Kulik, Kulik, & Bangert-Drowns 1990, Guskey & Pigott, 1988).
2.8 APPROACHES TO MASTERY LEARNING STRATEGY
Two genotype approaches to the use of mastery learning strategy are:
1. Group based teacher-paced approach.
2. Individual based, learner paced approach.
In group based teacher paced approach the instructions’ delivery and flow are controlled by the teacher. Students work co-operatively with their class-fellows. This model is based on Bloom's Mastery Learning. This approach has developed from within the field of education. According to research studies group based teacher paced approach has greater impact at the elementary and secondary school levels (Block & Anderson, 1975).
The second approach is based on keller’s (1968) personalized system of instructions. In this individual based learner paced approach, students learn independently and they themselves control the delivery and flow of instructions. It is designed for the situations where available time for learning is unrestricted. This approach is developed from the field of psychology and biology and has greater impact on college and university levels.
2.9 ESSENTIAL ELEMENTS OF MASTERY LEARNING
According to Guskey (1997, 2005), the two essential elements of mastery learning which have been outlined by the developers of mastery learning strategy are: 1) Feedback, corrective and enrichment; 2) instructional alignment.
2.9.1 Feedback, corrective and enrichment
Feedback, corrective and enrichment activities are essential for the Mastery Learning process because these activities differentiate and individualize instructions (Guskey, 1997, 2005) and make it a superior method. Each student, who faces difficulties, is provided with additional time and instruction that suit his/her characteristics. Gifted and talented students are provided with opportunities to extend their knowledge.
Feedback: Feedback is an essential element of the Mastery Learning instructional process. Based on formative classroom assessment, teacher provides diagnostic and prescriptive feedback. It reinforces learner to attain specified objectives, identify well learned skills/tasks, and find out areas of difficulties and needs of learners; thus addressing inequities in the instructional process.
Corrective Activities: Feedback does little to improve the learning process. Feedback paired with corrective activities can bring the required improvement. Correctives activities provide guidance and direction for remediation. As individual differences exist in students, therefore, Bloom (1971) argues that teacher must differentiate their teaching in order to help each student in learning. Guskey (2005) says that corrective must be designed qualitatively different from initial instructions according to the need of the students. An effective corrective involves the students differently by presenting the concept differently. No doubt the process of making corrections will be challenging for many teachers, but by sharing ideas, discussion of material and giving time to teacher to work collaboratively will greatly facilitate the process.
Enrichment Activities: In the Mastery Learning process, some students master the concepts in the first attempt, some in the second attempt and some in the third or fourth attempt. So for those, who have mastered the concept, enrichment activities are arranged. Enrichment activities provide students with exciting opportunities to utilize their time and expand their knowledge. Enrichment activities may include extra material on the same concept, self selected independent activities, tutoring pairs, or helping a teacher in preparing correctives activities.
2.9.2 Instructional Alignment
The second essential element of mastery learning is instructional alignment. Consistency in all instructional components is labeled as alignment (Bloom, 1971). To be truly effective, both elements of Mastery Learning must be combined.
According to Guskey (1997, 2005), teaching and learning process is generally perceived to have three major components; 1) learning goals or standards – a clear idea about what we want the learners to learn and be able to do; 2) instruction; and 3) results - in the form of competent learners. Competencies can be assess through proper assessment. Based on assessment, the teacher has to determine for whom the initial instruction was suitable and should be given enrichment activities; and for whom learning alternatives are required. Through these alternatives, all learners are hoped to reach the desired goal. Regardless of what is taught, what is the instructional process and how it was evaluated; Mastery Learning claims consistency or alignment in all components. For instance, if objectives are to acquire higher level skills such as applications, synthesis, or analysis, then mastery learning insists on instructional strategies that would give students’ opportunities to practice and their active engagement in those skills. It also requires specific feedback on students’ learning combined with remediation to correct learning errors. Finally, evaluating procedures should reflect those higher level skills as well.
This second component, i.e. instructional alignment put on teacher the responsibilities of taking some very important decisions. For example, they must decide about the most important and central concepts or skills for students to learn and understand. They also must decide about the evidence that properly reflects mastery of those concepts or skills by students. Normally, when a teacher administers a test or evaluates students, he informs the students about what they considered to be most important. Mastery Learning demands teachers to make these decisions more carefully and purposely.
2.10 MASTERY LEARNING PROCEDURE
The procedures for mastery learning as designed by Bloom (1968) as follow:
1) Break down the course or subject into smaller units
Subject content is divided into unit and subunits such as a textbook into chapters and chapters into small units having single idea or concept. These units are arranged sequentially, progressing from specific to complex and abstract concepts. A well defined unit /segment may involve a week or more of learning activities. It is compulsory for students to have mastery of the first unit before moving to the second unit.
2) Determine unit objectives and mastery level
Before starting a unit it is optimal to define expected objectives and mastery level for that unit. The student will move to the next when they will reach to the expected level of mastery in current unit.
3) Construct formative test for each unit
For each unit teacher should construct a formative test to determine student mastery for that specific unit. Before administering the test, students are informed that the test will be used to determine their mastery of a particular unit and will not be graded. The formative test reinforces those who have mastered the current unit and provide feedback about those who need more time and instructions. It points out areas where students feel difficulties and identifies points of modifications in teaching method.
4) Scoring of Formative tests
The use of grade can create tension and anxiety in students, therefore mastery and non-mastery is used while scoring test items. Teacher compares the performance of students with the previous test to make sure that they are doing well. The students’ performance on each test may be compared with the set criterion or norm for previous years to make sure that they are doing well.
5) Corrective learning procedure
The use of corrective learning procedure after discovering learning difficulties is very essential for Mastery Learning. The corrective procedure provides guidance and direction to students for remediating their learning difficulties. Correctives are designed differently from initial instructions. As variation exists in students, no single method for instructions works best for all. Therefore, teachers must differentiate their instructions during re-teaching process, to improve the interest and motivation of those in the class who are left behind.
Corrective learning procedures may include small group study sessions, peer tutoring, rereading specific topic, studying particular pages or workbooks or programmed materials, use of extra material, re-teaching etc.
To be very effective, correctives must be qualitatively different from the original instruction. They must provide students, who faced problems in learning, with an alternative method and additional time to learn. The best correctives present concepts in a different way and involve students differently than in the initial instructional process. Teachers incorporate different learning approaches, learning modalities, or forms of intelligence. Although the process of developing effective correctives may be challenging, many schools provide the teachers with time to work collaboratively, share ideas, materials, and expertise that really facilitates the process (Guskey, 2001).
Correctives are usually accompanied by “enrichment” or “extension” activities for those who master the concepts from the initial instructions. Enrichment activities provide the students exciting opportunities to expand their knowledge (Guskey, 2005).
6) Administration of second formative test
A second formative assessment is administered after the necessary corrective learning procedures. The purpose of the second formative test is to determine the effect of the corrective procedures on learners. If the majorities have attained mastery in second formative test, the class moves to the next unit.
2.11 VARIABLES OF MASTERY LEARNING
Mastery learning is a teaching strategy essentially derived from the work of Carroll (1963). The conceptual model of school learning, developed by Carroll (1963), provides theoretical base for the strategy. He sketched essential variables for mastery of the subject in his model of school learning. Bloom (1968) and his associates transformed the major variables of Carrol’s model in a workable strategy for Mastery Learning. These variables are discussed in detail below.
I. Aptitude for learning
For years, aptitude was viewed as a fixed and generic ability to carry out a variety of learning tasks. Carroll' (1963) came with a different view about aptitude. In his view aptitude was an index of the amount of time that a student would need to learn a subject to the prescribed level, rather than the index of the level to which a student could reach in a specified amount of time. In simple words aptitude is the amount of time that a student needs to master a particular task. Carroll’s model suggested that if every student was provided the time s/he needed and s/he actually spent the necessary time to learn the subject, then s/he might attain the prescribed level of learning. Carroll held that “time needed” and “time spent” were influenced by the students’ as well as instructions.
Working on Carroll’s logic Bloom argued that if aptitude was presumed as predictive of “time needed” to learn and not of “the level” to which a student could learn then it should be possible to fix the degree of learning expected from each student at some criterion level mastery performance. He argued that if student are distributed based on their aptitudes for learning a subject, then at the top of the distribution will be likely those (may be 5%) who would have a special aptitude for learning the subject. Similarly, at the bottom may be the same proportion (5%) with disabilities for learning the subject. For remaining 90% students, aptitude is predictive of the rate of learning rather than level of learning. Thus, according to Bloom, 95% of the students can achieve mastery if sufficient time for learning and appropriate help is provided.
II. Quality of instruction
Quality of instruction, according to Carroll (1963), is the extent to which the presentation, explanation and organization of elements of the learning task reaches the optimum for a particular student.
Individual differences in learning are obvious and proven form research results. Some students learn quite well when they study independently while others need highly structured learning environment. Some grasp the concept readily while other need concrete examples, illustrations and explanations. Some catch the idea in the first instance, while others need reinforcement and several repetitions of the explanation. The focus here is that the quality of instructions should be appraised in terms of its effects on individual student rather than on group. The strategy assumed that quality of instructions could best be defined in terms of clarity and appropriateness of cues for each student, the amount of active participation of each student and practice allowed to each one, and the quantity and variety of reinforcements exist for each student.
III. Ability to understand instructions
The ability to understand instructions makes the learner able to understand nature of the task given to him/her. This ability, according to Varughese (2002), depends upon instructional material and teaching skills of the instructor. A student having the ability to follow the teacher's communications and instructional material finds little difficulty in learning the subject. In our schools, verbal ability and reading comprehension determines the ability to understand instruction. These measures of language proficiency are essentially linked to achievement in the majority of subjects as verbal ability establishes some general ability that helps to understand instruction and learn from instructional materials.
In order to improve verbal ability, modifications in instruction are essential to meet the needs of individual student. According to Bloom (1964) majority of changes in verbal ability of the student can be brought at pre-school and elementary school levels and the rate of change decrease as s/ he gets older.
Verbal proficiency and ability to understand instruction can be improved by using different kinds of instructional material and techniques and by giving appropriate attention to individual students.
• Individualized instructional materials are very helpful for a student, especially for those who cannot grasp the concepts, ideas or a procedure in the textbook form. Orderly arranged units, instant feedback and reinforcement make learning easier. Individualized instructional materials may be used in the initial instructions or may be when learners come across difficulties during learning process.
• Small group study sessions, consisting of three, four or five students are very helpful for learners to overcome their learning difficulties in a cooperative manner. It gives freedom to each member to expose his difficulties and have them corrected in a dignified way without demeaning any one.
• Peer tutoring sessions, consisting of two students of different abilities. Usually a less able student is paired with an able student. The able student provides help through alternative explanations and applications to a less able student. This also strengthens the learning of the able students as he helps other to grasp an idea.
• Tutorial help is provided by someone other than the teacher. Tutor brings a fresh point of view about a concept or idea and deals learning difficulties of a student in different ways.
• Alternative books are provided by the teacher if students feel problems in understanding instruction of text books. Teacher determines whether alternative book explanations are more effective at that point or not.
• Audio-Visual methods such as filmstrips, slides, models, charts, etc. are also very effective that can be used by individual students who need concrete illustrations and vivid graphic explanations. Concrete experiences in the laboratory, academic games, puzzles and other attractive devices may be useful to improve the quality of instruction
• Workbook and Worksheets can provide drill and practice on particular tasks.
IV. Perseverance
Perseverance is defined by Carroll (1963) as the time a learner is willing or motivated to spend in learning. If the time provided to learn a particular task is less than the required time, the learner can’t achieve mastery level. Generally, learners’ attitude and interest have effects on perseverance. It is obvious that learners vary in their attitude and interest to learn a specific task and thus vary in the amount of perseverance for specific task. However, three factors contribute the amount of time needed.
• Quality of instruction, which includes organization and presentation of instructional material.
• Ability to understand instruction, which means how well the learner understands the language and material of instruction. It also includes the prerequisite skills for learning that task.
• Aptitude, which is defined simply as the time required by an individual to learn a particular task or skill to some pre-established level.
According to Varughese (2002) student's perseverance can be increased by the frequency of reward and evidence of success in a learning situation. As student achieves mastery of a given task, his perseverance is likely to increase. However, by improving the quality of instruction, the need for perseverance can be decreased to a great extent.
V. Time available for learning
Different students need different amount of time for learning. In the school, those who get appropriate time for learning, master the task smoothly and move forward while others stumble and accommodate difficulties as they proceed. Time spent on learning is the key element of mastery learning. According to Carroll (1963) the degree of learning is the function of time that a student spent in learning relative to that he requires to spend. If students are given the required amount of time for a particular task most of them will achieve mastery. The time required by the student is likely to be affected by his aptitudes, his verbal ability, the quality of instruction and help received inside and outside the class. The time required for mastery is different from individual to individual. In the group instructions these differences can be accommodated to a considerable extent by improving the quality of instruction and by the proper utilization of allotted time. The task of mastery learning strategy is to find ways of altering the time and ways of providing whatever time is needed by each student (Varughese, 2002).
2.12 ADVANTAGES OF MASTERY LEARNING STRATEGY
The application of Mastery Learning Strategy bears many advantages for teacher and students. Following are the some of the advantages of Mastery Learning Strategy:
• Students achieve prerequisite skills to proceed to the next unit.
• Teachers are well prepared as they do task analysis.
• Students are informed about objective before assigning activities.
• Reduce achievement gaps among students of the same class.
• Especially helpful for disadvantaged students.
2.13 DISADVANTAGES OF MASTERY LEARNING STRATEGY
Some disadvantages of Mastery Learning Strategy have also been drawn by the implementers of this plan.
• As all students require different time for learning so those who need less time for mastery have to wait for those who have not or to individualize instructions.
• This approach requires a variety of materials for re-teaching.
• Requires administration of several tests for each unit.
• Only objective tests are not sufficient for achieving objective of mastery learning, rather, a variety of tests are required in this approach.
2.14 RESEARCH STUDIES ABOUT MASTERY LEARNING
Over the years many research studies based on mastery learning have been conducted ranging in population from elementary school level to university level and in some cases using educational technology. Some research has been reviewed in the following lines.
1) Ejodamen and Raymond (2018) conducted a study entitled the “effect of mastery learning strategy on rural and urban students’ academic achievement in basic technology in Edo State, Nigeria”. A sample size of 155 (72 rural & 83 urban) drawn from 25 junior secondary schools of Edo State Nigeria. The design of the study was Quasi-experimental.
The study determined;
1. The pre-test and post-test mean achievement scores of rural and urban students taught through direct instruction strategy and those taught using mastery learning strategy.
2. The difference in the post-test mean achievement scores of rural students taught direct instruction strategy and those taught using mastery learning strategy.
3. The difference in the post-test mean achievement scores of urban students taught direct instruction strategy and those taught using mastery learning strategy.
4. The difference in the post-test mean achievement scores of rural and urban students taught Basic Technology Education through mastery learning strategy.
5. The interaction effect of students’ school location and instructional strategy on academic achievement of students.
Instrument used for data collection was Basic Technology Achievement Test (BTEAT) consisted of 50 multiple choice objective items.
The test items were drawn from the selected topics of the subject. BTEAT was used as pre-test before starting treatment and then at the end of treatment as post-test.
The Data was analyzed through Mean, t-test and Analysis of Covariance (ANCOVA). The result declared an enhanced academic achievement of rural and urban students taught using mastery learning strategy as compare to the students taught through direct instruction strategy. A significant difference was also found in the post-test academic achievement mean scores of rural and urban students taught through mastery learning strategy. The academic performance of the rural students was relatively higher than the urban students. The result declared that school location and instructional strategy have significant interaction effects on academic achievement of the students.
2) Hussain (2016) conducted a study on the “effect of Bloom’s mastery learning approach on students’ academic achievement in English at secondary level”. The core objective of the study was to investigate the effect of Bloom’s mastery learning approach on the academic achievement of 9th grade students in the subject of English. Other objectives were:
• To examine the effect of this approach on the academic achievement of students in different level of cognitive domain, and
• To explore the effects of this approach on the retention power of students in the subject of English.
A sample of 40 students was selected randomly. The study was experimental, pre-test post-test equivalent group design. Data was collected through achievement test consisted of 50 multiple choice questions.
Data was analyzed through mean, standard deviation, percentage and t-test. On the basis of analysis the researcher concluded that as compare to traditional approach, mastery learning is more successful and effective in the all six levels of cognitive domain and its effects on the retention power of students is greater than the traditional approach.
The researcher recommended that secondary school teacher should adopt mastery learning approach for teaching the subject of English and such studies should be conducted at different school levels in different subjects.
3) Lamidi, Oyelekan and Olorundare (2015) studied the effects of mastery learning strategy on students’ achievement in the concept of mole in the subject of chemistry. The research questions pursued in that research study were:
1. Is there any difference in the achievement of students exposed to mastery learning strategy and those not exposed to that strategy?
2. Is there any influence of gender on the achievement of students exposed to mastery learning instructional strategy?
3. Is there any influence of scoring level on the students’ achievement exposed to the mastery learning instructional strategy?
This design of the study was quasi-experimental. The pre-test and post-test control group design was used to test the hypotheses. Sample size was 110 (male and female). Intact classes were used for experiment. The instruments used for this study were:
1. Achievement Test on Mole Concept (ATMC), and
2. A researcher-designed mastery learning instructional material.
Comparison of achievements was made through paired samples t-test, and Analysis of Covariance (ANCOVA) was conducted to see the influence of gender and scoring level on students’ achievement.
After data analysis a significant difference was found in the achievement of students in favor of the experimental group. In respect of the second research question, no significant difference was found in the achievement of male and female students. In respect of third question it was found that low and medium scoring students score was improved.
Based on the findings, the researcher recommended that chemistry teachers should use mastery learning instructional strategy for teaching chemistry, school authorities should also help to create conducive environment to applying this strategy, and other educational bodies should organize training and workshops on mastery learning instructional strategy.
4) Yemi (2015) conducted a research study entitled, “Mastery Learning Approach (MLA): its effect on the students’ mathematics academic achievement”. It was quasi experimental study and the design adopted was Pre-test post test non-equivalent group design. Eighty subjects participated in this study. The hypotheses of the study were:
1. There is no significant difference in the pre-test and posttest mean scores of students exposed to mastery learning approach and those expose to traditional teaching method.
2. There is no significant relationship between the students’ attitudes towards the subject of mathematics and academic achievement in mathematics.
The instrument used for data collection was mathematics achievement test consisted of 30 items and Students Attitudinal Inventory (SAI) consisted of 16 items. Data was analyzed by ANCOVA. The results declared that mastery learning approach have significant effect on the academic achievement of students. Moreover, significant relationship was also found between the students’ attitudes toward mathematics and their academic achievement in mathematics. The researcher concluded that mastery learning approach is useful to enhance student achievement in the subject of mathematics.
5) Furo (2014) conducted a study on the effect of mastery learning approach on secondary school student’s achievement in chemistry in Rivers State Nigeria. The core objective was to compare the effect of mastery learning approach with conventional method in the subject of chemistry. The other objectives were:
1. To examine the effect of mastery learning approach on student achievement in the subject of chemistry.
2. To examine the effect of mastery learning approach on the academic achievement of the students of urban and rural areas.
3. To examine the effects of mastery learning approach on the academic achievement of male and female students of rural schools.
Sample was selected randomly. The sample size was 160. The design of the study was quasi-experimental.
The instrument used by the researcher for data collection was chemistry achievement test (CAT). Data was analyzed through ANCOVA. The results achieved after data analysis was in favor of mastery learning approach. The academic achievements of experimental groups were higher than conventional groups. The differences in the academic achievement of the students of urban and rural areas were found minimum. Moreover, no significant difference was found in the academic achievement of male and female students.
6) Udo and Udofia (2014) studied the effects of mastery learning strategy on students’ achievement in symbols, formulae and equations in chemistry. The objectives of the study were:
• To examine the effect of mastery learning on students’ achievement regarding the concepts of symbols, formulae and equations in the subject of chemistry.
• To compare the achievement scores of male and female students.
• To investigate whether mastery learning has any effect on the achievement of students having different interests.
• To see the interaction effects between independent (teaching method) and dependent variables (gender and interest).
• To observe the interaction effects between method of teaching and students' gender and their interest.
It was a quasi-experimental research study with pre-test post-test non-equivalent control group design. The target population was the students of grade 11th from fifteen public schools having co-education system. From among 15 schools, four intact classes from four schools have been selected as sample for the study.
The data collection tools were “Achievement Test” on chemistry and “Students’ Interest-scale”, developed by the researcher. The Achievement Test in the subject of chemistry consisted of 25 multiple choice questions, and Interest-scale consisted of 20 Likert scale items, developed to assess students’ interest in the subject of chemistry. Data was analyzed by ANCOVA. The findings declared mastery learning an effective teaching strategy as the performance of the students of the experimental group were significantly better than the students taught through conventional expository method. These findings further declared a better performance by male and female students of the experimental group. However, within experimental group the performance of male students was better than female students. Other findings of the study indicated high performance by those with high interest in both experimental and control groups. It was also found that the effects of teaching method were different at all levels of gender and was same at the students’ interest level.
The researcher concluded that teaching strategies are significant predictors of students' performance and he recommended mastery learning strategy for the teaching of chemistry.
7) A research study entitled, “effect of Mastery Learning Approach (MLA) on students’ achievement in Physics” was conducted by Adeyemo and Babajide (2014). This study was conducted in Lagos State. The design of the study was a non-randomized pre-test post-test control group design. One hundred and sixty participants of age range 14-18 were selected through stratified random sampling.
The objectives of the study were:
1. To compare the achievement of students’ exposed to mastery learning approach with those exposed to conventional teaching methods.
2. To determine relationship between students’ attitudes towards the subject of physics and their achievement.
The instruments used for data collection were:
1. Physics Achievement Test (PAT) consisted of 50 multiple choice questions.
2. A questionnaire consisted of 20 items.
The data was analyzed through t-test, Pearson Correlation and Analysis of Variance (ANOVA). The results declared that the performed of the experimental group was better than control group. It was also found that the students with positive attitudes towards the subject of Physics performed better than those with negative attitudes towards the subject.
8) Goreysh, Karger, Noohi, and Ajilchi (2013) conducted a study on the effect of combined mastery-cooperative learning on emotional intelligence, self-esteem and academic achievement in grade skipping. The Study was conducted in Tehran on first grade female students, randomly selected from among those students who, according to Ministry of Education, were suitable for grade skipping. The design of the study was qausi-experimental.
The core objective of the study was to examine the effects of two methods (Combined Mastery-cooperative Learning) in grad skipping. The other objectives were to investigate the effects of Combined Mastery-cooperative Learning on academic achievement, self-esteem and emotional intelligence of the students.
For assessing emotional intelligence and self esteem the researcher used “Emotional Intelligence” questionnaires of Bar-On (1997) and “Self-esteem” questionnaires of Coopersmith (1967).
The experimental period was 45 days with 11 hours per day. The data obtained was analyzed through one way ANOVA, and p-values less than 0.05 were considered statistically significant.
The researcher found that Combined Mastery- cooperative Learning method was resulted in a significant increase in self esteem and emotional intelligence, but in academic achievement no significant change was observed. The researcher further explained that no change in the academic achievement was due to the difficult course which was accompanied with grad skipping.
The results of the study also declared Mastery Learning combined with Cooperative Learning an effective method to promote mental health by creating a positive dependence, unity and cooperation among members of the group and developed healthier personalities. The researcher further added that this method helped students in self-awareness and empowered them to realize their own and other people’s needs and demands. This method also made students accountable for their actions and objectives by creating a sense of responsibility in them.
9) Sood (2013) studied the effect of mastery learning strategies on concept attainment in geometry among high school students. The objectives of the study were to compare Bloom’s Learning for Mastery (LFM) and Keller’s Personalized System of Instruction (PSI) with Conventional Method and to compare Bloom’s LFM and Keller’s PSI on concept attainment in geometry among secondary school students. For getting the objectives, a random sample of 105 students of 9th class was selected. They were then divided into three homogeneous groups. The first group was taught through Bloom’s LFM strategy, the second group was taught through Keller’s PSI and the third group was exposed to conventional method. The tool for data collection was self-developed “concept attainment test” in geometry. ANCOVA was used for data analysis.
The researcher found that both Bloom’s LFM and Keller’s PSI were significantly more effective as compared to conventional method of teaching in the attainment of geometrical concepts. By comparing Bloom’s LFM and Keller’s PSI, the researcher found Bloom’s LFM was significantly superior in attainment of the geometrical concept at secondary school level.
10) Abakpa & Iji (2013) studied the effect of mastery learning on students’ achievement in geometry in senior secondary school. The objectives of the study were to examine the effect of mastery learning on:
• Students’ achievement in geometry.
• Students’ achievement in geometry according to their ability.
• Students’ achievement in geometry according to gender.
• To investigate the interaction effect of teaching method on students’ ability.
• To investigate the interaction effect of teaching method on gender.
A quasi experimental non-equivalent control group design was used in that study. The Sample size was 270 selected from 26 senior secondary schools by multi stage sampling technique. Data was collected through Geometry Achievement Test. Geometry Achievement Test consisted of 50 items (40 multiple choice and 10 essay type questions). Data was analyzed through ANCOVA. Findings of study revealed high academic achievement by experimental group. It was also found that mastery learning improved the academic achievement scores of both male and female student and thus minimized gender differences. Other findings of the study declared that mastery learning effected low ability and high ability students equally and thus reduced achievement gaps.
11) Ezinwanyi (2013) used mastery learning approach for enhancing mathematics achievement of secondary school students.
The core objective of study was to examine the effect of Mastery Learning Approach on students’ achievement in mathematics at on Secondary School. The other objectives were:
• To compare the achievement scores of the mastery learning group with the collaborative mastery learning group.
• To compare the achievement scores of low and high ability students in the mastery learning group.
It was a quasi-experimental, pretest, post-test non-equivalent control group design. Three secondary schools were purposively selected. The whole class in each school was taken as a group. The sample size was 150. All the three groups were exposed to different teaching method. Group 1 was exposed to mastery learning approach, group 2 was exposed to collaborative mastery learning and group 3 was taught through conventional method. Experimental period was six weeks.
Data gathering tools were Mathematics Achievement Tests. Pre-test consisted of 55 items (50 multiple choice questions and 5 essay questions). Post-test consisted of 50 items (40 multiple choice questions and 10 essay type questions). Both pre-test and post-test were validated by mathematics experts. Data was analyzed by ANCOVA.
The findings of the study indicated that mastery learning improved students’ achievement in Mathematics. The findings also showed that mastery learning approach filled the gap between low ability and high ability students. However, no significant difference was noticed in the mean score of students in the two experimental groups.
The following recommendations were given by researcher:
• Teachers of mathematics should be encouraged to use mastery learning method in their class to promote students learning and improve students’ retention power of mathematics.
• Educational planners should plan additional instruction for the slow learners to master a concept and enrichment activities for the fast learners to keep them busy.
• Government should motivate and encourage teachers to ensure mastery of the concepts by all the students.
• Qualified teachers should be recruited to deal with the increasing population of secondary schools.
12) Moradi, Zarei and Zainalipour (2012) studied the Effects of Mastery Learning Method on academic self-concept, academic achievement, and achievement motivation. The target population for the study was grade 3 students of middle schools in Bandar Abbas city (Hormozgan Province) in academic year 2012. Through accessibility sampling 25 female students of middle school studying in grade III were selected. A semi-experimental design was employed in this study. The researcher used three types of instruments for data collection, these were:
1. Academic achievement test – it was used as before and after experiment for measuring academic performance of the students.
2. Academic self-concept questionnaire, a questionnaire consisted of 28 items prepared by Delavar (1993) to measure self-concept.
3. Achievement Motivation Questionnaire consisted of 29 statements, prepared by Hermens in 1970.
For data analysis, descriptive and inferential statistics were used. For calculating mean, standard deviation, and variance descriptive statistics were used. Inferential statistics were used for single and multiple factor co-variance analysis based on the hypotheses. Findings of study showed that achievement motivation, self-concept, and academic performance of the experimental group were considerably increased.
13) Nonye and Mgbemena (2012) studied the effect of mastery learning approach on the academic achievement of senior secondary school students. The study was conducted in Ogidi Education State, Nigeria. The design of the study was quasi-experimental pre-test, post-test design. The Sample size was 40. Twenty students were exposed to mastery learning approach and 20 students were taught through lecture method. The tool for data collection was Physics Achievement Test (PAT). The data collected through PAT was analyzed through mean, standard deviation and Z-test. The result revealed that the achievement of experimental group significantly better than the control group. The result also revealed that the achievement of female students was somewhat better than their male counterparts, but the difference was not significant at 0.05 level. The researcher concluded that mastery learning enhances students’ learning and he recommended that mastery learning should be encouraged in schools by teachers in place of the lecture method.
14) Shafie, Shahdan and Liew (2010) conducted a study titled “Mastery Learning Assessment Model (MLAM) in teaching and learning mathematics”. The Study was conducted in Malaysia on a private university student. The purpose of the study was to examine the effectiveness of MLAM in teaching and learning of mathematics. MLAM was based on repeating similar assessments through a mastery model. The sample was consisted of 30 students of bachelor class. Conventional and mastery learning procedure were introduced to the same students. The students were considered to master the specific unit when they scored 80% in assessment and those who scored less than 80% were given repeated assessment until they reached the specified criterion. Analyses were done by establishing a correlation between the scores of MLAM and the final exam. The findings declared a positive correlation between the two scores with a correlation value of 0.77. It was found that 90% of the students who have exposed to MLAM had successfully completed their course work and among them 70 % finally secured “A” grade and above in their final exam. However, 10% of the students who have gone through MLAM failed to get passing marks. The researcher argues that it was due to their negative and non-serious attitude towards the subject.
15) Damavandi and Kashani (2010) investigated the effect of mastery learning method on performance and attitude of the weak students of high school in the subject of chemistry. The researcher compared the effects of mastery learning method and common method on the performance and attitude of week students in high school.
It was an experimental study. From among 16 year old student 40 week students were selected, on the basis of placement test about pre-requisite concept of the subject. 40 students were then divided into two groups of 20. One group was exposed to experimental condition and the other was taught through common method.
The instruments used for data collection were:
1. Colorado Learning Attitudes: a questionnaire designed for measuring attitude of students toward the subject of chemistry.
2. Academic Achievement Test: it was designed by the researcher to measure performance of students in chemistry.
Data was analyzed through multivariate analysis of variance (MANOVA). Covariance-variance homogeneity test and Levine tests were executed on data. Unilateral variance analysis was used separately for testing the difference of groups in each one of the dependent variables.
The findings of the study indicated better performance and positive change in attitude of the week students exposed to the mastery learning method. The positive effect on students' attitude was found in some dimensions such as endeavor to understand concepts, application of knowledge, and relationship of knowledge to the real world. Mastery learning method also increased interest of the weak students in the subject of chemistry.
16) Davrajoo, Tarmizi, Nawawi and Hassan (2010) used mastery learning approach for enhancing algebraic conceptual knowledge. The objectives of the study were to examine the effectiveness of Algebraic Mastery Learning Module (AMaLM) in improving learners’ performance and to examine the relationship between anxiety and performance of low achiever in mathematics. The design of the study was quasi experimental. The sample consisted of 50 low achiever students of grade 11th from four schools.
The researcher used following instrument:
1) Diagnosis Module: consisted of two diagnostic tests, i.e. Algebraic Comprehension Test (ACT-1) and (ACT-II), to diagnose students’ knowledge about the application and understanding of algebra.
2) RMARS: a set of questionnaires specially developed to measure mathematics anxiety and attitude towards the subject of mathematics.
3) AMaLM: a tutoring module, specially developed, by taking in account learners’ differences, for the purpose of mastering algebraic concepts.
Data was analyzed through t-test and it was found that the experimental group showed statistically higher mathematics performance than the control group. The result also declared that low achievers were positively affected by the AMaLM. The researcher concluded that the average students will perform better when taught through AMaLM. Furthermore, if in general, students use AMaLM, they would show more improved achievement.
17) Broderick (2009) conducted a study with the title “mastery learning and its effect in ASSISTassistment”. The main objective of this study was to examine the effect of introducing mastery learning into the intelligent tutoring system ASSISTment on students’ learning. ASSISTment is online intelligent tutoring system developed by Professor Neil Heffernan and his team. ASSISTment , according to researcher, was the combination of “Assist” and “Assessment”. The other objectives of the study were to measure students’ self-discipline and proficiency. The instruments which were developed for measurement self-discipline and proficiency were GRIT and IRT.
• GRIT was a set of questions developed to measure student’s self-discipline. The students were required to reflect on and evaluate their self-control while answering the questions.
• IRT was data based ASSESTment that was updated periodically. It was consisted of mathematical problem developed to measure students’ proficiency in mathematics.
The 8th grade students from a local middle school were selected for the experiment. The experiment was performed in math lab where students worked on math problems daily. Pre-tests and post-tests on several different math units were administered over one month period. In between those tests, the students work on assignments. Experimental groups were given mastery learning assignments while control groups worked on their ordinary problem set. Students’ learning was measured from performance gains between pre and post-tests.
The score was calculated for each unit by taking difference of pre-test and post-test. Scores were then converted into Z-scores. Than average score for each student was calculated. The data was then analyzed through ANOVA.
Finding showed that the experimental groups learned significantly more than their counterparts in the control groups. Further, it was also noted that students with low self-discipline and low proficiency profited much more from Mastery Learning than their peers.
18) Wambugu and Changeiywo (2008) investigated the effects of mastery learning on students’ achievement in the subject of physics. The study has been conducted in Kenya. The specific objective was to compare mastery learning and regular teaching method on the basis of physics achievement of secondary school students. The single null hypothesis of the study was that there is no significant difference in the achievement of students in the subject of physics who were exposed to mastery learning approach and those who were exposed regular teaching method.
The design of the study quasi-experimental solomon’s four non-equivalent group design. The population of the study was secondary school students. Four schools having co-education were selected through purposive sampling. Each school was considered as a unit of sampling. The intact classes were taken as groups, two experimental and two control groups. The sample size was 161.
The instrument used for data collection was Physics Achievement Test (PAT). PAT was obtained from national examination council of Kenya, it was modified, and given to experts for validation. Its reliability was also tested in pilot study.
Teachers were trained to teach the experimental groups according to the manual, provided to them. Pre-test was given to one experimental and one control group in two schools, while post test is administered to all four groups in four schools. Treatment period was three weeks. Data was analyzed through t-test and ANCOVA. The result showed that students exposed to the mastery learning method achieved high scores as compared to those who were not exposed to that. The researcher concluded that mastery learning method, if implemented in four years of secondary education, will improve the performance of the students in physics.
19) Shanbhag (2007) conducted a study entitled “the effectiveness of mastery learning instructional strategy in attainment of competencies in mathematics”. Researcher held that learning activities should be based on psychological principles, and students’ characteristics should be considered properly, to achieve the required competency in the subject of mathematics. The objectives of the study were:
• To develop a mastery learning model based on concrete to abstract continuum.
• To make teaching instructions valid in term of content accuracy, language comprehension, and organization of material.
• To examine the effectiveness of mastery learning model on the learners’ liking towards the subject.
The true experimental design was used in this study. Pre-requisite skills were measured through criterion referenced achievement test and liking towards the subject of mathematics was measured through interviews, especially organized for the research objective. For data analysis the researcher used chi square test and z-test. The findings of the study showed that students taught through mastery learning model mastered all 15 competencies that researcher intended to achieve and students’ liking toward the subject was also increased as mastery learning model used play and participation of students in all activities.
20) A study conducted by Adeyemi (2007) to investigate the effectiveness of mastery learning approach in learning social studies established the superiority of mastery learning approach over conventional method. In this study the sample was selected purposively, consisted of 60 subjects. 60 students were then randomly divided into experimental and control groups. The design of the study was post-test only control group design. Data was collected through descriptive test from both groups and analyzed through t-test. The results of the study showed that teaching methods are very effective tools in presenting facts and ideas as the university students of the experimental group had a marginal advantage over the control group. The recommendations given by researcher were: a) teacher should be inventive in instructional plans in order to bring behavioral changes in learners, b) teacher should be trained properly and c) curriculum should be organizes according to the mastery learning g plane.
2.15 WHY THIS STUDY
Hundreds of studies have been conducted on the topic of Mastery Learning and almost every study has concluded positive effects of Mastery Learning on students’ different skills. According to David Mercer (1986) “Mastery learning has gone beyond the stage of debate and has been accepted by almost all educationists as a genuine approach in teaching learning process”, but in Pakistan we hardly find two or three studies on this topic. Therefore, researcher decided to introduce Mastery Learning in this country at secondary school level, the most important level of education that serve as base for higher education.
Mathematics is the most important subject as discussed in start of this chapter, but most of the students think it impossible to learn. Ellerton and Celemonts (1989) rightly said, “we believed the main lesson learned by most school leavers after year of being forced to study mathematics is that they can’t do it”. The most failure has been noticed in secondary school mathematics results, especially in public schools of KP. So researcher decided to apply Mastery Learning Strategy in the subject of mathematics at secondary school level in district Mardan of KP.
The study in hand was conducted in urban and rural areas of district Mardan with the aim to investigate the effects of Mastery Learning Strategy on students’ academic achievement, learning retention, attitude towards the subject of mathematics and achievement variations in these cultures of KP, Pakistan.
Almost all the studies conducted on Mastery Learning approach have used small groups of students as sample. The present study has selected the whole group of students studying in 9th grade in the selected public schools of district Mardan. The prominent feature of those students/participants was diversity in socio-economic status of their families.
CHAPTER – 3
RESEARCH METHODOLOGY
This chapter includes the detailed description of the methodology that the researcher adopted during the research process.
3.1 NATURE OF THE STUDY
The study was experimental in nature. The aim of the study was to investigate the effects of Mastery Learning Strategy on students’ academic achievements by comparing it with Conventional Teaching Method. The experiment was conducted in urban and rural areas of district Mardan of KP. The researcher selected the subject of mathematics for teaching. Further, two chapters of algebra from grade 9th mathematics were selected for this study.
3.2 POPULATION OF THE STUDY
The aim of the study was to investigate the effects of Mastery Learning Strategy on students’ achievements at secondary school level, so the target population of the study comprised all secondary school students of Government High Schools (GHS) and Government Higher Secondary Schools (GHSS) of district Mardan. In district Mardan total GHSS are 48 (Boys: 26, & Girls 22) and total GHS are 152 (Boys: 85 Girls: 67) (EMIS, 2016). In the urban areas the number of GHS is 23 (Boys: 16 & Girls: 7) and GHSS is 11 (Boys: 6 & Girls: 5). In the rural areas the number of GHS is 129 (Boys: 69 & Girls: 60) and GHSS is 37 (Boys: 20 & Girls: 17) (EMIS, 2016).
3.3 SAMPLE/PARTICIPANTS OF THE STUDY
Keeping in view the experimental nature of the study sample was selected through purposive sampling. Patton (1990) suggests purposive sampling when information rich cases or a wide range of variations is required. Author further adds that it facilitates comparison and allows logical generalization and maximum application of the information. Therefore, two schools of district Mardan namely, Govt. Girls High School No.1 Mardan and Govt. Girls Higher Secondary School Shehbaz Garhi, were selected. As the students of these schools had maximum variation in characteristics, therefore they were suitable for this study.
The students of Govt. Girls high School No.1 Mardan are mostly from Mardan city (urban area). They belong to middle and lower middle families. Some have educated parents. Their parents mostly earn their living through government jobs and small business.
The students of the Govt. Girls Higher Secondary School Shehbaz Garhi come from adjacent rural areas. They belong to the lower middle and poor families. Their parents are mostly uneducated. Their socioeconomic conditions are lower as compared to the students of GGHS, No.1 Mardan.
Two sections of 9th grade from each school constituted the sample of the study. The total sample size was 214 students. Usually administrative authority of school does not feel happy when a group of students is selected from a class for experimental purpose. They think it disturbance in class arrangement and teaching. So the whole sections were taken as groups for this study.
Table 3.1
Sample Size in GGHSS, Shehbaz Garhi Mardan and GGHS, NO: 1 Mardan
|GGHSS, Shehbaz Garhi Mardan |GGHS, NO:1 Mardan |
|Experimental |Control |Total |Experimental |Control |Total |
|Group |Group | |Group |Group | |
|59 |59 |118 |48 |48 |96 |
Table 3.1 shows the number of participants for the study. From GGHSS, Shehbaz Garhi School 118 students and from GGHS NO.1 Mardan 96 students participated in this study. GGHSS, Shehbaz Garhi had only two sections of grade 9th. Both were selected for the study. In GGHS NO.1 Mardan, there were 5 sections of grade 9th, two science and three sections of humanity group. From this school two science sections were selected as sample.
3.4 SELECTION OF TEACHERS
One Subject Specialist (SS) in the subject of mathematics was selected at GGHSS, Shehbaz Garhi to teach both groups in separate periods and One Secondary School Teacher (SST) was selected to teach both groups in GGHS, No.1 Mardan. These teachers were selected on the following basis:
• Both teachers were regular staff members in their respective schools.
• Both were qualified mathematics teachers.
• Both have professional qualifications of B.Ed and M.Ed and have attended refresher mathematics courses during their services.
• Both have more than 5 years experience of teaching mathematics to secondary classes.
• Both were willing to cooperate in this experimental study.
3.5 INSTRUCTIONS FOR TEACHERS
Selected teachers were academically and professionally qualified for teaching the subject of mathematic at secondary school level, but even then they were given a short training by the instructor at Regional Institute of Teacher Education (RITE). They were given full instructions about Mastery Learning Strategy (MLS) and Conventional Teaching Method. Lesson plans were thoroughly discussed with them. Both teachers were instructed to teach experimental groups through Mastery Learning and control groups through conventional method in their respective schools. Timetable for experimental groups was arranged with the permission of the principals of the respective schools. As mastery learning work on the philosophy of “take your time”, so 12 periods per week for experimental groups and 8 periods per week for control groups, for teaching the subject of mathematics, were arranged. Experimental period was two months and 20 days.
3.6 DEVELOPMENT OF LESSON PLANS
Lessons were planned by the researcher in collaboration with the SS mathematic and then were thoroughly discussed with mathematics’ master trainer in Provincial Institute of Teacher Education (PITE), and correction and omission was done. Four specimen lesson plans for experimental groups are given in appendices (Appendix-E; appendix-F; appendix-G; and appendix-H). Each lesson plan consisted of the following parts:
1. Introduction – consisted of information about subject, duration of period, methods to be adopted, and topic to be taught.
2. Objectives to be achieved.
3. Previous knowledge and its relation to the new topic.
4. Presentation and explanation by the teacher.
5. Students’ activities - cooperative learning, peer tutoring, individual exercise etc.
6. Formative assessment.
7. Corrective and enrichment activities.
8. Assessment/Closure
9. Home work.
The lessons for control groups were also planned by SS mathematics and researcher following traditional lesson planning steps. Two specimen lesson plans for conventional groups are given in appendices (Appendix-I & appendix-J). Each lesson plan consisted of the following parts:
1. Introduction – consisted of information about subject, duration of period, methods to be adopted, and topic to be taught.
2. Objectives to be achieved.
3. Presentation and explanation by the teacher.
4. Students’ activities ( imitation and exercise)
5. Home work
3.7 TEACHING STRATEGIES
Teaching strategies play a vital role in the teaching-learning process. In the absence of appropriate strategies we can’t expect to achieve educational objectives. In the present study explicit instruction, peer tutoring & cooperative learning strategies were used for the experimental groups. While control groups were taught through conventional method.
3.7.1 Explicit instruction
Explicit teaching is systematic and sequential. It involves teaching a skill or concept in a controlled environment using direct language. This type of instructions is focused on producing specific learning outcomes. Teacher clearly shows students, what to do and how to do it. In addition, the teacher checks for understanding of students, and at the end of each lesson revisits what the lesson has covered and ties it all together (Hattie, 2009). In the present study the following steps of explicit instruction, as evident in lesson plans were followed during the instructions in class room.
• Review relevant pervious learning and skills by asking few questions.
• Clear statement of goal and step-by-step presentation of new material
• Guided practice to ensure high rate of success and timely feedback
• Corrective and re-teaching when necessary
• Independent practice
• Assessment/Closure
3.7.2 Peer tutoring
In peer tutoring, two students work together on an instructional activity. The pairs of students can be of the same or different ability level. In the present study, students having high performance were paired with a lower performing student to review critical concepts and share knowledge, as evident in the lesson plans for experimental groups. Sometimes same ability students were also paired to extend their knowledge by sharing their learning experiences and ideas.
3.7.3 Cooperative learning
Cooperative learning is a teaching method that uses small, heterogeneous groups of mixed-ability. It is one of the recommended approaches that aim to achieve learning objectives by assisting each other in a group. In the present study this technique was used (as evident in lesson plans for experimental groups) to maximize the learning of each member of the group. Teachers gave a problem to the students, and provided them with opportunities to discuss, practice or review mathematical concepts.
3.7.4 Conventional Method
In conventional method teacher is at the center of all activities, teaches all students in the same manner and transmits knowledge that is prescribed in text book. Students passively receive information either by listening or taking notes. Mathematics is learned mainly by memorizing rules and procedures for solution of textbook problems.
In the present study the teacher provided clear, step-by-step demonstrations, as evident in lesson plan for control groups (Appendix-I & appendix-J), and students were provided with proper time to take notes and practice the procedure.
3.8 DESIGN OF THE STUDY
This was an experimental study as the major objective of the study was to investigate the effects of Mastery Learning Strategy by comparing it with Conventional Teaching Method, and experimental method is the logical and systemic method to test hypothesis. Pre-test, post-test non-equivalent group design (Quasi-Experimental design) was used. According to Best (1986) and Farooq (2001), this design is most excellent for classroom experiments when two groups are such naturally assembled as intact classes, that may be similar. Singh (2007) argues that quasi experiments are comparatively good in terms of internal validity.
Symbolic representation of the design is given in the figure below:
|O1 |X |O2 |
|O3 |C |O4 |
Figure 3.1: Pre-test post-test non-equivalent group design
(Source: Best, 1986)
O1 and O3 = Pre-test
O2 and O4 = Post-test
3.9 VARIABLES
Different kinds of variables involved in this study were:
a. Independent Variables
The two teaching strategies; mastery learning and conventional teaching were independent variables in this study.
b. Dependent Variables
Students’ achievements, learning retention, attitude towards mathematics and achievement gaps were the dependent variables.
c. Controlled Variables
Course content (selected chapters of grade 9th mathematics), students’ level (9th grade), physical facilities, one teacher for both experimental and control groups, sample size, average age of the sample, gender variation, methodological variation etc. were controlled variables.
d. Uncontrolled Variables
Many variables have hard effects on dependent variables, but could not be controlled due to many restrictions. In the present study these variables were: absence of some students during the experiment, anxiety, motivation, and interest of the students, socioeconomic conditions, parental attitude and home environment, previous effects of teaching in mathematics and present methods of teaching in other subjects.
3.10 RESEARCH TOOL
The following tools were used for data collection during this study:
1. Pre-test
2. Post-test-1
3. Post-test-2
4. Retention test
5. Attitude scale
3.11 RESEARCH TOOLS PREPARATION AND VALIDATION
3.11.1 Pre-test
A pre-test (appendix-A) was constructed in the algebra units of mathematics of grade 8th. As the name indicates, it was administered before treatment. The purpose of the test was to determine the cognitive entry behavior of the students and to address non-equivalency problems.
3.11.2 Validity and reliability of Pre-test
The pre-test was prepared by SS-mathematics. It was then critically analyzed by the mathematics’ experts (Appendix-N) in PGC, SSTs in GGHS, SS in GGHSS and trainers in Regional Institute of Teacher Education (RITE) and Provincial Institute of Teacher Education (PITE). After thorough discussion with experts on test item, 12 items were removed and 2 new items were included. It was then administered to 25 students of class 9th (Session 2013) of GGHSS, Shahdand. Based on pilot test results 5 more items were removed and 3 new items were included by experts. The test-retest method was applied to check reliability of the pre-test. The reliability coefficient of the pre-test was found 0.7.
3.11.3 Post-test-1
Post-test-1 (Appendix-B) was prepared by SS-mathematics in the first unit of grade 9th algebra. It was consisted of true/false questions; fill in the blanks; multiple choice questions and solution problems. It was administered to the students at the completion of first unite.
4. Post-test-2
Post-test-2 (Appendix-C) was prepared by SS-mathematics in the second unit of grade 9th algebra. It was also consisted of true/false questions; fill in the blanks; multiple choice questions and solution problems. It was administered to the students at the completion of second unite.
3.11.5 Validity and reliability of the post-tests
Two post-tests were prepared by SS in mathematics. These tests were validated through pilot testing and discussion with the mathematics experts (Appendix-N) in PGC, SSTs in GGHS, SS in GGHSS and trainers in PITE. The most difficult items were removed and some items were included according to experts’ suggestions. The test-retest method was applied to both post-tests to check reliability of the tests. For post-test-1 the reliability coefficient was 0.7 and for post-test-2 it was 0.8. Finally valid and reliable tests were approved.
3.11.6 Retention Test
A retention test (Appendix-D) was prepared by SS-mathematics. It was administered to both experimental and control groups after two months of the treatment. It was consisted of 5 true/false, 6 fill in the blanks, 14 multiple choice questions and 5 problem solving items. The total marks of the test were 50.
3.11.7 Validity and reliability of the retention test
Retention test was also prepared by the SS-mathematics. It was consisted of 30 items, drawn from post-test-1 and post-test-2. It was also approved after discussion with experts and pilot testing. The test-retest correlation coefficient of the two sets of measurements was 0.8.
3.1.8 Attitude Scale
To assess the change in attitude toward the subject of mathematics, an attitude scale Attitude Towards Mathematics Inventory (ATMI) prepared by Miss Martha Tapia and Sir George E. Marsh II (1996) was used (Appendix-K). It was given to all the students of experimental and control groups at the beginning and at the end of the treatment period. The sub-scales of ATMI are: self-confidence, value, enjoyment, and motivation.
Miss Martha Tapia is associate professor of mathematics education at Berry College, Georgia, where she teaches mathematics and mathematics education courses. Mr. George E. Marsh II is a professor of instructional technology in the institute of Interactive Technology at The University of Alabama. ATMI was received from Miss Marta Tapia by personal contact.
ATMI was translated into Urdu as the study was conducted in the government schools of KPK where medium of instruction is Urdu. It was first translated into Urdu by a language expert and then again translated into English by another language expert thus a correct translation of ATMI was obtained. Only item “9” was dropped in Urdu translation as it was translated differently by different experts.
3.11.9 Consistency, Reliability and Validity of the Attitude Scale
Internal consistency, reliability and validity of ATMI (Attitude Toward Mathematics Inventory) have been established by the inventors. This attitude scale (ATMI) originally consisted of 49 items measuring six domains of attitude, i.e. anxiety, enjoyment, value, motivation, confidence, and parent/teacher expectations, but after conducting an explanatory factor analysis using a sample of 545 high school students Tapia (1996) combined anxiety and confidence to form a single factor, self confidence. Parent/teacher expectation was dropped due to extremely low item-to-total correlation of this sub-scale (Tapia, 1996). The final scale comprised of four sub-scales. The sub-scales are: self-confidence (15 items), value (10 items), enjoyment (10 items), and motivation (5 items). Its internal consistency was calculated through Cronbach alpha coefficient, the score was 0.96. This showed a high degree of internal consistency. In order to increase the value of alpha 9 items out of 49 that had item-to-total correlation less than 0.50 were deleted. The alpha increased to 0.97. All 40 items had item-to-total correlation more than 0.50 suggested that all items of ATMI are measuring common trait (Tapia & Marsh, 2004).
Content validity of ATMI was established by relating the items to the variables: confidence, anxiety, value, enjoyment, and motivation. This organization was further explained by the four-factor model supporting different explanations for students’ self-confidence, enjoyment, value and motivation as basic dimensions of attitudes toward mathematics. There were 5 responses in front of each statement.
Validity and reliability have been established by applying it to 134 college students (Tapia & Marsh, 2002) and 545 high school students (Tapia & Marsh, 2004).
Ke (2008) tested ATMI with 160 students of fifth grade in a study of the application of cooperative, competitive, and individualistic goal structures in classroom use of computer mathematics games and found it reliable to assess the impact of his/her mathematics model on learner’s attitude.
Afrari (2012) examined the validity and reliability by applying ATMI on 269 (166 male & 103 females) middle grade students in United Arab Emirates (UAE) and concluded that ATMI can be used to determine the attitude of middle grade students with high reliability and validity.
Majeed, Darmawan and Lynch (2013) found ATMI as a viable scale to measure students’ attitude toward the subject of mathematics. Furthermore, they advocated that the validity and reliability estimates for ATMI are stable over the years after its first administration in 1996 and beyond the initial samples. The researchers further told that ATMI is predominantly useful for all teachers and researchers who want to monitor students’ attitude towards mathematics.
3.11.10 Distribution of items by sub-scales of Attitude Toward Mathematics Inventory (ATMI).
Total items of ATMI are 40 (appendix- K). 12 items are reversed in score. 15 items (9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 40) investigate self confidence; 10 items (1, 2, 4, 5, 6, 7, 8, 35, 36, 37, 39) examine value; 10 items (3, 24, 25, 26, 27, 29, 30, 31, 37,38) examine enjoyment; and 5 items (23, 28, 32, 33, 34) examine motivation of the students towards the subject of mathematics. According to Majeed, Darmawan and Lynch (2013) ATMI have a more distinct and cohesive factor structure than that of Fennema-Sherman Mathematics Attitudes Scales (1976). They relate it to the Neale’s (1969) definition of attitudes toward mathematics. Neale (1969) has an early contribution to the study of attitudes toward mathematics. Neale defined mathematical attitude as “a liking or disliking of mathematics, a tendency to engage in or avoid mathematical activity, a belief that one is good or bad at mathematics, and a belief that mathematics is useful or useless” (Neale, 1969.p. 632 as cited in Majeed, Darmawan & Lynch 2013).
A table showing the Relation of subscales of ATMI with Neale’s definition of mathematical attitude is given on next page.
Table 3.2
Relation of Subscales of ATMI with Neale’s definition of Mathematical Attitude
|Elements of Neale’s Definition |Sub-scales of ATMI |
|A belief that one is good or bad at mathematics |Self Confidence |
|A belief that mathematics is useful or useless |Value |
|A liking or disliking of mathematics |Enjoyment |
|A tendency to engage in or avoid mathematical activities |Motivation |
(Source: Majeed, Darmawan & Lynch 2013).
3.12 PROCEDURE OF THE STUDY
The research design was quasi-experimental. One group (one section of 9th grade) from each school was exposed to the experimental conditions while other to the conventional conditions. For this purpose two chapters were selected from the mathematic course of grade 9th i.e., chapter 4 and chapter 5. The selected chapters were divided into 20 small sub-units and objectives were determined for each sub-unit. For this purpose meetings were held with mathematics experts in PITE.
The formative evaluation test was designed for each subunit. This designing was based on Bloom’s taxonomy (1956). Second and third formative tests were prepared for those who could not achieve learning objectives in the first or second attempt. At the end of each chapter post-test was administered to all groups.
The procedure of the study for experimental groups is given in figure 3.2.
Figure 2.3: Procedure of the study
To assess attitude, Likert Scale attitude instrument (ATMI) was administered to control and experimental groups at the beginning and at the end of the treatment period. Likert scale is the most commonly used scale for measuring attitude (Gay, 2000). It is a five points bipolar response scale developed by Rensis Likert in 1932 (Singh, 2007). The responses of scale were (A) Strongly agree, (B) Agree, (C) Undecided, (D) Disagree, and (E) Strongly disagree.
After two months of the treatment a retention test was administered to both experimental and control groups.
3.13 ANALYSIS OF DATA
The data obtained through pre-test, achievement tests, retention test and ATMI was analyzed through Statistic Package of Social Sciences (SPSS) version 18.
3.13.1 Pre-test, Post-tests and retention test analysis
First of all the data obtained through pre-test, post-tests and retention test was properly entered into the SPSS (version 18) program. Independent samples t-test was performed on data in order to obtain mean, standard deviation, degree of freedom, t-value and p-value for each set of data. The null hypotheses of the research study were tested at 0.05 level through independent sample t-tests. For testing variability in the achievement scores of the students Levene’s test for equality of variance was performed on the data obtained through Pre-test, Post-tests and retention test.
3.13.2 Attitude Toward Mathematics Inventory (ATMI)
Data was collected through ATMI from all the four groups before and after the experimental period. ATMI was consisted of four factors/sub-scales, i.e. self-confidence, value, enjoyment and motivation. Total statements of ATMI were 40. Twelve items were reversed scored. The distribution of items according to the sub-scales was: self-confidence (15 items), value (10 items), enjoyment (10 items) and motivation (5 items). In the present study 39 items were used as item 9 was dropped due to its inappropriate translation in Urdu. For data collection the statements of each sub-scale/factor was separated, then score for each response was calculated. A mean score for all responses of a sub-scale was calculated before the treatment and then after the treatment. Then a paired samples t-test was performed on each set of data to test null hypothesis of the research study.
CHAPTER – 4
DATA ANALYSIS
This chapter deals with the analysis and interpretation of the data. The data were collected through pre-test, post-test-1, post-test-2, retention test and ATMI (Attitude Toward Mathematics Inventory). Data was analyzed through the Statistical Program of Social Sciences (SPSS) version 18. Independent sample t-test was applied to pre-test, post-test-1, post-test-2 and retention test scores and paired samples t-test was applied to ATMI after calculating mean scores of students’ scores.
Table 4.1 to table 4.12 and table 21 to table 22 are consisted of information about pre-test, post-tests and retention test analysis and interpretation. Table 4.13 to table 4.20 are consisted of information about attitude scale analysis and interpretation.
Table 4.1
Significant Difference between the Pre-test Mean Achievement Scores of Experimental and Control Groups of Rural Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |116 |13.29 |9.20 |0.634 |0.527 |
|Group II (Cont.) |59 |115.30 |14.41 |9.95 | | |
Table value of t at 0.05 = 1.9806
Table 4.1 shows the data analysis of pre-test scores of Group I (experimental) and Group II (control). There is statistically no significant difference between the mean achievement scores of experimental group (M = 13.29, S = 9.20) and control group (M = 14.41, S = 9.95), conditions; t(116) = .634, p = .527, α > .05.
The differences of mean scores (1.12) and standard deviations (0.75) are also very small. P-value is greater than 0.05, this shows similar abilities of both groups of rural area at pre-test stage.
Table 4.2
Significant Difference between the Pre-test Mean Achievement Scores of Experimental and Control Groups of Urban Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group III (Exp.) |48 |94 |17 |7.58 |.389 |.698 |
| Group IV (Cont.) |48 |93.99 |17.6 |7.63 | | |
Table value of t at 0.05 =1.9855
Table 4.2 shows the data analysis of pre-test scores of Group III (experimental) and Group IV (control). There is no significant difference between the mean achievement scores of experimental group (M = 17, S = 7.58) and control group (M = 17.6, S = 7.63), conditions; t(94) = .389, p = .698 > 05.
The differences of mean scores (0.6) and standard deviations (0.05) are very small. P-value is greater than 0.05 that shows no significant difference in the pre-test achievement score of the two groups. This shows similar abilities of both groups of urban area at pre-test stage.
Table: 4.3
Significant Difference between the Post-test-1 Mean Achievement Scores of Experimental and Control Groups of Rural Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |116 |30.32 |8.40 |5.03 |.000* |
|Group II (Cont.) |59 |114.33 |22.02 |9.48 | | |
Table value of t at 0.05 = 1.9806 * significant
Table 4.3 shows the data analysis of post-test-1 scores of Group I (experimental) and Group II (control). There is significant difference between the mean achievement scores of experimental group (M = 30.32, S = 8.40) and control group (M = 22.02, S = 9.48), conditions; t(116) = 5.03, p = .000 < .05.
Table: 4.4
Significant Difference between the Post-test-2 Mean Achievement Scores of Experimental and Control Groups of Rural Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |116 |39.58 |8.48 |10.04 |.000* |
|Group II (Cont.) |59 |113.738 |22.66 |9.77 | | |
Table value of t at 0.05 = 1.9806 *significant
Table 4.4 reveals statistically significant difference in the mean achievement scores of two groups at post-test-2. A highly significant difference is found between the mean achievement scores of experimental group (M = 39.58, S = 8.48) and control group (M = 22.66, S = 9.77), conditions; t(116) = 10.04, p = .000 < .05, the difference being highly in favor of experimental group.
On the basis of data analysis in tables 4.3 and 4.4, the sub-hypothesis (H1a0) stated as, “there is no significant difference in the achievement of the rural students taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected.
Table: 4.5
Significant Difference between the Post-test-1 Mean Achievement Scores of Experimental and Control Groups of Urban Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group III (Exp.) |48 |94 |34.87 |6.80 |6.419 |.000* |
| Group IV (Cont.) |48 |84.99 |24.02 |9.54 | | |
Table value of t at 0.05 = 1.9855 *Significant
Table 4.5 shows the data analysis of post-test-1 scores of group III (experimental) and group IV (control). There is significant difference between the mean achievement scores of experimental group (M = 34.87, S = 6.80) and control group (M = 24.02, S = 9.54), conditions; t(94) = 6.419, p = .000 < .05, the difference being highly in favor of experimental group.
Table: 4.6
Significant Difference between the Post-test-2 Mean Achievement Scores of Experimental and Control Groups of Urban Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group III (Exp.) |48 |94 |38.10 |6.74 |7.98 |.000* |
|Group IV (Cont.) |48 |81.97 |24.12 |10.09 | | |
Table value of t at 0.05 = 1.9855 *Significant
Table 4.6 also reveals statistically significant difference in the mean achievement scores of experimental group (M = 38.10, S = 6.74) and control group (M = 24.12, S = 10.09), conditions; t(94) = 7.98, p = .000 < .05, the difference being highly in favor of experimental group.
On the basis of data analysis in tables 4.5 and 4.6, the sub-hypothesis (H1b0) stated as, “there is no significant difference in the achievement of urban students taught through Mastery Learning Strategy and taught through Conventional Teaching Method” was rejected.
Testing Null Hypothesis 1
On analyzing the two sub-hypotheses (H1a0 and H1b0), it was found that the difference in the mean achievement scores of experimental and control groups in both areas (Rural and Urban) was statistically significant in favor of experimental groups. Hence the first null hypothesis (H10) stated as, “there is no significant difference in the achievement of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected. It was concluded that Mastery Learning Strategy improved academic achievements of the student as compared to Conventional Teaching Method.
Table: 4.7
Significant Difference in Short-term Retention Scores of Experimental and Control Groups of Rural Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |116 |34.95 |8.12 |7.844 |.000* |
|Group II (Cont.) |59 |113.921 |22.34 |9.30 | | |
Table value of t at 0.05 = 1.9806 *Significant
Table 4.7 shows the data analysis of short-term retention scores of Group I (experimental) and Group II (control). There is statistically significant difference in the mean scores and standard deviations of experimental group (M = 34.95, S = 8.12) and Control group (M = 22.34, S = 9.30), conditions; t (116) = 7.84, p = .000 < 0.05, the difference being highly in favor of experimental group.
Table: 4.8
Significant Difference in Long-term Retention Scores of Experimental and Control Groups of Rural Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |116 |36.20 |10.22 |10.4 |.000* |
|Group II (Cont.) |59 |115.89 |16.93 |9.91 | | |
Table value of t at 0.05 = 1.9806 * Significant
Table 4.8 shows the data analysis of long-term retention scores of Group I (experimental) and Group II (control). There is significant difference in the mean scores of experimental (M = 36.20, S = 10.22) and Control (M = 16.93, S = 9.91) groups, conditions; t (116) = 10.4, p = .000 < 0.05, the difference being highly in favor of experimental group.
On the basis of data analysis in tables 4.7 and 4.8, the sub-hypothesis (H2a0) stated as, “there is no significant difference in the learning retention of rural students taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected.
Table: 4.9
Significant Difference in Short-term Retention Scores of Experimental and Control of Urban Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group III (Exp.) |48 |94 |36.49 |6.46 |7.59 |.000* |
|Group IV (Cont.) |48 |83.69 |24.07 |9.32 | | |
Table value of t at 0.05 = 1.9886 *Significant
Table 4.9 shows the data analysis of short-term retention scores of Group III (experimental) and Group IV (control). There is statistically significant difference in the mean scores of experimental group (M = 36.49, S = 6.46) and Control group (M = 24.07, S = 9.32), conditions; t (84) = 7.59, p = .000 < 0.05, the difference being highly in favor of experimental group.
Table: 4.10
Significant Difference in Long-term Retention Scores of Experimental and Control Groups of Urban Area
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group III (Exp.) |48 |94 |36.64 |7.89 |8.450 |.000* |
|Group IV (Cont.) |48 |87.97 |20.81 |10.31 | | |
Table value of t at 0.05 = 1.9855 *Significant
Table 4.10 shows the data analysis of long-term retention scores of group III (experimental) and group IV (control). There is statistically significant difference in the scores of experimental (M = 36.64, S = 7.89) and Control (M = 20.81, S = 10.31) groups, conditions; t(94) = 8.45, p = .000 < 0.05, the difference being highly in favor of experimental group.
On the basis of data analysis in tables 4.9 and 4.10, the sub-hypothesis (H2b0) stated as, “there is no significant difference in the learning retention of urban students taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected.
Testing Null Hypothesis 2
On analyzing the two sub-hypotheses (H2a0 and H2b0), it was found that the difference in the retention scores of experimental and control groups in both areas (Rural and Urban) was statistically significant. Hence the null hypothesis (H20) stated as, “there is no significant difference in the learning retention of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected. It was concluded that Mastery Learning Strategy improved learning retention of the student as compared to Conventional Teaching Method.
Table: 4.11
Test of Variability in the Academic Achievement Scores of Experimental and Control Groups of Rural Area
|Levene’s test for equality of variance |F |Sig. |
|Pre-test |0.032 |0.858 |
|Post-test 1 |1.028 |0.313 |
|Post-test 2 |0.142 |0.707 |
Table 4.11 reveals the variability in achievement score of experimental and control groups of rural area. Levene’s test was applied on pre-test, post-test-1 and post-test-2 and p-value was found 0.858, 0.313, 0.707 respectively. Levene’s test was not found statistically significant as p-value is greater than 0.05.
The null hypothesis (H3a0) stated as, “there is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students of rural area”, was not rejected.
Table 4.12
Test of Variability in the Academic Achievement of Scores of Experimental and Control Groups of Urban Area
|Levene’s test for equality of variance|F |Sig. |
|Pre-test |0.182 |0.670 |
|Post-test 1 |3.862 |0.052 |
|Post-test 2 |3.777 |0.055 |
Table 4.12 reveals the variability in achievement scores of group III (experimental) and group IV (control). Levene’s test on pre-test, post-test 1 and post-test 2 was found 0.670, 0.052, and 0.055 respectively. Levenes test was not found significant as p-value is greater than 0.05.
The null hypothesis (H3b0) stated as, “there is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students of urban area”, was not rejected.
Testing Null Hypothesis 3
On analyzing the two sub-hypotheses (H3a0 and H3b0), Levene’s test was not found significant as p-value is greater than 0.05. It was concluded that variations in achievement scores of experimental and control groups were similar. Hence, the third null hypothesis (H30) stated as, “there is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students”, was not rejected. It was conclude that Mastery Learning Strategy did not reduce achievement gaps of experimental groups to significant level.
.
Table 4.13
Significant Difference in Self-confidence of the Rural Students of Experimental and Control Groups Before and After Treatment.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group I (Exp.) |Pre |2.597 |0.098 |13 |14.930* |.000* |
| |Post |3.512 |0.266 | | | |
|Group II (Cont.) |Pre |2.574 |0.159 |13 |0.329 |.747 |
| |Post |2.537 |0.188 | | | |
Table value of t at 0.05 = 2.2604 *Significant
Table 4.13 shows significant difference in the self-confidence of rural students of experimental groups before and after treatment.
` Group I has mean difference 0.915 (3.521-2.597) and difference in standard deviation is 0.229 (0.266-0.098) at 0.05 level is statistically significant as conditions; t(13) = 14.930 and p = .000 < 0.05.
Group II has mean difference 0.009 (2.537 – 2.574) and difference in standard deviations is 0.109 (0.188 – 0.042), conditions; t (13) = 0.329, p =.747 at 0.05 level, thus not significant.
Statistically significant difference was found in the self-confidence of the students of experimental group before and after treatment, i.e., their self-confidence increased after treatment; but no significant difference was found in the self-confidence of the students of control group before and after treatment.
Table 4.14
Significant Difference in the Views of Rural Students about the Value they give to the Subject of Mathematics Before and After Treatment.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group I (Exp.) |Pre |3.769 |0.630 |9 |4.440* |.002* |
| |Post |4.422 |0.332 | | | |
|Group II (Cont.)|Pre |3.968 |0.485 |9 |1.039 |.326 |
| |Post |3.920 |0.547 | | | |
Table value of t at 0.05 = 2.2621 *Significant
Table 4.14 shows significant difference in the views of rural students before and after treatment regarding valuing the subject of mathematics at secondary school level.
Group I has mean difference 0.653 (4.422 – 3.769) and difference in standard deviations is 0.298 (0.332 – 0.630) at 0.05 level, statistically significant as t(9) = 4.440 and p = .002.
Group II has mean difference 0.048 (3.920 – 3.968) and difference in standard deviations is 0.062 (0.547 – 0.485), conditions; t(9) = 1.039, p = 0.326 at 0.05 level, thus not significant.
So there is statistically significant difference in the views of urban students of experimental group regarding valuing the subject of mathematic before and after treatment, but no significant difference was found in the views of urban students of control group before and after treatment.
Table 4.15
Significant Difference in the Views of Rural Students Regarding Enjoying the Subject of Mathematics Before and After Treatment.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group I (Exp.) |Pre |2.756 |0.489 |9 |6.727* |.000* |
| |Post |3.374 |0.436 | | | |
|Group II (Cont.) |Pre |2.593 |0.441 |9 |0.714 |.493 |
| |Post |2.683 |0.818 | | | |
Table value of t at 0.05 = 2.2621 *Significant
Table 4.15 shows significant difference in the views of rural students before and after treatment regarding enjoying the subject of mathematics at secondary school level.
Group I has mean difference 0.618 (3.374 – 2.756) and difference in standard deviations is 0.291 (0.436 -0.489) at 0.05 level, statistically significant as t(9) = 6.727 and p = 0.000.
Group II has mean difference 0.09 (2.683 – 2.593) and difference in standard deviations is 0.398 (0.818 – 0.441), conditions; t (9) = 0.714, p = 0.493 at 0.05 level, thus not significant.
So there is statistically significant difference in the views of rural students of experimental group regarding enjoying the subject of mathematic before and after treatment, but no significant difference in the views of rural students of control group before and after treatment.
Table 4.16
Significant Difference in Students’ Motivation for the Subject of Mathematics Before and After Treatment in Rural Area.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group I (Exp.) |Pre |2.739 |0.887 |4 |3.196* |.034* |
| |Post |3.176 |0.610 | | | |
|Group II (Cont.) |Pre |2.668 |0.887 |4 |2.639 |.058 |
| |Post |2.719 |0.866 | | | |
Table value of t at 0.05 = 2.7764 *Significant
Table 4.16 shows significant difference in students’ motivation for the secondary school mathematics before and after treatment in rural area.
Group I has mean difference 0.437 (3.176 – 2.739) and difference in standard deviations is 0.138 (0.610 – 0.887) at 0.05 level is statistically significant as t(4) = 3.196 and p = 0.034.
Group II has mean difference 0.051 (2.719 – 2.668) and difference in standard deviations is 0.043 (0.866 – 0.887), conditions; t(4) = 2.639, p = 0.058 at 0.05 level, thus not significant.
Hence, there is statistically significant difference in the motivation of students of experimental group before and after treatment, while no such difference exist in the motivation of control group.
On the basis of data analysis in table 4.13, table 4.14, table 4.15 and table 4.16 the null hypothesis (H4a0) stated as, “there is no significant difference in the attitude of rural students towards the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected.
Table 4.17
Significant Difference in the Self-confidence of Urban Students of Experimental and Control Groups Before and After Treatment.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group III (Exp.) |Pre |2.203 |0.085 |13 |15.36* |.000* |
| |Post |3.321 |0.217 | | | |
|Group IV (Cont.) |Pre |2.008 |0.043 |13 |0.058 |.955 |
| |Post |2.009 |0.059 | | | |
Table value of t at 0.05 = 2.2604 *Significant
Table 4.17 shows significant difference in the self-confidence of students of urban area before and after treatment.
` Group III has mean difference 1.118 (3.321 – 2.203) and difference in Standard deviation is 0.132 (0.217 – 0.085), conditions; t(13) = 15.36 and p = 0.000 < 0.05, thus significant.
Group IV has mean difference 0.001 (2.009 – 2.008) and difference in standard deviations is 0.016 (0.059 – 0.043), conditions; t(13) = 0.058 and p = 0.955 > 0.05, thus not significant.
So there is statistically significant difference in the self-confidence of the students of experimental group of urban area before and after treatment, but no significant difference in the self confidence of the students of control group before and after treatment. The self-confidence of experimental group increased after treatment.
Table 4.18
Significant Difference in the Views of Urban Students about Value they give to the Subject of Mathematics Before and After Treatment.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group III (Exp.) |Pre |3.459 |0.310 |9 |6.427* |.000* |
| |Post |3.840 |0.161 | | | |
|Group IV (Cont.) |Pre |3.336 |0.313 |9 |1.741 |.116 |
| |Post |3.391 |0.269 | | | |
Table value of t at 0.05 = 2.2621 *Significant
Table 4.18 shows significant difference in the views of students of urban area before and after treatment regarding valuing the subject of mathematics.
Group III has mean difference 0.381 (3.840 – 3.459) and difference in standard deviation is 0.159 (0.161 – 0.310), conditions; conditions; t(9) = and p = 0.000 < 0.05 thus significant.
Group IV has mean difference 0.055 (3.391 – 3.336) and difference in standard deviations is .044 (0.313 - 0.269), conditions; t(9) = 1.741 and p = 0.116 > 0.05, thus not significant.
So there is statistically significant difference in the views of the urban students of experimental group regarding valuing the subject of mathematic before and after treatment, but no significant difference in the views of urban students of control group before and after treatment.
Table 4.19
Significant Difference in the Views of the Urban Students Regarding Enjoying the Subject of Mathematics Before and After Treatment.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group III (Exp.) |Pre |2.252 |0.502 |9 |7.664* |.000* |
| |Post |2.795 |0.384 | | | |
|Group IV (Cont.) |Pre |2.174 |0.600 |9 |0.957 |.364 |
| |Post |2.09 |0.429 | | | |
Table value of t at 0.05 = 2.2621 *Significant
Table 4.19 shows significant difference in the views of students of urban area before and after treatment regarding enjoying the subject of mathematics.
Group III has mean difference 0.543 (2.795 – 2.252) and difference in standard deviations is 0.118 (0.384 – 0.502), t (9) = 7.664 and p = 0.000 < 0.05, thus significant.
Group IV has mean difference 0.084 (2.09 – 2.174) and difference in standard deviations is 0.171(0.429 – 0.6), conditions; t(9) = 0.957 and p = 0.364 > 0.05, thus not significant.
So there is statistically significant difference in the views of urban students of experimental group regarding enjoying the subject of mathematic before and after treatment, but no significant difference in the views of students of control group before and after treatment.
Table 4.20
Significant Difference in Students’ Motivation for the Subject of Mathematics Before and After Treatment in Urban Area.
|Groups |Observations |Mean |SD |Df |t-value |p-value |
|Group III (Exp.) |Pre |2.122 |0.392 |4 |3.487* |.025* |
| |Post |2.566 |0.287 | | | |
|Group IV (Cont.) |Pre |2.091 |0.388 |4 |0.746 |.497 |
| |Post |2.122 |0.402 | | | |
Table value of t at 0.05 = 2.7764 *Significant
Table 4.20 shows significant difference in students’ motivation for the secondary school mathematics before and after treatment in urban area.
Group III has mean difference0.444 (2.566 – 2.122) and difference in standard deviations is 0.125 (0.267 – 0.392), conditions; t(4) = 3.487 and p = 0.025 < 0.05 thus significant.
Group IV has mean difference 0.031(2.122 – 2.091) and difference in standard deviations is 0.014 (0.402 – 0.388), conditions; t(4) = 0.746 and p = 0.497 > 0.05, thus not significant.
It was clear from the data analysis in table 4.20 that significant difference exist in the motivation of urban students of experimental group before and after treatment, while no such difference exist in the urban students of control group.
On the basis of data analysis in table 4.17, table 4.18, table 4.19 and table 4.20 the sub hypothesis (H4b0) stated as, “there is no significant difference in the attitude of urban students towards the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected.
Testing Null Hypothesis 4
On analyzing the two sub-hypotheses (H4a0 and H4b0), statistically significant difference was found in the attitude of the students of experimental groups in both areas (Rural and Urban) before and after treatment, but no significant difference was found in the views of students of control groups before and after treatment. The difference was highly in favor of post treatment.
Hence, the null hypothesis (H40) stated as, “there is no significant difference in the attitude of students towards the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method”, was rejected. It was concluded that Mastery Learning Strategy changed the attitude of the students positively towards the subject of mathematics.
Table 4.21
Significant Difference between the Post-Test-1 Mean Achievement Scores of Experimental Group-I (Rural) and Group-III (Urban)
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |105 |30.32 |8.4 |3.032 |.583 |
|Group III (Exp.) |48 |104.99 |34.87 |6.8 | | |
Table value of t at 0.05 = 1.9828
Table 4.13 reveals no significant difference between the mean achievement scores of Group I (M = 30.32, S = 8.4) and Group III (M = 34.87, S = 6.8), conditions; t(105) = 3.032, p = .583 < .05. Both experimental groups have almost similar achievement at post test 1.
Table 4.22
Significant Difference between the Post-Test-2 Mean Achievement Scores of Group-1 (Rural) and Group-III (Urban)
|Groups |N |Df |Mean |SD |T-value |P-value |
|Group I (Exp.) |59 |105 |39.58 |8.48 |.977 |.060 |
|Group III (Exp.) |48 |104.95 |38.1 |6.74 | | |
Table value of t at 0.05 =1.9828
Table 4.14 reveals no significance of difference in the mean achievement scores of Group I (M = 39.58, S = 8.48) and Group III (M = 38.1, S = 6.74), conditions; t(105) = .977, p = .060 < .05. Both experimental groups have almost similar achievement at post test 2.
Testing Null Hypothesis 5
On the basis of data analysis in table 4.21 and 4.22 the 5th null hypothesis (H50) stated as, “there is no significant difference in the academic achievement of students of urban and rural areas in the subject of mathematics taught through Mastery Learning Strategy” was accepted. It was concluded that Mastery Learning Strategy improved the academic achievement of the students in both areas (urban and rural).
CHAPTER – 5
SUMMARY, FINDINGS, DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS
5.1 SUMMARY
The present study was conducted to examine “The effect of Mastery Learning Strategy on Students’ Achievements in the Subject of Mathematics at Secondary School Level”. The focus of the study was to compare Mastery Learning Strategy with Conventional Teaching Method in the subject of mathematics at secondary school level. The major objectives of the study were: (1) To compare the effects of Mastery Learning Strategy with Conventional Teaching Method on students’ achievement in the subject of mathematics at secondary school level. (2) To investigate the effects of Mastery Learning Strategy on students’ long term and short term learning retention. (3) To compare the effects of Mastery Learning Strategy with Conventional Teaching Method on achievement gaps of the students of the same grade. (4) To investigate the effects of Mastery Learning Strategy on students’ attitude towards the subject of mathematics at secondary level. (5) To compare the academic achievements of urban and rural students taught through Mastery Learning Strategy.
The population of the study was 9th grade students of district Mardan. The 9th grade students of two female schools, namely GGHSS, Shehbaz Garhi and GGHS, No.1 Mardan were selected as sample through purposive sampling technique. There were two sections of 9th grade in GGHSS, Shehbaz Garhi and five sections of 9th grade (two sections of science students and three sections of arts students) in GGHS, No.1 Mardan. Both sections of GGHSS, Shehbaz Garhi and two science sections of GGHS, No.1 Mardan were selected as sample. One Secondary School Teacher from GGHS, No.1 Mardan and one Subject Specialist in mathematics was selected from GGHSS, Shehbaz Garhi in order to teach both sections in their respective schools. The researcher thoroughly discussed the objectives of the study with the selected teachers and proper training was given to the teachers about Mastery Learning Strategy and Conventional Teaching Method in RITE. The selected teachers were provided with lesson plans, teaching material and summative assessment tests. Formative tests and corrective activities were prepared by the teachers themslves. Time table was also provided to the teachers and permission of extra time was granted from respective principals for experimental groups. The principals were requested to work as “observers” and the researcher him/herself also did this job.
The tools for data collection were: pre-test, two post-tests, retention test, and ATMI. All tools were tested for validity and reliability.
Pre-test and ATMI were administered to all the four groups before treatment. At the completion of 1st unit, post-test-1 was administered to all groups. Then, after the completion of 2nd unit, post-test-2 was administered to all groups. At the completion of treatment period, ATMI was again administered to all the students of four groups. ATMI explored the attitude of the students’ before and after treatment. Short-term learning retention was calculated by taking the average of post-est-1 and post-test-2. After two months of the experimental period the retention test was administered and long-term learning retention score was calculated. It was noted that the students of experimental groups utilized more time than the students of control groups as the experimental groups got more opportunities fo rpractice and participation in learning activities. Data obtained through pre-test, post-test-1, post-test-2 and retention test were tested at 0.05 level through independent sample t-test. Levene’s test of variability was used to test variability in achievement of the students all groups. The data obtained through ATMI was also tested at 0.05 level by applying paired samples t-test. On the basis of findings, Mastery Learning Strategy was recommended for teaching of mathematics at secondary school level.
5.2 FINDINGS
The following findings are drawn on the basis of the data analysis of pre-test, post-tests and retention test:
1. At the pre-test stage it was found that the academic performance of Group I (experimental) was similar to the Group II (control). T-value was 0.634 less than 1.98 (t-tabulated value at 0.05 level) and P-value was 0.527 > 0.05 (table 4.1). Similarly Group III (experimental) was similar to Group IV (control) in academic performance (table 4.2) as the t-value was 0.389 less than 1.98 (t-tabulated value at 0.05 level) and p-value was 0.698 > 0.05.
2. A Significant difference was found in the mean academic achievement scores of Group I and Group II at post-test-1 stage. The t-value and p-value were found 5.03 and .000 respectively. As t-value was greater than 1.98 and p-value was less than 0.05, thus significant (table 4.3).
3. At post-test-2 stage significant difference was found between the mean academic achievement scores of Group I and Group II. T-value and p-value were found 10.04 and .000 respectively that were again significant at 0.05 level (table 4.4).
4. By comparing the achievement scores of Group III and Group IV significance of difference was in favor of the experimental group (Group III) at post-test-1 stage. The t-value was 6.419 and p-value was .000 < .05 (table 4.5).
5. At post-test-2 stage, significant difference was found between the mean academic achievement scores of Group III and Group IV. The t-value was 7.98 and p-value was .000 < .05 (table 4.6).
6. Significant difference was recorded when short-term learning retention scores of Group I and Group II were compared. The t-value was 7.84 and p-value was .000 < 0.05 (table 4.7).
7. Significant difference was recorded when long-term learning retention scores of Group I and Group II were compared. T-value was 10.4 and p-value was .000, which were significant at 0.05 level (table 4.8).
8. Significant difference was found in the short-term learning retention scores of Group III and Group IV as the t-value was 7.59 and p-value was .000 < 0.05 (table 4.9).
9. Significant difference was recorded in the long-term learning retention scores of Group III and Group IV. T-value was 8.450 and p-value was .000 < 0.05 (table 4.10).
10. Levene’s test for equality of variance was not found statistically significant at 0.05 level. By comparing the achievement scores of Group I and Group II the p-values for pre-test, post-test-1 and post-test-2 were found 0.858, 0.313, 0.707 respectively (table 4.11).
11. Levene’s test for equality of variance was also not statistically significant when applied to the achievement scores of Group III and Group IV. P-values for pre-test, post-test 1 and post-test 2 were calculated 0.670, 0.052, and 0.055 respectively (table 4.12).
12. By comparing the academic achievement scores of both experimental groups, i.e. Group I (Rural) and Group III (Urban), no significant difference was recorded. At post-test 1 stage, t-value and p-value was 3.032 and .583 respectively (table 4.21), while at post-test-2 stage, t-value and p-value was recorded as .977 and 0.060 respectively (table 4.22).
The following findings were drawn from the data analysis of attitude scale.
1. Self confidence of the experimental groups (Group I and Group III) was significantly increased after treatment period, while no significant difference was found in the self confidence of control groups (Group II and Group IV) before and after treatment. For Group I t-value was recorded 14.93 and p-value was .000 < 0.05, thus significant. The t-value and p-value for Group II were 0.329 and 0.747 respectively, therefore, not significant (4.13). Similarly, a significant difference was noted in self confidence of Group III after treatment when compared with fore treatment. For group III The t-value was 15.36 and p-value was .000 < 0.05, thus significant. The t-value and p-value for Group IV were 0.058 and 0.955 respectively. Those values were not significant at 0.05 level (table 4.17).
2. Significance of difference was found in pre and post views of experimental groups (regarding how much value students give to the subject of mathematics. The views of the experimental groups (Group I and Group III) changed positively after the treatment period while no significant change was recorded in views of control groups (Group II and Group VI). T-value and p-value of Group 1 were 4.44 and .002 respectively, and for Group II the t-value and p-value were 1.039 and 0.326 respectively (table 4.14). Similarly for Group III the t-value was 6.427 and p-value was .000, that showed significant difference at 0.05 level. For Group VI the t-value and p-value were 1.741 and 0.116 respectively (table 4.18).
3. A Significant difference was found in pre and post views of experimental groups when data about enjoying the subject of mathematics was analyzed. The views of the students of experimental groups (Group I and Group III) significantly changed while that of control groups (Group II and Group VI) remains unchanged. For Group I t-value and p-value were 6.727 and .000 respectively, therefore, significant. The t-value and p-value for Group II were 0.714 and 0.493 respectively, therefore, not significant (table 4.15). Similarly, For Group III the t-value was 7.664 p-value was .000 < 0.05, therefore, significant. For Group VI t-value was 0.957and p-value was 0.364, that were not significant at 0.05 level (table 4.19).
4. Statistically Significant difference was found in the views of the students of experimental groups (Group I and Group III) about motivation toward the subject of mathematics before and after treatment. While no significant difference was recorded in the views of students of control groups ( Group II and Group IV) about motivation toward the subject of mathematics before and after treatment. By comparing pre and post views Group I t-value and p-value were 3.196 and 0.034 respectively hence, significant at 0.05 level. The t-value and p-value for Group II were 2.639 and 0.058 respectively, thus statistically not significant (table 4. 16|). Similarly, by comparing pre and post views Group III t-value and p-value were 3.487 and .025 respectively, that were also significant at 0.05 level. While t-value and p-value for Group IV were 0.746 and 0.497 respectively, that were not significant at 0.05 level (4.20).
So it was found that the attitude of students exposed to Mastery Learning Strategy was positively changed and that of control groups do not changed by Conventional Teaching Method.
5.3 DISCUSSION
Teaching methods play an important role in teaching learning process. It is a planned and organized effort for bringing desirable behavioral changes in young generation. So to bring desirable changes in students’ behavior, proper methods of teaching, based on sound psychological principles of teaching, must be adopted.
Due to the importance of mathematics in social life, it is essential to improve the standard of teaching mathematics. It is not suitable to employ a single teaching method for all students having individual differences. As there are variations in students of the same class, so there should be variety in teaching process to meet these variations. It is indispensable to develop a program for teaching of mathematics that meets the needs of the whole group of learners. Mastery Learning adopts different teaching methods and techniques; and gives appropriate time of learning to each, thus making it possible for all learners to achieve instructional objectives. The founding father of Mastery Learning, Bloom (1968) was of the opinion that all students can achieve mastery of learning task if provided with proper instructions and time. Bloom outlined a teaching strategy that incorporates “feedback” and “corrective” procedures in class room teaching and labeled it Mastery Learning.
The current study aimed to compare the effectiveness of Mastery Learning Strategy with Conventional Teaching Method in the subject of mathematics at secondary school level. The sample consisted of four groups, two experimental and two control groups, at two different schools of district Mardan. Experimental and control groups of each school were similar in academic achievement as clear from the pre-test scores. The attitude of students towards the subject of mathematics was also similar at the beginning of the experiment.
Experimental groups were exposed to Mastery Learning instructions, while control groups were taught through Conventional Method. After the completion of one chapter post-test-1 and at the end of the experiment, post-test-2 was administered to all the students of experimental and control groups. The retention test was administered after two months of the experiment to all the students of four groupst. The data obtained was analyzed through SPSS version 18. After analysis, it was concluded that Mastery Learning Strategy improved students’ learning performance as compared to conventional method. These results are in line with those by Adeyemi (2007); Wambugu and Changeiywo (2008); Moradi, Zarei and Zainalipour (2012); Sood (2013), Udo and Udofia (2014); and Lamidi, Oyelekan and Olorundare (2015).
Hussain (2016) found mastery learning an effective approach in cognitive domain and its good effects on the retention power of students. The results of the present study also supports these findings. Mastery Learning Strategy raised students’ level of learning and improved their learning retention.
Bloom (1984) in his studies on group interaction has found that students exposed to mastery learning approach required more time to master advance material. In the present study too, more time was utilized by experimental groups, as they were not allowed to go ahead before mastering the previous level, but after one month of teaching the researcher found that the time for mastery gradually decreased, as their understanding increased and their ability to grasp knowledge grew sharpened.
At the end of the experiment a positive change was noticed in the attitude of students of experimental groups. Many students, who thought mathematics as an unimportant subject at the beginning were ready, at the end, to study mathematic in college classes too. Students’ self-confidence was increased and they were motivated toward solving mathematical problems at the end of experimental period. With the Mastery Learning Strategy they enjoyed solving mathematics’ problems. Their negative views about mathematics as a boring and dull subject were changed positively. They started taking interest in mathematics. These findings of the study show similarity with the findings of several studies previously conducted by different researchers. For example, a positive change in the attitude of weak students was found by Damavandia and Kashani (2010) and a positive change in the attitude of all students towards the subject of mathematics was observed by Davrajoo, Tarmizib, Nawawi and Hassand (2010). Shafie, Shahdan and Liew (2010), also found students more satisfied with mastery learning approach. Current study also supports the findings of Moradi, Zarei and Zainalipour (2012), who found that students’ achievement, motivation and self concept considerably increased by mastery learning.
The performance of many low achievers, when provided with more time and alternate teaching method, was improved during this study. These findings further support the earlier findings of Damavandia and Kashani (2010) and Davrajoo et al. (2010).
Contrary to the findings of Uchechi (2013) and Abakpa and Iji (2013) achievement gaps were not found significantly decreased when Levene’s test for equality of variance was applied, but a decrease was observed by teachers and observers. The results in table 4.12 are close to significant difference, so it can be presumed that achievement gaps may possibly be decreased if sample size is increased or if the experiment is continued for relatively a long period of time.
The above discussion declares that the results of the present study have extended prior work in two ways. First, it has extended the findings of researches conducted on the effects of mastery learning in different countries of the World by confirming that Mastery Learning Strategy is powerful to enhance academic achievement and positive change in the attitude of students towards the subject. Second, the findings of this study extended the findings of studies on same strategy. The findings that have been replicated in several experiments can be considered sound.
The consistency of results of this study and those conducted by other researchers in other parts of the World revealed that Mastery Learning Strategy overcome cultural barriers, as the application of mastery learning improved academic achievement, learning retention, and attitude toward the subject in Pakistani culture too. Moreover it equally benefited the students of urban and rural areas of Pakistan.
Students enjoyed peer tutoring and cooperative learning. They benefited more during these activities. It was observed that peer tutoring activities were very helpful for week students, they asked confidently from their peers when felt any problem in learning. It was also observed that low achievers were more motivated and their interest in the subject increased more. A gradual increase in self confidence also developed the power to overcome mathematics anxiety.
Teachers’ belief that only a small portion of a class can learn mathematic was also changed. They were happy to see the weak students of the class solving mathematics’ problems and enjoying that process. .
The present study was conducted on female students in urban and rural areas of district Mardan. And it was found that Mastery Learning Strategy have positive effects on the academic achievement, retention and attitude towards the subject of mathematics of the students. The study may have more or less the same effects if conducted on male students of district Mardan. So it can be generalized to all the male and female students of district Mardan.
5.4 CONCLUSIONS
The study in hand investigated the effects of mastery learning strategy on students’ achievement in the subject of mathematics at secondary school level. On the basis of data analysis, the following conclusions were drawn.
1. The 1st conclusion was drawn at pre-test stage that experimental and control groups in both schools were equal in academic achievement at the beginning of the experiment.
2. The 2nd conclusion was drawn on the basis of achievement scores of post-test-1 and post-test-2. Experimental groups (Group I and Group III) achieved high score as compare to their respective control groups (Group II and Group IV). Significant difference at 0.05 level was found when their mean academic achievement scores were compared. On the basis of data analysis the first null hypothesis (H10) that “there is no significant difference in the achievement of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method” was rejected and it was concluded that Mastery Learning Strategy is more effective than conventional methods, in the teaching of mathematics at secondary school level.
3. The 3rd conclusion was drawn on the basis of retention scores of the students. By comparing the short-term and long-term retention scores a significant difference was found in favor of experimental groups. Therefore, the null hypothesis (H20) that “there is no significant difference in the learning retention of students taught through Mastery Learning Strategy and taught through Conventional Teaching Method” was rejected and it was concluded that Mastery Learning Strategy is more effective in increasing the enhancing learning retention of the students.
4. The 4th conclusion was drawn on the basis of Levene’s test for equality of variance that was applied to post-tests. The variations in the achievement scores of the students of experimental and control groups were not significant. Hence, the 3rd null hypothesis (H30) that “there is no significant difference in the Mastery Learning Strategy and Conventional Teaching Method to reduce achievement gaps among the students” was not rejected and it was concluded that Mastery Learning Strategy did not reduced gaps of students significantly.
5. The 5th conclusion was drawn on the basis of data received through ATMI. By comparing pre and post data it was concluded that the attitude of experimental groups was positively changed toward the subject of mathematics, while no change was noted in the attitude of control groups. Therefore the 5th null hypothesis (H40) that “there is no significant difference in the attitude of students toward the subject of mathematic taught through Mastery Learning Strategy and taught through Conventional Teaching Method” was rejected.
6. The 6th conclusion was based on the comparison of achievement scores of both experimental groups, i.e. Group I (rural) and Group III (urban). No significant difference was recorded in the achievement scores of two experimental groups. The null hypothesis (H50) that “there is no significant difference in the academic achievement of students of urban and rural areas in the subject of mathematics taught through Mastery Learning Strategy” was accepted and it was concluded that Mastery Learning Strategy is equally effective for the students of urban and rural areas in the teaching of mathematics at secondary school level.
On the whole it was concluded that Mastery Learning is a powerful approach to enhancing academic achievement and learning retention of the students. Mastery Learning is also a powerful strategy for bringing positive change in the attitude of students toward learning mathematics.
5.5 RECOMMENDATIONS
The research study recommends the following suggestions:
1. As the results of the study have been found in favor of Mastery Learning Strategy, it is recommended for mathematics teachers to adopt Mastery Learning Strategy in their teaching.
2. The heads of all high and higher secondary schools may also play a vital role to motivate the teachers for adopting Mastery learning Strategy in teaching of mathematics.
3. This strategy may also be used in other subjects, especially in science subjects.
4. Mastery Learning may also be used at elementary level for concept development.
5. Regional Institutes for Teachers Education may incorporate Mastery Learning Strategy in their training programs for the achievement of set educational goals
6. The Provincial Institutes of Teacher Education may also develop pre-service and in-service teacher training programs in the light of mastery learning philosophy.
7. The Curriculum wing of education may also arrange workshops to develop corrective and enrichment material for different subjects at different school levels.
8. The curriculum developers may also play a vital role to develop curriculum based on Mastery Learning Strategy.
5.6 SUGGESTIONS FOR FUTURE WORK
The study in hand was limited only to the public sector female secondary schools of district Mardan due to limited resources and time constraints. Keeping in view the wide scope of the study and the utility of the teaching strategy, the following suggestions were made:
1. The study may be replicated so many times and likewise experimental treatment may be given to the students of other high and higher secondary schools of different districts of KP.
2. Students of private sectors may also be given such treatment in future studies.
3. The study may be extended to test other complex cognitive, affective and psychomotor variables.
4. The study may be extended to primary and middle level students as well.
5. The study may be replicated in other subjects as well.
6. The study may be replicated in other areas of Pakistan.
7. The study may be replicated in other cultures of the world.
REFERENCES
Abakpa, B.O. & Iji, C.O. (2011). Effect of mastery learning approach on senior secondary school student’s achievement in geometry. Journal of the Science Teachers’ Association of Nigeria, 46 (1), 165-176.
Adeyemi, B. A. (2007). Learning social studies through mastery learning approach. Educational Research and Review, 2 (4), 060-063.
Adeyemo, S. A. and Babajide, F. T. (2014). Effects of Mastery Learning Approach on Students’ Achievement in Physics. International Journal of Scientific & Engineering Research, 5(2), 910-920. Retrieved on Sep. 26, 2018 from: D. pdf
Advisory Committee on Mathematics Education (2011b). Mathematical Needs: The mathematical needs of learners. The Royal Society, London.
Aduda, D. (2003, February 27). Kenya Certificate of Secondary Education, Examination Results Released by Minister of Education. Daily Nation, Nairobi: Nation Media Group Ltd.
Afari, E. (2013). Examining the factorial validity of the attitudes towards mathematics inventory (ATMI) in the United Arab Emirates: Confirmatory factor analysis. International Review of Contemporary Learning Research, 2(1), 15-29.
Agwagah, N. V. (2004). Sustaining development in secondary school mathematics through constructivist framework: A model lesson plan. Journal of Mathematical Association of Nigeria, 29(1), 29-38.
Ainley, J., Graetz, B., Long, M. & Batten, M. (1995). Socioeconomic status and school education. Canberra: Australian Government Publishing Service.
Asante, K. (2012). Secondary students' attitudes towards mathematics. IFE Psychologia, 20(1), 121–133.
Australian Education Council (1990). A national statement on mathematics for Australian schools. Canberra: Curriculum Corporation.
Battista, M. T. (1999). The mathematical miseducation of America’s youth: Ignoring research and scientific study in education. Phi Delta Kappan, 80(6), 424-433.
Best, J. W. (1986). Research in education (5th ed.). N.J: Prentice-Hall, Englewood.
Bishara, S. (2015). Active and Traditional Teaching of Mathematics in Special Education. Creative Education, 6, 2313-2324. Doi: 10.4236/ce.2015.622238.
Block, J. H. (Ed.). (1971). Mastery learning: Theory and practice. New York: Holt, Rinehat and Winston.
Block, J. H., & Anderson, L. W. (1975). Mastery learning in classroom instruction. New York: Macmillan.
Block, J. H. (1979). Mastery learning - Current state of the craft. Educational Leadership, 37, 114-117
Bloom, B. S. & Krathwohl, D. R. (1956). Taxonomy of educational objective: The classification of educational goals, by a committee of college and university examiners. Hand book 1: Cognitive domain. New York, Longmans.
Bloom, B.S. (1964). Stability and change in human characteristics. New York: John Wiley & sons.
Bloom, B.S. (1968). Learning for mastery. Evaluation comments, 1(2), 1-12. Retrieved on November 5, 2011, from: programs.honolulu.hawaii.edu.intranet/files/upstf-student-success-bloom-1968.pdf
Bloom, B. S., Hasting, J. T., & Madaus, G. F. (1971). Handbook on formative and summative evaluation of student learning. New York: McGraw Hill.
Bloom, B. S. (1976). Human charteristics and school learning. New York: McGraw Hill.
Bloom, B. S. (1980). The new direction in educational research: Alterable variables. Phi Delta Kappan, 61, 382-385.
Broberic, Z. (2009). Mastery learning and its effects in assessment. Computer technical report series. Retrieved on December 25, 2011, from: techreports/pdf/10-09.pdf
Callahan, W.J. (1971). Adolescent attitudes towards mathematics, Mathematics Teacher, 64, 751-755.
Carroll, J. B. (1963). A model of school learning. Teacher college record, 64(8), 723-733.
Carroll, J. B. (1965). School learning over the long haul. In J. D. Krumboltz (Eds.), Learning and the educational process (pp.249-269). Chicago: Rand McNally.
Carroll, J. B. (1968). On learning from being told. Educational Psychologist, 5, 4-10.
Carroll, J. B. (1989). The Carroll model: A 25 years retrospective and prospective view. The Educational Researcher, 18(1), 26-31.
Changeiywo, J. M., Wambugu P. W. & Wachanga S.W. (2011). Investigation of students’ motivation towards learning secondary school physics through mastery learning approach. International Journal of Mathematics Science and Technology Education. 9(6), 1333-1350. Retrieved on Jan. 05, 2014 from:
Damavandi, M. E. & Kashani, Z. S. (2010). Effect of mastery learning method on performance, attitude of the weak students in chemistry. Proceda Social and Behavioral Sciences, 5, 1574-1579. doi:10.1016/j.sbspro.2010.07.327. Retrieved on October 13, 2011, from: 1877042810017015
Damodharan, V. S. & Rengarajan, V. (2007, March 31 to April 2). Innovative methods of teaching, In Proceedings of the Learning Technologies and Mathematics Middle East Conference,Saltan Qaboos University, Muscat, Oman. Retrieved on Aug. 15, 2013 from: . html
Davrajoo, E.,Tarmizi, R. A., Nawawi, M., Hassan, A. (2010). Enhancing algebraic conceptual knowledge with aid of module using mastery learning approach. Procedia - Social and Behavioral Sciences, 8, 362–369.
doi:10.1016/j.sbspro.2010.12.051.
Eagly, A. H., & Chaiken, S. (1993). The psychology of attitudes. Fort Worth, TX: Harcourt Brace Jovanovich college publshers.
Ejodamen, I. F. & Raymond, U. (2018). Effect of mastery learning strategy on rural and urban students’ academic achievement in basic technology in Edo State, Nigeria. World Journal of Research and Review, 6(6), 22-28.
Ellerton, N. & Clements, M. (1989). School mathematics: The challenge to change. Victoria: Deakin University Press.
Eshun, B. A. (2004). Sex-differences in attitude of students towards Mathematics in secondary schools. Mathematics Connection, 4, 1–13.
Etsey Y.K. & Snetzler, S. (1998, April 13-17). A Meta-analysis of gender differences in student attitudes toward mathematics. In Proceedings of the Annual Meeting of the American Educational Research Association.
Ezeugo N.C & Agwagah U.N.V (2000). Effects of concept mapping on students achievement in Algebra; implication for secondary school mathematics education in 21st century. Abacus; Journal of mathematical Association of Nigeria, 25(1), 1-12
Farooq, R.A (1980). A Comparative Study of Effectiveness of Problem Solving Approach and Traditional Approach of Teaching Social Studies to Secondary Schools Students (Unpublished doctoral dissertation). University of Punjab Lahore, Pakistan.
Farooq, R. A. (2001). Understanding Research in Education. Lahore: Z. A. printer.
Fennema, E., & Sherman, J. A. (1976). Fennema-Sherman mathematics attitudes scales: Instruments designed to measure attitudes toward the learning of mathematics by males and females. Journal for Research in Mathematics Education, 7(5), 324-326.
Ferguson, K. (2010). Inquiry Based Mathematics Instruction Versus Traditional Mathematics Instruction: The Effect on Student Understanding and Comprehension in an Eighth Grade Pre-Algebra Classroom (Master thesis, Cedarville University, Ohio, United States). Retrieved from: viewcontent.cgi?article=1025&context=education_theses
Fleener, M. J., Craven, L. D., & Dupree, G. N. (1997). Exploring and changing visions of mathematics teaching and learning: what do students think? Mathematics Teaching in the Middle School, 3, 40-43.
Furo, P. T. (2014). Effect of mastery learning approach on secondary school students achievement in Chemistry in Rivers State Nigeria. Chemistry and Materials Research, 6(9), 104-110. Retrieved on Feb. 02, 2015 from:
Galadima, I & Okogbenin, A. A. (2012). The effect of mathematical games on academic performance and attitude of senior secondary students towards mathematics in selected schools in Sokoto State. Abacus: The journal of the mathematical Association of Nigeria. 37(1), 30-37.
Gay, L. R. (2000). Educational research (5th Ed.). National Book Foundation Islamabad.
Ghani, N. A., Hamim, N., & Ishak, N. I. (2006). Applying mastery learning model in developing e-tuition science for primary school students. Malaysian Online Journal of Instructional technology,3(2), 43-49. Retrieved on Dec. 15, 2011, from:
Goreyshi, M. K., Kargar, F. R., Noohi, S., & Ajilchi, B. (2013). Effect of combined mastery- cooperative learning on emotional intelligence, self-esteem and academic achievement in grade skipping. Procedia – Social and Behavioral Sciences, 84, 470 – 474. doi:10.1016/j.sbspro.2013.06.586.
Government of Khyber Pakhtoonkhwa (2013). Result gazette secondary school certificate annual examination (9th). Board of Intermediate and Secondary Education Mardan, Pakistan.
Government of Khyber Pakhtoonkhwa (2013). Result gazette secondary school certificate annual examination (10th). Board of Intermediate and Secondary Education Mardan, Pakistan
Government of Khyber pakhtoonkhwa (2016). Annual Statistical Report of Government Schools: 2015-2016. Education Management Information System (EMIS). Elementary & Secondary Education Department, Peshawar
Government of Khyber Pakhtoonkhwa (2014). Result gazette secondary school certificate annual examination (10th). Board of Intermediate and Secondary Education Mardan, Pakistan.
Government of Pakistan, 2000. National Education Policy in 1998-2010. Ministry of Education Islamabad Pakistan. pp. 1-57.
Government of Pakistan (2002). Result gazette secondary school certificate annual examination (10th). Federal Board of Intermediate and Secondary Education Islamabad, Pakistan.
Government of Pakistan (2006). National curriculum for mathematics grade I-XII. Curriculum Wing, Ministry of Education, Islamabad, Pakistan. Retrieved on Aug. 15, 2013 from: 2006_eng.pdf
Government of Punjab (2002). Result gazette secondary school certificate annual examination (10th). Board of Intermediate and Secondary Education Rawalpindi, Pakistan.
Gowers. T, (2008). The Princeton companion to mathmatics. Princeton University Press.
Guskey, T. R. (1997). Implementing mastery learning (2nd ed.). Belmont, CA: Wadsworth.
Guskey, T. R. (2001). Mastery learning. In N. J. Smelser & P. B. Baltes (Eds.), International Encyclopedia of Social and Behavioral Sciences (pp. 9372-9377). Oxford, England: Elsevier Science Ltd.
Guskey, T. R. (2005). Formative classroom assessment and Benjamin S. Bloom: Theory, research, and implication. Paper presented at the annual meeting of the American educational research association, Montreal, Canada. Retrieved on November 15, 2011, from:
Guskey, T.R. & Pigott, T.D. (1988). Research on group-based mastery learning programs: A meta-analysis. The Journal of Educational Research. 81(4), 197-216.
Hartley, J. (1974). Programmed Instruction 1954–1974: A review. Programmed Learning and Educational Technology, 11, 278– 291.
Hattie, J. (2009). Visible Learning: A synthesis of over 800 meta-analyses relating to achievement. Milton Park, UK: Routledge.
Henderson, S.,& Rodrigues, S. (2008). Scottish student primary teachers’ levels of mathematics competence and confidence for teaching mathematics: Some implications for national qualifications and initial teacher education. Journal of Teaching: International research and pedagogy, 34(2), 93–107. Doi: 10.1080/02607470801979533
Herrera, T., Kanold, T.D., Koss, R.K., Ryan, P., & Speer, W.R. (2007). Mathematics teaching today: Improving practice, improving student learning (2nd ed.). Reston, VA: National Council of Teachers of Mathematics.
Hill, T. & Lewicki, P. (2005). Statistics: Methods and applications. (Statsoft version). Retrieved on January o5, 2011 from:
Hussain, I. (2016). Effect of Bloom’s mastery learning approach on students’ academic achievement in English at secondary level. Journal of Literature, Languages and Linguistics, 23, 35 – 43.
Ezinwanyi, I. U. (2013). Enhancing mathematics achievement of secondary school students using mastery learning approach. Journal of Emerging Trends in Educational Research and Policy Studies, 4(6): 848-854. Retrieved on Jan. 12, 2014 from:
Jusoh, W.N.H.W. & Jusoff, K. (2009). Using multimedia in teaching Islamic Studies. Journal of Media and Communication Studies, 1(5), 086-094.
Keller, F. S. (1968). Goodbye Teacher. Journal of Applied Behavior Analyses, 1, 79-89. doi:10.1901/jaba.1968.1-79
Keller, F. S., and Sherman, J. G. (1974). The Keller Plan handbook. Menlo Park, CA: W.A. Benjamin.
Khitab, U. (2011). The Development of low cost learning material for the teaching of chemistry at secondary level (Unpublished M.phil thesis). City University of Science & Information Technology Peshawar, Pakistan.
Kiani, M. N., Malik, S., & Ahmad, S. I. (2012). Teaching of Mathematics in Pakistan - Problems and Suggestions. Language in India, 12, 293-306. Retrieved on Jan. 20, 2014 from:
Klein M. (2000).Teaching mathematics against the grain, investigation for primary teachers. Social Science Press , Austeralia.
Kulik, C. C., Kulik, J., & Bangert-Drowns, R. L. (1990). Effectiveness of mastery learning programs: A meta-analysis. Review of Educational Research, 60(2), 265 – 299.
Kulik, J.A., Kulik, C.C. and Cohen, P.A. (1979) .A meta-analysis of outcome studies of Keller’s personalized system of instruction. American Psychologist, 34(4), 307-318.
Kulm, G. (1980). Research on mathematics attitude. In: R.J. Shumway (Eds.). Research in mathematics education (pp. 356-387). Reston, VA: National Council of Teachers of Mathematics.
Kumar S. & Ratnalikar D.N. (2004). Teaching of Mathematics. Anmol Publication PVT. LTD.
Kazu, I. Y., Kazu, H., & Ozdemir, O. (2005). The effects of mastery learning model on the success of the students who attended “Usage of Basic Information Technologies” course. Educational Technology & Society, 8 (4), 233-243.
Kurumeh M.S., Jimin N. & Mohammed, A.S. (2012). Enhancing senior secondary students’ achievement in algebra using inquiry method of teaching in Onitsha educational zone of Anambra State, Nigeria. Journal of Emerging Trends in Educational Research and Policy Studies,3(6), 863-868.
Lamidi, B. T., Oyelekan, O. S. & Olorundare, A. K. (2015). Effects of Mastery Learning Instructional Strategy on Senior School Students’ Achievement in the Mole Concept. Electronic Journal of Science Education, 19 (5), 1-20.
Majeed, A. A., Darmawan, G. N. and Lynch, P. (2013). A Confirmatory factor analysis of attitudes toward mathematics inventory (ATMI). The Mathematics Educator, 15(1), 121-135
Matthew, B. M. & Kenneth, G. O. (2013). A study on the effects of guided inquiry teaching method on students achievement in logic. International Researcher, 2(1), 135-140. Retreiveed on Aug. 12, 2014 from: M.MATTHEW%20gambia.pdf
Mbugua, Z. K.,Kibet, K., Muthaa, M. M. & Nkonke,G. R. (2012). Factors contributing to students’ poor performance in mathematics at kenya certificate of secondary education in Kenya: A case of Baringo County, Kenya. American International Journal of Contemporary Research, 2(6), 87 – 91.
Mercer, D. (1986). Mastery Learning. British Journal of In-Service Education, 12(2), 115-118. 10.1080/0305763860120212
Ministry of Education, Singapore (2006). Secondary mathematics syllabuses. Curriculum Planning and Development Division, Singapore.
Mizrachi, A .(2010). Active-Learning Pedagogies as a Reform Initiative, American Institutes for Research.
Moradi, S. Zarei, E. & Zainalipour, H. (2012). Effects of mastery learning method on academic self-concept, academic achievement, and achievement motivation in female middle school students in Bandar Abbas city. Academic Journal of Psychological Studies, 1(1), 1-7. Retrieved on May, 5, 2014, from: http:p//ajps.articles/AJPS,%201,%201,%201-7,%202012.pdf
Morrison, H.C. ( 1926). The practice of teaching in the secondary school. Chicago: University of Chicago Press.
Mustafa, J. (2011). Development of integrated activity based mathematics curriculum at secondary level in North West Frontier Province, Pakistan (Doctoral dissertation, depatrment of education, faculty of social sciences, International Islamic University Islamabad, Pakistan). Retrieved on Feb. 27, 2013 from: 7551/1/ 1027S.htm
Nafees, M. (2011). An Experimental Study on the Effectiveness of Problem-based versus Lecture-based Instructional Strategies on Achievement, Retention and Problem Solving Capabilities in Secondary School General Science Student (Doctoral thesis, department of education, faculty of social sciences, Institutional Islamic University Islamabad). Retrieved on Feb. 27, 2013 from: 7554/1/ 1030S.htm
Nastu, J. (2009). Project-based learning: Engages students, garners results. eSchool News Special Report, 1, 21-27. Retrieved on Feb. 27, 2013 from: 888.999.9319
Niss, M. (1996). International handbook of mathematics education. Kluwer Academic Publisher, Netherland.
Nonye, A. N. and Mgbemena, C. O. (2012).The effect of using mastery learning approach on academic achievement of senior secondary school II physics students. Elixir International Journal, 51, 10735-10737. Retrieved on Jan. 12, 2014 from: (2012)%2010735-10737.pdf
Obioma, G. O. (2005). Emerging issues in mathematics education in Nigeria with empahasis on the strategies for effective teaching and learning of word problems and algebraic expressions. Journal of Issues on Mathematics. The Annual Publication of Mathematics Panel of the Science Teachers Association of Nigeria. 8 (1), 1-8.
Odumosu, M.O., Oluwayemi, M.O. & Olatunde, T.O. (2012). Mathematics as a tool in technological acquisition and economic development in transforming Nigeria to attain vision 20: 2020. Proceedings of Annual Conference of the Mathematical Association of Nigeria (pp.199- 207).
Ojerinde, D. (1999). Mathematics in technological development focus on the next millennium: Implication for secondary education in Nigeria. A lead paper at The 36th Annual Conference of MathematicsAssociation of Nigeria (MAN), Nigeria.
Otunu-Ogbisi R.O. (2009): Societal Mathematics and Education Reform: An agent for positive Development changes in his New Age. Abacus: The Journal of the Mathematical Association of Nigeria, 36(1), 18- 26.
Ozden, M. (2008). Improving science and technology education achievements using mastery learning model. World Applied Science Journal, 5(1), 62-67. Retrieved on December, 20, 2011, from
Papanastasiou, C. (2002). Effects of background and school factors on Mathematics achievement. Educ. Res. Eval, 8(1):55-70
Pirie, S., & Kieren, T. (1992). Creating constructivist environments and constructing creative mathematics. Educational Studies in Mathematics, 23, 505-528.
Patton, M. Q. (1990). Qualitative evaluation and research methodology. (2nd ed.). Newbury Park, CA: Sage Publication.
Perveen, K. (2009). Comparative effectiveness of expository strategy and problem solving approach of teaching mathematics at secondary level (Doctoral thesis, Arid Agriculture University Rawalpindi, Pakistan). Retrieved on Feb. 27, 2013 from:
Sarita & Sarivasta (2005). Curriculum and instructions. Isha Books, Delhi.
Scopes, P. G. (1973) Maths in secondary schools. Cambridge university press.
Schenkel, B. (2009) The impact of an attitude toward mathematics on mathematics performance (Master thesis). Retrieved on Feb. 12, 2018 from: file?accession=marietta1241710279&disposition= inline Marietta College
Shafie, N., Shahdan, T. N. T., Liew, M. S. (2010). Mastery learning assessment model in teaching and learning mathematics. Procedia Social and Behavioral Sciences, 8, 294 –298. Doi:10.1016/j.sbspro.2010.12.040.
Shannon, P. (2004). Implementing Mathematical Manipulatives in the Elementary classroom. Retrieved on Jan, 12. 2014 from: t&rct=j&esrc=s&source=web&cd=6&ved=OCEIQjAF&url=http:%3A2F%2Fpublic
Sharan R. & Sharma M. (2009). Teaching of Mathematics. A.P.H. Publishing Corporation, New Delhi.
Sidhu, K. S. (2003). Teaching of Mthematics. Sterling Publishers, New Delhi.
Singh, K. (2007). Quantative Social Research Methods. SAGE Publications, New Delhi.
Skinner, B.F. (1954). The science of learning and the art of teaching. Harward Education Review,24,86-97.
Sood, V (2013). Effect of mastery learning strategies on concept attaintmentin geometry among high school students. International Journal of Behavioral and Movement Sciences. 02(02), 144 – 155. Retrieved on Jan. 13, 2014 from:
http//ijobsms.in/vol2issue.13bp16.pdf
Tapia, M. & Marsh II, G. E. (2004). An instrument to measure Mathematics Attitudes. Available on personal contact from: cho25344l.htm
Udo, M. E. & Udofia, T. M. (2014). Effects of mastery learning strategy on students’ achievement in symbols, formulae and equations in chemistry. Journal of Educational Research and Reviews. 2(3), 28-35. Retrieved on June, 10, 2014 From:
.
Usman, K.O (2002) The need to re-train in services mathematics teacher for the attainment of the objectives of Universal Basic Education. The academic journal of the mathematical Association of Nigeria (MAN), 27(1), 19 – 29.
Usmani, S and Dawani, K. (2013). Teaching Methods and their impact on performance of University students. South Asian Journal of Management Sciences, 7(1), 19-30. Reterieved on Jan. 20, 2014 from:
Varughese, A. (2002). Effect of mastery learning strategy on certain cognitive and Personality variables of secondary school students in Kerala (Doctoral thesis, School of Pedagogical Sciences, Mahatema Gandhi University, India). Retrieved on November 13, 2011 from: =full
Wambugue, P. W. & Changeiywo J. M. (2008). Effects of mastery learning approach on secondary school Students’ physics achievement. Eurasia Journal of Mathematics, Science & Technology Education, 4(3), 293-302. Retrieved on Nov. 1, 2011, from: .
Washburne, C. W. (1922). Educational measurement as a key to individualizing instruct~on and promotions. Journal of Educational Research, 5, 195-206.
Wong, B. S., & Kang, L. (2012). Mastery learning in the context of university education. Journal of NUS Teaching Academy, 2(4), 206-222. Retrieved on Jan. 20, 2014, from: htpp://nus.edu.sg/teachingacademy/article/mastery-learning-in-context-of-university-education-2/#sthash.iWZT3urw.dpuf
Yemi (2015). Mastery learning Approach (MLA): its effect on the students’ mathematics academic achievement. International Journal of Vocational Education and Training Research, 1(2). 22-26. Retrieved on March. 20, 2018, from:
APPENDIX – A
ACHIEVEMENT TEST
(Pre-test)
[pic]
Class: 9th
Subject: Mathematics
Prepared by
Miss Tayyba Taj
Subject Specialist Mathematics
INSTITUTE OF EDUCATION AND RESEARCH
UNIVERSITY OF PESHAWAR
ACHIEVEMENT TEST FOR Class 9TH (PRE-TEST)
Marks: 50 Time: 2 hours
Name of student:_______________________________
School:________________________________________
1: Multiple choice questions. (10)
1. Choose the correct order for the solution of the questions.
A) Parenthesis, addition/ subtraction, division/multiplication.
B) Parenthesis, division/multiplication, subtraction.
C) Division/multiplication, subtraction, Parenthesis.
D) Parenthesis, division/multiplication, addition/subtraction.
2. [pic]
(A) [pic] (B) [pic]
(C) [pic] (D) [pic]
3. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
4. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
5. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
6. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
7. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
8. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
9. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
10. [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
2: Match the columns. (10)
|Column A |Column C |Column B |
|(1) |[pic] | |(a) |[pic] |
|(2) |[pic] | |(b) |[pic] |
|(3) |[pic] | |(c) |[pic] |
|(4) |[pic] | |(d) |[pic] |
|(5) |[pic] | |(e) |[pic] |
|(6) |[pic] | |(f) |[pic] |
|(7) |[pic] | |(g) |[pic] |
|(8) |[pic] | |(h) |[pic] |
|(9) |[pic] | |(i) |[pic] |
|(10) |[pic] | |(k) |[pic] |
3: Solution Questions. (30)
1. Addition: [pic], [pic], [pic]
2. Subtraction: [pic], [pic]
3. Multiplication: [pic]
4. Division: [pic]
5. Simplification: [pic]
6. Factorizing: [pic]
APPENDIX – B
ACHIEVEMENT TEST
(Post-test-1)
[pic]
Class :9th
Subject: Mathematics
Prepared by
Miss Tayyba Taj
Subject Specialist Mathematics
INSTITUTE OF EDUCATION AND RESEARCH
UNIVERSITY OF PESHAWAR
ACHIEVEMENT TEST FOR 9TH CLASS (POST-TEST-1)
Marks: 50 Time: 2 Hours
Name of Student:___________________________________
School: ___________________________________________
1: Fill in the blanks. (05)
i) [pic]
ii) [pic]
iii) [pic]
iv) [pic]
v) [pic]=_______________________
2: Tick (() on true and cross (() on false one. (10)
i) [pic] is a polynomial. ( )
ii) [pic] is the simplest form. ( )
iii) [pic] is a polynomial of zero degree. ( )
iv) [pic] ( )
v) [pic] ( )
vi) [pic] ( )
vii) [pic] ( )
viii) [pic] ( )
ix) [pic] ( )
x) [pic] ( )
3: Choose the correct one. (10)
|1. |[pic] | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |None of them | |
|2. |[pic] | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |None of them | |
|3. |[pic] | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|4. |[pic] | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|5. |[pic] | | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |None of them | |
|6. |[pic] | | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|7. |[pic] | | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|8. |Reciprocal of [pic] is ___________ | | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |None of them | |
|9. |[pic]= ______________ | | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|10. |Reciprocal of [pic] is ______ | | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |None of them | |
4: Solve the following questions. (25)
I) If [pic] then find the value of [pic].
II) Solve: [pic]
III) Find the value of [pic] when[pic].
IV) Find the value of [pic]when [pic] and [pic]
V) Find the value of [pic]when [pic]and [pic]
APPENDIX – C
ACHIEVEMENT TEST
(Post-test-2)
[pic]
Class :9th
Subject: Mathematics
Prepared by
Miss Tayyba Taj
Subject Specialist Mathematics
INSTITUTE OF EDUCATION AND RESEARCH
UNIVERSITY OF PESHAWAR
ACHIEVEMENT TEST FOR CLASS 9TH (POST-TEST-2)
Marks: 50 Time: 2 Hours
Name of Student:___________________________________
School: ___________________________________________
1: Fill in the blanks. (05)
i) [pic]
ii) [pic]
iii) [pic]
iv) [pic]
v) [pic]
2: Tick (() the true and cross (() the false. (10)
i) [pic] ( )
ii) [pic] ( )
iii) [pic] ( )
iv) [pic] ( )
v) [pic] ( )
vi) [pic] ( )
vii) [pic] ( )
viii) [pic] ( )
ix) [pic] ( )
x) [pic] ( )
3: Choose the correct option from the following. (05)
|i. |Factors of [pic]are ____ | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |[pic] |
|ii. |Factorize: [pic]. | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|iii. |Factorize: [pic]. | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|iv. |Factors of [pic] are _____ | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|v. |Simplify: [pic] | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |[pic] | |
4: Factorize the following sentences. (30)
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
APPENDIX – D
RETENTION TEST
Class :9th
Subject: Mathematics
Prepared by
Miss Tayyba Taj
Subject Specialist Mathematics
INSTITUTE OF EDUCATION AND RESEARCH
UNIVERSITY OF PESHAWAR
RETENTION TEST FOR CLASS 9TH
Marks: 50 Time: 2 Hours
Name of Student:___________________________________
School: ___________________________________________
Q1. Fill in the blanks. (10)
vi) [pic]
vii) [pic]
viii) [pic]
ix) [pic]
x) [pic]
xi) [pic]
xii) [pic]
xiii) [pic]
xiv) [pic]
xv) [pic] =_____________________
Q2. Tick (() the true and cross (() the false. (10)
xi) [pic] ( )
xii) [pic] ( )
xiii) [pic] ( )
xiv) [pic] ( )
xv) [pic] ( )
xvi) [pic] ( )
xvii) [pic] ( )
xviii) [pic] ( )
xix) [pic] ( )
xx) [pic] ( )
Q3. Choose the correct option from the following. (10)
|i. |[pic] | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|ii. |[pic] | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|iii. |[pic] | | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |None of them |
|iv. |[pic] | | |
| |A. |[pic] | |B. |[pic] | |
| |C. |[pic] |D. |None of them | |
|v. |[pic] | | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|vi. |Factors of [pic]are ____ | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |[pic] |
|vii. |Factorize: [pic]. | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|viii. |Factorize: [pic]. | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. | | |
| | | | |[pic] | |
|iv. |Factors of [pic] are _____ | |
| |A. |[pic] |B. |[pic] | |
| |C. |[pic] |D. |[pic] | |
|v. |Simplify: [pic] | |
| |A. |[pic] |B. |[pic] |C. |[pic] |D. |[pic] | |
Q4. Solve the following questions. (10)
VI) Find the value of [pic] when [pic].
VII) Find the value of [pic]when [pic] and [pic]
Q5. Factorize the following sentences. (10)
i) [pic]
ii) [pic]
APPENDIX-E
LESSON PLAN 1 FOR EXPERIMENTAL GROUP
(Note: two or three periods on the same day or different days will be utilized)
Subject: Maths Topic: (a+b)2
Class: 9th
Methods: Mastery learning (Explicit instruction, Peer tutoring & Cooperative learning)
Time: 45 Minutes
Objective:
Factorization of ( a + b)2
Material:
A set of algebra tiles which includes:
• Tiles of one unit square i.e. "a" unit by "a" unit square.
• Rectangle tiles i.e. "a" unit by "b" unit rectangles
• Squares tiles i.e. "b" unit by "b" unit squares
‘b’ unite by ‘b’ unit square ‘a’ unit by ‘b’ unit rectangle ‘a’ unit by ‘a’ unit square
|Headings |Matter and Methods |B.B. Work |
|Previous knowledge |In order to check the previous knowledge of the students the teacher will ask | |
|Test: (5 Min) |some question. | |
| |Question |Expected Answer | |
|Q1 |What is binomial? |An algebraic expression of the sum or | |
| | |the difference of two terms. | |
|Q2 |How can we express a binomial in | a + b | |
| |simplest form? | | |
|Q3 |Write binomial in square form: |(a + b)2 | |
| | |( a - b)2 | |
|Declaration of the |On the basis of last question the teacher will declared the topic. |“The square of binomial”|
|Topic: | | |
|Initial instruction by |The teacher will start the lesson by demonstrating the square of binomial |( a + b)2 |
|teacher |i.e. ( a + b)2 |(a+b)(a+b) |
|Procedure-1 |The teacher will explain that the square of binomial will turn out into |a2+2ab+b2 |
| |trinomial. | |
|Explanation by the |The teacher will explain that the square of any binomial produces the |The square of any |
|teacher |following three terms: |binomial has the |
| |The square of the first term of binomial i.e. a2 |following form: |
| |Twice product of the two terms i.e. 2ab |a2+2ab+b2 |
| |The square of the second term i.e. b2 | |
|Students’ activity 1 |Then teacher will write few questions on blackboard and will invite the |(2x+3)2 |
| |students one by one for solution. |(4x+9)2 |
| | |(5x2+y)2 |
|Students’ activity 2 |Different ability students will be paired to work together. The students will | |
|(peer activity) |be given time to solve questions from exercise book and the teacher will walk | |
| |around to see the students’ progress. | |
|Formative assessment-1 |The students will be assessed by administering a short test. | |
| |If the teacher thinks that all the students have mastered the concept she will| |
| |proceed to the next formula otherwise she will explain the concept with | |
| |procedure-2. | |
|Teacher’s explanation |Now the teacher will explain the square of binomial by using tiles to make the| |
|Procedure-2 |concept more clear: | |
| |[pic] | |
| | | |
| | | |
| |[pic] | |
|Students’ activity 3 |Teacher |Students | |
| |The teacher will show the following shape | | |
| |and will ask the students some questions |1. Each side is equal to “a”, so | |
| |about it: |“a” by “a” or a2 | |
| |1. How can you express the first square? |2. Each side is equal to “b” so | |
| |2. How can you express the second square? |“b” by “b” or b2 | |
| |3. How can you express the two rectangles?|3. Both have one side equal to “a” | |
| |4. If you rearrange the squares and |and one side equal to “b” so it it | |
| |rectangles making up the larger square, |2ab. | |
| |what do you have? |4. | |
| | | | |
| | | | |
| | |(a + b)2 | |
|Formative assessment-2 |Assessment will be administered by the teacher to see how many students have | |
| |mastered the unit. | |
|Group activity |Students will be divided into mixed ability groups and corrective assignment | |
|/ Corrective |will be given to those who have not mastered the concept yet. | |
| |The most brilliant students will be given the task of group leaders. Leader | |
| |will guide the group in their corrective assignments. | |
|Enrichment assignments |Those who have mastered will be given enrichment assignments. | |
| |The students will be given practice in “how to square a number mentally”. | |
| | | |
| | | |
| | | |
| |Questions |Expected solution by students | |
| |Square 52 |Solution | |
| | |52 = 50+2 | |
| | |The square of 50+2 = (50+2)2 | |
| | |(50+2)=2500 + 200 + 4 = 2704. | |
| |Square 24 |[pic] | |
| |Square 48 |[pic] | |
|Home Work |The students who have mastered the unit will be given enrichment assignments | |
| |while those who have not mastered yet will be given corrective with necessary | |
| |instructions. | |
|Assessment |Assessment on unit will be administered next day. | |
|Extra Time |Those students who have not mastered the concept will be given extra time. | |
APPENDIX-F
LESSON PLAN 2 FOR EXPERIMENTAL GROUP
(Note: two or three periods on the same day or different days will be utilized)
Subject: Maths Topic: (a+b)3
Class: 9th
Methods: Mastery learning (Explicit instruction, Peer tutoring & Cooperative learning)
Time: 45 Minutes
Objective:
Factorization of ( a + b)3
Material:
Soft cubes that can be cut with knife
|Headings |Matter and Methods |B.B. Work |
|Previous knowledge |In order to check the previous knowledge of the students the teacher will | |
|Test:(5 Min) |ask some question. | |
| |Question |Expected Answer | |
|Q1 |What is binomial? |An algebraic expression of the sum or | |
| | |the difference of two terms. | |
|Q2 |How can express a binomial in | a + b | |
| |simplest form? | | |
|Q3 |Write binomial in cube form. | (a + b)3, (a – b)3 | |
|Declaration of the |On the basis of last question the teacher will declared the topic and its |The cube of sum |
|Topic: |objective. |( a + b )3 |
|Initial instruction by |The Teacher will explain (a + b)3 in simple way of multiplication. | |
|teacher | |( a + b)3 |
|Procedure-1 | |(a+b)(a+b)(a+b) |
| | |a³+3a²b+3ab²+b3 |
|Explanation by teacher |The teacher will demonstrate the cube of binomial with the help of | |
|Procedure-2 |different cubes, i.e. ( a + b)3 |( a + b)3 |
| |[pic] |a³+3a²b+3ab²+b3 |
| |The teacher will explain the cube of sum with the help of cubes. | |
|Students’ activity 1 |The students will be divided into groups and each group will be given a | |
|(Guided practice) |cubic soap and a knife and they will be asked to illustrate (a+b)3 by | |
| |cutting soap into parts of formula. i.e. a³+3a²b+3ab²+b3 | |
|Students’ activity 2 |The students will be given the following questions for solution: | |
|(Group activity) |(3x+ 6y)3 | |
| |(X2+x2y2)3 | |
| |(a+b) (a+b)2 | |
| |(x+1/x)3 | |
|Formative assessment 1|The teacher will assess the students by giving them individual assignments.| |
| |If she thinks that all the students have learned cube of sum then next | |
| |activity will be given to the students. | |
|Students’ activity 3 |Find the value of a3+b3 ,when a+b =9 and ab = 20 | |
| |Find the value of a3+b3 ,when a+b =5 and ab = 6 | |
| |Find the value of x3+1/x 3, when x+1/x= 2 | |
| |Find the value of x3+1/x 3, when x+1/x= 5/2 | |
|Formative assessment 2 |The teacher will assess the students by giving them short assessment. On | |
| |the basis of result she will proceed to the next step. | |
|Corrective |The students will be divided into mixed ability groups. Those who have |(a+b)3 |
| |mastered the concept will be given the task of guiding the group. |(x+y)3 |
| |Simple and then complex question will be given for solution. Teacher will |(x2+y2)3 |
| |give time to each group in understanding the concept. |(xy+yz)3 |
| |Simple multiplication and then the steps of formula will be followed. |(3a+8y)3 |
| |After group work individual assignment will be given to students. |X3+y3 = ? x+y = 5 xy = 8 |
| | |a3+b3 = ? a+b = 20 ab = 15 |
|Home Work |The students who have mastered the unit will be given enrichment | |
| |assignments while those who have not mastered yet will be given corrective | |
| |with necessary instructions. | |
|Assessment |Assessment on unit will be administered next day. | |
|Extra Time |Those students who have not mastered the concept will be given extra time. | |
APPENDIX-G
LESSON PLAN 3 FOR EXPERIMENTAL GROUP
(Note: two or three periods on the same day or different days may be utilized)
Subject: Maths Topic: (a-b)3
Class: 9th
Methods: Mastery Learning (Explicit instruction, Peer tutoring & Cooperative learning)
Time: 45 Minutes
Objective:
Factorization of (a - b)3
Material:
Soft cubes that can be cut with knife
|Headings |Matter and Methods |B.B. Work |
|Previous knowledge |In order to check the previous knowledge of the students the teacher will | |
|Test: (5 Min) |ask some question. | |
| |Question |Expected Answer | |
|Q1 |Write binomial in cube form: | (a + b)3, (a –b)3 | |
|Q2 |Expand ( a+ b)3 |a³+3a²b+3ab²+b3 | |
|Q3 |Expand ( a – b)3 |(a-b) (a-b) (a-b) | |
|Declaration of the |On the basis of last question the teacher will declared the topic and its |The cube of difference:(a - |
|Topic: |objective. |b)3 |
|Initial instruction by |The teacher will explain formula (a - b)3 by multiplying: | |
|the teacher |(a-b) , (a-b) & (a-b) |( a - b)3 |
|Procedure-1 | |(a-b)(a-b)(a-b) |
| | |a³-3a²b+3ab²-b3 |
| | |a³-3ab(a-b) -b³ |
|Student Activity -1 |The students will be given the following questions for solution: | |
|(peer activity) |(2x – 3y )3 | |
| |(x2 – z3)3 | |
| |(ax - by)3 | |
|Explanation by teacher |With the help of students the teacher will convert | “ The cube of difference” |
|Procedure-2 |the formula (a-b)³=a³-3a²b+3ab²-b³ in to |3D-1 |
| |(a-b)³=a³-3ab(a-b)-b³ for an illustration. | |
| |The teacher will explain the cube by taking the |[pic] |
| |drawing of cube: (a+b)³ = a³+3a²b+3ab²+b³ |3D-2 [pic] |
| |Above cube will be replaced by the difference a-b. | |
| |The edges are (a-b) + b with different combinations | |
| |(3D-1). | |
| |The term (a-b)³ is illustrated by the blue cube | |
| |(3D-2). | |
|Students’ activity 2 |The students will be provided with the printed 3D shapes and will be asked to color different parts of |
|(Group activity) |formula:(a-b)³ = a³-3ab(a-b)-b³ = a³-3a²b+3ab²-b³ |
| |[pic] |
|Explanation by the |The teacher will explain that the cube of any binomial produces the |The cube of any binomial has |
|teacher |following four terms: |the following form: |
| |The cube of the first term of binomial i.e. a3 |a³-3a²b+3ab²-b³ |
| |Three time product of the two terms i.e. 3a2b, in which first term will be|a³+3a²b+3ab²+b³ |
| |in square. | |
| |Three time product of the two terms i.e. 3ab2, in which second term will | |
| |be in square. | |
| |The cube of the second term i.e. b3 | |
|Students’ activity 2 |The students will be given time to solve questions from exercise book and |1. (2xy-y)3 |
|(Peer activity) |the teacher will walk around to see the students’ progress. |2. (x3-x2y2)3 |
| | |3. (x2y2 -5xyz) |
|Assessment by teacher |The students will be given assessment and those who need more time and | |
| |guidance will be given as per requirement. | |
|Students’ activity 3 |After assessment the students will be given complex questions for solution| |
| |individually. | |
| |Find the value of a3-b3 ,when a-b =-1 and ab = 6 | |
| |Find the value of a3-b3 ,when a-b =5 and ab = 6 | |
| |Find the value of x3-1/x 3, when x-1/x= 3/2 | |
| |Find the value of x3-1/x 3, when x-1/x= 8/3 | |
|Corrective + Enrichment| The students who need more time to master the concept will be given “corrective”. The students who |
| |master the concept will be given enrichment assignment. |
|Assessment by the |Assessment will be administered by the teacher to see how many students | |
|teacher |have mastered the unit. | |
|Home Work |The students who have mastered the unit will be given enrichment | |
| |assignments while those who have not mastered yet will be given corrective| |
| |with necessary instructions. | |
|Assessment |Assessment on unit will be administered next day. | |
|Extra Time |Those students who have not mastered the concept will be given extra time.| |
APPENDIX-H
LESSON PLAN 4 FOR EXPERIMENTAL GROUP
(Note: two or three periods on the same day or different days will be utilized)
Subject: Maths Topic: Factorization
Class: 9th
Methods: Explicit instruction, Peer tutoring & Cooperative learning
Time: 45 Minutes
Objective:
Enable the students to factorize a given polynomial.
Material:
• Tiles of 1 unit square.
• Rectangle tiles that are one unit wide and "x" units long.
• Square tiles that are "x" units long by "x" units wide.
[pic]
|Headings |Matter and Methods |B.B. Work |
|Previous knowledge Test: |In order to check the previous knowledge of the students the teacher will | |
|(5 Min) |ask some question. | |
| |Question |Expected Answer | |
|Q1 |What is factor? |One of the two or more numbers that | |
| | |when multiplied, produce a given | |
| | |product. | |
|Q2 |What are the factors of “9”? |3 X 3 | |
|Q3 |Show the different way you can |3 X 3 | |
| |represent these factors? |3 . 3 | |
| | |(3)(3) | |
|Declaration of the Topic:|On the basis of last question the teacher will declared the topic and its |Factorizing polynomial |
| |objective. | |
|Initial instruction by |The teacher will start the lesson by demonstrating the factors of 9 (3 X 3)|3 X 3 |
|teacher |with the help of tiles. |[pic] |
|Students’ activity 1 |Then teacher will ask the students to show the factors of “4” and “16” by | |
| |drawing the shapes in their note books. | |
|Repetition by the teacher|If the teacher thinks it necessary to repeat this idea she will repeat it | |
| |with several whole numbers until she think that students’ concept have been| |
| |cleared. | |
|Presentation by the |Te teacher will write a polynomial on the black board and will make factors|x2+4x+3 |
|teacher |of that. The students’ will also be involved while doing each step. |x2+3x+x+3 |
| | |x(x+3)+1(x+3) |
| | |(x+1) (x+3) |
| |Students will be divided into mixed ability groups and the following | |
|Students’ activity 2 |sentences will be given for factorization: | |
|(Group activity) |x2+ 6x +8 | |
| |x2+6x + 5 | |
| |x2+4x + 3 | |
|Students’ activity 3 |The students will be given the following factors for multiplication: | |
| |(x+4)(x+2) | |
| |(x+5)(x+1) | |
| |(x+1)(x+3) | |
|Students’ activity 4 |Teacher |Students | |
| |The teacher will show the following shape | | |
| |and will ask the students to show it in | | |
| |quadratic equation. | | |
| | | | |
| |[pic] |(x2 + 4x + 3) | |
|Formative assessment |A short assessment will be administered by the teacher to know how much | |
| |student have mastered the concept. | |
| |The teacher will explain again (x2 + 4x + 3) by using tiles. The teacher | |
|Explanation |will arrange the tiles indifferent forms and will explain factoring. | |
|by the teacher |[pic] | |
|Procedure 2 | | |
|Individual activity |The students will be given individual assignments. After checking | |
| |assignment the teacher will came to know that how many students have | |
| |mastered the unit. | |
| |Students having will be paired and will be assigned questions from exercise| |
|Corrective |for solution. The brilliant student in pair will be instructed to guide her| |
|(peer activity) |friend. | |
|Home Work |The students who have mastered the unit will be given enrichment | |
| |assignments while those who have not mastered yet will be given corrective | |
| |with necessary instructions. | |
|Assessment |Assessment on unit will be administered next day. | |
|Extra Time |Those students who have not mastered the concept will be given extra time. | |
APPENDIX-I
LESSON PLAN 5 FOR CONTROL GROUP
Subject: Maths Topic: (a+b)2, (a-b)2
Class: 9th Method: Conventional Method
Period + Time: 3rd, (40 Minutes)
Objectives:
To enable the students to:
1. Factorize ( a + b)2
2. Factorize ( a - b)2
Material:
Text book, Black board, and chalk
|Headings |Matter and Methods |B.B. Work |
|Declaration of the Topic:|The teacher will declared the topic and will write it on black |“The square of binomial” |
| |board. | |
|Presentation by teacher |The teacher will start the lesson by demonstrating the square of | |
|(5 Min) |binomial and will ask the students to note down in their note |( a + b)2 |
| |book. |(a)2+ 2 (a)(b) + (b)2 |
| | |a2+2ab+b2 |
|Presentation by teacher |Then teacher will write a question on blackboard and will solve | |
|(5Min) |it according to the demonstrated formula. |(2x+3)2 |
| | |(2x)2+ 2 (2x)(3) + (3)2 |
| | |4x2+12x+9 |
|Students Activity |The teacher will ask the students to solve next question from | |
|(10 Min) |exercise. | |
|Presentation by teacher |The teacher will demonstrating the square of binomial having | |
|(10 Min) |negative sign, and will ask the students to note down in their |( a -b)2 |
| |note book. |(a)2- 2 (a)(b) + (b)2 |
| |The teacher will then solve a question from exercise. And |a2-2ab+b2 |
| |students will note it in their note books. |______________________ |
| | |( 2x -5y)2 |
| | |(2x)2- 2 (2x)(5y) + (5y)2 |
| | |4x2-20xy+25y2 |
|Students Activity |The teacher will ask the students to solve next question from | |
|(10 Min) |exercise. | |
|Home Work |The student will solve the remaining question of exercise at | |
| |home. | |
APPENDIX-J
LESSON PLAN 6 FOR CONTROL GROUP
Subject: Maths Topic: (a+b)3 , (a-b)3
Class: 9th Method: Conventional Method
Period + Time: 3rd, (40 Minutes)
Objective:
To enable the students to:
1. Factorize ( a + b)3
2. Factorize ( a - b)3
Material:
Text book, Black board, and chalk
|Headings |Matter and Methods |B.B. Work |
|Declaration of the Topic: |The teacher will declared the topic and will write it on black board. |The cube of sum and the cube of difference |
| | |( a + b )3, ( a - b )3 |
|Presentation by teacher |The teacher will start the lesson by demonstrating the cube of sum on writing|( a + b)3 |
|(10 Min) |board. |(a)³+3(a)²(b)+3(a)(b)²+(b)3 |
| |The teacher will then solve a problem from exercise book, i.e. (3x+ 6y)3 |a³+3a²b+3ab²+ b³ |
| | | |
| | |a³ - 3ab (a-b) - b³ |
| | |3x³ - 3(3x)(6y) (3x-6y) – (6y)³ |
| | |27x³ - 54xy (3x-6y) – 216y³ |
|Students’ Activity |The students will note down each step from writing board in their note book. | |
|(5 Min) | | |
|Presentation by teacher |The teacher will demonstrate the cube of difference on writing board. |( a - b)3 |
|(10 Min) |The teacher will then solve a problem from exercise book, i.e. (2x- 3y)3 |(a)³- 3(a)²(b) +3(a)(b)²- (b)3 |
| | |a³ -3a²b +3ab² - b3 |
| | |______________________ |
| | |a³ - 3ab (a-b) - b³ |
| | |(2x)³ - 3(2x)(3y) (2x-3y) – (3y)³ |
| | |8x³ - 36xy (2x-3y) – 27y³ |
|Students Activity |The teacher will ask the students to solve next question from exercise book. | |
|(10 Min) | | |
|Presentation by teacher |The teacher will demonstrating the square of binomial having negative sign, | |
|(10 Min) |and will ask the students to note down in their note book. |( a -b)2 |
| |The teacher will then solve a question from exercise. And students will note |(a)2- 2 (a)(b) + (b)2 |
| |it in their note books. |a2-2ab+b2 |
| | |( 2x -5y)2 |
| | |------------------------ |
| | |(2x)2- 2 (2x)(5y) + (5y)2 |
| | |4x2-20xy+25y2 |
|Students Activity |The teacher will ask the students to solve next question from exercise. | |
|(10 Min) | | |
|Home Work |The student will be asked to solve the remaining question of exercise at | |
| |home. | |
APPENDIX-K
ATTITUDES TOWARD MATHEMATICS INVENTORY (ATMI)
Directions: This inventory consists of statements about your attitude toward mathematics. There are no correct or incorrect responses. Read each item carefully. Please think about how you feel about each item. Darken the circle that most closely corresponds to how the statements best describes your feelings. Use the following response scale to respond to each item.
PLEASE USE THESE RESPONSE CODES:
A – Strongly Disagree
B – Disagree
C – Neutral
D – Agree
E – Strongly Agree
1. Mathematics is a very worthwhile and necessary subject.
2. I want to develop my mathematical skills.
3. I get a great deal of satisfaction out of solving a mathematics problem.
4. Mathematics helps develop the mind and teaches a person to think.
5. Mathematics is important in everyday life.
6. Mathematics is one of the most important subjects for people to study.
7. High school math courses would be very helpful no matter what I decide to study.
8. I can think of many ways that I use math outside of school.
9. Mathematics is one of my most dreaded subjects.
10. My mind goes blank and I am unable to think clearly when working with mathematics.
11. Studying mathematics makes me feel nervous.
12. Mathematics makes me feel uncomfortable.
13. I am always under a terrible strain in a math class.
14. When I hear the word mathematics, I have a feeling of dislike.
15. It makes me nervous to even think about having to do a mathematics problem.
16. Mathematics does not scare me at all.
17. I have a lot of self-confidence when it comes to mathematics
18. I am able to solve mathematics problems without too much difficulty.
19. I expect to do fairly well in any math class I take.
20. I am always confused in my mathematics class.
21. I feel a sense of insecurity when attempting mathematics.
22. I learn mathematics easily.
23. I am confident that I could learn advanced mathematics.
24. I have usually enjoyed studying mathematics in school.
25. Mathematics is dull and boring.
26. I like to solve new problems in mathematics.
27. I would prefer to do an assignment in math than to write an essay.
28. I would like to avoid using mathematics in college.
29. I really like mathematics.
30. I am happier in a math class than in any other class.
31. Mathematics is a very interesting subject.
32. I am willing to take more than the required amount of mathematics.
33. I plan to take as much mathematics as I can during my education.
34. The challenge of math appeals to me.
35. I think studying advanced mathematics is useful.
36. I believe studying math helps me with problem solving in other areas.
37. I am comfortable expressing my own ideas on how to look for solutions to a difficult problem in math.
38. I am comfortable answering questions in math class.
39. A strong math background could help me in my professional life.
40. I believe I am good at solving math problems.
© 1996 Martha Tapia
APPENDIX-L
URDU TRANSLATION OF ATMI
نوٹ: ہر جملہ غور سے پڑھیں اور سوچیں کہ آپ اس بارے میں کیا محسوس کرتے ہیں۔ پھر جوابات کے خانے میں آپ کو جو جواب اپنی سوچ سے قریب تر محسوس ہو، اُس خانے میں([pic])کا نشان لگائیں۔
| |جملے |با لکل نہیں|نہیں |کچھ کہہ |ہاں |ہاں یقیناً |
| | | | |نہیں سکتی | | |
|۱) |ریاضی ایک کار آمد اور ضروری مضمون ہے۔ | | | | | |
|۲) |ریاضی کےسوالات حل کرنے میں مجھے بہت اطمینان ملتا ہے۔ | | | | | |
|۳) |ریاضی زہنی نشونما میں مدد دیتا ہےاورفردکو سوچناسکھاتاہے۔ | | | | | |
|۴) |میں ریاضی میںاپنی مہارت بڑھاناچاہتی ہوں۔ | | | | | |
|۵) |روز مرہ زندگی کیلئےریاضی ایک اہم مضمون ہے۔ | | | | | |
|۶) |ریاضی تمام لوگو کے مطالعے کیلئے نہایت اہم مضمون ہے۔ | | | | | |
|۷) |چاہے میں آگے کچھ بھی پڑھنےکافیصلہ کروں، ہائی سکول کی ریاضی کا| | | | | |
| |کورس میرے لئے بہت مدد گار ہوگا۔ | | | | | |
|۸) |سکول سے باہر بھی میں ریاضی کو بہت سے طریقوں سے استعمال کر | | | | | |
| |سکتی ہوں۔ | | | | | |
|۹) |ریاضی کا کام کرتے ہوئے میرا ذہن بالکل خالی ہو جاتا ہے اور میں| | | | | |
| |سوچنے سمجھنے کے قابل نہیں رہتی۔ | | | | | |
|۱۰) |ریاضی کا کام کرتے ہوئے میں گھبراہٹ کا شکار ہو جاتی ہوں۔ | | | | | |
|۱۱) |ریاضی کے سوالات مجھے بے چین اور بے آرام کر دیتے ہیں۔ | | | | | |
|۱۲) |میں ریاضی کے پیریڈ میں سخت ذہنی دباؤ محسوس کرتی ہوں۔ | | | | | |
|۱۳) |ریاضی کا نام سنتے ہی مجھے ناپسندیدگی کا احساس ہوتا ہے۔ | | | | | |
|۱۴) |یہ سوچنا ہی مجھے نروس کر دیتا ہے کہ مجھےریاضی کا سوال حل کرنا| | | | | |
| |ہے۔ | | | | | |
|۱۵) |مجھے ریاضی کے مضمون سے کوئی گھبراہٹ محسوس نہیں ہوتی۔ | | | | | |
|۱۶) |میں ریاضی کے کلاس میں بہت زیادہ خود اعتمادی محسوس کرتی ہوں۔ | | | | | |
|۱۷) |مجھے ریاضی کےسوالات حل کرنے میں زیادہ مشکل نہیں ہوتی۔ | | | | | |
|۱۸) |میں توقع کرتی ہوں کہ ریاضی کی کسی بھی کلاس میں اچھا کام | | | | | |
| |کرونگی۔ | | | | | |
|۱۹) |میں ریاضی کے پیریڈ میں اُلجھی ہوئی رہتی ہوں۔ | | | | | |
|۲۰) |ریاضی کے سوالات حل کرتے ہوئے مجھے ڈر سا محسوس ہوتا ہے۔ | | | | | |
|۲۱) |میں ریاضی بہت آسانی سے سیکھتی ہوں۔ | | | | | |
|۲۲) |مجھے خود پر بروسہ ہے کہ جدید ریاضی سیکھ سکتی ہوں۔ | | | | | |
|۲۳) |ریاضی کےمطالعے سے میں اکثر لُطف اندوز ہوتی ہوں۔ | | | | | |
|۲۴) |ریاضی ایک خشک اور بور مضمون ہے۔ | | | | | |
|۲۵) |ریاضی کے نئے سوالات حل کرنا مجھے پسند ہے۔ | | | | | |
|۲۶) |میں ایک مضمون لکھنے کے مقابلے میں ریاضی کے سوالات حل کرنے کو | | | | | |
| |ترجیح دُونگی۔ | | | | | |
|۲۷) |میں کالج میں ریاضی نہیں پڑھونگی۔ | | | | | |
|۲۸) |مجھے واقعی ریاضی پسند ہے۔ | | | | | |
|۲۹) |دوسری کلاسوں کے مقابلے میں مجھے ریاضی کی کلاس میں زیادہ خوشی | | | | | |
| |محسوس ہوتی ہے۔ | | | | | |
|۳۰) |ریاضی بہت دلچسپ مضمون ہے۔ | | | | | |
|۳۱) |میں ریاضی کا کام مطلوبہ مقدار سے زیادہ کرتی ہوں۔ | | | | | |
|۳۲) |میں ریاضی کا اتنا ہی کام لینے کی منصوبہ بندی کرتی ہوں جتنا | | | | | |
| |اپنی تعلیم کے دوران کر سکتی ہوں۔ | | | | | |
|۳۳) |ریاضی کی پیچدگی مجھے اچھی لگتی ہے۔ | | | | | |
|۳۴) |میرا خیال ہے کہ اختیاری ریاضی پڑھنا فائدہ مند ہے۔ | | | | | |
|۳۵) |مجھے یقین ہے کہ ریاضی مجھے دوسری جگہوں پر بھی مسائل حل کرنے | | | | | |
| |میں مدد دیگی۔ | | | | | |
|۳۶) |ریاضی کے مشکل سوالات کے حل کیلئے میں اطمینان سے اپنے خیالات | | | | | |
| |کا اظہار کرتی ہوں۔ | | | | | |
|۳۷) |میں ریاضی کی کلاس میں آسانی سے جوابات دیتی ہوں۔ | | | | | |
|۳۸) |ریاضی کی مضبوط بنیاد آئندہ پیشہ ورانہ زندگی میں مدد گار ہے۔ | | | | | |
|۳۹) |مجھے یقین ہےکہ میں ریاضی کے مسائل حل کرنے میں اچھی ہوں۔ | | | | | |
APPENDIX-M
EMAIL FROM MARTHA TAPIA
Dear Lubna,
You will need this information for the analysis.
Scoring Key
The following items are reverse items.
For your analysis, use the formula e.g item12 = 6 - item12 to get the correct value.
9. Mathematics is one of my most dreaded subjects.
10. My mind goes blank and I am unable to think clearly when working with mathematics.
11. Studying mathematics makes me feel nervous.
12. Mathematics makes me feel uncomfortable.
13. I am always under a terrible strain in a math class.
14. When I hear the word mathematics, I have a feeling of dislike.
15. It makes me nervous to even think about having to do a mathematics problem.
20. I am always confused in my mathematics class.
21. I feel a sense of insecurity when attempting mathematics.
25. Mathematics is dull and boring.
28. I would like to avoid using mathematics in college.
Subscales
Self-confidence : Items 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 & 40
Value: Items 1, 2, 4, 5, 6, 7, 8, 35, 36 & 39
Enjoyment: Items 3, 24, 25, 26, 27, 29, 30, 31, 37 & 38
Motivation: Items 23, 28, 32, 33 & 34
Hope this helps,
Martha Tapia
Martha Tapia, Ph.D.
Associate Professor
Department of Mathematics and Computer Science
Berry College
P.O. Box 495014
Mount Berry, GA 30149-5014
APPENDIX-N
LIST OF EXPERTS
|S. No |Name |Address |
|1 |Dr. Arshad Ali |Ph.D (Education), Professor, Institute of Education and Research, University of |
| | |Peshawar |
| |Mr. Gul Shad Khan |Sr. Instructor , PITE, Peshawar |
|3 |Dr. Bilal |Ph.D in Education, Sr. Instructor , PITE, Peshawar |
|4 |Mrs. Badeya Danish |Subject Specialist (Mathematics), Instructor in PITE, Peshawar. |
|5 |Mrs. Atia Amjed |M.Phil in Economics, Principal GGHS, Kaskoroona Mardan |
|6 |Mr. Amjed |Professor (Mathematics), Post Graduate College Madran. |
|7 |Mrs. Saeeda Akhtar |Principal GGHS, No: 1, Mardan. |
|8 |Miss Tayyaba Taj |Subject Specialist (Mathematics), GGHSS, Shehbaz Garhi Mardan. |
|9 |Mr. Riaz Khan |Lecturer , M.Phil Mathematics, The Mardan Model School Mardan. |
|10 |Mrs. Zainab Bibi |Principal GGHSS, Shehbaz Garhi Mardan. |
|11 |Mr. Alamgeer |Ph.D in Education, Master in Mathematics, Principal GHS, Hatyan |
|12 |Mr. Shah Zarin |Subject Specialist (Physics), Instructor in RITE, Mardan. |
APPENDIX-O
NO OBJECTION CERTIFICATE -1
Miss Lubna Toheed is allowed to conduct research study at GGHS, No. 1 Mardan and take experimental classes in 9th grade for the partial fulfillment of the degree of Ph.D in education.
____________________
Mst. Saeeda Akhtar
Principal
Govt.Girls High
School No.1 Mardan
APPENDIX-P
NO OBJECTION CERTIFICATE -2
Miss Lubna Toheed is allowed to conduct research study at GGHSS Shehbaz Garhi, Mardan and take experimental classes in 9th grade for the partial fulfillment of the degree of Ph.D in education.
____________________
Mst. Zainab Bibi
Principal
Govt.Girls Higher Secondary
School Shahbaz Garhi
APPENDIX-Q
EXPERIMENT COMPLITION CERTIFICATE-1
It is certified that Miss Lubna Toheed has successfully completed experimental study at GGHS, No.1 Mardan for the partial fulfillment of the degree of Ph.D in education.
____________________
Mst. Zainab Bibi
Principal
Govt.Girls Higher Secondary
School Shahbaz Garhi
APPENDIX-R
EXPERIMENT COMPLITION CERTIFICATE-2
It is certified that Miss Lubna Toheed has successfully completed experimental study at GGHSS, Shehbaz Garhi, Mardan for the partial fulfillment of the degree of Ph.D in education
____________________
Mst. Zainab Bibi
Principal
Govt.Girls Higher Secondary
School Shahbaz Garhi
APPENDIX- S
TEST SCORES OF GROUP-1 (EXPERIMENTAL)
|S.No |pre test |TEST-1 |TEST-2 |R-TEST |
|1 |10 |20 |25 |25 |
|2 |4 |15 |25 |24 |
|3 |20 |34 |40 |44 |
|4 |22 |40 |50 |50 |
|5 |0 |14 |34 |30 |
|6 |3 |15 |28 |30 |
|7 |18 |25 |45 |43 |
|8 |25 |30 |50 |50 |
|9 |2 |15 |23 |18 |
|10 |0 |20 |20 |15 |
|11 |15 |28 |35 |30 |
|12 |24 |30 |45 |45 |
|13 |12 |40 |45 |45 |
|14 |6 |25 |35 |33 |
|15 |8 |30 |40 |30 |
|16 |5 |18 |25 |20 |
|17 |12 |24 |30 |28 |
|18 |15 |30 |47 |46 |
|19 |16 |33 |45 |40 |
|20 |2 |20 |30 |25 |
|21 |24 |35 |48 |50 |
|22 |30 |45 |50 |50 |
|23 |23 |28 |35 |35 |
|24 |0 |15 |24 |20 |
|25 |12 |34 |44 |45 |
|26 |25 |30 |36 |35 |
|27 |18 |34 |35 |30 |
|28 |12 |24 |30 |25 |
|29 |19 |45 |50 |50 |
|30 |45 |50 |50 |50 |
|31 |28 |45 |50 |50 |
|32 |12 |33 |38 |34 |
|33 |8 |25 |30 |28 |
|34 |5 |26 |35 |30 |
|35 |3 |20 |34 |25 |
|36 |24 |39 |50 |45 |
|37 |9 |28 |42 |30 |
|38 |27 |45 |50 |50 |
|39 |6 |30 |40 |42 |
|40 |8 |28 |45 |25 |
|41 |10 |30 |38 |35 |
|42 |20 |40 |50 |48 |
|43 |0 |27 |30 |25 |
|44 |15 |30 |40 |45 |
|45 |6 |34 |44 |40 |
|46 |4 |27 |34 |25 |
|47 |18 |35 |50 |50 |
|48 |12 |30 |40 |40 |
|49 |18 |38 |47 |48 |
|50 |14 |40 |50 |40 |
|51 |22 |40 |50 |50 |
|52 |8 |33 |40 |30 |
|53 |15 |35 |45 |30 |
|54 |5 |27 |35 |25 |
|55 |18 |37 |44 |40 |
|56 |8 |28 |45 |45 |
|57 |16 |30 |45 |35 |
|58 |0 |28 |35 |25 |
|59 |18 |35 |40 |40 |
APPENDIX-T
TEST SCORE OF GROUP-2 (CONTROL)
|S.No |Pre-Test |TEST-1 |TEST-2 |R-TEST |
|1 |4 |10 |15 |10 |
|2 |20 |18 |20 |12 |
|3 |3 |15 |15 |5 |
|4 |0 |12 |16 |2 |
|5 |14 |25 |22 |15 |
|6 |15 |20 |22 |25 |
|7 |7 |15 |15 |10 |
|8 |18 |24 |12 |2 |
|9 |20 |30 |30 |15 |
|10 |2 |14 |15 |6 |
|11 |0 |6 |7 |0 |
|12 |24 |30 |45 |30 |
|13 |14 |22 |20 |15 |
|14 |15 |12 |15 |8 |
|15 |20 |15 |20 |10 |
|16 |30 |45 |40 |33 |
|17 |18 |20 |25 |20 |
|18 |8 |18 |20 |25 |
|19 |16 |20 |22 |12 |
|20 |14 |18 |20 |14 |
|21 |27 |30 |25 |20 |
|22 |28 |27 |18 |15 |
|23 |9 |18 |15 |0 |
|24 |12 |25 |25 |10 |
|25 |25 |30 |30 |26 |
|26 |6 |15 |14 |10 |
|27 |20 |30 |25 |24 |
|28 |12 |20 |20 |20 |
|29 |0 |12 |8 |0 |
|30 |23 |25 |30 |24 |
|31 |40 |50 |50 |45 |
|32 |3 |5 |8 |4 |
|33 |25 |33 |35 |30 |
|34 |9 |25 |25 |15 |
|35 |6 |9 |10 |8 |
|36 |10 |26 |30 |25 |
|37 |15 |22 |25 |20 |
|38 |14 |15 |20 |22 |
|39 |12 |30 |20 |15 |
|40 |6 |8 |10 |4 |
|41 |45 |46 |48 |45 |
|42 |12 |23 |14 |15 |
|43 |3 |15 |15 |15 |
|44 |2 |20 |30 |12 |
|45 |18 |30 |30 |20 |
|46 |19 |30 |34 |25 |
|47 |12 |24 |25 |24 |
|48 |4 |20 |24 |20 |
|49 |18 |35 |30 |30 |
|50 |14 |30 |32 |15 |
|51 |0 |10 |0 |20 |
|52 |18 |28 |25 |22 |
|53 |15 |28 |30 |25 |
|54 |6 |15 |15 |15 |
|55 |34 |32 |33 |30 |
|56 |20 |20 |25 |10 |
|57 |3 |12 |18 |12 |
|58 |28 |25 |25 |18 |
|59 |4 |10 |15 |10 |
APPENDIX-U
TEST SCORE OF GROUP-III (EXPERIMENTAL)
|S.No |Pre-Test |TEST-1 |TEST-2 |R-TEST |
|1 |20 |38 |40 |35 |
|2 |15 |30 |37 |35 |
|3 |35 |46 |48 |50 |
|4 |8 |29 |34 |30 |
|5 |25 |40 |42 |44 |
|6 |15 |30 |34 |30 |
|7 |12 |28 |30 |28 |
|8 |5 |25 |42 |36 |
|9 |28 |45 |40 |45 |
|10 |15 |37 |35 |32 |
|11 |17 |46 |48 |45 |
|12 |30 |45 |50 |48 |
|13 |25 |40 |37 |39 |
|14 |35 |45 |50 |50 |
|15 |15 |30 |32 |30 |
|16 |8 |28 |28 |30 |
|17 |25 |33 |35 |37 |
|18 |15 |24 |25 |18 |
|19 |20 |25 |27 |28 |
|20 |20 |36 |38 |37 |
|21 |17 |30 |35 |33 |
|22 |25 |40 |44 |45 |
|23 |15 |35 |35 |35 |
|24 |15 |44 |48 |40 |
|25 |18 |38 |40 |45 |
|26 |20 |43 |45 |45 |
|27 |11 |30 |40 |40 |
|28 |20 |38 |37 |37 |
|29 |19 |38 |40 |44 |
|30 |14 |44 |48 |47 |
|31 |18 |40 |43 |44 |
|32 |6 |26 |28 |24 |
|33 |20 |36 |37 |30 |
|34 |16 |30 |30 |33 |
|35 |18 |41 |45 |45 |
|36 |5 |26 |28 |28 |
|37 |18 |43 |45 |44 |
|38 |10 |40 |40 |38 |
|39 |22 |38 |40 |30 |
|40 |8 |25 |30 |25 |
|41 |0 |28 |30 |30 |
|42 |10 |27 |44 |40 |
|43 |24 |30 |34 |35 |
|44 |8 |27 |28 |20 |
|45 |17 |35 |40 |38 |
|46 |24 |40 |45 |43 |
|47 |22 |32 |36 |30 |
|48 |8 |30 |42 |44 |
PPENDIX-V
TEST SCORES OF GROUP-IV (CONTROL)
|S.No |Pre-Test |TEST-1 |TEST-2 |R-TEST |
|1 |25 |36 |35 |40 |
|2 |15 |10 |8 |15 |
|3 |10 |15 |20 |14 |
|4 |30 |40 |45 |45 |
|5 |20 |25 |24 |15 |
|6 |30 |32 |35 |30 |
|7 |15 |22 |24 |20 |
|8 |33 |30 |28 |20 |
|9 |15 |18 |12 |10 |
|10 |18 |40 |40 |30 |
|11 |10 |12 |24 |15 |
|12 |25 |30 |25 |25 |
|13 |25 |28 |30 |20 |
|14 |15 |26 |24 |22 |
|15 |12 |20 |22 |20 |
|16 |25 |34 |48 |50 |
|17 |25 |33 |30 |25 |
|18 |14 |18 |25 |20 |
|19 |12 |25 |15 |15 |
|20 |15 |18 |20 |15 |
|21 |25 |33 |35 |30 |
|22 |20 |22 |23 |15 |
|23 |18 |21 |23 |10 |
|24 |10 |20 |25 |20 |
|25 |15 |20 |20 |14 |
|26 |30 |40 |30 |25 |
|27 |30 |35 |32 |32 |
|28 |12 |20 |18 |20 |
|29 |35 |48 |50 |40 |
|30 |10 |19 |11 |10 |
|31 |8 |26 |25 |20 |
|32 |12 |20 |13 |10 |
|33 |30 |45 |40 |40 |
|34 |15 |15 |20 |12 |
|35 |15 |20 |15 |14 |
|36 |20 |28 |30 |35 |
|37 |16 |10 |20 |10 |
|38 |12 |18 |19 |25 |
|39 |14 |15 |12 |5 |
|40 |15 |18 |17 |17 |
|41 |6 |14 |8 |6 |
|42 |20 |27 |28 |20 |
|43 |16 |30 |10 |15 |
|44 |15 |22 |25 |20 |
|45 |15 |15 |20 |20 |
|46 |2 |5 |15 |5 |
|47 |10 |15 |10 |18 |
|48 |10 |20 |30 |25 |
-----------------------
Academic achievement
Aptitude (Time needed to learn)
Perseverance
(Time willingly spent in learning)
Quality of instructional events
Ability to understand instructions
Opportunity to learn
(Time to learn)
Mastered
Pre-test
ATMI
Unit 1
Formative Assessment A
Enrichment Activities
Correctives
Formative Assessment B
Post-test-1
Unit 2
Mastered
Non-mastered
Formative Assessment A
Correctives
Formative Assessment B
Enrichment Activities
Post-test-2
ATMI
Retention Test
Non-mastered
................
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