IMPACT OF IN-SERVICE TRAININGS ON THE PEDAGOGICAL SKILLS OF MATHEMATICS TEACHERS AT SECONDARY LEVEL IN PUNJAB

SUMAIRA KHAN

Reg.No. 06/IU.Ph.D/2013

DEPARTMENT OF EDUCATION

FACULTY OF EDUCATION

The Islamia University of Bahawalpur

PAKISTAN

2019

IMPACT OF IN-SERVICE TRAININGS ON THE PEDAGOGICAL SKILLS OF MATHEMATICS TEACERS AT SECONDARY LEVEL IN PUNJAB

By

SUMAIRA KHAN

Reg.No. 06/IU.PhD/2013

A thesis submitted in fulfillment of the requirements for the award of the degree of

Doctor of Philosophy

in

Education

Under the supervision of

DR. MUHAMMAD RAMZAN

DEPARTMENT OF EDUCATION

The Islamia University of Bahawalpur

FACULTY OF EDUCATION

PAKISTAN

2019

AUTHOR’S DECLARATION

I, Sumaira Khan Registration No. 06/IU.Ph.D/2013, Ph.D. Scholar of Session 2013-2017 in the Department of Education of the Islamia University Bahawalpur declare that the thesis entitled, “Impact of in-service trainings on the pedagogical skills of mathematics teachers at secondary level in Punjab” submitted by me in partial fulfillment of the requirement of Ph.D. in the subject of Education is my original work. I assure that it has not been submitted or published anywhere. It shall also not be submitted to obtain any degree to any other university or institution up till now.

SUMAIRA KHAN

FORWARDING CERTIFICATE

The research entitled “Impact of in-service trainings on the pedagogical skills of mathematics teachers at secondary level in Punjab.” is conducted under my supervision and the thesis is submitted to The Islamia University Bahawalpur in the partial fulfillment of the requirements for award of the degree of Doctor of Philosophy in Education with my permission.

Dr. Muhammad Ramzan

Supervisor

Certificate of Approval

This is to certify that the research work presented in this thesis, entitled “Impact of in-service trainings on the pedagogical skills of mathematics teachers at secondary level in Punjab.” was conducted by Mrs. Sumaira Khan under the supervision of Dr. Muhammad Ramzan.

No part of this thesis has been submitted anywhere else for any other degree. This thesis is submitted to the Department of Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Field of Education, Department of Education The Islamia University Bahawalpur.

Student Name: Sumaira KhanSignature: __________

Examination Committee:

a) External Examiner 1: Name Signature:___________

(Designation & Office Address)

…………………………………..

…………………………………..

b) External Examiner 2: NameSignature:___________

(Designation & Office Address)

…………………………………..

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c) Internal Examiner: NameSignature:___________

(Designation & Office Address)

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Supervisor Name:Signature:____________

Dr.Muhammd Ramzan

Name of HOD:Signature:_____________

Professor Dr. Nasreen Akhtar

THIS THESIS IS

DEDICATED TO

MY HUSBAND, WHOSE LOVE AND CARE ARE ASSESTS OF MY LIFE,

MY GREAT LOVING PARENTS

AND

MY SWEET DAUGHTERS VANEEZA, IZMA AND BAREENA WHO ARE BLESSINGS FOR US

ACHKNOWLEDGEMENTS

I am very thankful to the Allah Almighty the most compassionate and merciful for granting me able to achieve this big task. I wish to express my sincere appreciation and thanks to Dr. Muhammad Ramzan, the supervisor of my dissertation, who has contributed his lot of time and efforts for the accomplishment of the task. I also extend my gratitude to Prof. Dr. Akhtar Ali the Dean Faculty of Education for his kind directions and cooperation

I am humbly thankful to Dr. Shamim Haider Tirmazi for his timely guidance and encouragement. I am thankful to my class mates, Mr. Zubair Haider, Mr. Zafar, Ms.Shahista ,Mr. Sajjad, Mr. Naeem and Ms. Mahwish. My thanks will not be completed if I will not take the name of my friends Nusrat Tariq and Riffat Iqbal. I am really very thankful to my colleagues Samia Fahim and Ms.Shamim because without their undoubted help and guidance, I was unable to complete this task in time. I am thankful to all those whose work I benefitted. Finally, I also desire to express my gratefulness to my sisters Nadia Khan, Dr. Maria Khan and brothers Shahzad Ahmed Khan and Javed Ahmed Khan who helped me and prayed for my success.

SUMAIRA KHAN

Table of ContentsCHAPTER 11INTRODUCTION11.1 STATEMENT OF THE PROBLEM51.2 OBJECTIVES OF THE STUDY51.3. RESEARCH QUESTIONS61.4. SIGNIFICANCE OF THE STUDY61.5 DELIMITATIONS71.6. DEFINATION OF IMPORTANT TERMS7CHAPTER 29LITERATURE REVIEW92.1 Teacher Education92.2 Maths Education and its Significance122.2.1 Importance of Mathematics in other subjects142.2.2 Responsibilities of a Math teacher162.3 For Trainings, Effective Pedagogy of Mathematics182.3.1 Important Features ofTrainings (Characteristics of Effective Pedagogy of Mathematics)222.3.2 The subject matter preparation for effective training242.3.3 Importance of multiple solution approaches for effective pedagogy in developing math understandings272.3.4 Applications of effective mathematics pedagogy282.4 Mathematical understanding for secondary level math Teachers (MUST)292.4.1 Role of math teacher in promotion of classroom relationship322.4.2 Role of math teacher in making tool and its representation332.4.3 Role of math trainer in assessmentactivities352.5 Instructional models that work for maths teachersin trainings362.6 How disabilities can affect Maths achievement and their remedial methods392.7 Over view of trainings for maths teachers in the world432.7.1 Trainings of maths teachers in USA442.7.2 Trainings of maths teachers in Germany452.7.3 Trainings of maths teachers in China462.7.4 Comparison between U.S.A and China472.7.5 Trainings of maths teachers in India472.7.6 Importance of in-service training in Turkey492.8 Training scenario in Punjab (Pakistan)492.8.1 Mission and Vision of Staff Development Directorate (Punjab) / QAED (Quaid –e- Azam Academy for Education Development)502.8.2Role of DSD/QAED502.8.3 Role of the Directorate of Staff Development (DSD) in Teacher Training in Punjab502.8.4 Primary school teachers’ competency level in mathematics and in science subjects512.8.5 Frame work of CPD (Continuous professional Development) by DSD in Punjab Schools522.8.6 Comparison of students’ achievement in mathematics by trained and un-trained teachers (in Punjab)53CHAPTER 355RESEARCH METHODOLOGY553.1 Research design553.2 Population and Sampling553.3 Research instrument573.3.1 Development of Data Collection Tool583.3.2 Questionnaire for students623.3.3 Interview653.3.4 Observation Sheet653.5 Research assistants ‘training663.6 Reliability and Validity of Instruments663.7 Pilot Study663.8 Data Collection673.9 Data Recording673. 10 Data Analysis68CHPTER 469DATA ANALYSIS AND INTERPRETATION694.1 Demographic Characteristics of Respondents (Teachers)694.2 Factor-Wise Explanation of Teachers’ Data784.3 Interview1054.4 ObservationError! Bookmark not defined.4.4.1: Personal aspects of teachers1144.4.2: Teacher preparation and planning1154.4.3 Start of the lesson1174.4.4: Teachers’ performance1194.4.5. Student Activity1234.4. 6:Student-Teacher interaction1264.4.7-End of Lesson1274. 5 Demographic Characteristics of Respondents (Students)130CHAPTER 5152SUMMARY, FINDINGS, DISCUSSIONS, CONCLUSION AND RECOMMENDATIONS1525.1 Summary1525.2 Findings1535.3 Discussion1685.4 Conclusion1755.5 Recommendations177References180

LIST OF TABLES

Table 3.1Detail of Selected Sample57

Table 3.2 Detail of selected trial of teachers and students for each district57

Table 3.3: Factors wise reliability of teacher’s questionnaire58

Table 3.3.1: Factor wise statements of Approach of trainings59

Table 3.3.2.1: Factors wise reliability of student’s questionnaire.62

Table 3.3.2.3: Factor wise statements of the Understanding of Students63

sTable3.3.2.5: Factor wise statements about assessment of Students64

Table 4.1.3: Teaching Experience Wise Distribution of Teachers71

Table 4.1.5: Academic Qualification Wise Distribution of Teachers73

Table 4.1.8: Training received in last three years wise distribution of teachers76

Table 4.2.1: Frequency, mean and SD regarding as Approach of trainings78

Table 4.2.2: Frequency, mean and SD regarding Pedagogy as skills and strategies80

Table 4.2.3: Frequency, mean and SD regarding Pedagogy as understanding of students83

Table 4.2.4: Frequency, mean and SD regarding Pedagogy as Motivation84

Table 4.2.5: The Pearson Correlation Matrix85

Table 4.2.6: Comparing the Effect of Teaching Experience on Approach of Training86

Table 4.2.7: Comparing the Effect of Teaching Experience on pedagogy as skills and strategies86

Table 4.2.8: Comparing the Effect of Teaching Experience on pedagogy as understanding of students87

Table 4.2.9: Comparing the Effect of Teaching Experience on pedagogy as motivation87

Table 4.2.10: Comparing the Effect of Teaching Experience on for all factors88

Table 4.2.11: Comparing the Effect of academic qualification on Approach of Training88

Table 4.2.12 Comparing the Effect academic qualification on Teaching skills and strategies89

Table 4.2.13: Comparing the Effect of Academic Qualification on Understanding of students89

Table 4.2.14: Comparing the Effect of Academic qualification on pedagogy as motivation90

Table 4.2.15 Comparing the Effect of Academic qualification on all factors90

Table 4.2.16 Comparing theEffect of professional qualification on Approach of Training91

Table 4.2.17: Comparing theEffect of professional qualification on skills and strategies91

Table 4.2.18: Comparing theEffect of professional qualification on understanding of students92

Table 4.2.19: Comparing theEffect of professional qualification on pedagogy as motivation92

Table 4.2.20: Comparing theEffect of professional qualification on all factors93

Table 4.2.21: Comparing theEffect of name of training agency on approach of trainings93

Table 4.2.22: Comparing theEffect of Training agency on skills and strategies94

Table 4.2.23:Comparing theeffect of teaching experience on as understanding of students.94

Table 4.2.24: Comparing theeffect of teaching experience on approach of training of teachers95

Table 4.2.25: Comparing theEffect of training agency on all factors95

Table 4.2.26: Simple Linear Regression of Teachers’ Approach of Training and Teachers’ Skills and Strategies96

Table 4.2.27: Simple Linear Regression of Teachers’ Approach of Training and pedagogy as understanding of students96

Table 4.2.28: Simple Linear Regression of Teachers’ Approach of Training and Student’s motivation96

Table 4.2.29: t-test: Gender Wise Comparison of approach of trainings97

Table 4.2.30: t-test: Gender-wise comparison of Pedagogy as skills & strategies98

Table 4.2.31:t-test: Gender Wise Comparison of pedagogy as Understanding ofStudents98

Table 4.2.32 t-test: Gender Wise Comparison of Pedagogy as Motivation99

Table 4.2.33: t-test School-wise comparison of as approach of trainings99

Table 4.2.34 t-test School-wise comparison of Pedagogy as skills & strategies100

Table 4.2.35: t-test School-wise comparison of pedagogy as understanding of students100

Table 4.2.36: t-test: School -wise comparison of pedagogy as Motivation101

Table 4.2.37: t-test: Subject-wise comparison of as approach of trainings101

Table 4.2.38:t-test: Subject-wise comparison of Pedagogy as skills & strategies102

Table 4.2.39: t-test: Subject -wise comparison of pedagogy as understanding of students102

Table 4.2.40:t-test: Subject -wise comparison of pedagogy as Motivation103

Table 4.2.41: t-test: Date of joining -wise comparison of as approach of trainings103

Table 4.2.42: t-test: Date of joining -wise comparison of Pedagogy as skills & strategies104

Table 4.2.43:t-test: Date of joining -wise comparison of as understanding of students104

Table 4.2.44:t-test: Date of joining -wise comparison of pedagogy as Motivation105

Table 4.3.1 Impact of trainings on their attitude and teaching106

Table 4.3.2: Teachers ‘objectives of trainings107

Table 4.3.4: Agenda of trainings110

Table 4.3.5: Weaknesses and strengths of trainings112

Table 4.3.6: Suggestions for improvement of trainings113

Table 4.4.1.1: Pitch/flow of the voice114

Table 4.4.1.2: communication Style114

Table 4.4.1.4: Confidence115

Table 4.4.2.1: Lessons sequenced116

Table 4.4.2.2: Neatly presented lesson plan116

Table 4.4.2.3: Teaching aids Teaching aids (A.V. Aids)116

Table 4.4.3.3: Students’ understanding aboutSLOs (Student learning outcomes)118

Table 4.4.4.1: Command on subject119

Table 4.4.4.2: Class management119

Table 4.4.4.4: Appropriateness of the content120

Table 4.4.4.5: Continuity and coherence121

Table 4.4.4.6: Use of writing board121

Table 4.4.4.7: work sheet preparation121

Table 4.4.4.8: Supervision of students work122

Table 4.4.4.10: Student’s assessment procedure123

Table 4.4.5.2: Involvement in lesson124

Table 4.4.5.3: Participation in classroom activities124

Table 4.4.5.4: Group activities125

Table 4.4.5.5: Understanding and comprehension125

Table 4.4.6.2 Student- teacher communication126

Table 4.4.6.3 Classroom atmosphere127

Table 4.4.7.1 Appropriate summarization127

Table 4.4.7.2: Achievement of SLOs128

Table 4.4.7.3: Homework note books129

Table 4.4.7.4: Teaching methods129

Table 4.5.1 Gender Wise Distribution of Students130

Table 4.5.2: School -wise distribution131

Table 4.5.6: Frequency, Mean and SD regarding Understanding of Students136

Table 4.5.7: Frequency, mean and SD regarding Motivation137

Table 4.5.8: Frequency, Mean and SD regarding Assessment of Students139

Table 4.5.9: The Pearson Correlation Matrix141

Table 4.5.10: Comparing the Effect of Parents’ Education on Skills and Strategies141

Table 4.5.11 Comparing the Effect of parents’ education on students understanding142

Table 4.5.12: Comparing the Effect of parent’s education on motivation142

Table 4.5.13: Comparing the Effect of parent’s education on assessment of students143

Table 4.5.14: Simple Linear Regression of In-service Trainings and Skills and Strategies143

Table 4.5.15: Simple Linear Regression of in-service trainings and students’ understanding144

Table 4.5.16: Simple Linear Regression of Approach of in-service Trainings and Student’s motivation144

Table 4.5.17: Simple Linear Regression of In-service Trainings and student’s assessment145

Table 4.5.18: Gender Wise comparison with strategies and techniques145

Table 4.5.19: Gender Wise Comparison of student understands146

Table 4.5.20: Gender Wise Comparison of Motivation146

Table 4.5.21: Gender Wise Comparison of students’ assessment147

Table 4.5.22: School Wise Comparison of strategies and techniques147

Table 4.5.23: School Wise Comparison of students’ understanding148

Table 4.5.24: School Wise Comparison of students’ motivation148

Table 4.5.25: School Wise Comparison of students’ assessment149

Table 4.5.26: Class Wise Comparison of strategies and techniques149

Table 4.5.27: Class Wise Comparison of students understanding150

Table 4.5.28: Class Wise Comparison of students’ motivation150

Table 4.5.29: Class Wise Comparison of student’s assessment151

List of Figures

Figure 4.1.1: Gender wise distribution of teachers70

Figure 4.1.2: School wise distribution70

Figure 4.1.3: Teaching Experience Wise Distribution of Teachers71

Figure 4.1.4: Subject Wise Distribution of Teachers72

Figure 4.1.5: Academic Qualification Wise Distribution of Teachers73

Figure 4.1.6: Professional Qualification Wise Distribution Teachers74

Figure 4.1.7: Date of Joining Wise Distribution of Teachers75

Figure 4.1.8: Training received in last three years wise distribution of teachers76

Figure 4.1.9: Name of training agency wise distribution77

Figure: 4.3.1: Teachers ‘objectives of trainings106

Figure: 4.3.2: Teachers ‘objectives of trainings107

Figure: 4.3.3: Difference in teaching after trainings109

Figure: 4.3.4: Agenda of trainings111

Figure: 4.3.5: Weaknesses and strengths of trainings112

Figur4.3.6: Suggestions for improvement of trainings113

Figure 4.5.2: School Wise Distribution131

Figure 4.5.3: Parents’ education-wise distribution132

Figure 4.5.4: Class- wise distribution133

LIST OF ABBREVIATIONS

ANOVA: Analysis of Variance

CK: Content Knowledge

CPD: Continuous professional Development

DSD: Directorate of Staff Development

DTE: District Teacher Educator

GCET: Government College for elementary teachers

KCS: Knowledge of Content and Students

MSA: Multiple solution approaches

MUST: Mathematical understanding for secondary level math teachers

NACTE: National council for accreditation of teacher education

PCK: Pedagogical content knowledge

PITE: Punjab institute of teacher education

PK: Pedagogical knowledge

QAED: Quaid-e-Azam academy for educational development

SMK: Subject matter knowledge

SST: Secondary school teacher

TIMSS: Trends in international mathematics & science study

SD: Standard Deviation

Abstract

Teachers have a significant responsibility in the society and a trained teacher is considered to be familiar with students’ learning stages. A teacher always seeks for knowledge and in order to refresh his/her knowledge, and training during the service play an important role. In these trainings, teachers get chances to sharpen their skills, maximize their pedagogical knowledge and minimize their weak points present in their instruction. Present research investigated the impact of trainings on the pedagogical skills of mathematics teachers at secondary level in province Punjab. There was a descriptive study in nature and survey design of descriptive study was used to assemble data. Population of the study encompassed secondary schools’ teachers and their student. Sampling technique of multistage was used to assess the real representative sample from the population. Research sample was collected of 464 teachers and their 987 students from 10 districts of Punjab. Vast and deep reviews of related literature ease the researcher to make difference scales. Two questionnaire one for teacher and other for their students were developed, one semi conducted interview and an observation sheet for teachers were also developed. The questionnaire for teacher was consisted of four factors including Approach of trainings, Pedagogy as skills and strategies, pedagogy as understanding of students and pedagogy as motivation. The questionnaire for students was also consisted of four factors which are as follows: Skills and strategies, understanding of students, motivation and assessment of students. For a better understanding of students, questionnaire was converted into Urdu language and got opinion of the experts was included for the validation. For the analysis of the data, SPSS19th version was used. All scale and subscales demonstrated high and moderate reliability of data. For the statistical analysis of data, researcher used Standard Deviation, Mean, ANOVA, Independent Sample t-test Coefficients, Pearson Correlation and Regression Analysis. The finding told that there was a moderate positive correlation among all factors but it was found that association between approach of trainings and motivation was weak but positive and in case of students’ questionnaire, the factor skills and strategies have also weak but positive association with motivation. From the result, it was found that both male and female teachers had almost dissimilar opinion regarding as approach of trainings in skills and strategies and in motivation. A linear relationship among the variables of the study was found. The study projected additional research in the area to find out the impact of in-service trainings on elementary and primary school teacher of mathematics.

xiii

CHAPTER 1INTRODUCTION

Teacher is an imperative element for the success of any educational system. Teachers are scholars who aspire to build the career of the students and construct a favorable situation in which they instruct and prepare students for the development of a competent nation (Slavin, 2008). The all education system is designed for students and the role of teacher has a key position in whole system(Marks, 2000). Teachers are given an important task (Knuth, 2002; Kramarski & Mevarech, 2003). The teachers should update their knowledge and have good awareness of students’ psychology, teacher’s duty is not completed at this stage but he/she should be familiar with how to apply this knowledge. In order to achieve this task, teachers are supposed to educate with poise in their domains with the help of latest pedagogical skills which will suits to students and will be helpful in enhancing their capabilities. All of it depend on their qualification, training and approach towards their profession and how much they are interested in the wellbeing of the learner (Borg, 2011).

A trained teacher is considered to be familiar with students’ learning phases and also concerned with latest knowledge which may contribute in improving teaching and accepting social changes (Jackson & Bruegmann, 2009).There is a saying that teachers are born, not made can be reversed on the basis of latest in-service teaching trainings(Glover & Miller, 2001). In such workshops, teachers get a chance to sharpen their skills, maximize positive interest and minimize weak points exhibited in their instruction (Ma, 1999).The process of trainings may help in bridging the gaps between modern methodology and traditional teaching methods. In teaching profession, people come with faith, values, dedication to change themselves and to seek knowledge which may reflect from their behavior, styles and attitudes. Teachers have the objectives to get proper training so that they cope with the challenges in their profession (Ashcraft, 2002).

Studies show that in-service trainings in Pakistan have positive effects on teacher’s pedagogical skills. It has been come to know that there existed a remarkable difference between those teachers who have got in-service trainings and who have not (Ross & Bruce, 2007). The trainings ‘participants insisted to increase the duration of trainings and also some financial benefits (Saeed & Mahmood, 2002). Innovation in technology and new trends in education require competent teachers; they increase their competency level by utilizing maximum resources and techniques (Kebritchi, Hirumi, & Bai, 2010). It is expected that teacher generally documented children’s learning activities to enhance their abilities (Kramarski & Mevarech, 2003).

To meet the challenges of the society, it is essential to prepare teachers having the abilities to understand the modern trends. It seems that all these are missing in the educational curriculum, the organization and the instruction. The latest reforms in mathematics education are based on recognizing the important role of teacher of mathematics teaching at secondary school level and through in-service trainings, role of math teachers can be considered as curriculum innovators and implementers of these skills and knowledge (Watson, 2006). It is understood that initial training of teachers helps them in improving their pedagogical skills and content knowledge, but it is not sufficient to prepare teachers for the greater challenge where time limitation and pressure from summative assessment become hindrance for teachers (Jacob & Lefgren, 2004). To meet this challenge education department gives priority to in-service trainings (Kramarski & Mevarech, 2003).

The need for adequate preparation and professional development of teachers, conceptions and attitudes play an significant function in teaching learning procedure (Lewin, 2008). Three major factors for the quality of education of students are: creativity, social and emotional adjustability. The way to achieve this is the quality and efficiency of the teachers. It has been suggested that if teachers have negative attitude and beliefs, it directly effects on students ‘learning. These three factors are most debatable topic among general public and policy makers (Wilkins & Ma, 2003). It directly relates the quality of instruction in classrooms. For the better quality of teachers, a global trend to refresh the knowledge of teachers is common. For this purpose, different training programmes are conducted regularly under the supervision of Government sector and other donor agencies (Ali, Akhter, & Khan, 2010). In Pakistan, teacher education is given a highly attention (Halai, 2011)

Two important factors in teacher education are, input factors and process factors, in the perception of head teachers and teachers in input factors are teachers, resources and the community. Whereas process factors are related to proper interaction among teachers and students’ activities but it is unfortunate that in our classrooms, teachers still use conventional way of teaching, dictation and helping material in the shape of notes. Sometimes trainers fail to develop necessary skills and strategies in teachers Mostly trainers are not having required knowledge (Hussain, Ali, Khan, Ramzan, & Qadeer, 2011).

In-service trainings make a teacher more confident in circulating knowledge; they learn to link their understanding with latest methods and information. Mathematics provides a foundation for understanding of almost all subjects. It is common perception that knowledge related science and technology is incomplete for its description without mathematics and it helps in boosting the learning of other subjects.

In an education process, mathematics is considered as an important subject at school level. Math helps students in growing logical thinking by working like a tool which individuals can use in all spheres of life. Teaching process requires an understanding of latest techniques and it is difficult to get good results only with the help of traditional methods (Halai, 2006)

In order to analyze the effects of training in-service on the pedagogical skills, it is important to have deep observation and knowledge of implemented curriculum, methods of instructions, ways of assessment and the interpersonal role of teacher to get the better results (Tayyaba, 2010). The other important factor is mentoring; junior teachers are guided by their seniors. Instruction being used in any training plays an important role in assessing the effectiveness of teacher’s preparation (Baumert et al., 2010). The four components are important for in any teaching model. These are; Focus (objectives, target, and goals to be achieved), Syntax (Sequence of steps in organizing teaching), Social system (teacher role+ student role), Support of system (mechanism, equipment, teaching aids)

In National Mathematics Advisory Panel Report, successful mathematical teaching was discussed and it provide as additional support for the four methods of instruction (2008). For improvement of the structure of in-service trainings on mathematics teaching at school level, Govt. of Pakistan and especially Govt. of Punjab has serious concerns. For this purpose, different steps have been taken.

In Punjab, institutions PITE (Teachers Education Institute of Punjab) and DSD (Staff Development Directorate) are working for this purpose. Their main functions are to enhance the excellence of teachers and teaching and also to evaluate their performances, therefore, they regularly train the teachers of mathematics and science at secondary and primary level in all districts of Punjab by conducting subject oriented and pedagogical workshops (Jahangir, Saheen, & Kazmi, 2012). They put emphasis on integration of teacher education with field experiences which is important for better results. Learner-centered, activity based, and participatory learning experienced teachers’ trainings may helpful for teachers to clarify their concepts for good teaching (Perveen, 2010).

In teaching and learning process, understanding self and others, developing habits and capacity for self-directed learning, involving with subject content, and developing professional skills in pedagogy, observation, documentation, analysis and clear interpretation may be result oriented. Preparation of mathematics teachers in Punjab is a continuous process at secondary and primary school level. Persons having conventional degrees are bound to get the professional teaching degrees to become a teacher. After induction in education department they get the trainings from especially established public sector institutions. It is a process that continues in all their service which helps them to groom and to prepare for accepting the changing requirements of the world. Teachers in-service training programs are supposed to enhance the capabilities of teachers with the altering trends of curriculum (Shah, Kiani, Mahmood, & Hussain, 2011). In 36 districts of Punjab Province, 6941 higher secondary and secondary schools are working and there are 32 Elementary Teachers Training Institutions providing in service training to public sector teachers.

Teachers training programs are controlled by DSD (Staff Development Directorate) (Javed, Juan, & Nazli, 2012). In spite of all these initiatives, there is a need to evaluate that whether trainings are fulfilling their objectives which are especially designed for the teachers of mathematics at secondary level of Punjab schools, if not then what are the basic causes for this failure and at what extent, for this researcher apply the validated tool and try to achieve the set objectives

1.1 STATEMENT OF THE PROBLEM

Teachers of mathematics preparation, especially in teacher education institutions are normally overlooked in Pakistan. It is hard to judge their effectiveness and their continuous professional development, which is the sole source for their improvement. In Punjab PITE (Institute of Teacher Education Punjab) and DSD (Staff Development Directorate)/QAED (new name of DSD i.e. Quaid e Azam Academy for Educational Development) are conducting a series of trainings to enhance and refresh the knowledge of Mathematics teachers. It seems necessary to assess that are these professional trainings imparting latest knowledge relevant to pedagogy? Therefore, the current study is designed to analyze the Impact of In-Service Trainings on the Pedagogical skills of Mathematics Teachers at Secondary stage in Province Punjab.”

1.2 OBJECTIVES OF THE STUDY

Objectives of the study are as follows:

1. To study the effectiveness of in- service trainings on the teaching skills of mathematics teachers.

2. To analyze teaching methods applied by teachers for teaching of mathematics

3. To observe the pedagogical skills and attitude of teachers towards students during teaching in their classes.

4. To evaluate the teachers’ satisfaction on in-service trainings. What are different agencies participating in teaching trainings and their apparent and hidden objectives.

5. To locate the problems of math teachers during and after getting in-service trainings

1.3. RESEARCH QUESTIONS

The research study is an attempt to answer the following questions

1. Do in-service trainings have effectiveness on the teaching skills of mathematics teachers?

2. What the teaching methods are applied for teaching of mathematics?

3. What are the pedagogical approach and expertise of teachers toward the student in process of teaching in their classes?

4. What is the teacher’ satisfaction level on in-service trainings?

5. What are the problems faced by maths’ teachers during and after getting in-service trainings?

1.4. SIGNIFICANCE OF THE STUDY

After the review from literature, researcher is convinced that this research would be important in recognition of the area where changes are necessary. The methodology of research provided, point of view of SST’s of mathematics.

This study would be beneficial for mathematics trainers who impart trainings, this study would provide a base to remove flaws of trainings indicated by maths teachers and ultimately students will be benefited. The study will be useful for the planners of teacher-training programs in developing better curriculum and methodology of math for secondary school teachers. The study would also be helpful in getting overview of the developed countries about pedagogy adopted by their math teachers, which may give insight to teacher and training institutional heads involved in preparing mathematics teachers in Punjab. The findings of this study would be helpful to inculcate useful changes in curriculum and teaching of mathematics. This study might be useful for maths teachers who got in- service trainings. This research will be helpful in analyzing whether teachers improved their instructional strategies, methodologies and pedagogical knowledge for their students. The study will be useful for the planners of teacher-training programs in developing better curriculum and methodology for moths’ teachers.

The study would be supportive in getting overview of the developed countries about pedagogy adopted by their maths teachers, which may give insight to those who conduct training sessions in preparing Maths teachers in Punjab (Pakistan). The findings of this study would help to inculcate useful changes in training agenda. In Pakistan still teacher’s trainings are not giving proper results at the same ratio as expected which are reflected by the outcomes of students learning. There is need to develop a model for trainings of Maths teachers which reflects the combination of old and new strategies. In order to improve the deficiencies, present in teaching of mathematics at secondary level is important to focus on, professional development trainings. If professional trainings are well planned and managed ultimately improve the pedagogical skills of Math teachers.

1.5 DELIMITATIONS

Study was bordered to only math teachers (female and male) working as math teachers at secondary school level in Punjab and students remained under their guidance. Sample was collected from six districts of lower Punjab and four districts of upper Punjab.

1.6. DEFINATION OF IMPORTANT TERMS

Pedagogy: The science or art of teaching

Pedagogical Skills: Different methods of teaching, communication skills, Counselling with students, making link between curriculum and real words adopted by a teacher in classrooms are Called pedagogical skills.

Secondary School Teacher: Teachers (male and female) teaching maths subject to 9th and 10th classes

DSD/QAED: DSD means Directorate of Staff Development. Its function is to enhance and get better result of teaching by conducting trainings of all subjects especially for science and math subjects of Punjab. On July 2017 its name is changed into QAED i.e. Quaid e Azam Academy for Educational Development.

PITE: PITE (Institute of Teacher Education Punjab). Formerly the conduction of trainings was the responsibility of PITE and after that DSD assisted the PITE for trainings in Punjab.

CHAPTER 2LITERATURE REVIEW

This chapter illustrates the pertinent literature about the given topic in a detailed and sequential manner. The researcher reviewed the literature regarding the effectiveness of professional trainings on the pedagogical skills of mathematics teachers and also observed the change in teaching attitude caused by In-service trainings. All the relevant aspects like teacher education, Mathematics education its need, significance & strategies to overcome the fear of the subject, importance of in-service trainings, and overview of in-service trainings in Punjab (Pakistan) are discussed here.

2.1 Teacher Education

The teacher education is expected to improve their beliefs and performance in pedagogy. Teacher education is smoothly moving from quantitatively to quality education (Slavin, 2008). At each and every level teacher education is being discussed, but still there are little researches which show impact of teacher’ trainings on the betterment of students (Borg, 2011). If we are serious to enhance the quality of students, it must be given emphasis on teacher education quality. That’s why there is a need of good trainings aligning with the curriculum and quality books. The methods of teaching of each subject demands different techniques and strategies (Mewborn, 2003). Teacher education is directly associated with students’ behavior i.e. for students, there are so many topics that they cannot search for themselves, sometimes without teachers they are helpless and they need guidance so, communicating is the important part of teacher education.

In most of the countries where teacher education is being studied, this subject i.e. Communication skills is taught as a compulsory subject (Baumert et al., 2010). It is the important feature of teacher education. Knowledge must be delivered in a systematic manner, in this way teaching becomes interesting. Some people believed that knowledge can be transferred from one person to another. Actual teaching is based in persuading the student. Teacher act as an instrument in helping a student and this is a way for students to understand what a teacher wants to teach.

Teacher Education provides a base for teachers to learn more and more about children psychology. For this purpose, different topics have been included in the agenda of trainings related to child psychology. Students get encouragement and stimulation from their teachers. From trainings teachers understand the value of right things in right manner at right time. They are especially told that time, resources and energy should not be wasted (Harris & Sass, 2011). In poor and under developed countries they learn how to make low-cost material as A.V. aids for instruction. This makes learning easy, interesting and effective for them. Unfortunately, during the training of teachers, it is the most neglected part. Along with effective teaching, there should be a system of support both in their motivational and emotional aspects. This will improve the level of success of students and their self-efficacy but unfortunately their emotional training is not up to the mark (Baylor, Shen, Warren, & Park, 2004). At some extent, these days’ teachers’ education system is facing problems of incompatibility and unrelated syllabus in their curriculum. There is a prominent gap between the lives how students live and they learn in schools and colleges. Students and teachers are living in the age of technology and no doubt they are surrounded by this technology in their lives. Although schools and college have tried to keep their speed as the rate of change of in students’ lives but still there is a lot of gap because especially students’ lives in multitasking all-around technology motivated and energetic world. But this concept of driven technology is still missing at some extent in schools, colleges and in teachers’ education. If administration of teachers’ education is serious in quality and quantity, they should assure that prospective and in-service teachers have equally access to latest technology regardless of their economical background (Cochran-Smith & Fries, 2001). Then it might be possible to bridge out these mentioned gaps between modern methodology and existing traditional teaching methods.

If a teacher is our centre of focus then automatically student will be. Last few decades are busier than ever to find out new teaching trends, techniques and cultural adaptability in teacher education. There is a need to sought out and conceptualize a proper model. That model will suit which is a combination of latest technology and tradition but based on the geographical regional requirement by providing good professionals. So, one can say that it is not necessary that a model which is successful in developed countries just like UK, USA and Australia and it may also successful in under developed countries because they have their own problems and priorities (Fuchs et al., 2009). Now we can negate the saying that teachers are born, not made on the basis of latest techniques of teaching training. Now in developed countries, it is believed that if we want to improve the quality of teaching ultimately, we should work on to improve the quality of teacher and try to focus his capacity building (Darling-Hammond, 2000).

Effects of Teachers’ Efficiency on their Teaching

For effectiveness of teachers, Kennedy (2008) has identified following three areas

1. Cognitive resources

1. Presentation in the class room

1. Effects on learners

Special features of quality of teaching are, these three aspects which provide a frame work (Kennedy, 2008). Although two programmes such as No Child Left Behind and Race to the Top have changed the quality of teaching (Jahng, 2011). According to Kennedy, teacher inputs which can be viewed as tests taken by teachers and students performance reflect limited portion of teacher efficiency. Researchers are focusing on teachers’ performance, along with the attitude and behavior of teachers (Ushioda, 2008). Latest performances of training on in-service persisting the spotlight on this area.

Main difference between previous and present form of classroom practices needs to focus on how these behaviors may change. Present practice-based teacher education movement shows similarity from three other periods in history (McDonald, Kazemi, & Kavanagh, 2013). Anyhow, three major differences exist that differentiate present movement from its previous one: (a) Higher order skills goals for students rather than basic skills of past movements,(b) student teacher must learn about in teaching uncertainty and skills to manage it and (c) more focus on the subject matter as opposed strictly in past(Knight et al., 2015). On the basis of above said researches, teacher quality requires three types of qualities- resources, presentation, and outcome on students to reflect a center on their learning. There are many subjects which have been focused in teacher education but Mathematics education has an important place in teacher education. There are many researches which indicate that mostly students are frightened from maths students face many problems in understanding math (Yeager et al., 2016)

Researchers believe that they have to search for the facts of mathematics that should be taught to teachers with an open view of mathematics content, not just adding more content to teachers preparation programme (Wood, 2004).Teachers have to be experienced two constructing views about learning mathematics, one from university/ college mathematics courses and the other from mathematics education courses in teacher education institutions (Siemon, Virgona, Lasso, Parsons, & Cathcart, 2004).It is observed that across the countries in teacher education programs, there are differences with respect to subject matter preparation.

The difference depends on how much the country can invest in teacher preparation programme. However, still in developed countries just like USA the focused problem is what and how much of mathematics must be taught and significant effective teaching in teacher preparation programme still exists. Another research, focusing on SMK (subject matter knowledge) and valuable teaching internationally, has identified similar problems and challenges for different countries ( Davis, Hersh, & Marchisotto, 2011).According to Shulman(1986), for teaching maths which is a complex activity, teachers need to have proper content knowledge of subject, pedagogical content knowledge and knowledge of curriculum

2.2 Maths Education and its Significance

From the early 1900s at elementary level effectiveness of teaching of mathematics has been studied. Mathematics which is in general considered to be one of the dull and dry subjects in the school curriculum. If the teacher is successful in locating and arousing interest of the child depending upon his mental level, he can help the child to a greater extent in gaining more and more knowledge effectively (Abelev et al., 2006).

There is no doubt that Maths is multidimensional and composed of a lot of interest. No one can avoid Maths in his life if he/she wants to lead a happy and successful life, we should take interest in Maths. It is the duty of a Maths teacher to put all his effort to make the subject easy and interesting for students. Future teachers should be given proper education to make this subject understandable for their students. Teachers should learn different strategies and techniques to make their teaching more effective and livelier. It is generally observed that increasing and maintaining students’ interest in this subject, is a difficult task for teachers, different students have different choices, mostly are fearful about this subject (Swars, Daane, & Giesen, 2006). It is the duty of teachers to find out the interest of the students by providing the situation relevant to the subject, if one succeeds in motivating and developing interest of their students, he will be in this position to be called as a successful teacher.

Now days, performance of teacher has a big question mark because new teachers have lack of enthusiasm, latest knowledge and techniques. In these days, different policies in education department are being implementing to enhance the practical capabilities of the teacher educators. There is the main responsibility on the shoulders of teachers to make their instruction effective and interesting. Therefore, evaluation of teacher is more important than evaluation of the student (Hill, Schilling, & Ball, 2004).

Near about in all subjects: mathematics provides the base. So, there is no doubt to say that science and technology is incomplete for its description without math. The nation which wants to put itself in the race of developed countries, it gives importance to math subject in their curriculum at each level. It also helps in boosting the learning level of other subjects. In education process at each level, we cannot deny the importance of mathematics (Houston, 2007) If one student is good, in mathsmost chances are there that he/she will be good in other subjects also, because Maths help him in improving his logical thinking. Maths works like a tool which one mostly uses in all spheres of his life. In the real world, we cannot get good results with the help of only traditional methods. To cope with this situation, teachers derive a number of solutions from the students to reach at conclusions. In this way teacher tries to improve their concepts unconsciously. This is one of the best ways

2.2.1 Importance of Mathematics in other subjects

It is fact that in our daily life Maths provides us support. Use of Maths is all-around, our day start and ends with activities related to Mathematical thinking and reasoning. In daily routine, we deal with measurement of food, cloth, account of income and expenditure such type of problems demand proper knowledge of Arithmetic (Blum & Niss, 1991). If we go further and talk about the professions related to daily life, for example cobbler, mason, carpenter, fruit and vegetable seller merchant etc. all shows that Math is correlated to all these activities and professions(Leedy, LaLonde, & Runk, 2003).Even for each and every individual knowledge of basic mathematics is positively correlated to these activities and professions. Now we will see connection of Mathematics with other subjects (Holmes, Sreenivasan, Glass, Kohn, & Sastry, 2002).

A famous saying that Maths is the mother of all sciences has now become a fact because at root level of each subject, there is Math. According to Roger Bacon “Mathematics is the gate way and key of all sciences, Maths and physics have positive correlation. The formulas, rules, laws, principles and equations get the shape because of Mathematics. For explanation of Friction, Boils law, Charles law, Expansion of solids, liquids and Metals needs help from Mathematics Algebra involves in explaining and formation of equations All the chemical bonding and their expressions cannot be completed without Mathematics.

Maths has correlation with Chemistry. Simply if we discuss the structure of water it needs Maths for explanation as two molecules of Hydrogen and 1 molecule of Oxygen needs to form (Lindberg, 1982). Their molecular weight also be discussed with the help of Moths. To balance chemical equations, knowledge of Mathematics. Different fields of Biology as Bio-chemistry, Bio-technology, and Bio-physics etc. need Mathematics for explanation. Nutrition’s value, laboratory tests, blood composition, plants structure all needs mathematics. Structure of cells of different compounds can be explained well with Mathematics we can also see Engineering as an application of Mathematics. Derivation of all formulas based on Mathematics, near about all branches of Engineering have based on Applied Mathematics (Nilsen, Angell, & Grønmo, 2013). Agriculture directly deals with Maths; even from bowing seeds to ripe crops each step needs Maths. Ratio of fertilizer in different crops, time period and to keep account of watering all is only possible with Maths. Correlation of Maths has high positive value (Parr, Edwards, & Leising, 2006)

According to Pythagoras who said well “Where there is money, there are numbers.” Mathematics is the science of facts and truth. Keats said that “Truth is beauty.” In Mathematics a child draws and constructs so many different shapes, graphs, figures and sketches (Chandrasekhar, 1987). This all can be done well if a child has knowledge of drawing and Arts. In spite of that it is also noted that for drawing good geometrical figures, good drawing is needed. And in the case of fine Arts knowledge of Mathematics can be applied (Nasr & De Santillana, 1968)

1. Drawing and painting with evenness.

1. Important topics of Arithmetic, Ratio and proportions are the key elements in making sketches.

1. Musical notes work on the principles of Mathematics.

1. Figures and sketches are measured with use of Mathematics.

A great mathematician Leibnitz said that music is a modern secreted exercise in arithmetic of a mind unconscious dealing with numbers. Thus, we can say that there is a positive correlation among mathematics and fine arts and drawing. It is essential for teachers to have an ability to understand the developed trends in teaching to cope with upcoming challenges of education. This can be possible only after modification in curriculum, design of organization structure and methods of teaching Like in other professions, Doctors, Engineers, Accountants, Architects and lawyers, knowing their objects are not enough but skill and mastery in their profession is must. Although s, there is a big gap between a teaching profession and others. Because in teaching, a teacher learns from his experiences with students of different mental level and aptitude Alfred said that most of the professional solve the problems of their clients but they have no ability to tell their clients how to figure out their problems, but a teacher has a specialty to guide his students to solve their problems to motivate the students’ learning towards mathematics (Stone, 1992).

2.2.2 Responsibilities of a Math teacher

At primary level, teacher especially adopts activity based and interesting methods in making the concepts of his students. He manipulates different procedures and strategies to make the learning of his students effective. In this way students actively participate in learning and this make their learning easy and interesting (Geiger, Forgasz, Tan, Calder, & Hill, 2012). But it is a long debate that whether Secondary school teachers only have proficiency in subject knowledge or also command on how to teach? Or they know the best way, is to integrate both.

Most of the time by the reaction of his student, a teacher understands that at what level of understanding a student is? Responses of students are very important for a teacher; these responses help a teacher to improve his methodology. It is not the era of cramming, one has to explain what he has learnt (Watson & Geest, 2005). In social and physical environment, a child where he lives almost reacts in some different ways. From his birth, he reacts in harmful and in faithful way. Teaching helps a child in adjustment. This can be done in two ways to modify the environment or to strengthen the child. Teaching always helps a child to become a useful and an efficient member of the society. If we strengthen the child this will really help the society in forwarding with rapid progress (Ainley, Pratt, & Hansen, 2006)

Maths teacher has an ability to maximize positive interest and minimize the weak points of their students influenced by their home and community in which they are brought up. A teacher can be evaluated through the goal and standards which he sets for his students. If a teacher succeeds in developing the emotional and liberty aspect of the student by providing them liberty, affection and freedom (Carroll, Forlin, & Jobling, 2003). Personal relationship among teacher and students affects positively on growth of children. It is the liability of teacher to prepare his learners, to meet the challenges of future. Teaching helps an immature child to develop physically and emotionally in his life.

Teacher affects students’ personality in both ways i.e. formal and in-formal and due to these, students have proper involvement in their studies and listen their teachers attentively. Although teacher is an agent of formal teaching but teacher also act as an agent of informal teaching as his parents, sisters, brothers and his friends. It is strongly believed that a good maths teacher knows about latest methods, techniques and strategies. He is also familiar with their students’ likes and dislikes and he is expected to understand the objectives which he has to achieve. At a time, his/her teaching acts as a science and art. It treats with logically and dramatically (Folkestad, 2006).

Researchers have strong point of view that teachers competency about their subject is not enough for effective teaching but the attitude and experiences of the teachers shared by them in their class rooms are more important(Thames & Ball, 2010).Important talent of the school mathematics teachers are, good mathematical concepts, procedural fluency and skills to handle the situation. National council of teachers of Mathematics stressed that in any educational program, teacher is the most important part. For implementation of all educational process, teacher’behaviour towards their subject is most important. Teacher is mostly responsible at all stages.”(NCTE,1998).

Desire of teachers is to enhance their teaching capacity to become an effective educator in the field of education, so they should be trained and educated in encouraging and constructive atmosphere. For students at early stage teacher’s behavior matters a lot, Teachers’ hope that they will educate and prepare young students in a very good way, so it is the duty of administration to introduce such type of trainings to develop their social consciousness and modify their frame of mind, Marry(2010) said that teachers should be trained confidently to understand the new pedagogical skills, attractive behavior according to students’ specific requirements. They should know the learning cycle and stages, fretful and socially engaged knowledge of their students. In this way they can help in improving of teaching process and social change (Cochran-Smith, 2005)

If teachers have proper knowledge of their subject, they improve also skill. This will make a teacher more confident and competent in his field of teaching, for good teaching these steps are important and necessary (Cochran-Smith & Lytle, 2006). In circulating knowledge, they link their knowledge with latest methods and information. Loughran (2006) states for a good teacher understanding of teaching is very important. Teacher should be fully aware that what, how and why they articulate their experiences in teaching and learning process (Loughran, 2006).

A teacher educator should be able to fulfill the needs of all approaches. Nowadays, we have skillful teachers with latest recourses and technology. Technology should not be considered as a split tool; it should act as a part of teaching and integral part in effective teaching. Teacher educator institutions prepare their students in such a way that they will be able to meet the challenges of future. They should be given opportunities to teach in latest technology equipped classrooms. This is way to get practical experience of teaching and responsibility of quality teaching. (Hiebert, 1997)

2.3 For Trainings, Effective Pedagogy of Mathematics

For effective mathematics trainings, clear goals should be set that suit learning development. Tasks should be implemented that encourage reasoning. Successful teaching involves students in resolving and discussing tasks resulted as different solutions, which makes them logical(Aaronson, Barrow, & Sander, 2007). Effective mathematics trainings slots in trainees, in making connections between mathematical representations for deep understanding. Learning is both conscious and un-conscious process. By analyzing and comparing student approaches, healthy discussion among trainees will be started. Effective trainings of mathematicsare focused on purposeful questions to assess trainees’ reasoning and sense making about important mathematical concepts.

Effective teaching of mathematics constructs fluency with understanding. It will support in creativestruggle in learning the mathematics. Effective pedagogy of mathematics demands trainees progress in thinking (Garrison & Kanuka, 2004). Effective Learning Environment is important to modify a trainer role as a facilitator and the duty of a trainer should be to write down about the trainees’ activities so that they could take lead. Instructional designs, basically are teaching models, they explain the process that produce and identify learning environment where, interact to achieve specific objectives(Joyce, Weil, & Calhoun, 1986)With the help of developing some basic habits, a math teacher can keep his math teaching on the right track(Ding, Li, Piccolo, & Kulm, 2007)

Principle1: First make it sense

Principle2: keep in mind the Goals

Principle3: make out assessment tool

Principle4: passion with mathematics

Principle1: First make it sense

A trainer should struggle to understand the base of mathematics concepts and simple ways to solve them. It is not a simple way it needs a lot of hard work and efforts. Sometimes a trainer may understand in little time, the different ways to teach difficult concepts, whereas sometimes it may take many years. Spiraling is an interesting i.e. clearing the concept in many steps or simple in many years to introduce the meaning full concept (Hanley, 2005). In which ways, a problem should be solved is called its procedures. The trainees can know to solve but they should be given sometime. Someone can learn “how” without understanding, only by practicing or cramming. When someone tries to understand without understanding, it may easily forget(Burns, 2004). To find out a link between a concept and its solution is tricky.

The Chinese teachers also focused on both, concept and procedural fluency along with conventional methods but more emphasis on practice which proved their contribution to teaching mathematics content (An, Kulm, & Wu, 2004) Conceptual and procedural understanding in fact helps each other, But there should be always concept first and then procedure. Chinese believed that fluency is important but without understanding procedural fluency has no importance. As described in a thumb rule that teacher should not leave any topic without confirming from each trainer that he/she has fully understanding of the concept and trainer should confirm by asking them a question or by giving some example solved by the trainee.

Principle 2:

A teacher should be fully aware of his goals and sub goals (Myung, 2000). He is considered to understand the psychology of the trainees so that how much work they can do. Some simple goals are described below so that a teacher can assess, where is he?

· Facilitate the trainers so that he can understand problems related to daily life or problems related to science and mathematics.

· trainees should be ready to understand the complex mathematics. Because of this goal, teacher will create interest in maths for trainees (Kovas & Bogdanova, 2014).

· Teaching deductive reasoning. It is a broad goal but in teaching geometry to high classes, it is necessary.

· Passion is very important for a maths’ trainee. He should learn to love it. It is also a strong goal; it leads that student at the very least have no negatively feelings about mathematics

In USA teachers are very keen in making different activities and also they worked on inquiry based questions to clear the concepts and critical thinking of trainers(An et al., 2004)A maths trainer should understand that he cannot restrict his/her trainees to only for text book or for curriculum but his students have full knowledge of given concepts.

Principle 3:

Nowadays a teacher of mathematics uses different tools which are plenty in numbers. At very first level, come a black/white board, pencils, protractor, compass, ruler, and eraser and of course book. After that, teachers have software, different activities, and lessons. Then there will be workbooks, fun books, work texts and books. There is manipulative abacus, measuring cups, scales, algebra tiles, games related to maths and so on. There are so many choices for selection of tools. It’s frightening a trainee to what to do (Bitter, 2014). It can be resolved as a trainer just has to start somewhere, may be with the basics, and then add to his “tool box” to increase slowly. as he got opportunity. Using many tools at the same time is not important but it is important to use more suitable tool according to situation to attain best results. Many studies have been conducted to find out curriculum as a tool for a math trainer (Collopy, 2003).

Essential Tools for Training

1. The black/white board. It will be convenient for everyone

The text or any related book and defined curriculum. Both are for the help of teacher but teacher should not be bound to these because teacher has flexibility to rearranging the order, skip pages, supplement material etc. (Van de Walle, Karp, & Williams, 2007).

1. Manipulative are physical bits and pieces; the students get practice of difficult concept. Some manipulative is very helpful for example a 100- bead basic abacus, some kind of fraction manipulative which are really helpful in understanding the concept of fraction. Drawing pictures are also very good example of them (Antle, Droumeva, & Ha, 2009).

1. Geometry and measuring tools are also essential in teaching. Now dynamic geometry software is also in use of teacher but this software cannot replace compass and ruler constructions.

1. Games are good for drilling basic facts and there is a game that ‘value 1000 worksheets. Now internet is full of online math games. For teaching algebra and calculus, some graphing software are being using (Drier, 2001).

Principle 4:

A trainer is, who shows the techniques with his attitude. Mathematics is mostly used in our daily life by using reasoning. Teacher –learner relationship demanded trustworthy and democratic environment, the important element is not to give over knowledge to students but to improve the capacity of students, thinking without focusing on their ages (Delong, 2015). Trust between teacher and learner is very important. (Barber, 1994). There are some ideas to create interest in maths lesson.

1. Let it make sense. This without help can make a difference and can create an interest in students.

1. Fun maths books should be used, just like “puzzle books, or some interesting exercises related to maths along with schools’ book should be needed. Some stories books related to these topics should also be used to teach maths concept (Moyer, 2001)

1. Include some math history

1. Explain to your students while using mathematics in daily life that why are you using.

2.3.1 Important Features of Trainings (Characteristics of Effective Pedagogy of Mathematics)

Many researchers have been done in the case that what kind of pedagogical practices help the trainers (Anthony & Walshaw, 2009). According to them, trainees are actively engaged with mathematics. They cannot take only some values of effective mathematics pedagogy in segregation but should be explained as a part of a whole that can influence trainees’ learning but practice should be part of it. All trainees should be involved in healthy discussion and this will increase trainees’ thinking process. Effective trainers are convinced for providing chances to all trainees with opportunities to effort with the mathematics. Effective trainers show expectations and obligation on trainees that they can give a better solution of the problem (McDonough & Clarke, 2003).

If we examine the links between pedagogical practices and trainees’ outcomes in the field of social and academic spheres of life, it is simple to understand that maths is all around us. It shapes an individual that how he will be able to solve the problems. There is also a need to focus on histories, cultures and practices. This will give them confidence to face these problems. It is a best way to recognize the effective teaching of mathematics.

Trainers may present themselves as significant source for enhancing trainees’ mathematical skills (Cobb & Hodge, 2002) They influence the ways in which trainees think (Anthony & Walshaw, 2009). By sewing reasonable settings, successful trainers give notice to all their academic needs. The constructive behavior develops increases trainees ‘comfort level.

To make the base of Shulmans ideas, other researchers (Ekawati & Kohar, 2017) insisted that without content knowledge, pedagogy and curriculum knowledge, maths teacher lacks relevant tools for practice. It forms the support for explaining the various mathematical solutions. Contemporary researchers like (Rooselyna, 2015) used PCK(pedagogical content knowledge) or (Teaching knowledge of Mathematics) MKT as a theoretical frame for teaching mathematics. Solving problems in different ways MSA (multiple solution approaches) requires MKT that incorporates the content knowledge. MSA can work as providing a base that gives many solutions to solve a problem.

When entire class of trainees takes part in discussion, it can provide an opportunity for broader interpretations and understanding. It can help them in facing the challenge and thinking on possible solutions of that problem. Trainers can play important role in the discussion with their trainees, by giving an opportunity to each student for discussion and for reaching at appropriate solution by listening and giving respect to one another solution. Class discussion is an integral part of maths trainer’s techniques and strategies. A dedicated trainer plan training format related experiences that allow trainers to construct their present skills and experiences. The main role of a trainer in maths class is to monitor trainee’s discussion and to evaluate different point of views. This will help a trainer to understand the mental level and ability of the whole class. Effective trainer create place for their knowledge and interest. This shows the interest of teacher in his class by assessing the students’ competencies, language, and understanding level, so trainer struggle for the better understanding of trainees(Michael Johnson Nabie, Raheem, Agbemaka, & Sabtiwu, 2016).

2.3.2 The subject matter preparation for effective training

Among mathematics education researchers, rising discussion is about effective trainings of mathematics, the subject matter preparation that is what and how a lot of content knowledge is needed? Works of Lee Shulman (1986), Deborah Ball and her colleagues (1990 and 2008), Darling-Hammond (2006) and others were the main inspiration to recognize the right balance between content and pedagogy (Kar). A mathematician should never fail to remember that framing its instruction is very important to go with the needs of future of newly appointed teachers (Wittmann, 1981).

Policymakers, Teacher educators, School administrators and trainers are facing a general problem what makes teacher more effective in educating students (Haylock, 2007). Teacher preparation, knowledge of instruction and learning, subject matter knowledge (SMK), teacher experience and qualifications have a great contribution to develop teachers as more effective (Rollnick & Mavhunga, 2016). However, there are two important means that is (i) teacher knowledge about subject matter and (ii) pedagogical knowledge. In the past they were totally treated separately. Lee Shulman is considered as the most prominent person to address this dichotomy. He was the first to introduce the concept of pedagogical content knowledge (PCK). In teacher education programme there always seems an apprehension between content knowledge and pedagogical knowledge (Shúilleabháin).

Trainers need to have proper information about the subject what they have to teach but well beyond (Kar). On the other hand, there are some strong views that trainers should focus on material, they want to teach but more focused on delivering that material i.e. pedagogical (Hill, Ball, & Schilling, 2008). Though, most researches put stress on streamlining the two approaches Therefore for definition and explanation of SMK, PCK Le Shulman tried to discussed in the international perspective. The issue of teacher’s knowledge of mathematics particularly finding the balance between content and pedagogical knowledge has been a hot issue for several decades (Even, 1993).

A common approach for teachers’ preparation programme is, to take a set of topics Algebra, Number theory, discrete mathematics, Data handling, Area, Perimeter etc. for in-service teachers (Nillas, 2010) Policy makers and administrators are in the point of view that more mathematics will help them to become good teachers. Whereas some researches showed a weak relationship between courses taken by students and their performances on exams. This does not support the more courses more mathematics. However famous scholar Freudenthal stressed on understandings of topics in more depth in conventional curriculum.

If a trainer wants to improve the base of mathematics of his trainees he should focus on both depth and breadth of topics. Therefore, pedagogical content knowledge term is best to introduce the depth and breadth of mathematics (Shulman, 1986). Pedagogical knowledge based on experience it simply addresses how to teach? Whereas content knowledge tells what to teach? Both are the demand of mathematics curriculum and teachers own temperament. Shulman (1986) discussed the three important issues.

First is regarded how teachers teach i.e. the way of their teaching “we study the research on teaching and discover its implications for practice” (Nillas, 2010) There are three types of propositional knowledge. Theoretical knowledge, convenient experience and moral reasoning. Second one is based on well documented material. Third knowledge is strategic knowledge which comes in action when a teacher faces his students in actual classroom situation and teacher has to face problems directly “collide with the ideology (Shulman, 1987).

It means we can say that PCK (pedagogical content knowledge) is another shape of realistic knowledge which a trainer used to guide their trainees in background of classroom. PCK is directly related how we explain formulas and concepts. Actually, pedagogical techniques are those techniques which makes the delivery of a concept easy or difficult. For teachers pedagogical and content knowledge both are important. They are interrelated so at a time both are important. Effective mathematics is a complex subject. Much of the complexity lies from defining what is effective as well as what type of practices makes a teacher effective. Still link between PCK and effective mathematics is still not understood (McCrory, Floden, Ferrini, Mundy, Reckase, & Senk, 2012).

However, for effective mathematics some standards can be established. By developing a positive atmosphere of learning, trainees should be engaged in purposeful discussion, this will prove as a source for investigating individual problems of students to stimulate learning. Subject matter knowledge has its own importance; it is the first step of the ladder because if someone is confident in content knowledge, he can learn the procedures/methods to explain that knowledge. To inquire and gain control over real world problems mathematics is a powerful means. If a teacher combines SMK and PCK, certainly effects on student’s learning.

Philosophical approaches and common sense assure that if educator has full control on his topic and this knowledge may help his students to learn their subject (D. Ball). On the contrast, if teacher has limited knowledge, he will unable able to pass accurate knowledge and also will not be able to challenge the queries of his students. “Delicately, teacher form their practice, the types of questions they ask, the ideas they support, the kind of responsibilities they give to their students (Maoto & Wallace, 2006). Mathematics teachers need to genuinely recognize the mathematical ideas that are essential to the grade level of the students they want to teach(Darling-Hammond, 2010). It is not enough for a teacher to have only procedural fluency but also have an ability to represent and join mathematical ideas so that students may figure out and realize the power and diversity of these ideas (Maoto & Wallace, 2006).

Teachers also need to understand students’ thinking process to help them to understand questions such as anything I do in one side of the equation; I must do another side of the equation in order to balance it. So, what is wrong if I add to numerator of both fractions and get? Why? Or why should I cram quadratic formula when I can use calculator to find the roots of decimal places therefore mathematics teachers have knowledge beyond routine knowledge. During, classroom teaching, teachers focus on fundamental connections between what they are learning and what they are teaching, it will be only possible for him when he has solid foundation of his subject knowledge. Today the biggest challenge in developed and under developed countries is what type of courses should be taught in teacher educator institutions for their professional development. In general, most teacher educator program just focus either on content or pedagogy. Only a small number of programs focus on the both (Worden, 2015). Therefore, more research is needed about what works for teachers to develop an effective mathematics teaching. In different states of USA, for elementary teachers more focused be put on methodology instead of more content. Whereas at secondary level much focused is on content.

2.3.3 Importance of multiple solution approaches for effective pedagogy in developing math understandings

It was indicated by classroom studies (Michael Johnson Nabie et al., 2016) that if teacher introduces multiple solutions opportunities, it opens the solutions options that make sense to them. In 2010 Cramer, Monson, Whitney, & Wyberg’s studied that when 6thgrade, were given problems related to fractions, they tried to solve them by using different strategies like graph, blocks pictures etc. students used pictures to interpret the given problem of fraction. This shows that students used multiple solution approaches (Leikin & Levav-Waynberg, 2007). Their work showed that Multiple Solutions Approaches developed the students’ creativity and improve their logical thinking instead of cramming and it also help students to understand that Math is embodied helps in building the network of ideas (Michael Johnson Nabie et al., 2016). On other side, some researches (Ainsworth, 2006) showed disagreement toward MSA. They proved that sometimes mere accesses to multiple solutions not necessarily lead to improvement, it can also mislead and de tracks the student’s approach.

Professionals and Researchers who consider multiple perspective in math’s learning support that multiple solution approaches as an alternative to develop mathematical understanding, creativity, skills, logical thinking, interest and ability to find for alternatives.MSA allows students to understand the basic mathematical ideas to further improvement in higher mathematics (D. L. Ball & Bass, 2000). Sum up the worth of MSA declaring that you can learn more from solving a problem in many ways than. MSA also improves the performance of learner.

Wide-ranging literature (Michael Johnson Nabie et al., 2016) propose that what is learnt by students in their classrooms. Limited teacher’s knowledge (Lampert, Beasley, Ghousseini, Kazemi, & Franke, 2010) can obstruct the use of multiple solutions in the classroom. Test scores and results put pressure on teachers to provide students multiple opportunities to construct multiple frames of references. Literature review (Michael J Nabie, Akayuure, & Sofo, 2013) on problem solving recommend that multiple solution approaches also help in solving the daily life problems, and Maths is one of them to solve their daily life problems, therefore in order to increase their mathematical understanding MSA is one of the best strategies. MSA and MKT have strong relation because if one is good in maths knowledge he can present different solutions and approaches to solve a problem. Multiple solutions approaches can be encouraged through open-ended questions, problem solving, projects and investigations. To understand that there are so many alternatives for a problem (Stohlmann, Moore, Cramer, & Maiorca, 2015)

2.3.4 Applications of effective mathematics pedagogy

From different research studies it is found that successful teachers always try to facilitate their students, by providing different opportunities for students’ learning and their engagement. They focus on developing interrelationships which ultimately help students to develop their cultural and mathematical identities (Glover, Miller, Averis, & Door, 2007). These teachers have high expectations from their students about enhancing students’ ability to think, reason, communication and reflection, this relationship helps in developing student’s mathematical competencies and distinctiveness. Students eager to learn in somebody's company, Teachers can make and create an atmosphere in which every student can feel respect, safe and get involved by doing self-activities. Most of the researches indicates that, do not support students in such a way that they totally depend upon their teachers. This will damage the creativity of students (An et al., 2004).

Effective teachers encourage such an atmosphere in which students compel to think, instead of providing solutions they encouraged to think (Stein, Grover, & Henningsen, 1996). Every day classroom practice enables students to solve these problems in different and easy ways. They use clear techniques, such as they express how will they solve or share their pattern. It is important that students should differ, if they do not agree with some solution (Stein, Engle, Smith, & Hughes, 2008). Teachers support their students in working together and giving respect to other ideas and opinion to establish common understandings. Teacher motivates students to listen carefully each other’s views and help them to conclude on common points (Bray, 2013). These activities of teachers shift students’ attention from procedural rules to making logic to mathematics. This makes students not only to think about answers but the ways to lead towards the answers (Ottmar, Rimm-Kaufman, Larsen, & Berry, 2015). Students try to understand mathematical language. It is also a significant role of a teacher to promote mathematical terms and language; it will help in removing the hesitation from math subject (Kaya & Aydın, 2016).

2.4 Mathematical understanding for secondary level math Teachers (MUST)

Math educators at the Pennsylvania state university and the University of Georgia presented MUST framework. It is their joint collaboration which is supported by the National Science foundation. The researchers have formed and developed a set of situations that are appropriate to secondary level math classroom (Hill et al., 2008). It is an important step towards helping in-service teachers in correcting their concepts by giving relevant examples. Another framework KCS (Knowledge of contents and students, this frame work helps students in conceptualizing, identifying, measuring and improving teachers pedagogical content knowledge defined by Schulman (Loewenberg Ball, Thames, & Phelps, 2008).It helps in designing and improving teacher’s knowledge in effective instruction and it also determine teacher’s ability in inspiring students to learn mathematics.

It can be assessed that neither the subject matter nor the pedagogical knowledge only can address the deficiencies and challenges of our mathematics teacher education programme. Now recent researches are interested in developing a framework that tries to streamline the two lines of thoughts (Hoover, Mosvold, Ball, & Lai, 2016). Here are some frameworks that address the SMK, pedagogical knowledge and practices for effective mathematics teaching. Since the area is wide, it will be discussed secondary mathematics teaching Mathematical understanding for secondary Teaching (MUST) Framework.

The MUST framework was with a desire to characterize mathematical knowledge for teaching at the secondary school level. The framework was inspired by the work of Deborah Ball and her colleagues at university of Michigan. One important characteristics of MUST framework is that it was developed out of classroom performances and examples were drained from variety of classroom practices and circumstances. The must framework includes the practices of practicing teachers. Thus, it could be classified as a bottom up approach. The framework focuses exclusively on mathematical understanding at secondary school level. MUST framework includes three components: (1) mathematical proficiency, (2) mathematical activity, and (3) mathematical work of teaching.

1. Mathematical proficiency based on conceptual understanding to procedural fluency that teachers need to promote in their students. The mathematical proficiency of teachers needs to be well ahead of secondary level that covers the mathematics in both elementary and college level (Leong, 2013).

2. Mathematical activity is the work of “doing mathematics (Kim, 2013). The framework proposes a more conscious and elaborated grasp of activities, the teachers take up for their students to be taught in mathematics classroom.

3. Mathematical work of teaching is an understanding of the mathematical thoughts of students, mainly the nature, psychology and their misconceptions (Hill, Ball, & Schilling, 2008).

It means teacher have to be able in finding students’ complexity and obstacles, so that he can address these problems. This is a best approach in choosing relevant examples and illustrations during teaching. About a mathematics topic, both in preparation and teaching, a teacher must be aware about misconceptions of their students (Hill, Ball, & Schilling, 2008). According to the frame, KCS content knowledge is knotted with thinking of students. Definition of KCS is based on academic and experimental work on teacher knowledge. KCS includes both subject matter knowledge and pedagogical knowledge for teaching MKT model. However, researchers argue that KCS is different from MKT model.

According to KCS model, a teacher may be very good in having content knowledge but poor in how to teach. Whereas KCS model provides in predicting how students think? Or how he relates to mathematical design? According to researchers, there are four intertwining categories of teaching mathematics. The categories are 1) Subjective understanding, 2) Classroom collectivity, 3) Curriculum formation, 4) Mathematical stuff. First two are fall out in knowing and, therefore treated as dynamic, the last two are termed as dynamic and it takes seconds to transform dynamic knowledge and for transfer of stable knowledge it requires decades (Davis & Simmt, 2006). Throughout their work in teaching for mathematics Davis and Smmit recommend that 1) a little bit fluency is needed for the above four discussed features. 2) Mathematics which is selected for teachers, these four aspects can direct towards appropriate emphasis for developing courses.

John Ferrini-Mundy and her colleagues at Michigan State University set up the term knowledge for Algebra teaching (KAT) in presenting a two-dimensional framework, linking teaching and kind of knowledge, to conceptualize the knowledge for (algebra) teaching (McCrory, Floden, Ferrini-Mundy, Reckase, & Senk, 2012). The framework measures the teachers highly developed mathematical knowledge and knowledge faster to teaching by analyzing three types of teaching practices: 1) decompressing, 2) trimming, 3) linking.

1. Decompressing attaches basic meanings to symbols/variables/algorithms that we use in an unconscious way. It means putting more emphasis on basic knowledge than cramming or procedural fluency.

1. Trimming means orderly holding of mathematical concepts, also involve trimming of related content while ordering important mathematical features.

1. Bridging means creating connections between teacher’s goals and students’ understanding.

The goal of teacher is looking for to assemble the relevant ideas of school algebra to abstract algebra, connecting one area of school of mathematics to another. To analyze the instructional materials KAT framework is also used (Hoover et al., 2016), just like other qualitative and quantitative studies work. It can be concluded that content knowledge with respect to effective teaching is still a new thought. Near about a decade or two decade before at elementary or secondary level, subject matter was taken for granted or treated as “Common sense” (Ball & McDiarmid, 1990). But Ball et al. (1990 and 2008), Davis and Simmt(2006), and Evan(1993) believe that teachers have also gaps in their knowledge just like students.

We have to understand that teachers are coming from that institutions where they not often developed understanding of content knowledge, this can be a cause at the back of inadequate SMK and student’s low achievement levels. T