Investigation of Competence in Numeracy Skills Amongst

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INVESTIGATION OF COMPETENCE IN NUMERACY SKILLS AMONGST

FORM I ENTRANTS IN TANZANIA: A CASE OF SCHOOLS IN EASTERN INSPECTORATE ZONE

VICTOR LWEYEMAMU MWESIGA

A THESIS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN EDUCATION OF THE OPEN UNIVERSITY OF TANZANIA

2015

CERTIFICATION

The undersigned certify that they have read and hereby recommend for acceptance by The Open University of Tanzania, a thesis titled: Investigation of Competence in Numeracy Skills amongst Form I Students in Tanzania: A Case of Schools in Eastern Inspectorate Zone, in fulfilment of the requirements for the degree of Doctor of Philosophy.

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Prof. C.K. Muganda

(Supervisor)

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Date

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Prof. R.W.P. Masenge

(Supervisor)

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Date

COPYRIGHT

No part of this thesis may be reproduced, stored in any retrieval system, or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise without prior written permission of the author or The Open University of Tanzania in that behalf.

DECLARATION

I, Victor Lweyemamu Mwesiga, declare that this thesis is my own original work and that it has not been and will not be presented to any other University for similar or any other degree award.

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Signature

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Date

DEDICATION

This work is dedicated to my father; the late Ta Revelian L. Mwesiga, who during the time of writing the first draft of this thesis, I was besides his bed at Hyderabad Apollo Hospital in India. His anxiety to know the progress of my study was a great encouragement to accomplish this work.

ACKNOWLEDGEMENTS

First, I would like to thank my Supervisors: Prof. C. K. Muganda and Prof. R. W. P. Masenge. Without their guidance, critique and encouragement, completion of this work would have not been possible. My appreciation goes especially to Prof. R. W. P. Masenge, my advisor on technical matters in mathematics. I extend my thanks to Dr. F. M. Mulengeki and other Faculty of Education staff at The Open University of Tanzania. Their assistance was very helpful.

My employer, the Ministry of Education and Vocational Training is acknowledged for granting me permission to pursue this highest degree programme. I am thankful to both the Chief School Inspector (Headquaters) and Zonal Chief Inspector (Dar es Salaam Zone) for all the assistance they provided me. I appreciate the responsive co-operation I received from heads of schools, teachers, and students in the sampled schools, as their collaboration enabled me to collect the required data.

I am very grateful to my mother; Ma Ernestina Mukalweika, my Uncle Mr. Thadeo Mtembei, my young sisters and brothers for their love, encouragement and prayers. I also, recognise the role and contributions of my wife Gerades Kokuaisa and our sons Muberwa, Bashange, Ishemo and Mugisha. The significance of their contributions is in their acceptance to take additional family chores and responsibilities, hence allowing me sufficient time to spend on my studies. They accepted reducing the time allocated for spending with either a husband or a father.

Finally, I owe much thanks to my colleagues at the Dar es Salaam Zonal School Inspectorate Office who assisted me in one way or another. However, I am solely responsible for any shortcomings that are in this work.

ABSTRACT

The study has employed a descriptive research design and elements of both qualitative and quantitative methods to accomplish the fundamental goal of investigating the competence of Form I entrants in numeric skills. The focus was on identifying students’ errors on primary school mathematics, the associated misconceptions and causes. Consequently, a remedial approach was designed for intervention. The test, questionnaire and a focus-group discussion guide were employed for data collection. While errors were explored by a scrutiny of students’ work sheets, discussion guide was utilized to ascertain the conjectures of underlying misconceptions and causes. A questionnaire was used to capture views of teachers on the ability of Form I entrants in mathematics, and interventions taken. The major findings were that: most students had inadequate numeracy knowledge and skills; they committed both conceptual and procedural errors; schools had no system of identifying students’ mathematical learning difficulties for intervention; and there was no an effective remedial approach for correcting mathematical misconceptions amongst Form I entrants. The study recommends: (i) A system of testing Form I entrants on competencies in numeracy knowledge and computation skills, that are requisite for understanding secondary school mathematics. (ii) Introduction of a remedial programme to all Form I entrants by using a remedial approach developed in this study. (iii) Use of teaching strategies which allow diagnosis of learning misconceptions and remedial activities during the lesson. (iv) Immediate scoring of assignments, identification of errors and provision of feedback through extensive corrections. (v) A review of policies on teacher training and textbooks to address the issue of teaching/learning of mathematics at primary and secondary school levels.

TABLE OF CONTENTS

CERTIFICATION ii

COPYRIGHT iii

DECLARATION iv

DEDICATION v

ACKNOWLEDGEMENTS vi

ABSTRACT vii

TABLE OF CONTENTS viii

LIST OF FIGURES xvi

ACRONYMS AND ABBREVIATIONS xvii

CHAPTER ONE 1

1.0 GENERAL INTRODUCTION 1

1.1 Introduction 1

1.2 Background to the Problem 1

1.3 Statement of the Problem 11

1.4 Objectives of the Study 12

1.4.1 General Objective 12

1.4.2 Specific Objectives 12

1.5 Research Questions 13

1.6 Significance of the Study 13

1.7 Theoretical Framework 14

1.8 Conceptual Framework 16

1.9 Limitation of the Study 21

1.10 Delimitations of the Study 21

1.11 Definitions of Underlying Terms 22

1.11.1 Competence 23

1.11.2 Numeracy 23

1.11.3 Errors 24

1.11.4 Misconceptions 24

1.11.5 Conjectures 25

1.12 Organization of the Study 25

1.13 Concluding Remarks 27

CHAPTER TWO 28

2.0 REVIEW OF RELATED LITERATURE 28

2.1 Introduction 28

2.2 How Students Learn Mathematics 28

2.3 Basic Causes of Students’ Learning Difficulties 29

2.4 Mathematical Errors and Misconceptions in Numeric Skills 31

2.5 Diagnosis in the Context of Mathematics 34

2.6 Methods of Diagnosing Learning Difficulties 35

2.7 Remediation in the Context of Mathematics 37

2.8 Studies Done Outside Tanzania 39

2.9 Studies Done in Tanzania 47

2.10 Knowledge Gap 52

2.11 Concluding Remarks 52

CHAPTER THREE 54

3.0 RESEARCH DESIGN AND METHODOLOGY 54

3.1 Introduction 54

3.2 Research Design and Approach 54

3.2.1 Research Design 54

3.2.2 Research Approach 55

3.3 Area of the Study 56

3.4 Population, Sample Size and Sampling Procedure 57

3.4.1 Target Population 57

3.4.2 Sampling Procedure 57

3.4.2.1 Selection of Sample Schools 57

3.4.2.2 Selection of Students 59

3.4.2 .3 Selection of Teachers 59

3.4.3 Sample Size 60

3.5 Types and Sources of Data 60

3.5.1 Data Collection Techniques 61

3.5.1.1 Test 61

3.5.1.2 Questionnaire 62

3.5.1.3 Documentary Review 63

3.5.1.4 Focus-Group Discussion 63

3.6 Data Analysis Plan 64

3.7 Reliability, Internal and External Validity 65

3.7.1 Reliability 65

3.7.2 Validity 66

3.8 Ethical Dimensions 67

3.9 Concluding Remarks 68

CHAPTER FOUR 69

4.0 DATA PRESENTATION, ANALYSIS AND DISCUSSION 69

4.1 Introduction 69

4.2 Competence of Form I Entrants in Mathematics 69

4.2.1 Performance of Form I Entrants in PSLE-Mathematics Subject 70

4.2.2 Teachers’ Views on Performance of Form I Entrants in Mathematics and Efforts to Help Them 72

4.3 Difficult Level of Test Items 75

4.4 Diagnosis of Errors, Misconceptions and Causes 81

4.4.1 Computation of Units of Measurements 82

4.4.1.1 Converting Units of Area, Volume and Capacity 83

4.4.1.2 Computing Units of Length and Time 86

4.4.2 Estimating Length/Height, Weight and Time 88

4.4.3 Computations Involving Percentages 91

4.4.4 Computations with Angles 96

4.4.5 Number Patterns 99

4.4.6 Simple Algebra 100

4.4.7 Use of BODMAS Rule 105

4.4.8 Subtraction of Integers 107

4.4.9 Division of Numbers 111

4.4.10 Multiplication of Numbers 117

4.4.11 Subtraction of Numbers 125

4.4.12 Addition of Numbers 133

4.5 Summary and Discussion of Major Findings 139

4.5.1 Types of Errors and Misconceptions on Computations of Units of Measurements 139

4.5.2 Types of Errors and Misconceptions on Simple Algebra 141

4.5.3 Types of Errors and Misconceptions in Computations of Fractions and Decimals 143

4.5.4 Types of Errors and Misconceptions on Computations of Percentages 147

4.5.5 Types of Errors and Misconceptions on Computations of Whole Numbers 148

4.5.6 Types of Errors and Misconceptions in Using the BODMAS Rule 150

4.5.7 Types of Errors and Misconceptions in Subtraction of Integers 151

4.6 The Pattern Emerging from the Major Findings 152

4.7 Remedial Approach 159

4.7.1 Effective Format of Remedial Lesson 159

4.7.2 Sample Remedial Lesson 166

4.8 Concluding Remarks 174

CHAPTER FIVE 175

5.0 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 175

5.1 Introduction 175

5.2 Summary 175

5.3 Conclusions 178

5.4 Recommendations 179

5.4.1 Recommendations for Action 179

5.4.2 Recommendation for Further Study 180

REFERENCES 181

APPENDICES 191

LIST OF TABLES

Table 1.1: Zones Performance in Mathematics FTSEE for Years 2003 -2007 by Means 4

Table 3.1: Sample size 60

Table 4.1: Range of row scores of Form I entrants in PSLE-mathematics subject in five secondary schools 70

Table 4.2: Teachers’ views on ability of Form I entrants in mathematics 73

Table 4.3: Percentages of Students’ Failure on Competence Tested in the Task 76

Table 4.4: Students’ Failure Rates on Addition, Subtraction, Multiplication and Division of Numbers 80

Table 4.5: Student Failure Rates (%) By Work, Errors, Causes of Errors and Misconceptions on Converting Units of Area, Volume and Capacity 84

Table 4.6: Student Failure Rates (%) By Work, Errors, Causes of Errors and Misconceptions on Computation of Units of Time, and Length 87

Table 4.7: Student Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Estimating Length, Time and Weight 89

Table 4.8: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Computing Percentages 93

Table 4.9: Students Failure Rates (%) By Errors, Causes of Errors and Misconceptions on Calculation of Angles 97

Table 4.10: Students Failure rates (%) by Errors, causes of Errors and Misconceptions on Patterns of Numbers 99

Table 4.11: Students Failure Rates (%) by Errors, Causes of Errors And Misconceptions On Simple Algebra 102

Table 4.12: Students Failure rates (%) by Work, Errors, Causes of Errors and Misconceptions on Using BODMAS Rule 106

Table 4.13: Students Failure Rates (%) by Work, Errors, Cause of Errors and Misconceptions on Subtraction of Integers 108

Table 4.14: Students Failure rates (%) by Work, Errors, Causes of Errors and Misconceptions on Dividing Decimal Numbers 112

Table 4.15: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Division of Integers 114

Table 4.16: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Dividing Fractions 116

Table 4.17: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions in Multiplication of Decimals 119

Table 4.18: Students Failure Rates (% ) by Work, Errors, Causes of Errors and Misconceptions on Multiplication of Fractions 121

Table 4.19: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Multiplication of Decimals and Mixed Numbers 122

Table 4.20: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Multiplication of Whole Numbers 123

Table 4.21: Students Failure Rates (%) By Work, Errors, Causes of Errors and Misconceptions on Multiplication of Decimals and Powers of 10 124

Table 4.22: Students’ Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Subtraction of Mixed Numbers 127

Table 4.23: Students Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Subtraction of Integers 129

Table 4.24: Students’ failure rates (%) by work, errors, causes of errors and misconceptions in subtraction of decimals numbers 131

Table 4.25: Students failure rates (%) by work, errors, causes of errors and misconceptions on addition of fractions 135

Table 4.26: Students’ Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions on Addition of Decimal Numbers 137

Table 4.27: Students’ Failure Rates (%) by Work, Errors, Causes of Errors and Misconceptions in Addition of Whole Numbers 138

Table 4.28: Specified Difficult Content and Corresponding Categories of Common Errors and Misconceptions 154

LIST OF FIGURES

Figure 1.1: Remediation of Misconceptions and Errors in Numeric Skills 17

ACRONYMS AND ABBREVIATIONS

BODMAS Brackets of Division, Multiplication, Addition and Subtraction

CSEE Certificate of Secondary Education Examination

FTSEE Form Two Secondary Education Examination

INSET In Service Training

MAT Mathematical Association of Tanzania

MA-TEST Mathematics Achievement Test

MOEC Ministry of Education and Culture

MOEVT Ministry of Education and Vocational Training

NECTA National Examinations Council of Tanzania

PSLE Primary School Leaving Examination

SACMEQ Southern and Eastern Africa Consortium for Monitoring Educational Quality

SPSS Statistical Package for Social Sciences

STIP Science Teaching Innovation Programme

TC Teachers College

TIE Tanzania Institute of Education

TSS Takwimu za Shule za Sekondari

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CHAPTER ONE

1.0 GENERAL INTRODUCTION

1.1 Introduction

This chapter presents the background to the problem, statement of the problem, objectives, research questions, significance, conceptual framework, limitations and delimitation of the study as well as definitions of underlying terms.

1.2 Background to the Problem

According to Walshaw and Anthony (2009), understanding of mathematics influence decisions making in all areas of life internationally i.e. private, social, and civil life. Thus, mathematics education is a key to increasing the post-school and citizenship opportunities of young people. Many students struggle with mathematics and find it difficult as they continually encounter difficulties in learning. It is therefore imperative that, we find out what we can do to break this pattern.

The focus of this study is on students’ acquisition of basic mathematical knowledge and skills. Showing the importance of mathematics worldwide, UNESCO (2011:10) states that:

“Mathematics is omnipresent in today’s world, notably in the technological subjects surrounding us, and in exchange and communication processes – but it is generally invisible. Hence there is a lack of awareness of the importance of developing a mathematics culture beyond basic knowledge relating to numbers, measurements and calculations. It is important that basic education removes this invisibility, especially because the needs attached to so called mathematical literacy go well beyond the needs traditionally associated with basics computational knowledge”.