MATHEMATICS ANXIETY AS A VARIABLE IN THE CONSTRUCTIVIST APPROACH TO THE TEACHING OF SECONDARY SCHOOL MATHEMATICS by PETER LEONARD HAWKEY submitted in accordance with the requirements for the degree of DOCTOR OF EDUCATION in the subject DIDACTICS at the UNIVERSITY OF SOUTH AFRICA PROMOTER: PROF M P VAN ROOY NOVEMBER 1995 ACKNOWLEDGEMENTS I dedicate this thesis to my wife Gillian and my two daughters Lisa and Sarah who have all been very patient and supportive whilst I have given much of my attention to my studies and research. I acknowledge, with extreme gratitude, the contribution of my promoter, Professor M.P. Van Rooy, whose knowledge and guidance provided me with confident direction and whose dedication and diligence were an example to me. I also acknowledge the friendly and efficient services of Anne East (typing), Elaine Klitguard (statistical analysis) and Professor Mike Thurlow (professional advice). SUMMARY Mathematics anxiety is a personal characteristic which is widespread and continuing. It has a debilitating effect on mathematics performance and contributes to perceptions and attitudes that perpetuate a dislike for mathematics and a lack of confidence when dealing with mathematical problems. An investigation of relevant literature on mathematics anxiety identifies sources and symptoms and emphasises a need for a comprehensive approach to remediation. The historical development of an appropriate measuring instrument is documented and statistical evidence is used to create a mathematics anxiety rating scale suitable for measuring anxiety levels of secondary school pupils and student teachers. The extensive literature interest, research publications and remedial programmes emphasise the problem of mathematics anxiety and thus the need for a comprehensive approach to remediation. Mathematics teaching and curriculum design is expounded to provide the necessary direction to the alleviation of mathematics anxiety. General perspectives on curriculum design are discussed and a cyclical systems approach is recommended. Elements of this approach are detailed and are linked to important personal characteristics to add a humanistic and socio-cultural view of curriculum design in mathematics. The didactic viability of constructivism as an approach to mathematics curriculum design is investigated. Constructivism embodies a philosophy and a methodology which addresses the critical aspects influencing mathematics anxiety. Classroom topics and activities are reviewed in terms of a constructivist approach and the sources of mathematics anxiety are discussed from a constructivist perspective. A longitudinal case study of pupils during their five years at secondary school as well as a study involving student teachers was undertaken. Mathematics performance, perceptions, attitudes and levels of anxiety were investigated by means of tests, questionnaires, and mathematics anxiety rating scales. The statistical results of this research provide evidence to support a comprehensive approach to the remediation of mathematics anxiety. Constructivism is seen as the synthesis of critical aspects of teaching and curriculum development which will stem the perpetuation of mathematics anxiety. Constructivism provides the didactic approach to develop each individual's intellectual autonomy and mathematics power, by instilling a problem solving attitude and a self-confidence when doing mathematics. KEY TERMS: mathematics anxiety; constructivist approach; mathematics teaching; mathematics curriculum; secondary school mathematics; systems approach; situation analysis; classroom environment; understanding; self-confidence; communication; social interaction; problem solving; radical constructivism; social constructivism; intellectual autonomy; mathematics power. 570.772 HAWK \~\\1\\\\\1\ 01644951 Student Number: 251-127-4 I declare that MATHEMATICS ANXIETY AS A VARIABLE IN THE CONSTRUCTIVIST APPROACH TO THE TEACHING OF SECONDARY SCHOOL MATHEMATICS is my own work and that all the sources that I have used or quoted have been indicated and acknowledged by means of complete references. SIGNATURE DATE Mr P.L. Hawkey MATHEMATICS ANXIETY AS A VARIABLE IN A CONSTRUCTIVIST APPROACH TO THE TEACHING OF SECONDARY SCHOOL MATHEMATICS. CHAPTER ONE Page Introduction and orientation 10 1 . 1 Reasons for this research 10 1 . 1 . 1 The emergence of interest in mathematics anxiety 10 1 .1 .2 Changing concerns of curriculum design 11 1.1. 3 The emphasis on teaching methodology in mathematics 13 1 . 2 Statement of the problem 15 1 . 3 Analysis of the problem 17 1 .3.1 The problem of mathematics anxiety 17 1.3.2 The development of a mathematics curriculum 18 1.3.3 Implementing reform in mathematics instruction 19 1 .4 Definition of terms 20 1.4.1 Mathematics anxiety 20 1.4.2 A variable 20 1.4.3 A constructivist approach 20 1.4.4 Teaching of secondary school mathematics 21 1 . 5 Aims and objectives of the study 23 1. 5.1 Providing a clear understanding of mathematics anxiety as a variable in mathematics teaching 23 1.5.2 To provide an overview of mathematics teaching and curriculum design in mathematics 23 1.5.3 Providing a view of the constructivist approach to the teaching and learning of mathematics 24 1.5.4 To conduct empirical research on mathematics anxiety and achievement to support literature views 25 1.5.5 To analyse and synthesise the implications of the literature and empirical research for mathematics teaching 25 1 . 6 Programme of study 26 1 .6.1 Literature study of mathematics anxiety 26 1.6.2 Literature study of teaching and the curriculum 27 1 1.6.3 Literature study of constructivism 28 1.6.4 Empirical research procedures 29 1.6.5 Conclusions 30 CHAPTER TWO Mathematics anxiety as a variable in mathematics teaching and learning 31 2.1 General perspective 31 2.1.1 Why mathematics anxiety? 31 2.1.2 What is mathematics anxiety? 33 2.1.3 The importance of mathematics anxiety 34 2.1.3.1 Mathematics anxiety is widespread 35 2.1.3.2 Mathematics anxiety affects performance 35 2.1.3.3 Mathematics anxiety causes mathematics avoidance 38 2.1.3.4 Vicious circle effect 39 2.2 Research on mathematics anxiety 40 2.2.1 Measuring mathematics anxiety 41 2.2.1.1 The Three-Number Anxiety Item from the Taylor Scale 41 2.2.1.2 The National Longitudinal Study of Mathematics Abilities 42 2.2.1.3 Fennema and Sherman Scales 42 2.2.1.4 Mathematics Anxiety Rating Scale {MARS) 43 2.2.2 Sources of mathematics anxiety 45 2.2.2.1 Socio-cultural factors 45 2.2.2.2 Emotive factors 47 2.2.2.3 Cognitive factors 50 2.2.2.4 Educational or school factors 52 2.2.3 Symptoms of mathematics anxiety 56 2.2.3.1 Non-task orientated behaviour 56 2.2.3.2 Dependence on teacher 57 2 2.2.3.3 Non-social behaviour 57 2.2.3.4 Physical reactions 58 2.3 Implications for this study 59 2.4 Summary and synthesis 61 CHAPTER THREE Mathematics teaching and curriculum design in mathematics 64 3.1 Introduction 64 3.2 A general perspective on curriculum design 65 3.2.1 A traditional view 66 3.2.2 A comprehensive view 68 3.2.3 The systems approach 69 3.2.4 A process approach 71 3.2.5 Synthesis of curriculum design 72 3.3 Perspective on the mathematics curriculum 73 3.3.1 General perspectives 75 3.3.1 .1 Situation analysis 76 3.3.1.2 Aims and objectives 78 3.3.1.3 Choice of subject matter 79 3.3.1 .4 Teaching methods 81 3.3.1.5 Evaluation 84 3.3.1 .6 A synthesis of mathematics curriculum components 85 3.3.2 Approaches to curriculum design in mathematics 86 3.3.2.1 Cognitive psychology 86 3.3.2.2 A humanistic approach 87 3.3.2.3 A child-centred approach 89 3.3.2.4 The affective domain 90 3.3.2.5 The socio-cultural domain 91 3.3.2.6 Synthesis of mathematics curriculum design 92 3 3.3.3 Mathematics anxiety and the mathematics curriculum 93 3.3.3.1 Situation analysis and mathematics anxiety 94 3.3.3.2 Aims and objectives and mathematics anxiety 105 3.3.3.3 Choice of subject matter and mathematics anxiety 105 3.3.3.4 Teaching methods and mathematics anxiety 106 3.3.3.5 Evaluation and mathematics anxiety 106 3.3.3.6 A synthesis of the mathematics curriculum and mathematics anxiety 107 3.3.4 Constructivism and the mathematics curriculum 108 3.4 Mathematics teaching and mathematics anxiety 110 3.4.1 Mathematics anxiety and critical variables in mathematics teaching 1 1 1 3.4.1.1 Classroom environment 111 3.4.1.2 Understanding 112 3.4.1 .3 Self-confidence 1 1 3 3.4.1.4 Communication 114 3.4.1.5 Social interaction 1 1 5 3.4.1.6 Problem solving 1 1 7 3.4.2 A synthesis of mathematics teaching and mathematics anxiety 1 1 8 3.5 Constructivism and anxiety in mathematics teaching and learning 120 3.6 Summary 121 CHAPTER FOUR The constructivist approach to teaching and learning mathematics 126 4. 1 Constructivism in perspective 126 4.1.1 Radical constructivism 129 4.1.2 Social constructivism 130 4 4.2 Why constructivism? 133 4.2.1 The evolution of constructivism 133 4.2.2 Beyond the cognitive view 136 4.2.3 The need for constructivism 137 4.3 A constructivist view of mathematical development 141 4.3.1 Classroom environment 142 4.3.2 Understanding 144 4.3.3 Communication 146 4.3.4 Social interaction 148 4.3.5 Intellectual autonomy and self-confidence 154 4.3.6 Problem solving skills 156 4.4 Reflection and constructivism 158 4.5 Representation and constructivism 160 4.6 Implementing a constructivist approach in the mathematics classroom 1 62 4.6.1 Extra classroom activities at secondary school 163 4.6.1.1 Providing a sense of numbers 165 4.6.1.2 Dealing with novel problems 169 4.6.1.3 Allowing pupils to reflect their methods 174 4.6.2 Teaching