i LEARNERS? MATHEMATICAL REASONING WHEN GENERALIZING FROM NUMBER PATTERNS IN THE GENERAL EDUCATION AND TRAINING PHASE By WILLIAMS CHAPASUKA NDLOVU STUDENT NUMBER: 0616576G SUPERVISOR: PROFESSOR MARGOT BERGER A research report submitted to the Wits School of Education, Faculty of Science, University of the Witwatersrand in partial fulfilment of the requirements for the degree of Master of Science by combination of coursework and research. Johannesburg, South Africa June 2011 ii Copyright Notice The copyright of this research report vests in the University of the Witwatersrand, Johannesburg, South Africa, in accordance with the University?s Intellectual Property Policy. No portion of the text may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, including analogue and digital media, without prior written permission from the author and the University. Extracts of or quotations from this report may, however, be made in terms of Sections 12 and 13 of the South African Copyright Act No. 98 of 1978 (as amended), for non-commercial or educational purposes. Full acknowledgement must be made to the author and the University. iii ABSTRACT This study aims to explore GET learners? mathematical (algebraic) reasoning when generalizing from number patterns. Data was collected in a former model C school in greater Johannesburg area by means of a questionnaire based task involving number patterns. The mathematical reasoning of the grade 9 participants when generalizing from number patterns was examined within a commognitive framework. According to this perspective, thinking is a special activity of communication in which a participant of a discourse engages. The participants? responses to questions in the questionnaire based task were classified according to particular aspects of the discourse they used, specifically routines (strategies) and visual mediators. The participants? generalization routines were further classified into one of the three main categories; numeric, figural and pragmatic generalizations. The analysis focused on how the learners? derived rules for the nth term and their justifications for their responses. The results of this study strongly support the notion that students? algebraic reasoning when generalizing in number patterns is intertwined with their choices of routines and mediators. Most learners used recursive routines while a few used explicit routines (classified and categorized as numeric routines) and number-mediators. Also, most participants found it easier to informally verbalize their generalizations. However participants? spoken justifications of their written and spoken responses often did not match their use of routines and visual mediators. As such, an awareness and appreciation (by teachers) of students? diverse use of routines and mediators when generalizing from number patterns could have direct pedagogical implications in the mathematics classrooms. KEYWORDS Algebra, Generalization, Commognition, Thinking, Communication and Reasoning iv DECLARATION I declare that this research report is my own unaided work. It is being submitted for the degree of Master of Science at the University of the Witwatersrand, Johannesburg South Africa. It has not been submitted before for any degree or examination at any other University. ________________________ (Signature of Candidate) 7th day of June in the year 2011 v DEDICATION In loving memory of my late Parents: Enala & Chapasuka vi ACKNOWLEDGEMENTS First and foremost I thank the Almighty GOD for giving me life and a good health. And also, I would like to thank the following people for their contributions towards my work: My supervisor, Professor Margot Berger, for many hours of advice and encouragement, often under difficult circumstances, in which she helped me to develop and maintain the focus and yet enabled me to take ownership of this work. I am most privileged to have worked with such an understanding and exceptional person. My former lecturers; Professor Jill Adler from University of Witwatersrand and Dr. Mercy Kazima from the University of Malawi ? Chancellor College for encouraging me that I can do it. Your efforts and intellect to see the potential in others have not gone unnoticed. Thanks. Ngiyabonga. I wish to pass my special thanks to the Principal of the school involved in this study, the 29 Grade 9 learners, who willingly agreed to participate in my study, spending so much time with them in their classroom. I am particularly grateful to L5, L9, L15, L17, L23 and L28, who willingly gave of their time to further participate in the interviews. I am grateful to the National Research Foundation, the Department of Education and the University of the Witwatersrand for the financial assistance provided for my study. Last but not least, I wish to express my sincere gratitude to my wife Jean Pemphero, for her love, patience and understanding throughout this academic journey. And also to our lovely daughter, Faith Dinna, for continuously asking when are you going to finish studying and become a Doctor or Professor? Such kinds of questions always bring a new projectile into my life. My brothers, sisters and friends thank you for the support and encouragement. I am proud of you all. God Bless. vii TABLE OF CONTENTS Page Research title ................................................................................................................................... i Copyright notice .............................................................................................................................. ii Abstract and keywords ................................................................................................................... iii Declaration .................................................................................................................................... iv Dedication ....................................................................................................................................... v Acknowledgements ......................................................................................................................... vi Table of contents ........................................................................................................................... vii References ....................................................................................................................................... x Appendices ..................................................................................................................................... xi List of tables .................................................................................................................................. xii List of figure ................................................................................................................................. xiii Abbreviation ............................................................................................................................ ?..xiv CHAPTER 1: INTRODUCTION 1.1. Introduction 1 1.2. Background to the study 1 1.3. Research Problem 3 1.3.1. Statement of Purpose 3 1.3.2. Research Questions 3 1.3.3. Number patterns 4 1.3.4. Limitations of the study 4 1.4. Rationale of the study 4 1.4.1. Algebra in the School Curriculum 5 1.4.2. Curriculum Reforms and Personal Experience 6 1.5. The structure of the report 6 1.5.1. Chapter 1 6 1.5.2. Chapter 2 7 1.5.3. Chapter 3 7 1.5.4. Chapter 4 7 1.5.5. Chapter 5 7 1.5.6. Chapter 6 8 1.5.7. Chapter 7 8 CHAPTER 2: THEORETICAL FRAMEWORK 2.1. Theoretical Framework 9 2.1.1 Commognitive Perspective 9 2.1.2 Definition of Commognition 10 2.1.3 Description of Commognitive tenets 10 2.1.4. Mathematics as a Discourse 12 2.1.5. Summary of Commognitive Theory 15 viii 2.2. Literature Review 17 2.2.1. Number Patterns 17 2.2.2. Algebraic Generalization 18 2.2.3. Algebraic Thinking 25 2.2.3.1. Reasoning through Generalization 25 2.2.3.2. Learners? Thinking Process 26 2.2.3.3. Cognitive Gap between Algebra and Arithmetic 28 2.3. Conclusion 30 CHAPTER 3: RESEARCH DESIGN & METHODOLOGY 3.1. Introduction 31 3.2. Research approach 31 3.3. Research methodology 32 3.3.1. Case study 32 3.3.2. Research Instruments 32 3.3.2.1. The Questionnaire based task 32 3.3.2.2. Discussion of items in the task 34 3.3.2.3. Piloting the task 37 3.3.2.4. Questionnaire task based interviews 38 3.3.2.5. The transcriptions 39 3.4. Validity and Reliability Issues 39 3.4.1. Validity and Reliability 39 3.4.2. Trustworthiness 40 3.4.2.1. The Instruments 40 3.4.2.2. The use of audio tapes 41 3.4.3. Data Interpretation 41 3.4.4. Researcher effect 41 3.4.5. Researcher bias 41 3.5. Ethical Issues 42 3.5.1. Ethics of the study 42 3.5.2. Confidentiality 42 3.6. Conclusion 42 CHAPTER 4: ANALYTICAL FRAMEWORK 4.1. Introduction 43 4.2. Classification of learners? routines 43 4.3. Description of routine categories 45 4.3.1. The routines categories 45 4.3.2. The routines sub-categories 46 4.3.2.1. Numeric Generalization ? Recursion 46 4.3.2.2. Numeric Generalization ? Explicit 47 ix 4.3.2.3. Figural Generalization 48 4.3.2.4. Pragmatic Generalization 49 4.4. Aspects of Generalization 50 4.4.1. Globalizing 51 4.4.2. Local (Extending) 51 4.5. Indicators of Generalization Categories 51 4.5.1. Numerical Generalization 51 4.5.2. Figural Generalization 52 4.5.3. Pragmatic Generalization 53 4.6. Mathematical Visualization 53 4.6.1. Visual Mediators. 54 4.6.2. Description of the Visual Mediators 54 4.6.3. Indicators of Mediators 55 4.7. Conclusion 56 CHAPTER 5: ANALYSIS AND INTERPRETATION OF DATA 5.1. Quantitative Analysis 57 5.1.1. Learners? Generalization Approaches 57 5.1.2. Learners? Written Responses 58 5.1.2.1. Level Descriptors. 58 5.1.2.2. Learners? Choices of Routines and Mediators 59 5.1.3. Learners use of Mediators 63 5.1.4. Summary of the Previous Discussion 68 5.2. Qualitative Analysis 68 5.2.1. Interview Approach. 71 5.2.2. The Interviewed Participants 71 5.2.3 Learners? Generalization Routines 73 5.2.4. Learner Interview Analysis 75 5.2.4.1. Low Ability Learners 75 5.2.4.1. i) Researcher Comments (L5) 78 5.2.4.1.ii) Researcher Comments (L9) 82 5.2.4.2. Medium Ability Learners 83 5.2.4.2.i) Researcher Comments (L15) 85 5.2.4.2.ii) Researcher Comments (L17) 89 5.2.4.3. High ability Learners 90 5.2.4.3.i) Researcher Comments (L23) 93 5.2.4.3.ii) Researcher Comments (L28) 97 5.2.5. Discussion of the Interviews (episodes of the transcriptions) 98 5.3. Summary of analysis and findings 99 x CHAPTER 6: RESULTS AND DISCUSSION 6.1. Participants? Common Routines 101 6.2. Participants? Common Mediators 102 6.3. Learners? Algebraic Reasoning 105 6.4. Learners? Difficulties 107 6.5. Learners? Mathematical Communication 111 6.6. Comparison of the Interviewed Learners 113 6.7. Summary and Conclusion 116 CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS 7.1. Introduction 117 7.2. Detailed findings 118 7.3. Reflections of the study 122 7.4. Limitations of the study 123 7.5. Implications of the study 124 7.6. Recommendations 125 7.7. Conclusions 125 REFERENCES 127 xi APPENDICES A1 ??????????? ?Questionnaire-based task 134 A2 ....................................................... Ethics documentation 138 B ............................................................ Question 1 QRASS 145 C ............................................................ Question 2 QRASS 146 D ............................................................ Question 3 QRASS 147 E ..??????????????..Question 4 QRASS 148 F .................................................... Interview Transcriptions 150 xii LIST OF TABLES 1 ????????????. Linking the GET and FET Mathematics LO?s 3 2 ................................................. Definitions of generalization 19 3 ................................................. Aspects of generalizations 22 4 ................................................. Research Questions and Instruments 39 5 ................................................. Classification of routine categories 44 6 ................................................. Summary of Learners? Mediators (LVM) 55 7 ................................................. Overall routine utilization in the study 60 8 ................................................. Overall numerical generalization routines 61 9 ................................................. Summary of Mediators 63 10................................................ Questions 1 ? 4 Responses 66 11................................................ Examples of Learners? solutions 67 12 ............................................... Summary of the Interviewees? responses 70 13 ............................................... Routines versus Mediators in generalization 73 14................................................ Examples of Learners? Mediators 74 15 ............................................... Learners? generalization routines 119 16 ............................................... Choice of generalization routines by learner ability 120 xiii LIST OF FIGURES 1 ................................................. Tiles Groups 12 2.................................................. Example of Growing Patterns 18 3 ................................................. Tabular Mediation (Representation) 50 4.................................................. Diagrammatic-Tabular Representations (Mediation) 53 xiv ABBREVIATIONS GET Education and Training GD Gauteng Department of Education FET Further Education and Training OBE Outcome Based Education RNCS Revised National Curriculum Statement C2005 Curriculum 2005 NCS National Curriculum Statement DoE Department of Education LR Learners? Routines LVM (LM) Learners? Visual Mediators QRASS Question Response and Summary Sheet LD Level Descriptors TLA Task Level of Attainment MALATI Mathematics Learning and Teaching Initiative SO Specific Outcomes AS Assessment Standards LO Learning Outcomes AF Analytical Framework NCTM National Council for Teachers Mathematics