Enhancing mathematics teachers’ mediation of a selected object of learning through participation in a learning study: the case of functions in Grade 10 VASEN PILLAY A THESIS SUBMITTED TO THE FACULTY OF HUMANITIES, UNIVERSITY OF THE WITWATERSRAND, JOHANNESBURG, IN FULFILMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY JOHANNESBURG 2013 Copyright Notice The copyright of this thesis 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 University. Extracts of or quotations from this thesis 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. An electronic version of this thesis may be found via the Library’s webpage (www.wits.ac.za/library). For permission requests, please contact the University Legal Office or the University Research Office (www.wits.ac.za). ii Declaration I declare that this thesis is my own unaided work. It is being submitted for the degree of Doctor of Philosophy at the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination at any other University. ________________________________ Vasen Pillay th 12 day of November in the year 2013 iii Abstract This thesis explores the potential of learning study as a teacher professional development model. The learning study is underpinned by variation theory and the judicious use of examples, with the goal of enhancing teachers’ mediation of a selected object of learning. The topic of functions at grade 10 provides the context of the content in which the learning study was implemented. More specifically, the object of learning identified by the teachers was to improve the learners’ ability to identify and name the class of function given its algebraic representation. In the teaching of mathematics, examples form a key resource for teachers to introduce concepts. Within the context of this study the judicious use of examples means that examples are selected and sequenced in a particular fashion so as to create an example space which keeps the critical feature of the object of learning in focus and thereby provides learners with opportunities to discern the object of learning. In selecting and sequencing examples, I draw on principles inherent in variation theory, particularly the idea of varying one aspect whilst keeping other aspects invariant. The learning study cycle reported on in this thesis comprised four lessons. The critical feature for the selected object of learning emerged after the second lesson in the cycle and was focused on only in the last lesson of the cycle. Focusing on the highest power of the independent variable emerged as the feature that enabled learners to discern the object of learning. Describing the enacted object of learning forms the substance of the data analysed in the thesis, with analysis following the production of data from the lesson transcripts. Data production was accomplished by drawing on Bernstein’s theory of the pedagogic device, and specifically the evaluative rule since it concerns itself with the transmission of criteria as to what counts as valid knowledge. To construct a more general account of the operation of evaluative judgement I follow the work of Davis who recruits Hegel’s theory of judgement. Hegelian judgement is used to elaborate Bernstein’s evaluative rule and in doing so, a theory of learning is backgrounded. In this study, I bring a theory of learning into focus by drawing on the principles of variation theory as it provides tools that enable a reading of the opportunities the pedagogy creates that enable learners to see ‘something’ in a certain way. This study demonstrates and so confirms that irrespective of the context in which you work: i) when a critical feature is identified, ii) examples are carefully selected with the critical feature in mind, and iii) if the critical feature is in focus for the teacher, increasingly more learners will begin to discern the intended object of learning. iv Keywords: Critical feature Examples Functions Learning study Object of learning Teacher Professional Development Variation Theory v To my wife, Saras, and my daughter, Kimesha vi Acknowledgements First and foremost, I am greatly indebted to my supervisor, Professor Jill Adler, for her very humane qualities, for her encouragement, enthusiasm and expertise throughout this study, and for skilfully mediating my ownership of this work. I am most fortunate to have had the honour of working with such an exceptional person. I wish to thank the four teachers for their willingness to participate in this study, their generosity of welcoming me into their classrooms and their eagerness in sharing their thoughts and ideas. I am grateful to Professor Ulla Runesson for the interest that she has shown in this work, for her valuable discussions and for her encouragement, enthusiasm and expertise during the study. I would like to express my sincerest thanks to the Sasol Inzalo Foundation for their generous funding of this study. I would also like to say thank you for the many opportunities provided to network with other individuals who critically engaged with my work and helped focus my thinking. I am also grateful to the National Research Foundation (NRF) for partially funding this study through the wider Wits Maths CONNECT project for secondary schools, directed by Professor Jill Adler. Without the support from the Sasol Inzalo Foundation and the NRF this study would not have been possible. Any opinion, findings or recommendations expressed in this thesis are those of the author and do not necessarily reflect the views of either funder. The members of the Wits Maths CONNECT project for secondary schools (Craig Pournara, Erlina Ronda, Moneoang Leshota, Ntsiki Luxomo, Patricia Nalube, Regina Essack, Shadrack Moalosi, Sitti Patahuddin and Thulisile Nkambule) have productively engaged with and interrogated aspects of this work. I am immensely grateful to this team as their ideas were productive for my development of ideas pertaining to this thesis. Thank you to Nomonde Mda and Lorraine Tlhwaele, who helped me as a doctoral student with everything I could possibly ask for. To my parents Athie and Ganes, and to Visva, Deenash and Levandri, thank you for your continuous support, encouragement and understanding. Finally, to Saras and Kimesha, you lived through the journey and production of this thesis. Thank you for your understanding and unwavering support during the time of writing this thesis. vii Table of Contents COPYRIGHT NOTICE .................................................................................................................... ii DECLARATION ............................................................................................................................... iii ABSTRACT ....................................................................................................................................... iv ACKNOWLEDGEMENTS............................................................................................................. vii ABBREVIATIONS ........................................................................................................................... xi LIST OF FIGURES ......................................................................................................................... xii LIST OF TABLES .......................................................................................................................... xiv CHAPTER 1 INTRODUCTION TO THE STUDY AND RESEARCH QUESTION ................ 1 1.1 INTRODUCTION ............................................................................................................................ 1 1.2 THE PROBLEM STATEMENT AND RESEARCH QUESTION ................................................................ 6 1.3 THE ‘GAP’ IN WHICH THIS STUDY IS LOCATED ............................................................................. 8 1.4 FURTHER RATIONALE FOR UNDERTAKING THIS STUDY .............................................................. 11 1.5 STRUCTURE OF THE THESIS ........................................................................................................ 13 CHAPTER 2 REVIEW OF LITERATURE RELEVANT TO THE STUDY ........................... 15 2.1 INTRODUCTION .......................................................................................................................... 15 2.2 TEACHER PROFESSIONAL DEVELOPMENT ................................................................................. 15 2.2.1 Mathematics teachers’ learning as a form of CoP or PLC .............................................. 16 2.2.2 Lesson study as a form of a PLC....................................................................................... 19 2.2.3 From lesson study to learning study ................................................................................. 22 2.3 FUNCTIONS ................................................................................................................................ 26 2.3.1 Functions – their importance ............................................................................................ 26 2.3.2 Functions – in the curriculum ........................................................................................... 28 2.3.3 Functions – a pedagogical perspective ............................................................................. 30 2.4 EXEMPLIFICATION AND ITS SIGNIFICANCE ................................................................................. 33 2.5 PROBLEM STATEMENT AND FOCUS QUESTIONS REVISITED ......................................................... 41 2.6 SUMMARY ................................................................................................................................. 41 CHAPTER 3 THEORETICAL UNDERPINNING OF THE STUDY ....................................... 44 3.1 INTRODUCTION .......................................................................................................................... 44 3.2 PEDAGOGY CONDENSES IN EVALUATION ................................................................................... 45 3.3 HEGELIAN JUDGEMENT ............................................................................................................. 47 3.3.1 The Judgement of Existence .............................................................................................. 47 3.3.2 The Judgement of Reflection ............................................................................................. 47 3.3.3 The Judgement of Necessity .............................................................................................. 48 3.3.4 The Judgement of the Notion ............................................................................................ 49 3.4 LEARNING AS EXPERIENCING VARIATION .................................................................................. 50 3.5 SUMMARY ................................................................................................................................. 53 CHAPTER 4 METHODOLOGY AND DESIGN ......................................................................... 55 4.1 INTRODUCTION .......................................................................................................................... 55 4.2 THE METHODOLOGICAL APPROACH ........................................................................................... 55 4.3 TEACHERS IN THE STUDY – SETTING THE SCENE ........................................................................ 57 4.4 THE PROCESS – IMPLEMENTING THE LEARNING STUDY .............................................................. 62 4.5 DATA COLLECTION STRATEGIES EMPLOYED .............................................................................. 64 4.5.1 Video recording of lessons ................................................................................................ 65 4.5.2 Teacher questionnaires ..................................................................................................... 66 4.6 MY ROLE AS RESEARCHER ......................................................................................................... 70 viii 4.7 SUMMARY ................................................................................................................................. 71 CHAPTER 5 DATA PRODUCTION, RIGOUR AND ETHICS ................................................ 73 5.1 INTRODUCTION .......................................................................................................................... 73 5.2 MOVING FROM THE THEORETICAL TO THE EMPIRICAL ............................................................... 74 5.3 DEVELOPING THE EXTERNAL LANGUAGE OF DESCRIPTION ........................................................ 75 5.3.1 The Evaluative Event ........................................................................................................ 76 5.3.2 Existence ........................................................................................................................... 79 5.3.3 From existence to reflection .............................................................................................. 83 5.3.4 Indicators of reflection ...................................................................................................... 83 5.3.5 From reflection to grounding the notion ........................................................................... 92 5.4 ISSUES OF RELIABILITY AND VALIDITY IN THIS STUDY ............................................................... 99 5.5 ETHICAL CONSIDERATIONS ...................................................................................................... 101 5.6 REVISITING THE RESEARCH QUESTIONS ................................................................................... 103 CHAPTER 6 THE INTENDED OBJECT OF LEARNING ..................................................... 106 6.1 INTRODUCTION ........................................................................................................................ 106 6.2 OVERVIEW OF THE PRE-TEST ................................................................................................... 106 6.3 LEARNERS’ PERFORMANCE IN THE PRE-TEST ........................................................................... 108 6.4 IDENTIFYING THE OBJECT OF LEARNING .................................................................................. 110 6.6 THE POST-TEST ....................................................................................................................... 116 CHAPTER 7 AN OVERVIEW OF THE LEARNING STUDY AND A DESCRIPTION OF THE LESSONS .............................................................................................................................. 121 7.1 INTRODUCTION ........................................................................................................................ 121 7.2 OVERVIEW OF THE LEARNING STUDY ...................................................................................... 121 7.3 LESSON 1 ................................................................................................................................. 123 7.4 LESSON 2 ................................................................................................................................. 128 7.5 LESSON 3 ................................................................................................................................. 133 7.6 LESSON 4 ................................................................................................................................. 135 CHAPTER 8 THE ENACTED OBJECT OF LEARNING ....................................................... 140 8.1 INTRODUCTION ........................................................................................................................ 140 8.2 THE ENACTED OBJECT OF LEARNING – LESSON 1 ................................................................... 144 8.3 THE POST-LESSON DISCUSSION – LESSON 1 ............................................................................. 155 8.4 THE ENACTED OBJECT OF LEARNING – LESSON 2 ................................................................... 156 8.5 POST-LESSON 2 DISCUSSION AND THE EMERGENCE OF A CRITICAL FEATURE ........................... 169 8.6 THE ENACTED OBJECT OF LEARNING – LESSON 3..................................................................... 173 8.7 THE POST-LESSON DISCUSSION – LESSON 3 ............................................................................. 185 8.8 THE ENACTED OBJECT OF LEARNING – LESSON 4 ................................................................... 188 8.9 THE POST-LESSON DISCUSSION – LESSON 4 ............................................................................. 201 8.10 SUMMARY ............................................................................................................................. 205 CHAPTER 9 THE LIVED OBJECT OF LEARNING .............................................................. 207 9.1 INTRODUCTION ........................................................................................................................ 207 9.2 THE POST-TEST ........................................................................................................................ 207 9.3 THE DELAYED POST-TEST ........................................................................................................ 212 9.4 SHIFTS IN TEACHERS’ THINKING .............................................................................................. 225 9.4.1 Shifts in Teacher J’s thinking .......................................................................................... 226 9.4.2 Shifts in Teacher S’s thinking.......................................................................................... 227 9.4.3 Shifts in Teacher L’s thinking ......................................................................................... 228 9.4.4 Shifts in Teacher T’s thinking ......................................................................................... 229 ix 9.5 SUMMARY ............................................................................................................................... 230 CHAPTER 10 CONCLUSION ..................................................................................................... 231 10.1 INTRODUCTION ...................................................................................................................... 231 10.2 THE VALUE OF DOING THIS STUDY ......................................................................................... 233 10.3 RECOMMENDATIONS – ATTEMPTING TO ADDRESS SOME POLICY DEMANDS .......................... 238 10.4 LIMITATION OF THIS STUDY AND IDEAS FOR FURTHER RESEARCH ......................................... 242 BIBLIOGRAPHY: ......................................................................................................................... 243 APPENDIX A: ETHICS CLEARANCE ..................................................................................... 249 APPENDIX B: LESSON PLANS AND WORKSHEET ............................................................ 250 APPENDIX C: LESSONS CHUNKED INTO EVENTS ........................................................... 258 x