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Dynamic Digital Technologies for Dynamic Mathematics.Indd Dynamic Digital Dynamic Digital Technologies for Dynamic Mathematics for Dynamic Digital Technologies Technologies for This report summarises the outcomes of the Nuffi eld Foundation funded 2014–17 project ‘Developing teachers’ mathematical knowledge for Dynamic Mathematics teaching and classroom use of technology through engagement with key mathematical concepts using dynamic digital technology’. The Nuffi eld Foundation is an endowed charitable trust that aims to improve social Implications for teachers’ well-being in the widest sense. It funds research and innovation in education and social policy and also works to build capacity in education, science and social science research. knowledge and practice The Executive Summary of this report is also available as a separate publication, ISBN 9781782772248. Dr Alison Clark-Wilson is a Principal Research Fellow at the UCL Institute Hoyles Celia and Clark-Wilson Alison of Education and holds a Master’s degree and PhD in mathematics education. A former secondary maths teacher, she is very active in European mathematics education research communities, as a member of the executive of BSRLM and as a leader of the technology working groups at the biennial European Congress on Mathematics Education. She is a Chartered Mathematics Teacher and a Fellow of the Institute of Mathematics and its Applications. Professor Dame Celia Hoyles holds a fi rst class honours degree in mathematics and a Masters and Doctorate in mathematics education. She taught mathematics in London schools before moving into higher education, and is now Professor of Mathematics Education at the UCL Institute of Education. She was the UK Government’s Chief Adviser for Mathematics (2004–07) and Director of the National Centre for Excellence in the Teaching of Mathematics (2007–13). She was awarded an OBE in 2004 and made a Dame Commander of the Order of the British Empire in 2014. ISBN 978-1-78277-221-7 UCL Institute of Education Press 20 Bedford Way Alison Clark-Wilson London WC1H 0AL and Celia Hoyles 9 781782 772217 www.ucl-ioe-press.com Final Report IOE_Dynamic Digital Technologies for Dynamic Mathematics_5p.indd 1 17/08/2017 11:57 Dynamic Digital Technologies for Dynamic Mathematics Implications for teachers’ knowledge and practice Alison Clark-Wilson and Celia Hoyles Final Report April 2017 UCL Institute of Education Press The Nuffield Foundation is an endowed charitable trust that aims to improve social well-being in the widest sense. It funds research and innovation in education and social policy and also works to build capacity in education, science and social science research. The Nuffield Foundation has funded this project, but the views expressed are those of the authors and not necessarily those of the Foundation. More information is available at www. nuffieldfoundation.org First published in 2017 by the UCL Institute of Education Press, University College London, 20 Bedford Way, London WC1H 0AL www.ucl-ioe-press.com © Alison Clark-Wilson and Celia Hoyles 2017 British Library Cataloguing in Publication Data: A catalogue record for this publication is available from the British Library ISBNs 978-1-78277-221-7 (paperback) 978-1-78277-222-4 (PDF eBook) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of University College London. Typeset by Quadrant Infotech (India) Pvt Ltd Printed by CPI Group (UK) Ltd, Croydon, CR0 4YY Cover image © the authors Contents CONTENTS 3 ACKNOWLEDGEMENTS 6 LIST OF FIGURES 7 LIST OF TABLES 9 TERMINOLOGY AND ACRONYMS USED IN THE REPORT 11 ABOUT THE AUTHORS 12 EXECUTIVE SUMMARY 13 1 AIMS AND BACKGROUND 13 2 RESEARCH QUESTIONS AND METHODOLOGY 13 3 SUMMARY OF FINDINGS 15 3.1 Development of teachers’ Mathematical Pedagogical Technology Knowledge 15 3.1.1 Algebraic variable 15 3.1.2 Linear functions 15 3.1.3 Geometric similarity 15 3.1.4 Teachers’ lesson planning 16 3.1.5 Lesson observations of landmark activities 16 3.1.6 Teacher engagement 17 3.2 Design features of the professional development toolkit (PD toolkit) 17 4 DISCUSSION 17 4.1 Development of teachers’ Mathematical Pedagogical Technology Knowledge 17 4.2 Design of professional development activities and support 20 5 IMPLICATIONS FOR POLICY AND PRACTICE 22 5.1 Implications 22 5.2 Recommendations for policy 23 5.3 Recommendations for practice 23 MAIN REPORT 24 1 AIMS AND BACKGROUND 24 1.1 Brief history of Cornerstone Maths 25 3 Alison Clark-Wilson and Celia Hoyles 1.2 Research methodology 26 1.2.1 Sample of schools and teachers 26 1.2.2 The professional development cycles 27 1.2.3 Landmark activities 27 1.2.4 Adapted lesson study approach 30 1.3 Conceptualising mathematical knowledge for teaching with technology (MPTK) 30 1.3.1 Analysing lesson plans to gain insight into MPTK 34 1.3.2 Research ethics 34 1.4 Research questions 34 1.5 Teacher sample and their classroom settings 35 1.5.1 The professional and mathematical backgrounds of the teachers 35 1.5.2 Teachers’ classroom settings 36 2 MAIN FINDINGS 37 2.1 Teachers’ MPTK prior to their classroom use of dynamic mathematical technology 37 2.1.1 The teachers’ orientations towards dynamic mathematical technology 37 2.1.2 Teachers’ mathematical pedagogical technology knowledge 39 2.1.2.1 Algebraic variable 39 2.1.2.2 Linear functions 45 2.1.2.3 Mathematical (geometric) similarity 53 2.2 Teachers’ espoused MPTK as seen through their lesson plans for landmark activities 60 2.2.1 Algebraic variable 61 2.2.2 Linear functions 64 2.2.3 Geometric similarity 66 2.2.4 Development of teachers’ lesson plans over time 68 2.3 Teachers’ MPTK made visible through lesson observations/interviews for the landmark activities 70 2.3.1 Algebraic variable: Lesson observations 71 2.3.2 Linear functions: Lesson observations 71 2.3.3 Geometric similarity: Lesson observations 72 4 Contents 2.4 Lesson observations of landmark activities 72 2.5 General comments on lesson organisation and management 74 2.6 Case studies of teachers 75 2.7 Factors affecting teachers’ engagement 75 2.7.1 The focus on mathematics 75 2.7.2 Alignment with personal goals and ambitions 75 2.7.3 School roles and responsibilities 76 2.7.4 Departmental culture 76 3 THE PROFESSIONAL DEVELOPMENT TOOLKIT (PD TOOLKIT) 77 3.1 Design features of the PD Toolkit 77 3.1.1 Getting started 80 3.1.2 Curriculum units 80 3.1.3 Departmental case studies: Embedding DMT 81 3.1.4 Evidence base for different audiences 82 3.1.5 Project community 82 3.2 Evaluation of the PD Toolkit within school-based settings in London 82 4 DISCUSSION 83 4.1 Pedagogical knowledge 86 4.2 Mathematical Knowledge and Mathematical Knowledge for Teaching 86 4.3 Personal orientations 87 4.4 Technology Instrumental Genesis 88 4.5 Mathematical Pedagogical Technology Knowledge 89 5 IMPLICATIONS FOR POLICY AND PRACTICE 90 5.1 Implications 90 5.2 Recommendations for policy 91 5.3 Recommendations for practice 91 6 REFERENCES 92 APPENDIX A: LINKS WITH DFE-FUNDED MATHS HUBS 95 APPENDIX B: PROJECT ADVISORY GROUP 96 5 Alison Clark-Wilson and Celia Hoyles APPENDIX C: CASE STUDIES OF THREE TEACHERS 97 Case study 1: Sasha 97 Case study 2: Phoebe 103 Case study 3: Chris 109 APPENDIX D: ESSENTIAL PD ACTIVITIES TO USE DMT FOR DYNAMIC MATHEMATICS 115 APPENDIX E: EVALUATION OF THE PD TOOLKIT IN THREE SCHOOLS 117 School A 117 School B 119 School C 120 APPENDIX F: LIST OF PROJECT PUBLICATIONS 123 Acknowledgements This report summarises the outcomes of the Nuffield Foundation funded 2014–17 project ‘Developing teachers’ mathematical knowledge for teaching and classroom use of technology through engagement with key mathematical concepts using dynamic digital technology’ (see www. nuffieldfoundation.org/developing-teachers-mathematical-knowledge- using-digital-technology). The Nuffield Foundation is an endowed charitable trust that aims to improve social well-being in the widest sense. It funds research and innovation in education and social policy and also works to build capacity in education, science and social science research. The Nuffield Foundation has funded this project, but the views expressed are those of the authors and not necessarily those of the Foundation. More information is available at www.nuffieldfoundation.org The authors thank the Nuffield Foundation, the project advisory group and the teachers who worked with us as partners in the project. 6 List of figures Figure 1: Numbers of teachers involved in the project and their level of involvement (n = 209) 26 Figure 2: Landmark activity: Patterns and expressions 28 Figure 3: Landmark activity: Linear functions 29 Figure 4: Landmark activity: Geometric similarity 29 Figure 5: Adapted lesson study approach 30 Figure 6: Components of [Mathematical] Pedagogical Technology Knowledge (Thomas and Hong, 2013) 31 Figure 7: A mapping of the features of landmark activity lesson plans to provide insight into a teacher’s MPTK 33 Figure 8: Contrasting classroom arrangements (photos all taken from the ‘front’ of the classroom) 36 Figure 9: Software prompt to name an algebraic variable to represent the ‘pattern number’, showing the slider that can be dragged to change its value 41 Figure 10: Distribution of scores (n = 74, x = 3.0, SD = 1.25) 42 Figure 11: Lisa’s tiles (Hoyles and Healy, 2003) 43 Figure 12: Which is larger, 2n or n+2? 44
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