WHAT COUNTS IN TEACHING MATHEMATICS

WHAT COUNTS IN TEACHING MATHEMATICS

Self-Study of Teaching and Teacher Education Practices Volume 11 Series Editor John Loughran, Monash University, Clayton, Australia Advisory Board Mary Lynn Hamilton, University of Kansas, USA Ruth Kane, Massey University, New Zealand Geert Kelchtermans, University of Leuven, Belgium Fred Korthagen, IVLOS Institute of Education, The Netherlands Tom Russell, Queen s University, Canada For further volumes: http://www.springer.com/series/7072

WHAT COUNTS IN TEACHING MATHEMATICS Adding Value to Self and Content Edited by Sandy Schuck University of Technology Sydney, Sydney, Australia Peter Pereira DePaul University, Chicago, USA 123

Editors Sandy Schuck University of Technology Sydney PO Box 222 Lindfield NSW 2070 Australia sandy.schuck@uts.edu.au Peter Pereira DePaul University 2320 Kenmore Chicago IL 60614 USA ppereira@condor.depaul.edu ISSN 1875-3620 ISBN 978-94-007-0460-2 e-isbn 978-94-007-0461-9 DOI 10.1007/978-94-007-0461-9 Springer Dordrecht Heidelberg London New York Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Series Editor s Foreword When the International Handbook of Self-Study of Teaching and Teacher Education Practices (Loughran, Hamilton, LaBoskey, & Russell, 2004) was launched at the American Educational Research Association Annual conference in 2004, the discussant, Denis Phillips of Stanford University posed an interesting question for the S-STEP community. Denis asked how the field might further develop and progress in the wake of the achievement of publishing a Handbook that so thoroughly captured the research literature and ideas around self-study. He was of the view that in publishing the Handbook the S-STEP community might feel as though the job had been done and that furthering the core intent of self-study might be overtaken by an approach to gate-keeping that could limit the influence of self-study on the very field it was designed to foster teaching and teacher education. This book by Schuck and Pereira, like that of Crowe s (2010) in Social Studies, offers another clear and strong response to Phillips insightful challenge to the S-STEP community. Schuck and Pereira have assembled an impressive group of mathematics educators, each of whom offers new ways of thinking about the teaching and learning of mathematics as a consequence of their involvement in self-study. The catalyst for their writing was a concern to help our students develop mathematical pedagogical content knowledge as well as subject content knowledge... [because our] soon to be teachers of maths in primary and secondary schools need to recognise, and know how to reduce, the conceptual difficulties that often arise for school students. These authors efforts in so doing are captured in the chapters of this book in ways that demonstrate an abiding commitment to teaching and learning about mathematics teaching that is the core business of mathematics teacher education. The outcome of a serious consideration of a self-study methodology in researching the teaching and learning of mathematics is clearly evident in each of the chapters. Each of the authors demonstrates how they had to challenge their own conceptions of mathematics teaching and learning in order to develop deeper understandings of their own practice because, ultimately, they were concerned to genuinely challenge their own students approach to, and understanding of, mathematics. It is this focus on their students learning that has been so important in shaping their own learning about practice and is clearly an outcome derived from a thoughtful approach to researching practice through self-study. v

vi Series Editor s Foreword Because self-study has been a guiding methodology to the research in this book, it is not surprising that other key aspects of teaching and learning also emerge. For example, the book is loosely divided into two parts. The first part offers insights gained as a consequence of learning though mentoring and collaboration a key aspect of self-study. The second part delves into the tensions and conflicts inherent in challenging students attitudes and beliefs about mathematics and mathematics teaching. Through these two separate but closely related organizing principles, a coherent, thoughtful and honest account of salient issues in researching the teaching and learning of mathematics is portrayed for the reader in a most accessible and meaningful way. Schuck and Pereira have certainly assembled a core group of mathematics educators whose work demonstrates quality and scholarship in ways that offer many invitations and opportunities for others to build upon. This book brings to the surface the value of learning with, and from, others through collaboration, mentoring and challenging the status-quo. Key concepts of reflection (Dewey, 1933) and framing and reframing (Schön, 1983) continually emerge in the chapters as the authors draw attention to the need to recognize and reconsider taken-for-granted aspects of practice that are so frequently overlooked or ignored in practice. In so doing, the ability to look again into the heart of teaching and learning in order to purposefully develop long-lasting pedagogical relationships in teacher education emerges as a theme that binds these chapters together into a coherent whole. The editors describe self-study as challenging, unsettling and uncomfortable. However, as they make clear, and as each of the chapters in this book more than illustrates, in accepting the challenge to look into practice in new ways, the sense of being unsettled and uncomfortable clearly leads to new learning that enhances the pedagogical experience for all of those involved. A crucial outcome of self-study is to build on and expand our knowledge of teaching and learning about teaching in ways that matter for the profession. In this book, Schuck and Pereira have done just that by drawing on the experiences of a range of authors who, through their common focus on mathematics, help to offer breakthroughs in a field in ways that genuinely model innovative approaches to studying mathematics teaching and learning. This book is a fine addition to the growing body of literature of subject-specific self-study. It illustrates how understanding and responding to the self is important, but of itself, not sufficient for developing a pedagogy of teacher education (Korthagen, et al., 2001; Loughran, 2006; Ritter, 2007). Despite the superficial interpretations of those not involved in self-study but who often have opinions about the work as being too concerned with the self, this book makes clear that looking beyond the self is central to knowledge growth and development in ways that do make a difference for teaching and learning about mathematics teaching. In a previous self-study Schuck (2009) highlighted the value of listening to, and learning from, her students of teaching. Likewise, Pereira (2005) has illustrated how to learn from the tensions and dilemmas associated with learning about how to become a teacher of mathematics. Together they have created a formidable editorial team and have developed a quality product that in years to come will be seen as a seminal work in the field of self-study of teacher education practices in mathematics.

Series Editor s Foreword vii References Crowe, A. (Ed.) (2010). Advancing social studies education through self-study methodology: The power, promise, and use of self-study in social studies education. Dordrecht, The Netherlands: Springer. Dewey, J. (1933). How we think. Lexington, MA: D.C. Heath and Company. Korthagen, F. A. J., Kessels, J., Koster, B., Langerwarf, B., & Wubbels, T. (2001). Linking theory and practice: The pedagogy of realistic teacher education. Mahwah, NJ: Lawrence Erlbaum Associates Publishers. Loughran, J. J. (2006). Developing a pedagogy of teacher education: Understanding teaching and learning about teaching. London: Routledge. Loughran, J. J., Hamilton, M. L., LaBoskey, V. K., & Russell, T. (Eds.) (2004). International handbook of self-study of teaching and teacher education practices. Dordrecht, The Netherlands: Kluwer Academic Publishers. Pereira, P. (2005). Becoming a teacher of mathematics. Studying Teacher Education, 1(1), 69 83. Ritter, J. K. (2007). Forging a pedagogy of teacher education: The challenges of moving from classroom teacher to teacher educator. Studying Teacher Education: A journal of self-study of teacher education practices, 3(1), 5 22. Schön, D. A. (1983). The reflective practitioner: How professionals think in action. NewYork: Basic Books. Schuck, S. (2009). How did we do? Beginning teachers teaching mathematics in elementary schools. Studying Teacher Education, 5(2), 113 123. J. John Loughran

Contents 1 What Counts in Mathematics Education?... 1 Sandy Schuck and Peter Pereira Part I Collaborations and Critical Friends 2 Tensions of Mentoring Mathematics Teachers: Translating Theory into Practice... 13 Paul Betts 3 Team-Teaching About Mathematics for All Collaborative Self-Study... 29 Hafdís Guðjónsdóttir and Jónína Vala Kristinsdóttir 4 Growing Possibilities: Designing Mathematical and Pedagogical Problems Using Variation... 45 Cynthia Nicol 5 Resisting Complacency: My Teaching Through an Outsider s Eyes... 61 Sandy Schuck Part II Examining Our Practice: Conflicts, Dilemmas and Incongruities 6 How Students Teach You to Learn: Using Roundtable Reflective Inquiry to Enhance a Mathematics Teacher Educator s Teaching and Learning... 77 Robyn Brandenburg 7 Making Sense of Students Fractional Representations Using Critical Incidents... 93 Nell B. Cobb 8 Reforming Mathematics Teacher Education Through Self-Study. 111 Joanne E. Goodell ix

x Contents 9 Opportunities for Learning A Self-Study of Teaching Statistics in a Mathematics Learning Centre... 127 Sue Gordon 10 Reconstructing Teachers of Mathematics... 145 Peter Pereira 11 Are We Singing from the Same Songbook?... 161 Anne Prescott 12 Adding Value to Self and Content in Mathematics Education: Working in a Third Space... 177 Peter Pereira and Sandy Schuck Author Index... 187 Subject Index... 191

Contributors Paul Betts Faculty of Education, University of Winnipeg, Winnipeg, Canada, p.betts@uwinnipeg.ca Robyn Brandenburg School of Education, University of Ballarat, Ballarat, VIC, Australia, r.brandenburg@ballarat.edu.au Nell B. Cobb DePaul University, Chicago, IL, USA, ncobb@depaul.edu Joanne E. Goodell Cleveland State University, Cleveland, OH, USA, j.goodell@csuohio.edu Sue Gordon The University of Sydney, Sydney, Australia, sue.gordon@sydney.edu.au Hafdís Guðjónsdóttir School of Education, University of Iceland, Reykjavík, Iceland, hafdgud@hi.is Jónína Vala Kristinsdóttir School of Education, University of Iceland, Reykjavík, Iceland, joninav@hi.is Cynthia Nicol Department of Curriculum and Pedagogy, University of British Columbia, Vancouver, BC, Canada, cynthia.nicol@ubc.ca Peter Pereira School of Education, DePaul University, Chicago, IL, USA, ppereira@condor.depaul.edu Anne Prescott Centre for Research in Learning and Change, University of Technology Sydney, Sydney, Australia, anne.prescott@uts.edu.au Sandy Schuck Centre for Research in Learning and Change, University of Technology Sydney, Sydney, Australia, sandy.schuck@uts.edu.au xi

About the Authors Paul Betts is an Associate Professor at the University of Winnipeg. He taught mathematics in the public school system for 7 years before moving into a lecturer position at Brandon University in 1999. While at Brandon, he completed most of his PhD at the University of Regina in mathematics curriculum. He began a position at the University of Winnipeg in 2003 as an Assistant Professor specializing in mathematics education. In 2005, he successfully completed his PhD dissertation, which considered pre-service teachers experiences with the nature of mathematics. He recently earned tenure and promotion at the University of Winnipeg. His current research focuses on (novice) mathematics teacher professional learning and the potential of philosophies of mathematics for the reform of mathematics education. p.betts@uwinnipeg.ca Robyn Brandenburg is a Senior Lecturer in the School of Education at the University of Ballarat, Victoria, Australia. Her learning and teaching interests include a focus on developing pre-service teacher (PST) conceptual and mathematical pedagogical content knowledge, and reflective practice of teacher educators and PSTs. Her research is largely based on self-study methodology which is underpinned by systematically examining assumptions about learning and teaching as a means of enhancing pedagogy and practice. r.brandenburg@ballarat.edu.au Nell B. Cobb is an Associate Professor of Mathematics Education at DePaul University, School of Education in Chicago Illinois. She is responsible for teaching middle and elementary mathematics education courses at the undergraduate and graduate levels. Nell also teaches content courses on mathematics for elementary teachers in the Mathematical Sciences Department. She is the co-director of the newly designed Masters of Science in Middle School Mathematics Education (MSME) Program and also serves as the Secondary Education Program Leader in the School of Education, Department of Teacher Education. Nell has coordinated the Algebra Project Teacher Resource Materials Team and she has coordinated a number of Algebra Project Professional Development Institutes. She has presented at several conferences and published articles in the areas of mathematics teacher competencies and general mathematics education. ncobb@depaul.edu xiii

xiv About the Authors Joanne E. Goodell is an Associate Professor at Cleveland State University, where she teaches mathematics education and qualitative research methods courses, and directs the CSUTeach STEM teacher preparation program. She started her teaching career in Australia as a secondary mathematics teacher. Her research interests encompass self-study, gender issues and reform in mathematics education. j.goodell@csuohio.edu Sue Gordon is Senior Lecturer at the Mathematics Learning Centre and Honorary Lecturer in the Faculty of Education and Social Work at The University of Sydney, Australia. Her teaching supports students learning of mathematics and statistics at her university with a focus on statistics for Psychology. Her research is concerned broadly with teaching and learning in higher education including statistics education. Current projects include co-editing a Special Edition of the Statistics Education Research Journal on qualitative approaches in statistics education research, investigating university teachers conceptions of student diversity in their classes and their views of effective teaching in a range of professional areas, and exploring the experiences of coordinators, teachers and students in mathematics bridging courses. sue.gordon@sydney.edu.au Hafdís Guðjónsdóttir is Associate Professor in Special Education and Inclusive Pedagogy at the University of Iceland, School of Education. She graduated from Iceland Teacher College as a teacher in the year 1973. She earned a B.A. in Special Education from Iceland University of Education 1990, M.A. in Special Education from University of Oregon, Eugene 1993 and PhD in Special Education from the same school in the year 2000. For 25 years she taught as a general and special education teacher at compulsory schools. From the year 2000 she has taught at the University of Iceland School of Education. She has also collaborated with colleagues around the world, consulted with educators in Latvia, been a part of the European Agency for Development in Special Needs Education research group and the program chair for Self-study of Teacher Education Sig at AERA for 3 years. Her main research interest is inclusive education, pedagogy, teacher development and professionalism, teacher research, and self study of teaching and teacher education practices. hafdgud@hi.is Jónína Vala Kristinsdóttir is an Assistant Professor of Mathematics Education at the University of Iceland, School of Education. She worked for 20 years as a general classroom teacher in an elementary school. Teaching is her primary profession and she has been active in curriculum planning and writing in mathematics for elementary schools. She teaches undergraduate and graduate courses on mathematics education and education in general. Her research interests are in mathematics learning and teaching in diverse classrooms. Current research and writing include teachers development in mathematics teaching in diverse classrooms and a self-study of mathematics for all learners. joninav@hi.is Cynthia Nicol is an Associate Professor of mathematics education in the Department of Curriculum and Pedagogy at the University of British Columbia, Canada. Her research focus is in the areas of teacher education, mathematics

About the Authors xv education, Aboriginal education, and culturally responsive research ethics and teaching practices. She is particularly interested in the methodologies of participatory action research, self-study research and community-based action research for the transformative possibilities they offer in researching mathematics, teacher and Aboriginal education. Her current research projects explore the nature of problem-based learning, culturally responsive education, and place-based learning. cynthia.nicol@ubc.ca Peter Pereira (Editor), a retired Associate Professor in the School of Education at DePaul University in Chicago, has taught courses in mathematics, mathematics education, and curriculum. He is currently teaching a geometry course for middle school mathematics teachers focusing on developing mathematical habits of mind. He has published papers on deliberative curriculum theory, mathematics education, computer programming, dynamic geometry, and mathematics teacher development. He is on the international advisory board of Studying Teacher Education and his current scholarship concerns the complex process of becoming a mathematics teacher. He received a Master s degree from Harvard University and did his doctoral work at the University of Chicago. Prior to coming to DePaul, he taught mathematics in high schools in the United States and in England and was an instructor for mathematics teachers in India for two summers. ppereira@condor.depaul.edu Anne Prescott is a Senior Lecturer at the University of Technology Sydney. Anne is the coordinator of the Teacher Professional Development Program in the Australian Centre for Child and Youth: Culture and Wellbeing and is a member of the Centre for Research on Learning and Change (L&C) in the Faculty of Arts and Social Sciences. Anne was a mathematics teacher in secondary schools before becoming a university lecturer. She coordinates the mathematics education subjects in the secondary education program at UTS. Anne s research interests are in student misconceptions, and the issues confronting pre-service and beginning teachers of mathematics as they enter the profession. anne.prescott@uts.edu.au Sandy Schuck (Editor) is Associate Professor in Education at the University of Technology Sydney. Sandy is the leader of the research stream, Pedagogical Practice and Innovation, which is part of the University Research Strength, the Centre for Research on Learning and Change (L&C). She is the Coordinator of Education Higher Degrees by Research for the Faculty of Arts and Social Sciences. Sandy was a mathematics teacher in secondary schools before becoming a university lecturer. She coordinates the mathematics education subjects in the primary education program at UTS. Sandy s research interests are in three main areas: affective aspects of mathematics education, pedagogy with innovative technologies, and mentoring and induction of early career teachers. She has been engaged in self-study of her practices in teaching mathematics for over a decade. She is co-editor of two books on self-study of teacher education practices and has written numerous papers about her self-study work. sandy.schuck@uts.edu.au