Spatial Reasoning in the Early Years Over the past several years, “spatial reasoning” has gained renewed prominence among mathematics educators, as spatial skills are proving to be not just essential to mathematical understanding but also strong predictors of future success beyond the classroom in (cid:192)elds such as science, technology, and engineering(cid:17) (cid:37)y e(cid:91)ploring both primary and emergent dimensions, Spatial Reasoning in the Early Years helps de(cid:192)ne the concept of spatial reasoning and provides compelling evidence of the need for a clear focus within early education speci(cid:192)cally(cid:17) (cid:55)he authors review the research, look across current theories, and investigate implications for contemporary school mathematics pedagogy as they identify areas of inquiry necessary to bring a stronger spatial reasoning emphasis into the classroom(cid:17) (cid:55)he book contains many classroom(cid:16) or workshop(cid:16)based vignettes, highlighting the comple(cid:91)ity of spatial reasoning in educational practice, providing an in(cid:16)depth analysis of spatial reasoning as it applies to classroom practice, and o(cid:895)ering new ways of framing lessons to help young students hone their spatial reasoning abilities(cid:17) (cid:55)he book concludes with a forward(cid:16)looking agenda that contributes to developing a greater understanding of the role spatial reasoning plays in educational conte(cid:91)ts and beyond(cid:17) (cid:54)upported by plentiful visual representations, Spatial Reasoning in the Early Years skillfully integrates the conceptual and the concrete, making this te(cid:91)t a dynamic and accessible resource(cid:17) Brent Davis is (cid:51)rofessor and (cid:39)istinguished (cid:53)esearch (cid:38)hair of the (cid:58)erklund (cid:54)chool of (cid:40)ducation at the (cid:56)niversity of (cid:38)algary, (cid:36)lberta, (cid:38)anada(cid:17) (cid:55)he Spatial Reasoning Study Group is a transdisciplinary team of researchers from across (cid:49)orth (cid:36)merica with its hub at the (cid:56)niversity of (cid:38)algary(cid:17) (cid:44)ts members work in and across mathematics education, mathematics, psychology, curriculum studies, and cognitive science(cid:17) (cid:55)he group currently includes (cid:38)atherine (cid:39)(cid:17) (cid:37)ruce, (cid:37)everly (cid:38)aswell, (cid:47)issa (cid:39)(cid:183)(cid:36)mour, (cid:37)rent (cid:39)avis, (cid:48)ichelle (cid:39)refs, (cid:46)rista (cid:41)rancis, (cid:39)avid (cid:43)allowell, (cid:61)achary (cid:43)awes, (cid:39)onna (cid:46)otsopoulos, (cid:47)ynn (cid:48)c(cid:42)arvey, (cid:45)oan (cid:48)oss, (cid:60)ukari Okamoto, (cid:51)aulino (cid:51)reciado, (cid:49)athalie (cid:54)inclair, (cid:39)iane (cid:55)epylo, (cid:45)ennifer (cid:54)(cid:17) (cid:55)hom, and (cid:58)alter (cid:58)hiteley(cid:17) This page intentionally left blank Spatial Reasoning in the Early Years Principles, Assertions, and Speculations Brent Davis and the Spatial Reasoning Study Group (cid:41)irst published (cid:21)(cid:19)(cid:20)(cid:24) by Routledge (cid:26)(cid:20)(cid:20) (cid:55)hird (cid:36)venue, (cid:49)ew (cid:60)ork, (cid:49)(cid:60) (cid:20)(cid:19)(cid:19)(cid:20)(cid:26) and by Routledge (cid:21) (cid:51)ark (cid:54)quare, (cid:48)ilton (cid:51)ark, (cid:36)bingdon, O(cid:91)on, O(cid:59)(cid:20)(cid:23) (cid:23)R(cid:49) Routledge is an imprint of the Taylor & Francis Group, an informa business (cid:139) (cid:21)(cid:19)(cid:20)(cid:24) (cid:55)aylor (cid:9) (cid:41)rancis (cid:55)he right of the editor to be identi(cid:192)ed as the author of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 7(cid:27) of the (cid:38)opyright, (cid:39)esigns and (cid:51)atents (cid:36)ct (cid:20)(cid:28)(cid:27)(cid:27)(cid:17) (cid:36)ll rights reserved(cid:17) (cid:49)o part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers(cid:17) Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identi(cid:192)cation and e(cid:91)planation without intent to infringe(cid:17) Library of Congress Cataloging in Publication Data (cid:36) catalog record for this book has been requested(cid:17) (cid:44)(cid:54)(cid:37)(cid:49): (cid:28)7(cid:27)(cid:16)(cid:20)(cid:16)(cid:20)(cid:22)(cid:27)(cid:16)7(cid:21)(cid:28)(cid:19)(cid:22)(cid:16)(cid:25) (cid:11)hbk(cid:12) (cid:44)(cid:54)(cid:37)(cid:49): (cid:28)7(cid:27)(cid:16)(cid:20)(cid:16)(cid:20)(cid:22)(cid:27)(cid:16)7(cid:28)(cid:21)(cid:19)(cid:23)(cid:16)(cid:22)(cid:11)pbk(cid:12) (cid:44)(cid:54)(cid:37)(cid:49): (cid:28)7(cid:27)(cid:16)(cid:20)(cid:16)(cid:22)(cid:20)(cid:24)(cid:16)7(cid:25)(cid:21)(cid:22)7(cid:16)(cid:20) (cid:11)ebk(cid:12) (cid:55)ypeset in Palatino and (cid:48)yriad Pro by (cid:37)rent (cid:39)avis Contents Contributors vii Acknowledgments ix Section 1. What is spatial reasoning and why should we care? 1 1. What is spatial reasoning? 3 Walter Whiteley, Nathalie Sinclair, Brent Davis 2. The development of spatial reasoning in young children 15 Yukari Okamoto, Donna Kotsopoulos, Lynn McGarvey, David Hallowell 3. Developing spatial thinking 29 Zachary Hawes, Diane Tepylo, Joan Moss Section 2. If spatial reasoning is so important, why has it taken so long to be noticed? 45 4. A history and analysis of current curriculum 47 Brent Davis, Michelle Drefs, Krista Francis 5. Spatial knowing, doing, and being 63 Jennifer S. Thom, Lissa D’Amour, Paulino Preciado, Brent Davis Section 3. What are the curricular and pedagogical implications of spatial reasoning? 83 6. Spatializing the curriculum 85 Catherine D. Bruce, Nathalie Sinclair, Joan Moss, Zachary Hawes, Beverly Caswell 7. Motion and markings 107 Lynn McGarvey, Nathalie Sinclair, Jennifer S. Thom, Donna Kotsopoulos 8. Interactions between three dimensions and two dimensions 121 Krista Francis, Walter Whiteley Section 4. And so? What kind of research agenda might we need to pursue? 137 9. Spatializing school mathematics 139 Brent Davis, Yukari Okamoto, Walter Whiteley References 151 Name Index 169 Subject Index 173 This page intentionally left blank Contributors Catherine D. Bruce is (cid:36)ssociate Professor in the (cid:54)chool of (cid:40)ducation at (cid:55)rent (cid:56)niversity, Ontario, (cid:38)anada(cid:17) (cid:54)he studies children(cid:183)s learning of mathematics including fractions, algebra and the development of spatial reasoning(cid:17) Beverly Caswell is the (cid:39)irector of the Robertson Program for (cid:44)nquiry(cid:16)based (cid:55)eaching in (cid:54)cience and (cid:48)athematics at the (cid:56)niversity of (cid:55)oronto, Ontario, (cid:38)anada(cid:17) (cid:54)he works with educators to design learning environments that promote deeper understanding of concepts and equity(cid:17) Lissa D’Amour is (cid:36)ssistant Professor in the (cid:58)erklund (cid:54)chool of (cid:40)ducation, (cid:56)niversity of (cid:38)algary, (cid:36)lberta, (cid:38)anada(cid:17) (cid:54)he studies the interplay of an(cid:91)iety and attachment security in dynamic teacher(cid:178)learner systems(cid:17) Brent Davis is Professor and (cid:39)istinguished Research (cid:38)hair in (cid:48)athematics (cid:40)ducation in the (cid:58)erklund (cid:54)chool of (cid:40)ducation, (cid:56)niversity of (cid:38)algary, (cid:36)lberta, (cid:38)anada(cid:17) (cid:43)e researches the educational relevance of the comple(cid:91)ity sciences, focusing in particular on teachers(cid:183) disciplinary knowledge of mathematics(cid:17) Michelle Drefs is (cid:36)ssistant Professor and (cid:39)irector of (cid:55)raining for the (cid:54)chool and (cid:36)pplied (cid:38)hild Psychology program in the (cid:58)erklund (cid:54)chool of (cid:40)ducation, (cid:56)niversity of (cid:38)algary, (cid:36)lberta, (cid:38)anada(cid:17) (cid:43)er research interest is in the assessment of early numerical competencies(cid:17) Krista Francis is (cid:36)ssistant Professor and (cid:39)irector of the (cid:44)mperial Oil (cid:54)cience (cid:55)echnology (cid:40)ngineering and (cid:48)athematics (cid:11)(cid:44)O(cid:54)(cid:55)(cid:40)(cid:48)(cid:12) initiative with the (cid:58)erklund (cid:54)chool of (cid:40)ducation, (cid:56)niversity of (cid:38)algary, (cid:36)lberta, (cid:38)anada(cid:17) (cid:43)er research focus is (cid:46)(cid:178)(cid:20)(cid:21) (cid:54)(cid:55)(cid:40)(cid:48) teacher professional learning(cid:17) David Hallowell is a doctoral candidate in Education at the University of California (cid:54)anta (cid:37)arbara(cid:17) (cid:43)e holds (cid:48)(cid:17)(cid:36)(cid:17) degrees in both Education and Philosophy(cid:17) (cid:43)is research focuses on young children(cid:183)s spatial reasoning(cid:17) Zachary Hawes is a Research O(cid:896)cer at the (cid:39)r(cid:17) Eric (cid:45)ackman (cid:44)nstitute of Child (cid:54)tudy, a research center and laboratory school of the University of (cid:55)oronto, Ontario, Canada(cid:17) (cid:43)is research e(cid:91)amines the development of children(cid:183)s spatial and numerical cognition, with a focus on the e(cid:895)ects of classroom(cid:16)based interventions(cid:17) viii SPATIAL REASONING IN THE EARLY YEARS Donna Kotsopoulos is Associate Professor of Education and the director of the (cid:48)athematical (cid:37)rains (cid:47)aboratory at (cid:58)ilfrid (cid:47)aurier University, Ontario, Canada(cid:17) (cid:43)er research focus is on the human sciences(cid:17) (cid:54)he studies mathematical cognition and learning across the lifespan(cid:17) Lynn McGarvey is a Professor of (cid:48)athematics Education at the University of Alberta, Alberta, Canada(cid:17) (cid:43)er research focuses on mathematical reasoning of young children with emphasis on the topics of algebraic thinking and spatial reasoning(cid:17) Joan Moss is Associate Professor in Department of Applied Psychology and (cid:43)uman Development at the University of (cid:55)oronto, Ontario, Canada(cid:17) (cid:43)er research has focused on young students(cid:183) development of rational number, early algebra, and spatial reasoning(cid:17) Yukari Okamoto is Professor of Education at the University of California (cid:54)anta (cid:37)arbara(cid:17) (cid:54)he speciali(cid:93)es in children(cid:183)s numerical, spatial, and biological reasoning(cid:17) (cid:54)he received her Ph(cid:17)D(cid:17) from (cid:54)tanford University and was a (cid:54)pencer (cid:41)oundation Postdoctoral (cid:41)ellow(cid:17) Paulino Preciado is Assistant Professor and Director of the (cid:48)ath (cid:48)inds (cid:44)nitiative in the (cid:58)erklund (cid:54)chool of Education, University of Calgary, Alberta, Canada(cid:17) His research focuses on the collaborative design of tasks and lessons aimed at promoting deep mathematical understanding through intellectual engagement(cid:17) Nathalie Sinclair is Professor in the (cid:41)aculty of Education and a Canada Research Chair in (cid:55)angible (cid:48)athematics (cid:47)earning at (cid:54)imon (cid:41)raser University, (cid:37)ritish Columbia, Canada(cid:17) (cid:54)he is the author co(cid:16)author of Mathematics and the Body: Material Entanglements in the Classroom, among other books(cid:17) Diane Tepylo is mathematics teacher and a Ph(cid:17)D(cid:17) candidate in the Department of Applied Psychology and Human Development at the (cid:44)nstitute for (cid:54)tudies in Education, University of (cid:55)oronto, Ontario, Canada(cid:17) Her research focuses the role of spatial reasoning in mathematics and teacher(cid:16)centered professional learning(cid:17) Jennifer S. Thom is Associate Professor at the University of (cid:57)ictoria (cid:37)ritish Columbia, Canada(cid:17) Her research focuses on mathematics in the early years as well as the embodied and collective nature of mathematics and mathematical understanding(cid:17) Walter Whiteley is Professor of (cid:48)athematics and (cid:54)tatistics(cid:17) He is a member of the (cid:42)raduate Programs in (cid:48)athematics and a member of Education, in Computer (cid:54)cience and (cid:44)nterdisciplinary (cid:54)tudies, (cid:60)ork University, Ontario, Canada(cid:17) He is a researcher in discrete applied geometry (cid:11)in engineering, sciences, and mathematics(cid:12), and a geometry educator for future teachers and for future mathematicians(cid:17) Acknowledgments (cid:55)he (cid:54)patial Reasoning (cid:54)tudy (cid:42)roup (cid:11)(cid:54)R(cid:54)(cid:42)(cid:12) is a transdisciplinary team(cid:17) (cid:44)ts members work in and across mathematics education, mathematics, psychology, curriculum studies, and cognitive science(cid:17) (cid:55)he group currently includes Catherine D(cid:17) (cid:37)ruce, (cid:37)everly Caswell, (cid:47)issa D(cid:183)Amour, (cid:37)rent Davis, (cid:48)ichelle Drefs, (cid:46)rista (cid:41)rancis, David Hallowell, (cid:61)achary Hawes, (cid:54)teven (cid:46)han, Donna (cid:46)otsopoulos, (cid:47)ynn (cid:48)c(cid:42)arvey, (cid:45)oan (cid:48)oss, (cid:60)ukari Okamoto, Paulino Preciado, (cid:49)athalie (cid:54)inclair, Diane (cid:55)epylo, (cid:45)ennifer (cid:54)(cid:17) (cid:55)hom, and (cid:58)alter (cid:58)hiteley(cid:17) (cid:55)he (cid:54)R(cid:54)(cid:42) has met regularly since (cid:21)(cid:19)(cid:20)(cid:21)(cid:17) (cid:55)his book is its (cid:192)rst major print publication, following up on a series of presentations and symposia at major conferences (cid:11)e(cid:17)g(cid:17), (cid:37)ruce et al(cid:17), (cid:21)(cid:19)(cid:20)(cid:22)(cid:30) (cid:54)inclair (cid:9) (cid:37)ruce, (cid:21)(cid:19)(cid:20)(cid:23)(cid:12)(cid:17) (cid:47)ike those conference engagements, this te(cid:91)t is better described as a “co(cid:16)authored publication” than a “joint presentation” or an “edited collection(cid:17)” Unfortunately, we are unable to signal the e(cid:91)tent of members(cid:183) contributitions to each part of this book, but we are keen to underscore that the listed authors for the chapters are convenient (cid:192)ctions(cid:17) (cid:55)hey indicate who took major responsibilities, but they do not re(cid:193)ect the depths of the shared thinking and collaborative e(cid:895)orts(cid:17) (cid:55)he work of the (cid:54)R(cid:54)(cid:42) is made possible through the support of the (cid:44)mperial Oil (cid:54)cience, (cid:55)echnology, Engineering, and (cid:48)athematics (cid:44)nitiative(cid:17) (cid:44)O(cid:54)(cid:55)E(cid:48) is housed in the (cid:58)erklund (cid:54)chool of Education at the University of Calgary(cid:17) (cid:55)he cover image is from an original photograph provided by (cid:54)teven (cid:46)han(cid:17) (cid:58)e would like to e(cid:91)press our appreciation to (cid:45)onna (cid:47)a (cid:45)oy, (cid:48)eghna (cid:54)oni, and (cid:60)u (cid:61)hang at the University of California (cid:54)anta (cid:37)arbara and to the (cid:48)ath (cid:23)(cid:19)(cid:19)(cid:19) project group at (cid:60)ork University for their helpful suggestions(cid:17)