Combinatorics and Reasoning Representing, Justifying and Building Is
Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By
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Mathematics Education Library VOLUME 47 Managing Editor A.J. Bishop, Monash University, Melbourne, Australia
Editorial Board M.G. Bartolini Bussi, Modena, Italy J.P. Becker, Illinois, U.S.A. M. Borba, Rio Claro, Brazil B. Kaur, Singapore C. Keitel, Berlin, Germany G. Leder, Melbourne, Australia F. Leung, Hong Kong, China D. Pimm, Edmonton, Canada K. Ruthven, Cambridge, United Kingdom A. Sfard, Haifa, Israel Y. Shimizu, Tennodai, Japan O. Skovsmose, Aalborg, Denmark
For further volumes: http://www.springer.com/series/6276
Carolyn A. Maher · Arthur B. Powell · Elizabeth B. Uptegrove (Editors)
Combinatorics and Reasoning Representing, Justifying and Building Isomorphisms
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Editors Dr. Carolyn A. Maher Rutgers University Graduate School of Education 10 Seminary Place New Brunswick, NJ 08901 USA [email protected]
Dr. Arthur B. Powell Rutgers University Department of Urban Education 110 Warren Street Newark, NJ 07102 USA [email protected]
Dr. Elizabeth B. Uptegrove Felician College Department of Mathematical Sciences 223 Montross Avenue Rutherford, NJ 07070 USA [email protected]
Series Editor: Alan Bishop Monash University Melbourne 3800 Australia [email protected]
ISBN 978-0-387-98131-4 e-ISBN 978-0-387-98132-1 DOI 10.1007/978-0-387-98132-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010926950 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
This book is dedicated to the Kenilworth students who participated in the longitudinal study and from whom we continue to learn so much. We thank you for your continuing commitment, abundant trust, and generous sharing of how mathematical ideas and ways of reasoning are built.
Preface
Our research project on mathematical learning focuses on the accomplishments of a cohort group of learners from first grade though high school and beyond, concentrating on their work on a set of combinatorics tasks. We describe their impressive mathematical achievements over these years. We illustrate in detail the processes by which students learn to justify solutions to combinatorics problems that were challenging for their age and grade level. Based on transcribed video data and learners
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