Research in Mathematics Education Series editors Jinfa Cai James A. Middleton More information about this series at http://www.springer.com/series/13030
Patricio Felmer Erkki Pehkonen Jeremy Kilpatrick Editors Posing and Solving Mathematical Problems Advances and New Perspectives
Editors Patricio Felmer University of Chile Santiago, Chile Erkki Pehkonen University of Helskini Helsinki, Finland Jeremy Kilpatrick University of Georgia Athens, USA Research in Mathematics Education ISBN 978-3-319-28021-9 ISBN 978-3-319-28023-3 (ebook) DOI 10.1007/978-3-319-28023-3 Library of Congress Control Number: 2016933779 Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
Introduction Systematic research on problem solving in mathematics can be seen to have begun over 70 years ago with the work of George Pólya, whose most famous publication was likely the book How to Solve It (Pólya, 1945 ). Today there is a huge literature on mathematical problem solving that includes research studies, descriptions, surveys, and analyses. Among the most influential publications have been (and still are) the book by Mason, Burton, and Stacey ( 1985 ); the book by Schoenfeld ( 1985 ); and the paper by Kilpatrick ( 1987 ). The Mason et al. ( 1985 ) book emphasizes the importance of creativity and highlights the many cul-de-sacs in problem solving as well as the importance of a solver s persistence. The book by Schoenfeld ( 1985 ) is a well-known sourcebook. Younger researchers call it the black book of problem solving. Kilpatrick s ( 1987 ) paper underlines the connection between problem solving and problem posing, giving special emphasis to problem formulation. These publications form part of the foundation on which this book rests. The chapters in the book are based on presentations at the final workshop of a comparative research project from 2010 to 2013 between the University of Chile and the University of Helsinki. The project, whose title was On the Development of Pupils and Teachers Mathematical Understanding and Performance when Dealing with Open-Ended Problems, was initiated by Prof. Erkki Pehkonen (Helsinki) and Prof. Leonor Varas (Santiago). In 2009, the Chilean CONICYT (Comisión Nacional de Investigación Científica y Tecnológica) and the Finnish Academy opened a cooperative program in educational research. Profs. Pehkonen and Varas worked together on an application for a research grant whose leading idea was pupils development with open-ended problem solving. The project was funded and operated for 3 years. The final workshop, an integral part of the joint research project, was originally designed as a forum to discuss the main results of the project. However, with support from the Center for Advanced Research in Education (CIAE) and the Center for Mathematical Modeling (CMM), both at the University of Chile, a grant was obtained that enabled the workshop to be expanded well beyond the project participants. The grant supported the invitation of more than 20 international specialists in the field of mathematical problem solving to join the workshop. In the selection of additional participants, we tried to get a broad group v
vi Introduction of specialists from different parts of the world. After the workshop, all presenters were offered an opportunity to contribute a chapter to the book, and almost all accepted the invitation. Each paper was blind reviewed by two people in most cases an author of a different chapter, but in some cases an outside reviewer. The program of the 4-day problem-solving workshop at the University of Chile (Santiago) in December 2013 was as follows: Tuesday 10 Wednesday 11 Thursday 12 Friday 13 9:00 9:45 Yan Ping Xin United States Leonor Varas Chile 9:45 10:30 Peter Liljedahl Canada Salomé Martinez Chile Teachers workshop (CF) 11:00 11:45 Masami Isoda Japan Hähkiöniemi Finland Andras Ambrus Hungary Teachers workshop (CF) 11:45 13:00 Jeremy Kilpatrick United States Jinfa Cai United States John Mason England Markku Hannula and Liisa Näveri (CF) Finland 15:00 15:45 Erkki Pehkonen Finland 15:45 16:30 José Carrillo Spain 17:00 17:45 Rosa Leikin Israel 17:45 18:30 Bernd Zimmermann Germany Torsten Fritzlar Germany Susan Leung Taiwan Patricio Felmer Chile Yew Hoong Leong Singapore Wim van Dooren Belgium Markku Hannula Finland Valentina Giaconi and María Victoria Martínez (CF) Chile Alejandro López and Paulina Araya (CF) Chile Closing ceremony with music from Los Bosquinos Band In the case of several authors, usually the first one gave the presentation. The book is divided into three parts: (I) Problem Posing and Solving Today; (II) Students, Problem Posing, and Problem Solving; and (III) Teachers, Problem Posing, and Problem Solving. Part I begins with the summary of the role of mathematical textbooks in problem posing by Jinfa Cai et al. In the next paper José Carrillo and Jorge Cruz discuss the role of problem posing and solving. Affect is also an important factor in problem solving; this is dealt with by Valentina Giaconi et al. in the frame of Chilean elementary students. Nicolas Libedinsky and Jorge Soto Andrade examine the cooperation between affect and problem solving. Jeremy Kilpatrick opens a new aspect in problem solving, discussing problem solving and inquiry. The section is closed by Bernd Zimmermann who looks at the history of mathematics and reveals interesting problems. The section review is given by John Mason.
Introduction vii Part II begins with Jinfa Cai s and Frank Lester s overview on problem-solving research results. Then András Ambrus and Krisztina Barczi-Veres consider the situation of problem solving in Hungary, especially from the viewpoint of average students. Torsten Fritzlar explains the results of an exploratory problem implemented by him. The next paper is from Erkki Pehkonen et al. who describe a new data gathering method used in the Chile Finland research project. Manuel Santos-Trigo and Luis Moreno-Armella have used technology in order to foster students experiences in problem solving. In the chapter of Tine Degrande et al., the modeling aspects of problem solving are under focus. Yan Ping Xin deals with model-based problem solving. Here Masami Isoda has written the section review. Part III begins with John Mason s considerations where he examines the concept of problem from a new viewpoint. The paper of Patricio Felmer and Josefa Perdomo- Díaz discusses Chilean novice teacher in problem solving. Leong Yew Hoong et al. deal with problem solving in the Singaporean curriculum. Problem posing in the elementary school program is examined by Shuk-kwan S. Leung. Edward A. Silver discusses problem solving in teachers professional learning. Peter Liljedahl explains on the conditions of teaching problem solving. The section review is given by Kaye Stacey. Finally we would like to thank a lot of peoples for their helping hands. Especially we are grateful for those anonymous reviewers who helped us to improve the chapters in the book. But above all we thank Gladys Cavallone for her huge job in practically organizing the workshop at the university and her efficient handling of the papers of the book. Santiago, Chile Helsinki, Finland Athens, USA Patricio Felmer Erkki Pehkonen Jeremy Kilpatrick References Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123 147). Hillsdale, NJ: Erlbaum. Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically. Bristol: Addison-Wesley. Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Contents Part I Problem Posing and Solving Today How Do Textbooks Incorporate Mathematical Problem Posing? An International Comparative Study... 3 Jinfa Cai, Chunlian Jiang, Stephen Hwang, Bikai Nie, and Dianshun Hu Problem-Posing and Questioning: Two Tools to Help Solve Problems... 23 José Carrillo and Jorge Cruz Affective Factors and Beliefs About Mathematics of Young Chilean Children: Understanding Cultural Characteristics... 37 Valentina Giaconi, María Leonor Varas, Laura Tuohilampi, and Markku Hannula On the Role of Corporeality, Affect, and Metaphoring in Problem-Solving... 53 Nicolás Libedinsky and Jorge Soto-Andrade Reformulating: Approaching Mathematical Problem Solving as Inquiry... 69 Jeremy Kilpatrick Improving of Mathematical Problem-Solving: Some New IDEAS from Old Resources... 83 Bernd Zimmermann Part 1 Reaction: Problem Posing and Solving Today... 109 John Mason Part II Students, Problem Posing, and Problem Solving Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research... 117 Frank K. Lester Jr. and Jinfa Cai ix
x Contents Teaching Mathematical Problem Solving in Hungary for Students Who Have Average Ability in Mathematics... 137 András Ambrus and Krisztina Barczi-Veres Memorable Diagonals : Exploratory Problems as Propositions for Doing Mathematics... 157 Torsten Fritzlar Pupils Drawings as a Research Tool in Mathematical Problem- Solving Lessons... 167 Erkki Pehkonen, Maija Ahtee, and Anu Laine The Use of Digital Technology to Frame and Foster Learners Problem-Solving Experiences... 189 Manuel Santos-Trigo and Luis Moreno-Armella Proportional Word Problem Solving Through a Modeling Lens: A Half-Empty or Half-Full Glass?... 209 Tine Degrande, Lieven Verschaffel, and Wim Van Dooren Conceptual Model-Based Problem Solving... 231 Yan Ping Xin Reaction: Students, Problem Posing, and Problem Solving... 255 Jeremy Kilpatrick Part III Teachers, Problem Posing, and Problem Solving When Is a Problem? When Is Actually the Problem!... 263 John Mason Novice Chilean Secondary Mathematics Teachers as Problem Solvers... 287 Patricio Felmer and Josefa Perdomo-Díaz Infusing Mathematical Problem Solving in the Mathematics Curriculum: Replacement Units... 309 Yew Hoong Leong, Eng Guan Tay, Tin Lam Toh, Khiok Seng Quek, Pee Choon Toh, and Jaguthsing Dindyal Mathematical Problem Posing: A Case of Elementary School Teachers Developing Tasks and Designing Instructions in Taiwan... 327 Shuk-Kwan S. Leung Mathematical Problem Solving and Teacher Professional Learning: The Case of a Modified PISA Mathematics Task... 345 Edward A. Silver
Contents xi Building Thinking Classrooms: Conditions for Problem-Solving... 361 Peter Liljedahl Reaction: Teachers, Problem Posing and Problem-Solving... 387 Kaye Stacey Index... 393