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Teaching Secondary School Mathematics Through Storytelling

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Teaching Secondary School Mathematics Through Storytelling TEACHING SECONDARY SCHOOL MATHEMATICS THROUGH STORYTELLING by Chandra Balakrishnan B.A., Simon Fraser University 2000 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Faculty ofEducation © Chandra Balakrishnan 2008 SIMON FRASER UNIVERSITY Spring 2008 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without permission ofthe author. APPROVAL Name: Chandra Balakrishnan Degree: Master of Science Title of Thesis: Teaching Secondary School Mathematics Through Storytelling Examining Committee: Chair: Cecile Sabatier, Assistant Professor, Faculty of Education Peter Liljedahl, Assistant Professor, Faculty of Education Senior Supervisor Rina Zazkis, Professor, Faculty of Education Committee Member Stephen Campbell, Assistant Professor, Faculty of Education External Examiner Date Defended/Approved: ~Q...Iilt'L...n...30"\.J..........~_~"'-T='.o"",C'Cft'::..l......Io....&;;lo~ _ ii SIMON FRASER UNIVERSITY LIBRARY Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the "Institutional Repository" link of the SFU Library website <www.lib.sfu.ca> at: <http://ir.lib.sfu.ca/handle/1892/112>) and, without changing the content, to translate the thesis/project or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without the author's written permission. Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence. While licensing SFU to permit the above uses, the author retains copyright in the thesis, project or extended essays, including the right to change the work for subsequent purposes, including editing and publishing the work in whole or in part, and licensing other parties, as the author may desire. The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive. Simon Fraser University Library Burnaby, BC, Canada Revised: Fall 2007 Sl1\1()N r=nASER UNIVEHSI'rV TH1NK.ING Of THE WORLO STATEMENT OF ETHICS APPROVAL The author, whose name appears on the title page of this work, has obtained, for the research described in this work, either: (a) Human research ethics approval from the Simon Fraser University Office of Research Ethics, or (b) Advance approval of the animal care protocol from the University Animal Care Committee of Simon Fraser University; or has conducted the research (c) as a co-investigator, in a research project approved in advance, or (d) as a member of a course approved in advance for minimal risk human research, by the Office of Research Ethics. A copy of the approval letter has been filed at the Theses Office of the University Library at the time of submission of this thesis or project. The original application for approval and letter of approval are filed with the relevant offices. Inquiries may be directed to those authorities. Bennett Library Simon Fraser University Burnaby, BC, Canada Last revision: Summer 2007 ABSTRACT It has long been understood that stories can be used to teach elementary and middle school students mathematical ideas. The objective ofthis research was to examine how stories can be integrated into mathematics instruction at the secondary level. As a cognitive tool, story can play an especially significant role in engaging students' imaginations. Nine separate studies were conducted to examine what effect different types ofstories have on student learning and to provide mathematics instructors with an array ofexamples ofhow stories can be used at the high school level. Keywords: mathematics; imagination; stories; story; secondary; narrative III DEDICATION For my father and mother, Perumal and Gowry Balakrishnan. Everything I am is a result ofthe sacrifices you made and the unfailing love you surrounded me with. I am blessed beyond measure to have you for my parents. You made me feel like I can accomplish anything. For my children, Sage, MacKenzie, and Elle. Thank you for inspiring me with your stories, your brilliance, and your wonderful senses ofhumour. Thank you for your love, patience, and understanding. You are the loves ofmy life and all ofthis was for you. For my brothers and sister and their spouses, Roger, Desi, Anita, Fiona, and Kelly. Thank you for your support and encouragement. iv ACKNOWLEDGEMENTS I would like to thank Dr. Peter Liljedahl for his patience and guidance through this entire process. Not only was the research that he and Dr. Zazkis conducted a cornerstone for this thesis, but he always knew just what to say to get me back on track. could not have asked for a better advisor and mentor. I would like to thank Dr. Rina Zazkis for helping me see the learning of mathematics in creative new ways and for always taking the time out ofher busy schedule to talk to me about my research. I would like to thank Henry Fair, Henry Bugler, Jenny-Sue Calkins, and Elva McManus for believing in me and encouraging me to go back to school and pursue my academic dreams. I would like to thank Brent Start, Doug Cariou, Carson and Nicole Power, Brian Patel, Angie Levesseur, Janet Laluc, Karen Blom, Aaron McKimmon, Sacha and Janet Page, and Mike and Tisha Wade for their support and for being there for me through difficult times. v TABLE OF CONTENTS Approval ii Abstract iii Dedication iv Acknowledgements v Table of Contents vi List ofFigures xi List ofTables xii 1 Introduction 1 2 Story and the Imagination 8 2.1 Imagination and Mathematics 8 2.2 Imagination and Emotions 9 2.3 Story as a Cognitive Tool 10 2.4 Story and Culture 10 2.5 Story and Emotion 11 2.6 Story and Meaning 12 2.7 Story and Individuals 13 2.8 Story and Children 14 3 The Role ofStory in the Learning ofMathematics 15 3.1 Renewed Interest in Stories 15 3.2 Stories to Ask a Question 17 3.2.1 Instructional Implications 19 3.3 Stories to Accompany a Topic .20 3.3.1 Archimedes 20 3.4 Stories to Introduce an Idea 22 3.4.1 Descartes' Fly 23 3.4.2 Karl Freidrich Gauss 25 3.5 Stories to Intertwine with a Mathematical Topic 27 3.5.1 Princess Dido 27 3.5.2 Kathy's Wardrobe 31 3.5.3 Pure Mathematical Narrative 33 3.5.4 Numberville 35 3.6 Stories to Explain a Concept 36 3.6.1 The Twelve Diamonds 38 3.7 Stories to Introduce an Activity 39 VI 4 Shaping Stories to Maximize Imaginative Engagement .44 4.1 Binary Opposites 45 4.1.1 Archimedes Revisited 48 4.1.2 Princess Dido Revisited 50 4.1.3 Hippasos and the Pythagoreans 53 4.2 Metaphor 56 4.2.1 The Shield Metaphor 58 4.2.2 Metaphor in Fiction 62 4.2.3 Flatland 65 4.3 Knowledge and Human Meaning 68 4.3.1 Historical Narrative or Historical Fiction? 69 4.4 Real-life or Fantasy Contexts 75 4.4.1 The Story ofTan 79 4.4.2 Gulliver's Travels 81 4.5 Rhyme, Rhythm, and Pattern 84 4.5.1 Anno's Mysterious Multiplying Jar, Antaeus, and One Grain ofRice 86 4.5.2 Mathematically-Structured Story 88 4.6 Jokes and Humour 90 4.7 Orally Told Story or the Read Story 94 5 Research Questions 98 5.1 The Imagination 98 5.2 Stories and the Imagination 98 5.3 Stories and Mathematics 99 5.4 Research Question One 100 5.5 Research Question Two 101 6 Methodology: Participants, Setting, Data, and Theoretical Frameworks l03 6.1 Participants and Setting 103 6.1.1 Location ofthe Studies 104 6.1.2 Mathematics 8 Honours 104 6.1.3 Mathematics 9 Honours 105 6.1.4 Mathematics 9 105 6.1.5 Principles ofMathematics 10 106 6.2 Data 107 6.2.1 The Legend ofChikara Ni Bai 108 6.2.2 The Pythagorean Problem 108 6.2.3 The Four 4s 108 6.2.4 Skull Island 108 6.2.5 The Skippers 108 6.2.6 Kelloggs Needs Your Help 108 6.2.7 The Wicked Sons 109 6.2.8 Mathematically-Structured Story 109 6.2.9 The Story of Princess ...J3 109 6.3 Theoretical Framework 109 6.3.1 Egan's Theoretical Framework 11 0 6.3.2 Zazkis and Liljedahl's Theoretical Framework .111 vii 7 Study I: The Legend ofChikara Ni Bai 117 7.1 Treatment 117 7.1.1 Target and Problematic Issue 117 7.1.2 Cognitive Tools 117 7.1.3 Presentation ofProblem and Conclusion 120 7.1.4 Participants 121 7.1.5 The Story 121 7.2 Analysis 123 7.2.1 Attention to Patterns 123 7.2.2 Change ofBase 126 7.2.3 Realistic and Fantasy Contexts 129 8 Study II: The Pythagorean Problem 137 8.1 Treatment 137 8.1.1 Target and Problematic Issue 137 8.1.2 Cognitive Tools 138 8.1.3 Presentation ofProblem, Extending the Problem Situation, and Conclusion 139 8.1.4 Participants 141 8.1.5 The Story 141 8.2 Analysis 141 8.2.1 Binary Opposites, Knowledge and Human Meaning, and Jokes and Humour 141 8.2.2 Level ofEngagement.
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