USING SUBORDINATION TO TEACH AND LEARN SUBORDINATION TO TEACH AND LEARN MATHEMATICS by Jennifer Law ... Using Subordination to Teach and Learn Mathematics ... Expression Scattergories

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<ul><li><p>USING SUBORDINATION TO TEACH AND LEARN MATHEMATICS </p><p> by </p><p> Jennifer Law </p><p>B.Sc., University of British Columbia, 1996 B.Ed., University of British Columbia, 1997 </p><p>THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF </p><p> MASTER OF SCIENCE </p><p>In the Secondary Mathematics Education Program </p><p>Faculty of Education </p><p> Jennifer Law 2011 SIMON FRASER UNIVERSITY </p><p>Summer 2011 </p><p>All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. </p><p>Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, </p><p>particularly if cited appropriately. </p></li><li><p>APPROVAL </p><p>Name: Jennifer Law </p><p>Degree: Master of Science </p><p>Title of Thesis: Using Subordination to Teach and Learn Mathematics </p><p>Examining Committee: </p><p> Chair: Elizabeth Marshall, Associate Professor Faculty of Education, SFU </p><p> ___________________________________________ </p><p> Dr. Nathalie Sinclair, Associate Professor Faculty of Education, SFU </p><p>Senior Supervisor </p><p> ___________________________________________ </p><p> Dr. Peter Liljedahl, Associate Professor Faculty of Education, SFU </p><p>Supervisor </p><p> ___________________________________________ </p><p> Dr. Rina Zazkis, Professor Faculty of Education, SFU </p><p>External Examiner </p><p>Date Defended/Approved: July 27, 2011 </p><p> ii</p></li><li><p>Partial Copyright Licence </p></li><li><p>STATEMENT OF ETHICS APPROVAL </p><p>The author, whose name appears on the title page of this work, has obtained, for the research described in this work, either: </p><p>(a) Human research ethics approval from the Simon Fraser University Office of Research Ethics, </p><p>or </p><p>(b) Advance approval of the animal care protocol from the University Animal Care Committee of Simon Fraser University; </p><p>or has conducted the research </p><p>(c) as a co-investigator, collaborator or research assistant in a research project approved in advance, </p><p>or </p><p>(d) as a member of a course approved in advance for minimal risk human research, by the Office of Research Ethics. </p><p>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. </p><p>The original application for approval and letter of approval are filed with the relevant offices. Inquiries may be directed to those authorities. </p><p> Simon Fraser University Library </p><p>Simon Fraser University Burnaby, BC, Canada </p><p> Last update: Spring 2010 </p></li><li><p>ABSTRACT </p><p>The practice tasks we find in mathematics are often problematic, both for student </p><p>motivation and for student learning. Often, nothing is learned because the student is </p><p>involved in mechanistic repetition. Drawing on the idea that only awareness is educable, </p><p>Dave Hewitt argues for an approach to practice that avoids thoughtless repetition by </p><p>shifting the focus of the activity. His subordination tasks focus attention on a more </p><p>engaging task, while still enabling practice. This action research study involves </p><p>developing subordination tasks for principles of mathematics 10 students and investigates </p><p>whether such tasks can be designed and implemented across the curriculum, and used in </p><p>every class. Six tasks were created and implemented. An analysis of the results shows </p><p>that in addition to the three characteristics of subordination defined by Hewitt, tasks need </p><p>to consider the classroom milieu, be accessible, engaging, and relevant, practice an </p><p>appropriate skill, and provide immediate feedback. </p><p> iii</p></li><li><p>DEDICATION </p><p>For my husband, Terry, who has always given me </p><p>unconditional love, patience and support. </p><p>______________________ </p><p>For my children, Matthias and Nathalie, whose smiles and laughter light up my life. </p><p> iv</p></li><li><p>ACKNOWLEDGEMENTS </p><p>I would like to express my deepest appreciation to Dr. Nathalie Sinclair for her </p><p>patience, encouragement and guidance from the beginning of this journey. I would also </p><p>like to thank Dr. Peter Liljedahl for his insights and support. My thanks also go out to Dr. </p><p>David Pimm who introduced me to the article that became the focus of my research. </p><p>I would like to thank my fellow classmates of the Mathematics Education </p><p>Program. In particular, I am grateful to Dan Kamin who encouraged me to keep going, </p><p>helped to edit parts of my first draft, and gave me advice whenever I needed it. </p><p> v</p></li><li><p>TABLE OF CONTENTS </p><p>Approval ............................................................................................................................ ii </p><p>Abstract............................................................................................................................. iii </p><p>Dedication ......................................................................................................................... iv </p><p>Acknowledgements ........................................................................................................... v </p><p>Table of Contents ............................................................................................................. vi </p><p>List of Figures................................................................................................................... ix </p><p>List of Tables ..................................................................................................................... x </p><p>Chapter 1 : Introduction .................................................................................................. 1 Background................................................................................................................................ 1 Rationale .................................................................................................................................... 2 </p><p>Chapter 2 : Literature Review......................................................................................... 5 Using Traditional Methods....................................................................................................... 5 Teaching and Learning with Understanding........................................................................ 10 Mathematical Tasks............................................................................................................... 15 Practice in Mathematics ......................................................................................................... 17 </p><p>Order of Practice Problems................................................................................................... 18 Variation in Practice Exercises ............................................................................................. 20 Teacher-Initiated Learner-Generated Examples ................................................................... 21 </p><p>Subordination of Teaching to Learning................................................................................ 23 Conclusion................................................................................................................................ 26 </p><p>Chapter 3 : Theoretical Framework and Methodology .............................................. 28 Research questions.................................................................................................................. 28 Theoretical Framework.......................................................................................................... 28 </p><p>Non-mathematical Example of the Use of Subordination .................................................... 30 Mathematical Examples of Using Subordination ................................................................. 32 </p><p>Using subordination to introduce notation ..................................................................................... 32 Using subordination to practice definitions.................................................................................... 35 Using subordination to introduce conventions ............................................................................... 37 </p><p>Engagement .......................................................................................................................... 39 Methodology ............................................................................................................................ 40 </p><p>General Setting ..................................................................................................................... 40 Classroom Structure and Instruction..................................................................................... 41 Participants ........................................................................................................................... 42 Task Creation........................................................................................................................ 42 Research Procedure .............................................................................................................. 44 </p><p>Chapter 4 : RESULTS AND ANALYSIS by TASK.................................................... 48 Task 1 Stacking Squares...................................................................................................... 49 </p><p>Background........................................................................................................................... 49 Rationale ............................................................................................................................... 50 Task Development ................................................................................................................ 52 </p><p> vi</p></li><li><p>Task Description: Outline of the lesson plan........................................................................ 53 Why Stacking Squares is a Subordination Task ................................................................... 54 Results and Analysis............................................................................................................. 56 </p><p>Observations................................................................................................................................... 56 Analysing Engagement .................................................................................................................. 60 Is This a Good Subordination Task? .............................................................................................. 62 </p><p>Task 2 Expression Scattergories ......................................................................................... 65 Background........................................................................................................................... 65 Rationale ............................................................................................................................... 65 Task Development ................................................................................................................ 67 Task Description: Outline of the lesson plan........................................................................ 69 Why Expression Scattergories is a Subordination Task....................................................... 70 Results and Analysis............................................................................................................. 71 </p><p>Observations................................................................................................................................... 71 Analyzing Engagement .................................................................................................................. 74 Is This a Good Subordination Task? .............................................................................................. 75 </p><p>Task 3 Whats My Number?............................................................................................... 77 Background........................................................................................................................... 77 Rationale ............................................................................................................................... 78 Task Development ................................................................................................................ 79 Task Description: Outline of the lesson plan........................................................................ 81 Why Whats My Number is a Subordination Task................................................................ 83 Results and Analysis............................................................................................................. 84 </p><p>Observations................................................................................................................................... 84 Analyzing Engagement .................................................................................................................. 86 Is This a Good Subordination Task? .............................................................................................. 88 </p><p>Task 4 Function Factory...................................................................................................... 90 Background........................................................................................................................... 90 Rationale ............................................................................................................................... 91 Task Development ................................................................................................................ 92 Task Description: Outline of the lesson plan........................................................................ 93 Why Function Factory is a Subordination Task................................................................... 94 Results and Analysis............................................................................................................. 96 </p><p>Observations................................................................................................................................... 96 Analyzing Engagement .................................................................................................................. 98 Is This a Good Subordination Task? ............................................................................................ 101 </p><p>Task 5 Whats My x? ......................................................................................................... 103 Background......................................................................................................................... 103 Rationale ............................................................................................................................. 104 Task Development .............................................................................................................. 105 Task Description: Outline of the lesson plan...................................................................... 106 Why What...</p></li></ul>