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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  • USING SUBORDINATION TO TEACH AND LEARN MATHEMATICS

    by

    Jennifer Law

    B.Sc., University of British Columbia, 1996 B.Ed., University of British Columbia, 1997

    THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

    MASTER OF SCIENCE

    In the Secondary Mathematics Education Program

    Faculty of Education

    Jennifer Law 2011 SIMON FRASER UNIVERSITY

    Summer 2011

    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.

    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,

    particularly if cited appropriately.

  • APPROVAL

    Name: Jennifer Law

    Degree: Master of Science

    Title of Thesis: Using Subordination to Teach and Learn Mathematics

    Examining Committee:

    Chair: Elizabeth Marshall, Associate Professor Faculty of Education, SFU

    ___________________________________________

    Dr. Nathalie Sinclair, Associate Professor Faculty of Education, SFU

    Senior Supervisor

    ___________________________________________

    Dr. Peter Liljedahl, Associate Professor Faculty of Education, SFU

    Supervisor

    ___________________________________________

    Dr. Rina Zazkis, Professor Faculty of Education, SFU

    External Examiner

    Date Defended/Approved: July 27, 2011

    ii

  • Partial Copyright Licence

  • 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, collaborator or research assistant 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.

    Simon Fraser University Library

    Simon Fraser University Burnaby, BC, Canada

    Last update: Spring 2010

  • ABSTRACT

    The practice tasks we find in mathematics are often problematic, both for student

    motivation and for student learning. Often, nothing is learned because the student is

    involved in mechanistic repetition. Drawing on the idea that only awareness is educable,

    Dave Hewitt argues for an approach to practice that avoids thoughtless repetition by

    shifting the focus of the activity. His subordination tasks focus attention on a more

    engaging task, while still enabling practice. This action research study involves

    developing subordination tasks for principles of mathematics 10 students and investigates

    whether such tasks can be designed and implemented across the curriculum, and used in

    every class. Six tasks were created and implemented. An analysis of the results shows

    that in addition to the three characteristics of subordination defined by Hewitt, tasks need

    to consider the classroom milieu, be accessible, engaging, and relevant, practice an

    appropriate skill, and provide immediate feedback.

    iii

  • DEDICATION

    For my husband, Terry, who has always given me

    unconditional love, patience and support.

    ______________________

    For my children, Matthias and Nathalie, whose smiles and laughter light up my life.

    iv

  • ACKNOWLEDGEMENTS

    I would like to express my deepest appreciation to Dr. Nathalie Sinclair for her

    patience, encouragement and guidance from the beginning of this journey. I would also

    like to thank Dr. Peter Liljedahl for his insights and support. My thanks also go out to Dr.

    David Pimm who introduced me to the article that became the focus of my research.

    I would like to thank my fellow classmates of the Mathematics Education

    Program. In particular, I am grateful to Dan Kamin who encouraged me to keep going,

    helped to edit parts of my first draft, and gave me advice whenever I needed it.

    v

  • TABLE OF CONTENTS

    Approval ............................................................................................................................ ii

    Abstract............................................................................................................................. iii

    Dedication ......................................................................................................................... iv

    Acknowledgements ........................................................................................................... v

    Table of Contents ............................................................................................................. vi

    List of Figures................................................................................................................... ix

    List of Tables ..................................................................................................................... x

    Chapter 1 : Introduction .................................................................................................. 1 Background................................................................................................................................ 1 Rationale .................................................................................................................................... 2

    Chapter 2 : Literature Review......................................................................................... 5 Using Traditional Methods....................................................................................................... 5 Teaching and Learning with Understanding........................................................................ 10 Mathematical Tasks............................................................................................................... 15 Practice in Mathematics ......................................................................................................... 17

    Order of Practice Problems................................................................................................... 18 Variation in Practice Exercises ............................................................................................. 20 Teacher-Initiated Learner-Generated Examples ................................................................... 21

    Subordination of Teaching to Learning................................................................................ 23 Conclusion................................................................................................................................ 26

    Chapter 3 : Theoretical Framework and Methodology .............................................. 28 Research questions.................................................................................................................. 28 Theoretical Framework.......................................................................................................... 28

    Non-mathematical Example of the Use of Subordination .................................................... 30 Mathematical Examples of Using Subordination ................................................................. 32

    Using subordination to introduce notation ..................................................................................... 32 Using subordination to practice definitions.................................................................................... 35 Using subordination to introduce conventions ............................................................................... 37

    Engagement .......................................................................................................................... 39 Methodology ............................................................................................................................ 40

    General Setting ..................................................................................................................... 40 Classroom Structure and Instruction..................................................................................... 41 Participants ........................................................................................................................... 42 Task Creation........................................................................................................................ 42 Research Procedure .............................................................................................................. 44

    Chapter 4 : RESULTS AND ANALYSIS by TASK.................................................... 48 Task 1 Stacking Squares...................................................................................................... 49

    Background........................................................................................................................... 49 Rationale ....................................................................