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  • PRESERVICE TEACHERS’ UNDERSTANDINGS OF DIVISION AND PROPORTIONAL

    RELATIONSHIPS WITH QUANTITIES

    by

    IBRAHIM BURAK OLMEZ

    (Under the Direction of Andrew Izsák)

    ABSTRACT

    The purpose of this study was to investigate how preservice teachers’ understandings of

    multiplication and division support and constrain their understandings of ratios and proportional

    relationships in terms of quantities. Six preservice teachers were selected purposefully based on

    their performances in a previous course. An explanatory case study with multiple cases was used

    to make comparisons within and across cases. Two semi-structured interviews were conducted

    with each pair. The results revealed that preservice teachers’ understandings of partitive and

    quotitive division, and reliance on only multiplicative relationships shaped in critical ways how

    they reasoned about ratios. Preservice teachers who did not explicitly identify different meanings

    for division were found to have difficulty maintaining one of two distinct perspectives on ratio.

    The results also showed that additive relationships such as repeated addition and the phrase “for

    every” might have constrained teachers from keeping the two perspectives distinct.

    INDEX WORDS: Division, Multiplication, Ratios and proportional relationships, Preservice teachers

  • PRESERVICE TEACHERS’ UNDERSTANDINGS OF DIVISION AND PROPORTIONAL

    RELATIONSHIPS WITH QUANTITIES

    by

    IBRAHIM BURAK OLMEZ

    BS., Bogazici University, Turkey, 2011

    A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment

    of the Requirements for the Degree

    MASTER OF ARTS

    ATHENS, GEORGIA

    2014

  • © 2014

    Ibrahim Burak Olmez

    All Rights Reserved

  • PRESERVICE TEACHERS’ UNDERSTANDINGS OF DIVISION AND PROPORTIONAL

    RELATIONSHIPS WITH QUANTITIES

    by

    IBRAHIM BURAK OLMEZ

    Major Professor: Andrew Izsák

    Committee: Sybilla Beckmann-Kazez Kevin C. Moore

    Electronic Version Approved:

    Maureen Grasso Dean of the Graduate School The University of Georgia May 2014

  • iv

    DEDICATION

    I dedicate this work to

    My parents, Muhammet and Feride Olmez

    My brother, Sener Bahadır

    My sister, Elif Tugce, and

    My wife, Safiye Bahar Olmez

    For their endless love and support

  • v

    ACKNOWLEDGEMENTS

    I would like to first thank my major professor, Andrew Izsák, for his guidance and

    encouragement throughout my M.A. study. His comments and edits greatly improved this

    document. I also thank him for allowing me to be a member of the Proportional Reasoning of

    Middle Grades Pre-service Teachers (PROMPT) project for two years. I note that the data in this

    study were collected for the PROMPT Mini Grant (PROMPT-mini) project, which was funded

    by the Office of STEM Education at the University of Georgia.

    I would like to express my appreciation to Sybilla Beckmann-Kazez and Kevin C. Moore

    for agreeing to be on my committee as well as for their valuable feedback and suggestions.

    I extend special thanks to the Fulbright Scholarship for providing me invaluable experience

    and support in my academic career at the University of Georgia. I am also thankful to the

    Department of Mathematics and Science Education for giving me the opportunity to pursue my

    M.A. degree in such a prestigious program surrounded by outstanding professors in the field.

    I would like to thank all my friends at home, specifically Mehmet, Salih, Bugra, Gokhan,

    and here, Selcuk, Sedat, and Oguz, for their guidance and friendship during my graduate studies.

    I owe gratitude to my parents, Feride and Muhammet, for always supporting my decisions

    and encouraging me to move beyond limits. I will always be grateful for the love and

    opportunities they have provided for me. Also, I would like to thank my brother Sener Bahadir

    and my sister Elif Tugce for always keeping me in their hearts.

    Finally, deepest thanks for my wife Safiye Bahar for all her love and support during my

    continued education.

  • vi

    TABLE OF CONTENTS

    Page

    ACKNOWLEDGEMENTS.............................................................................................................v

    LIST OF TABLES....................................................................................................................... viii

    LIST OF FIGURES ....................................................................................................................... ix

    CHAPTER

    1 INTRODUCTION .........................................................................................................1

    Background and Rationale .......................................................................................3

    Research Questions ..................................................................................................4

    2 LITERATURE REVIEW ..............................................................................................6

    Partitive and Quotitive Division ..............................................................................6

    Conceptual Framework ............................................................................................7

    Teachers and Students’ Reasoning of Division .......................................................8

    Theoretical Framework ..........................................................................................13

    3 METHODOLOGY ......................................................................................................16

    Methods..................................................................................................................16

    Setting ....................................................................................................................16

    Participants.............................................................................................................18

    Data Collection ......................................................................................................18

    Analysis..................................................................................................................21

  • vii

    4 FINDINGS...................................................................................................................24

    Case 1: Lisa and Tess.............................................................................................27

    Case 2: Chip and Amber ........................................................................................43

    Case 3: Paul and Amy............................................................................................64

    Cross-case Analysis................................................................................................79

    5 DISCUSSION AND CONCLUSIONS .......................................................................82

    Conclusions............................................................................................................82

    Implications............................................................................................................87

    Future Research .....................................................................................................88

    Concluding Comment ............................................................................................88

    REFERENCES ..............................................................................................................................90

  • viii

    LIST OF TABLES

    Page

    Table 1: Overview of Multiplicative Relationships.......................................................................14

    Table 2: Interview Tasks................................................................................................................19

    Table 3: Preservice Teachers’ Use of Division and Perspective in Each Task .............................26

  • ix

    LIST OF FIGURES

    Page

    Figure 1: Amy’s Drawing of Ratio Table ......................................................................................21

    Figure 2: Lisa’s Drawing of Strip Diagram ...................................................................................22

    Figure 3: Combining the Two Perspectives...................................................................................23

    Figure 4: Lisa’s Drawing of Ratio Table .......................................................................................27

    Figure 5: (a) Lisa’s Drawing of Strip Diagram and (b) Tess’s Drawing of Strip Diagram ...........30

    Figure 6: (a) Tess’s Division Representation and (b) Tess’s Drawing of Ratio Table ..................33

    Figure 7: (a) Lisa’s Drawing of Strip Diagram and (b) Tess’s Distribution Strategy ...................35

    Figure 8: (a) Lisa’s Drawing of Strip Diagram and (b) Tess’s Drawing of Strip Diagram ...........39

    Figure 9: (a) Chip’s Drawing of Double Number Line and (b) Amber’s Division by Batches ....44

    Figure 10: Amber’s Drawing of Strip Diagram Based on Chip’s Ideas ........................................

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