coming to understand angle and angle measure: a design-based research curriculum study using ...

295 Pages·2013·4.76 MB·English

COMING TO UNDERSTAND ANGLE AND ANGLE MEASURE: A DESIGN-BASED
RESEARCH CURRICULUM STUDY USING CONTEXT-AWARE UBIQUITOUS
LEARNING

Helen Crompton

A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in
partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of
Education

Chapel Hill
2013

Approved by:
Susan N. Friel
Cheryl M. Bolick
Gemma Mojica
Catherine Scott
Jeffrey A. Greene
© 2013
Helen Crompton
ALL RIGHTS RESERVED

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ABSTRACT
HELEN CROMPTON: Coming to understand angle and angle measure: A design-based
research curriculum study using context-aware ubiquitous learning
(Under the direction of Susan N. Friel)

This study uses design-based research (DBR) to develop an empirically-substantiated
instructional theory about students’ development of angle and angle measure, with real-world
connections and technological tools through the use of context-aware ubiquitous learning.
The research questions guiding this research are:
1. How do students come to understand angle and angle measure through the use of
real-world connections and technology enabled learning tasks?
2. What are effective means of support to facilitate students’ understanding of angle
and angle measure?
A conjectured local instruction theory was developed from a thorough review of the literature
in chapter two. This review encompassed research-based developmental trajectories and
effective instructional supports for promoting students’ understanding of angle and angle
measure. It was conjectured that context-aware u-learning was a good support for students
coming to understand angle and angle measure. Context-aware u-learning in this study
involves the use of real-world connections and a Dynamic Geometry Environment.
The local instruction theory was subject to a cyclical iterative process of anticipation,
enactment, evaluation, and revision (Gravemeijer & van Eerde, 2009). This process
contributes to the theories of how students come to understand angle and angle measures,
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while also developing a set of instructional activities which can be utilized and adapted by
educators to meet the needs of the students in their classrooms. The instructional sequence
was implemented in one classroom-based teaching experiment in the first macro cycle of the
DBR process. A second macro cycle was implemented using revised instructional materials
in one classroom-based teaching experiment.
Findings indicate that context-aware u-learning is a valuable mathematical context for
introducing students to angle and angle measure. Common misconceptions about angle can
be avoided as the students study angles in the real world which presents them with angles
with rays of different length and in various orientations. Good foundations were built by
having the students consider angle by the generalizable properties and over the seven days
the students showed good movement across the van Hiele levels of geometric thinking. A
revised local instruction theory is presented as a result of the findings from this study.
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DEDICATION

This dissertation is dedicated to my mother, who gave me steadfast encouragement
and enthusiasm for my studies.

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ACKNOWLEDGEMENTS
First and foremost, I would like to thank my husband Stephen. Without his support
and sacrifice, I would never have been able to pursue my goals and dreams for furthering my
studies. Thank you for believing in me and encouraging me throughout my studies. We did
it! I would like to offer my sincere appreciation to my family. I would like to thank my Dad
for raising me to have faith and understand that with Him all things are possible. To my
sisters, Ruth and Allison, thank you for giving me advice and allowing me to talk for hours
about my ideas and projects.
To Dr. Susan Friel, my dissertation advisor, thank you for your patience, wisdom, and
belief in this project. You set me on the right track and I am proud to have been your student.
To my dissertation committee, Cheryl Bolick, Gemma Mojica, Catherine Scott, and Jeff
Greene, I would like to express my thanks and appreciation for taking the time and effort to
contribute knowledge and expertise for successful completion of this project. To John
Traxler, my external advisor and friend, thank you for your wisdom and ideas.
I owe a great debt of gratitude to Joe and Susan Stockwell who made my PhD dreams
a reality. Without your friendship, this would not have been possible. To my co-researcher
Bryan Fede, thank you for joining this journey and giving me your time. I would like to say
thank you to the Principal and teachers who allowed me to come into their classrooms to
conduct my studies. My final thank you goes to my friends at UNC-CH; you kept me
motivated and provided me with a wonderful support system.
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CONTENTS
CHAPTER I: INTRODUCTION ...............................................................................................1
The Purpose of this Study ..................................................................................................6
CHAPTER TWO: REVIEW OF THE LITERATURE .............................................................8
Angle and Angle Measure .................................................................................................9
How children come to understand angle .................................................................10
How children come to understand angle measure...................................................15
Curricular expectations for learning about angle and angle measure ..............................19
Support for Learning about Angle and Angle Measure ..................................................21
Real-world connections ...........................................................................................23
Technology: Dynamic Geometry Environments ....................................................26
Mobile learning: Context-aware ubiquitous learning .............................................33
CHAPTER THREE: DESIGN-BASED RESEARCH ............................................................43
Design-Based Research ...................................................................................................43
Local instruction theory ..........................................................................................46
Theoretical Framework: Conjectured Local Instruction Theory .....................................48
Instructional materials .............................................................................................52
The content and structure of the lessons. ........................................................59
CHAPTER FOUR: METHODOLOGY ..................................................................................67
Participants ......................................................................................................................67
Research Team ................................................................................................................69
Design-Based Research Protocol for this Study ..............................................................69
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Data Collection and Analysis ..........................................................................................72
Pre and post instruction clinical interviews ............................................................73
Co-researcher and witness classroom observations ................................................77
Whole class and small group video .........................................................................77
Daily mini cycle reflection ......................................................................................77
Artifact collection....................................................................................................78
Researcher reflection journal ..................................................................................78
Retrospective analysis .............................................................................................78
CHAPTER FIVE: FINDINGS .................................................................................................81
How Students Come to Understand Angle and Angle Measure .....................................83
Level one: Visual level of geometric thinking. .......................................................84
Level two thinking in Lessons Three and Four ..............................................95
Level three: Informal deduction level of geometric thinking. ................................98
Summary of lessons five and six and student responses ................................99
Level two thinking during Lessons Five and Six .........................................100
Evidence of level three thinking in Lessons Five and Six ............................103
Pre and Post Instruction Interviews ...............................................................................106
Summary by Student – Pre and Post Instruction Interviews .........................................107
Interviews: Macro cycle one. ................................................................................107
Summary of macro cycle one ........................................................................................119
Interviews: Macro cycle two. ................................................................................121
Summary of macro cycle two ...............................................................................134
Effective Means of Support to Facilitate Understanding of Angle and Angle Measure137
Context-aware ubiquitous learning .......................................................................137
Academic tasks .............................................................................................146
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Mathematical discussion ...............................................................................151
Changes to instructional materials ........................................................................153
Summary of changes made to instructional materials in TE1 ......................153
Summary of changes made to instructional materials in TE2 ......................154
CHAPTER SIX: CONCLUSION ..........................................................................................156
Revised Local Instruction Theory .................................................................................158
Changes to the instructional sequence. .................................................................166
Limitations and Future Research ...................................................................................171
APPENDIX A ........................................................................................................................176
Second Level: Analysis .................................................................................................177
Third Level: Informal Deduction ..................................................................................178
APPENDIX B ........................................................................................................................179
Lesson One ....................................................................................................................179
Lesson Two ...................................................................................................................185
Lesson Three .................................................................................................................192
Lesson Four ...................................................................................................................200
Lesson Five ....................................................................................................................208
Lesson Six .....................................................................................................................219
APPENDIX C ........................................................................................................................224
Student Recruitment Script – Face to Face ...................................................................224
Student Assent ...............................................................................................................226
Parental Consent ............................................................................................................229
Teacher Recruitment Script – Email .............................................................................234
Teacher Consent ............................................................................................................235
APPENDIX D ........................................................................................................................240
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Angle Interview Protocol ..............................................................................................240
Angle Activities .............................................................................................................240
Activity 2a: Identifying and Defining Angles ...............................................................243
Activity 2b: Identifying and Defining Angles ...............................................................245
Activity 3: Sorting Angles .............................................................................................248
Activity 4: Angle Measure ............................................................................................252
Activity 5: Angle Relationships ....................................................................................254
APPENDIX E ........................................................................................................................258
APPENDIX F.........................................................................................................................267
APPENDIX G ........................................................................................................................268
REFERENCES ......................................................................................................................269

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