How online small groups co-construct mathematical artifacts to do collaborative problem solving

251 Pages·2009·4.16 MB·English

by  Çakır, Murat Perit

How Online Small Groups Co-construct Mathematical Artifacts
to do Collaborative Problem Solving

A Thesis
Submitted to the Faculty
Of
Drexel University
by
Murat Perit Çakır
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
May 2009

ii

Dedications

To my parents, Güler and Nurettin Çakır
for their love, support, encouragement and patience.
iii

Acknowledgements

This dissertation would not have been possible without the support of many people. It is
truly a product and a testament to the group cognitive phenomena it has attempted to
study. I am indebted to Dr. Gerry Stahl for his mentorship, support and encouragement
through the years I have been working towards this thesis. None of this would have been
possible without the leadership and vision of Dr. Stahl at the Virtual Math Teams project.

I am grateful to Dr. Alan Zemel for initiating me to the realm of Ethnomethodology and
helping me to develop the analytical skills that allowed me to conduct this study. Other
members of the dissertation committee, Dr. Rosina Weber, Dr. Mick Khoo, Dr. Jason
Silverman, and Steve Weimar helped me to significantly improve this work with their
insightful comments and suggestions. I am grateful for the time and effort they devote in
reading the manuscript and for the invaluable feedback they provided.

Other members of the Virtual Math Teams project have also been highly influential on
my work. I would like to thank Dr. Wes Shumar, Johann Sarmiento, Nan Zhou, Ramon
Toledo, Annie Fetter, Ian Underwood, Dr. Fatos Xhafa, Dr. Jan-Willem Strijbos, Dr.
Stefan Trausan-Matu, Dr. Martin Wessner, Martin Mühlpfordt, and Dr. Elizabeth Charles
for their intellectual support and friendship.

Last, but definitely not least, I would like to thank all my family members, friends, and
colleagues at the ISchool @ Drexel who made this experience meaningful, enjoyable and
rewarding for me.
iv

TABLE OF CONTENTS

LIST OF TABLES............................................................................................................vii!
LIST OF FIGURES.........................................................................................................viii!
ABSTRACT.......................................................................................................................xi!
CHAPTER 1. INTRODUCTION.....................................................................................13!
CHAPTER 2. RESEARCH CONTEXT...........................................................................19!
2.1. The Role of Math Education in the Age of Participation......................................19!
2.2. Review of Instructional Technology......................................................................22!
2.2.1. Computer-Aided Instruction (CAI).................................................................23!
2.2.2. Intelligent Tutoring Systems (ITS)..................................................................26!
2.2.3. Constructivist Learning Environments (Logo-as-Latin).................................30!
2.2.4. Computer-Supported Collaborative Learning (CSCL)...................................32!
2.2.4.1. Social Theories of Learning.....................................................................33!
2.2.4.2. Evolution of Research Methods for Studying Collaborative Learning....38!
2.2.4.3. CSCL Applications in Math Education....................................................40!
2.2.5. Summary of IT Research.................................................................................43!
2.3. Review of CSCL Studies of Multimodal Interaction.............................................49!
2.3.1. The Problem of Social Organization in CSCL................................................49!
2.3.2. Approaches in CSCL to Analyzing Multimodal Interaction...........................53!
2.3.3. The Unit of Analysis........................................................................................63!
CHAPTER 3. RESEARCH DESIGN...............................................................................68!
3.1. Institutional Context: The Math Forum @ Drexel................................................68!
v

3.2. VMT Project: A Design-based Approach to CSCL System Development...........71!
3.3. The VMT Online Collaborative Learning Environment.......................................75!
3.4. Data........................................................................................................................82!
3.5. Methodology: Ethnomethodological Conversation Analysis................................86!
3.5.1. Historical Background....................................................................................88!
3.5.2. Illustration of EM/CA Methodology...............................................................98!
3.5.2.1. Mathematical Artifacts.............................................................................99!
3.5.2.2. Identification of Members’ Methods.....................................................100!
3.5.2.3. Communicative Affordances.................................................................106!
3.5.2.4. Co-construction of Math Artifacts.........................................................114!
CHAPTER 4. CASE STUDY OF A VIRTUAL MATH TEAM...................................118!
4.1. Characterization of Group Interaction.................................................................119!
4.1.1. Co-construction of the triangular grid.........................................................120!
4.1.2. Introduction of the hexagonal array.............................................................125!
4.1.3. Achievement of indexical symmetry through referential work.....................127!
4.1.4. Decomposition of the hexagonal array into partitions.................................133!
4.1.5. Joint discovery of a counting method...........................................................136!
4.1.6. Constitution of a new math task....................................................................139!
4.1.7. Co-construction of a method for counting sticks..........................................146!
4.1.8. Collective noticing of a pattern of growth....................................................154!
4.1.9. Resolution of referential ambiguity via visual proof....................................157!
4.1.10. Re-initiating the discussion of the algebraic formula.................................167!
4.1.11. Co-reflecting on the joint achievement of the team....................................171!
vi

4.1.12. Overcoming the problem of overlapping sticks..........................................173!
4.1.13. Derivation of the formula for the number of sticks.....................................179!
4.1.14. The wiki summary of the team....................................................................182!
4.2. Findings...............................................................................................................185!
4.2.1. Availability of the Production Process.........................................................185!
4.2.2. Mutability of Chat and Whiteboard Contents...............................................186!
4.2.3. Monitoring Joint Attention in an Online Environment.................................188!
4.2.4. Methods for Referencing Relevant Artifacts in the Shared Visual Field......189!
4.2.5. Coordination of Whiteboard Visualizations and Chat Narratives...............191!
4.2.6. Chat versus Whiteboard Contributions as Persistent Referential Resources194!
4.2.7. Persistence of Chat and Management of Parallel Threads..........................196!
4.2.8. Past and Future Relevancies Implied by Shared Mathematical Artifacts....199!
4.2.9. Joint Management of Narrative, Graphical and Symbolic Realizations......201!
CHAPTER 5. DISCUSSION..........................................................................................205!
5.1. Grounding through Interactional Organization....................................................205!
5.2. Implications for CSCL Design and Pedagogy.....................................................214!
CHAPTER 6. CONCLUSION.......................................................................................222!
REFERENCES...............................................................................................................232!
VITA...............................................................................................................................249!
vii

LIST OF TABLES

Table 2.2.1: Summary of Instructional Technology.........................................................44
Table 3.2.1: Design phases of the VMT service...............................................................73
Table 3.3.1: Description of whiteboard features...............................................................81
viii

LIST OF FIGURES

Figure 3.3.1: The VMT Chat Environment (as of Spring 2007)......................................76!
Figure 3.3.2: The VMT Wiki environment.......................................................................77!
Figure 3.3.3: The VMT Lobby.........................................................................................77!
Figure 3.3.4: A message-to-message reference................................................................78!
Figure 3.3.5: A message-to-whiteboard reference............................................................79!
Figure 3.3.6: Whiteboard controls....................................................................................80!
Figure 3.3.7: MathML examples supported by VMT Chat..............................................82!
Figure 3.4.1: Task description for Spring Fest 2006........................................................84!
Figure 3.5.1: First stages of Davidcyl's drawing activity...............................................101!
Figure 3.5.2: Davidcyl introduces the 4th column and pastes a copy of the
whole shape................................................................................................102!
Figure 3.5.3: Davidcyl uses copy/paste to produce the next stage of the pattern...........102!
Figure 3.5.4: Davidcyl’s drawing of the 6th stage...........................................................103!
Figure 3.5.5: The state of the VMT environment when Davidcyl posted his
chat message.............................................................................................104!
Figure 4.1.1: Six stages of 137's drawing actions obtained from the Replayer tool.
The timestamp of each stage is displayed under the corresponding image.
Snapshots focus on a particular region on the whiteboard where the
relevant drawing activity is taking place...................................................121
Figure 4.1.2: The evolution of Qwertyuiop's drawing in response to 137’s request......123!
Figure 4.1.3: The state of the whiteboard when Qwertyuiop’s drawing reached
its 12th stage in Figure 4.1.2......................................................................124!
Figure 4.1.4: Snapshots from the sequence of drawing actions performed by 137........129!
Figure 4.1.5: Jason uses the referencing tool to point to a stage of the hexagonal
array..........................................................................................................131!
ix

Figure 4.1.6: 137 splits the hexagon into 6 regions........................................................134!
Figure 4.1.7: A reconstruction of the first three iterations of the geometric pattern......137!
Figure 4.1.8: Green lines and arrows produced by 137..................................................142!
Figure 4.1.9: Qwertyuiop repositions the green lines on the left. Shortly after 137
increases their thickness.............................................................................144!
Figure 4.1.10: 137 adds an elongated hexagon in orange...............................................147!
Figure 4.1.11: Qwertyuiop points to the triangle which contains the sticks to be
counted for the stage indexed by sidelength=2. The green lines
enclosed by the reference correspond to 1+2=3 sticks............................151!
Figure 4.1.12: 137 adds an orange segment to the drawing............................................152!
Figure 4.1.13: 137 adds a red hexagon inside the partition the team has been
oriented to................................................................................................155!
Figure 4.1.14: 137 changes the color of the green lines inside the red hexagon
to blue.......................................................................................................160!
Figure 4.1.15: Qwertyuiop highlights the triangle by using the referencing tool...........162!
Figure 4.1.16: Qwertyuiop moves the lines added by 137 away....................................163!
Figure 4.1.17: Qwertyuiop repositions the red lines to mark a part of the larger
triangle. Then he adds two horizontal lines in green, parallel to the
existing green line. Finally, he adds 3 more lines in purple. Since
Qwertyuiop uses a thinner brush to draw the green and purple lines,
they are difficult to see.............................................................................164!
Figure 4.1.18: 137 increases the thickness of the newly added green and purple lines.
The final state of the diagram presents a visual proof that 3 sets of
collinear lines do not overlap with each other.........................................165!
Figure 4.1.19: 137 highlights 2 horizontal lines in orange following his proposal at
7:55:44 (line 829).....................................................................................170!
Figure 4.1.20: The state of the whiteboard when 137 began his exposition
at 8:01:31 (line 854).................................................................................175!
Figure 4.1.21: 137’s drawing that followed his posting at 8:01:31 (i.e. line 854).
The triangles added in blue follow the chat posting that proposes the
multiplication of what is marked with orange by 3................................175!