preservice teachers’ patterns of metacognitive preservice teachers’ patterns of...

PRESERVICE TEACHERS’ PATTERNS OF METACOGNITIVE BEHAVIOR DURING

MATHEMATICS PROBLEM SOLVING IN A DYNAMIC GEOMETRY ENVIRONMENT

by

ANA KUZLE

(Under the Direction of James W. Wilson)

ABSTRACT

As an experienced and passionate problem solver for years, I wanted to better understand

the metacognition that students exhibit when solving nonroutine geometry problems in a

dynamic geometry environment. In this study, dynamic tool software—namely, the Geometer’s

Sketchpad—was used by the participants. My intention was to focus on participants’ decision

making, reflection, reasoning, and problem solving as well as to understand what situations and

interactions in a dynamic geometry environment promote metacognitive behavior.

Case studies were conducted of two mathematics education preservice teachers who had

previously completed a semester of college geometry and had prior experience working in

Geometer’s Sketchpad. Artigue’s (2002) instrumental approach and Schoenfeld’s (1981) model

of episodes and executive decisions in mathematics problem solving were used to identify

patterns of metacognitive processes in a dynamic geometry environment. Data sources for this

study consisted of think-aloud protocols, individual interviews after each problem-solving

session, students’ written solutions, researcher’s observation notes, video files of problem

solving sessions and a final interview. All collected data were analyzed using the constant

comparative method for both the within-case and the cross-case analysis.

Problem solving of the two participants was described through identifying the

metacognitive processes within each problem-solving episode, and associating them with the

Geometer’s Sketchpad use. During the reading, understanding, and analysis episodes, the

participants engaged in monitoring behaviors such as sense making, drawing a diagram, and

allocating potential resources and approaches that helped make productive decisions. During the

exploring, planning, implementation, and verification episodes, the participants made decisions

to access and consider knowledge and strategies, make and test conjectures, monitor the

progress, and assess the productivity of activities and strategies and the correctness of an answer.

Geometer’s Sketchpad played an important role in supporting these metacognitive processes.

Their use of metacognitive questions helped prompt a metacognitive activity. The effectiveness

of solution approaches was dependent on the presence of managerial decisions. Cognitive

problem-solving actions not accompanied by appropriate metacognitive monitoring actions

appeared to lead to unproductive efforts. Redirection and reorganizing of thinking in productive

directions occurred when metacognitive actions guided the thinking and when affective

behaviors were controlled.

INDEX WORDS: Problem solving, Metacognition, Nonroutine geometry problems,

Preservice teachers, Dynamic geometry software

PRESERVICE TEACHERS’ PATTERNS OF METACOGNITIVE BEHAVIOR DURING

MATHEMATICS PROBLEM SOLVING IN A DYNAMIC GEOMETRY ENVIRONMENT

by

ANA KUZLE

B.Sc., University of Zagreb, Croatia, 2005

M.Sc., University of Zagreb, Croatia, 2007

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

Fulfillment of the Requirements for the Degree

DOCTOR OF PHILOSOPHY

ATHENS, GEORGIA

2011

© 2011

Ana Kuzle

All Rights Reserved

PRESERVICE TEACHERS’ PATTERNS OF METACOGNITIVE BEHAVIOR DURING

MATHEMATICS PROBLEM SOLVING IN A DYNAMIC GEOMETRY ENVIRONMENT

by

ANA KUZLE

Major Professor: James W. Wilson

Committee: Patricia S. Wilson

Jeremy Kilpatrick

Electronic Version Approved:

Maureen Grasso

Dean of the Graduate School

The University of Georgia

December 2011

iv

ACKNOWLEDGEMENTS

I would like to thank Dr. James Wilson, Dr. Patricia Wilson, and Dr. Jeremy Kilpatrick for their

time and patience while serving on my doctoral committee. Thank you for your insightful

comments and questions that opened my eyes about teaching and learning mathematics. I wish to

thank Dean Maureen Grasso and the graduate school for the Dissertation Completion Award.

I am immensely grateful to my two participants, Wes and Aurora, who gave up their free

time to participate in this study. Without you there would be no study. I am indebted to both of

you for gaining insight about how students solve problems in a technology environment.

I also want to thank my family and friends who have been with me through this

experience. Mom, Dad, and Grandma: thank you for your support and instilling in me the

importance of education. Doug, Laurel, and Ana: thank you for the motivational chats, laughs,

and for being my friends. A special thank you goes to my best friends back home, Iva and Stipe,

whom I could always count on. Thank you for lifting my spirit when I doubted myself!

My sincere gratitude goes to Dr. James Wilson for guiding me throughout my experience

at Georgia. Thank you for your time, patience, reading numerous drafts of this report, and

guidance in improving my dissertation. Thank you for believing in me and never giving up on

me in helping me find my own voice as a mathematics educator.

Finally, I would like to thank my partner and my best friend Nico. Thank you for

encouraging me to pursue my dreams when I doubted myself the most. You were my voice of

reason! Without you I would have not had the strength to achieve this endeavor. Thank you for

listening to me, for visiting me, for always putting a smile on my face, and for loving me!

v

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS........................................................................................................... iv

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

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

CHAPTER

1 BACKGROUND AND DESCRIPTION OF THE PROBLEM....................................1

Background ..............................................................................................................2

Purpose Statement..................................................................................................10

Research Questions................................................................................................11

Overview of the Study ...........................................................................................12

2 LITERATURE REVIEW ............................................................................................13

Grounding Problem Solving and Metacognition in Constructivist Theory ...........13

Mathematical Problem Solving..............................................................................16

Metacognition ........................................................................................................27

Research on Metacognitive Aspects of Problem Solving......................................38

Problem Solving in Dynamic Geometry Environments ........................................46

Theoretical Framework..........................................................................................51

3 METHODOLOGY ......................................................................................................53

Research Design.....................................................................................................53

Verbal Reports .......................................................................................................54

vi

Summary of the Pilot Study...................................................................................57

Participant Selection ..............................................................................................59

Data Collection Procedures....................................................................................60

Mathematical Problems .........................................................................................62

Timeline for Data Collection .................................................................................66

Data Analysis .........................................................................................................67

Validity and Reliability..........................................................................................70

Chapter Summary ..................................................................................................71

4 THE CASE OF WES.......................................................................................