TEACHERS’ KNOWLEDGE OF STUDENT THINKING AND THEIR INSTRUCTIONAL PRACTICES IN ALGEBRA by AYHAN KURSAT ERBAS (Under the Direction of James W. Wilson) ABSTRACT This study was designed to address and contribute to our emerging knowledge and understanding of teachers’ knowledge of student difficulties and related issues with instructional practices in algebra. The perspectives suggested by the constructs of teaching in context, cognitively guided instruction, and pedagogical content knowledge influenced the theoretical orientations of the present study. Since “knowing” is a variety of separate entities, I found the distinction of knowing as knowing-about (i.e., knowing-that, knowing-why, knowing-how) and knowing-to useful to look at different degrees of knowing. Qualitative case study research design and methodologies were used in generating data collected from two inservice mathematics teachers of first year algebra (one eighth-grade and one ninth-grade teacher) who were selected purposefully. Data collection strategies included conducting audio-recorded semi-structured interviews, making video-recorded classroom observations, and collecting archival documents. Data stories about each case included thick descriptions of each participant’s beliefs, knowledge, and practices concerning student thinking. Findings revealed that even though both teachers presented an awareness and recognition of students’ thinking and difficulties in terms of “knowing-that,” their knowledge in terms of “knowing-why” and “knowing-how” was narrow and even problematic in some cases. Such insufficient knowledge might have limited the teachers’ pedagogical content knowledge of student thinking in terms of “knowing-to” and hampered the teachers when acting in the moment. Issues other than conceptual, cognitive, and epistemological problems characterized both teachers’ knowledge of and beliefs about general sources of students’ difficulties in terms of “knowing-why.” Those issues were: lack of arithmetical and geometrical knowledge, lack of motivation, lack of experience with nontraditional curricula, lack of practice in similar type of problems, carelessness, and inability to understand and apply definitions. However, the teachers were able to give explanations for students’ difficulties and mistakes in specific concepts they were teaching. Both case studies revealed that textbook dependence was central to the teachers’ practices at different stages of instruction such as when planning lessons, assigning homework, or assessing students’ learning. This dependence served as a blocking factor for teachers in trying to get more elaboration and knowledge of student thinking. Commonalities and differences arise for individual reasons for textbook dependence. INDEX WORDS: algebra, mathematics education, teacher knowledge, beliefs, student thinking, student difficulties, inservice mathematics teachers, qualitative research, case study TEACHERS’ KNOWLEDGE OF STUDENT THINKING AND THEIR INSTRUCTIONAL PRACTICES IN ALGEBRA by AYHAN KURSAT ERBAS B.S., Middle East Technical University, Turkey, 1997 M.S., Middle East Technical University, Turkey, 1999 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 2004 © 2004 Ayhan Kursat Erbas All Rights Reserved TEACHERS’ KNOWLEDGE OF STUDENT THINKING AND THEIR INSTRUCTIONAL PRACTICES IN ALGEBRA by AYHAN KURSAT ERBAS Major Professor: James W. Wilson Committee: Shawn Glynn Jeremy Kilpatrick Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia August 2004 iv DEDICATION To my parents Memet and Guldane, my lovely wife Betul, my teachers and friends, for their constant support, encouragement, and wisdom. v ACKNOWLEDGEMENTS I would like to thank my major professor, Dr. James W. Wilson, for his high expectations, constant support, and encouragements throughout my graduate work. He has been a true friend and a great mentor. I appreciate all the time and effort he put into this work and my scholarship. My special thanks go to my committee members, Dr. Shawn Glynn, Dr. Jeremy Kilpatrick, Dr. Clint McCrory, and Dr. Denise Mewborn, I thank them for their constant support, encouragement, valuable comments, and critiques to make this work better. I am very grateful to my parents, Memet and Guldane, for their lifelong support and commitment to their children. They never gave up on me. They have been my inspiration in my hardest times. I also thank my brother Selami and my sister Sumeyye for always being with me in their hearts. How can I forget my lovely wife, Betul? I am grateful for her constant support and sacrifices, as she has considered me as the center of her world. I am sorry that I could not spend enough time with her in the past year. Thanks to the faculty, staff, and graduate students in the Mathematics Education Department and Learning and Performance and Support Laboratory in the College of Education at the University of Georgia, who always had time to talk and provide support when I needed it. I particularly thank Dr. Patricia Wilson, Dr. Chandra Orrill, and Dr. Michael Hannafin for their constant support. Special thanks to past and present InterMath project members and Nicholas Holt for their friendship and support. vi I thank my Master of Science degree professor, Dr. Yasar Ersoy, for his scholarship, encouragement, and support until this day. I also would like to thank the Republic of Turkey Ministry of National Education for their financial assistance during my Doctor of Philosophy degree studies. So many friends, so little space. I would like to thank all of my friends whose names cannot all be written here across the borders of countries all over the world for all the support they have provided for the completion of this work and their lifelong friendship. Last but not least, I thank the teachers who agreed to participate in this study. I express my utmost gratitude for your commitment, time, and wisdom. I feel honored to work with you. Without you, this work could not have been done. vii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.............................................................................................................v LIST OF TABLES...........................................................................................................................x LIST OF FIGURES ....................................................................................................................... xi CHAPTER 1 INTRODUCTION .........................................................................................................1 Problem Statement and Research Questions.............................................................2 Rationale for the Study..............................................................................................3 2 CONCEPTUAL AND THEORETICAL ORIENTATION.........................................12 A Conceptual and Theoretical Framework .............................................................12 Focus on Student Thinking......................................................................................14 Issues in Teacher Knowledge..................................................................................18 Teacher Beliefs........................................................................................................26 Teacher Goals..........................................................................................................28 3 REVIEW OF LITERATURE ......................................................................................29 Teachers’ Conceptions and Knowledge of Student Thinking in Algebra...............29 Teaching and Learning of Radical Expressions......................................................44 Teaching and Learning of Graphing and Graphs ....................................................47 Student Difficulties in Algebra................................................................................53 viii 4 METHODOLOGY ......................................................................................................68 Design of the Study: A Qualitative Approach.........................................................68 Participant Selection................................................................................................69 Participants and Research Contexts ........................................................................72 Data Collection........................................................................................................79 Data Analysis ..........................................................................................................92 Validity, Reliability, and Trustworthiness ..............................................................95 5 THE CASE OF MS. SANDS.....................................................................................102 Ms. Sands’s Beliefs About Mathematics and Algebra..........................................102 Ms. Sands’s Beliefs About and Practices in the Teaching and Learning of Algebra .................................................................................................................105 Ms. Sands’s Beliefs About Middle School Students’ Thinking and Difficulties in Algebra 1 .........................................................................................................124 Ms. Sands’s Knowledge of and Approaches to Students’ Thinking in Unit 3: Burning