AN ABSTRACT OF THE DISSERTATION OF William F. Buckreis for the degree of Doctor of Philosophy in Mathematics Education presented on August 30, 1999. Title: Elementary Mathematics Teacher Subject Matter Knowledge and Its Relationship to Teaching and Learning. Abstract approved: Redacted for Privacy V Margaret L. Niess The purpose of this investigation was to explore how differences in an elementary mathematics teacher's subject matter knowledge structure impact classroom teaching and student learning. The study included two phases. Phase 1 focused on the selection of a single case. An open-ended questionnaire and interview were used to identify the subject matter knowledge structure for addition, subtraction, multiplication, and division of three elementary teachers. One teacher was selected who demonstrated clearly different levels of knowledge for multiplication and division. An additional interview provided information on the teacher's specific climate for teaching mathematics and details about the unit on multiplication and division to be observed. Phase 2 included daily classroom observations for approximately one hour each day of a seven-week unit on multiplication and division. Informal interviews were conducted with the teacher throughout the unit to better understand the lessons and allow the teacher an opportunity to clarify statements and actions. A final teacher interview occurred after the last classroom observation. At the conclusion of the observations, the students were assessed to determine their knowledge of multiplication and division based on the teacher's unit objectives. And six students, representing the range of class performance, were interviewed to provide additional insights into the students' learning. The teacher's subject matter knowledge of multiplication was strong but her knowledge of division was faulty and incomplete on several topics including the different meanings of division, the conceptual underpinnings of division procedures, the relationships between symbolic division and real life problems, and the idea of divisibility. Although the translation of the teacher's subject matter knowledge was complex, it seemed to be directly related to classroom teaching and students' learning. The teacher's narrow understandings were associated with an incomplete developing of the full range of division situations. Although the students had significantly more success on the post assessment problems involving multiplication than on those involving division (understandable since the teacher spent more time teaching multiplication than division), a more worrisome concern was that the students in this study exhibited serious misconceptions associated with the meanings of division, division computation, and notions of divisibility. ELEMENTARY MATHEMATICS TEACHER SUBJECT MATTER KNOWLEDGE AND ITS RELATIONSHIP TO TEACHING AND LEARNING By William F. Buckreis A Dissertation Submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Completed August 30, 1999 Commencement June 2000 Doctor of Philosophy dissertation of William F. Buckreis presented August 30, 1999 APPROVED Redacted for Privacy Major Professor, representing Mathematics Education Redacted for Privacy Chair of Department of Science and Mathematics Education Redacted for Privacy Dean of Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request. Redacted for Privacy William F. Buckreis, Author ACKNOWLEDGEMENTS I wish to express my sincere gratitude to those who have provided their help and encouragement throughout this study. First, completing this study would not have been possible without the support and encouragement of my committee. I wish to extend my gratitude to each member: Dr. Maggie Niess, Dr. Norm Lederman, Dr. Tom Dick, Dr. Dianne Erickson, and Dr. David Griffiths. I am especially grateful to Maggie Niess for her wisdom, guidance, patience, and belief in me. I could not have done it without her. Next, I would like to thank Meg McGee (the teacher in the study) for allowing me to assess her knowledge, observe her class, and learn so much about teaching and learning from her and the students. Thanks, also, for making it an enjoyable experience. Finally, and most importantly, I would like to express my deepest gratitude and love to my wife, Marie, and son, Sean. I am richly blessed to have them in my life. Their continuous loving support and encouragement provided me the strength and courage I needed to keep going when times were tough. Without them, this dissertation would never have been possible. lowe them so much. Thanks, Marie and Sean! T ABLE OF CONTENTS CHAPTER I: THE PROBLEM ........................................................... 1 Introduction ........................................................................ 1 Statement of the Problem .........................................................5 Significance of the Study ................. ".......................................7 CHAPTER II: REVIEW OF THE LITERATURE .....................................9 Introduction ........................................................................9 Subject Matter Knowledge of Mathematics ................................... 10 Subject Matter Knowledge in Classroom Teaching ..........................31 Discussion and Conclusions....................................................38 Recommendations ..............................................................40 CHAPTER III: DESIGN AND METHOD............................................ .42 Introduction.......................................................................42 Subjects ...........................................................................43 Method ............................................................................44 Phase 1: Selecting the Single Case ................................... .44 Phase 2: Classroom Observations ....................................45 T ABLE OF CONTENTS (Continued) Sources of Data ..................................................................47 QuestionnairelInterview................................................47 Semi-structured Interview .............................................49 Classroom Observations ............................................... 50 Classroom Documents .................................................5 1 Researcher's Journal ................................................... 51 Informal Interviews.....................................................52 Post Assessment ........................................................ 52 Researcher ............................................................... 54 Analysis of Data..................................................................57 Analysis of Questionnaire Data........................................57 Analysis of Classroom Observation Data ............................ 58 Analysis of Post Assessment Data....................................60 Research Questions .....................................................61 CHAPTER IV: ANALYSIS OF DATA ................................................63 Introduction.......................................................................63 Meg: Teaching is Eclectic .......................................................65 Academic and Professional Profile ...................................65 Self-Described Subject Matter Knowledge ..........................71 Classroom Profile.......................................................85 Multiplication Development... ..........................................92 Division Development ................................................ 107 Student Learning Profile.... , ......................................... l16 Summary ........................................................................ 135 TABLE OF CONTENTS (Continued) CHAPTER V: DISCUSSION AND CONCLUSIONS............................. 144 Introduction ..................................................................... 144 Meg's Subject Matter Knowledge Structure ................................ 145 Subject Matter Knowledge and Its Relationship to Teaching and Learning ..................................................... 149 Teaching and Learning................................................ 150 Time..................................................................... 155 Summary .......................... ".................................... 156 Limitations of the Study ........................................................ 157 Implications and Recommendations for Mathematics Teacher Education .................................................................... 159 REFERENCES .......................................................................... 164 APPENDICES ........................................................................... 169 List of Figures R~ ~ 1. Researcher's Subject Matter Know ledge Structure ........................56 2. Meg's Subject Matter Knowledge Structure ................................72 3. Relationship Between Addition and Subtraction ...........................74 4. Addition Algorithm ............................................................75 5. Subtraction Algorithm .........................................................75 6. Addition and Subtraction of Fractions.......................................76 7. Multiplication as Repeated Addition