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Mathematical Knowledge for Teaching Teachers: the Case of Multiplication and Division of Fractions Syracuse University SURFACE Mathematics - Dissertations Mathematics 2011 Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions Dana E. Olanoff Syracuse University Follow this and additional works at: https://surface.syr.edu/mat_etd Part of the Mathematics Commons Recommended Citation Olanoff, Dana E., "Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions" (2011). Mathematics - Dissertations. 64. https://surface.syr.edu/mat_etd/64 This Dissertation is brought to you for free and open access by the Mathematics at SURFACE. It has been accepted for inclusion in Mathematics - Dissertations by an authorized administrator of SURFACE. For more information, please contact [email protected]. Abstract This study attempts to answer the question, What is the mathematical knowledge required by teachers of elementary mathematics content courses in the area of multiplication and division of fractions? Beginning in the mid-1980s, when Shulman (1986) introduced the idea of pedagogical content knowledge, researchers have been looking at the knowledge needed to teach in a variety of different content areas. One area that has garnered much of the research is that of mathematics. Researchers have developed frameworks for what they call mathematical knowledge for teaching, but there has been little work done looking at the knowledge requirements for teachers of teachers. This study attempts to fill this gap by determining some aspects of a framework for the mathematical knowledge required to teach prospective elementary teachers multiplication and division of fractions. In order to determine aspects of a framework for mathematics teacher educator knowledge in relation to multiplication and division of fractions, I interviewed, observed, and audiotaped three experienced teacher educators in different educational settings to determine the mathematical work of teaching prospective teachers fraction multiplication and division. My analysis focused on three of major tasks that came out of the work: introducing fraction multiplication, helping students make sense of fraction division, and assessing student understanding. Each of these tasks played a major role in the work of the teacher educators, and the knowledge required to perform these tasks was evident in varying degrees in each teacher educator. After analyzing the three mathematical tasks and the knowledge required by them, I was able to determine some components of a framework for the mathematical knowledge needed for teaching teachers multiplication and division of fractions. These aspects include: understanding multiple representations of fraction multiplication and division and how these representations relate to each other, to whole number ideas, and to the algorithms, deciding which aspects of the topics will help prospective teachers make the connections that they will need in order to teach these topics, especially since time often plays a factor in what gets taught in mathematics content classes for prospective teachers, setting specific goals of exactly what one wants one’s students to know, rather than having a general goal of wanting prospective teachers to develop conceptual understanding of a topic, and being able to design and use assessments effectively to help decide if one is achieving one’s goals. While each of the aspects described above are components of a framework for the mathematical knowledge needed by teacher educators, the three teacher educators in my study all lacked or were unable to demonstrate some of the knowledge components that would have helped them to meet their goals, despite having a wealth of experience teaching and designing mathematics content courses for prospective elementary teachers. One possible reason for this is that each of the teacher educators in my study were basically alone in their departments, without opportunities to collaborate or discuss these ideas with anyone else. These results suggest a need for better professional development for teacher educators in the field of mathematics education. Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions By Dana E. Olanoff B.A. William Smith College, 1999 M.S. Michigan State University, 2003 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Education in the Graduate School of Syracuse University May 2011 Copyright 2011 Dana E. Olanoff All rights reserved Table of Contents Abstract……………………………………………………………….………….....i Acknowledgements………………………………………………………………..viii List of Tables, Figures, and Appendix……….……………………………………xi List of Relevant Abbreviations…………………………………………………….xii Chapter 1: Introduction…………………………………………………………….1 Rationale…………………………………………………………………...2 Historical Perspective……………………………………………………...3 Content Area……………………………………………………………….8 Research Questions……………………………………………………….11 Chapter 2: Literature Review……………………………………………………..12 Research on Teacher Knowledge………………………………………....12 Frameworks for Teacher Knowledge……………………………..12 Synthesizing the Frameworks……………………………………..35 Development of Teacher Knowledge…………………………………......36 Learning From Experienced Teachers…………………………….37 Inservice Teacher Development…………………………………..39 Prospective Teachers’ Knowledge Development…………………42 Summary………………………………………………………….44 Research on Teacher Educators…………………………………………..45 Characteristics of Teacher Educators……………………………..46 Mathematics Content Courses for Elementary Teachers…………48 Looking at Teacher Educator Knowledge………………………...50 Teacher Educator Development…………………………………..56 Summary………………………………………………………….59 Research on Multiplication and Division of Fractions……………………60 Division and Multiplication……………………………………….63 Teachers’ Knowledge and Understanding of Fractions…………...70 v What Should Teachers’ Understandings of Multiplication and Division of Fractions Look Like?.................................................76 Enhancing Teachers’ Rational Number Knowledge……………..78 Theoretical Frameworks in the Development of Rational Number Knowledge………………………………...…………...81 The Implication of These Ideas for Teacher Educators…………..85 Chapter 3: Methods……………………………………………………………….87 Respondents………………………………………………………………88 Data Collection……………………………………………………………90 Data Analysis…………………………………………………………......94 Chapter 4: The Teacher Educators……………………………………………….96 Tom—Introduction………………………………………………………96 Mathematics Content Course…………………………………….97 General Goals for Course………………………………………..98 Typical Classroom Session…………………………………......100 Goals for Multiplication/Division of Fractions…………………102 Knowledge of Students………………………………………….106 Stephanie—Introduction………………………………………………...107 Mathematics Content Course……………………………………109 General Goals for Course……………………………………….110 Typical Classroom Session………………………………….......111 Goals for Multiplication/Division of Fractions………………….113 Knowledge of Students………………………………………….117 Karen—Introduction…………………………………………………….122 Mathematics Content Course……………………………………124 General Goals for Course……………………………………….125 Typical Classroom Session…………………………………......128 Goals for Multiplication/Division of Fractions…………………131 Knowledge of Students…………………………………………135 vi Summary………………………………………………………………138 Chapter 5: The Mathematical Tasks of Teaching Teachers…………………..141 Introducing Fraction Multiplication…………………………………..141 Teacher Educators Introduction to Fraction Multiplication…..144 Mathematical Knowledge in the Task of Introducing Fraction Multiplication………………………………………153 Helping Students Make Sense of Fraction Division…………………..158 Teacher Educators’ Teaching of Fraction Division…………...160 Mathematical Knowledge in the Task of Helping Students Make Sense of Fraction Division…………………..………..169 Assessing Student Understanding……………………………………..173 Teacher Educators’ Designing and Using Assessments………175 Mathematical Knowledge in the Task of Assessing Student Understanding………………………………………………..182 Summary………………………………………………………………………184 Chapter 6: Mathematical Knowledge for Teaching Teachers: Where Do We Go From Here?............................................................................................186 Aspects of Mathematical Knowledge Required for Teaching Teachers..188 How Does One Develop MKTT?............................................................191 Limitations and Future Research…………………………………...….195 References……………………………………………………………………..201 Appendix……………………………………………………………………….223 vii Acknowledgements I certainly would not have made it through the dissertation process without the help and support of so many people. Thank you so much to all of my friends, family, and colleagues who have stood by me, encouraged me, and never let me fail. In particular, I would like to first and foremost thank my husband, Stefan. I feel like the luckiest woman in the world to have you for a husband. I am sure that I would not have a dissertation if it were not for you. Thank you for being there all the time to take care of me, cook meals, do laundry, shop for groceries, keep the house in a relatively neat state, etc., all while I was reading, writing, and transcribing. Thank you most for reminding me that if it were easy, everyone would have a PhD, and for never losing faith in me, even if I lost it in myself at times. Thank you also to my parents and grandparents, sisters, cousins, and family friends for encouraging me and believing in me. I am certainly blessed to have such a wonderfully supportive family, and I remember
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