# Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions

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 Inservi ce 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 M ultiplication……………………………………… .63

Teachers’ Knowledge and Understanding of Fractions…………. ..70

vi

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

Re spondents……………………………………………………………… 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 — Intr oduction……………………………………………………. 122

Mathematics Content Course…………………………………… 124

General Goals for Course………………………………………. 125

Typical Classroom Session…………………………………... ...128

Goals for Multiplication/Division of Fractions………………… 131 Knowledge of Student s………………………………………… 135

vii

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 a nd Future Research…………………………………...….1 95

Reference s……………………………………………………………………..201

Appendi x……………………………………………………………………….223

viii

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 how lucky I am every day to have you all around. Thank you to all the members of the mathematics and science education seminar. Anyone who read a draft of part of this paper and gave me helpful comments, I appreciate it so much. You definitely helped make this a better product in the end, and it is always nice to know that there are so many people who are in the same boat as you are. Thank you to Helen Doerr and Jodelle Wuertzer Magner for talking with me about fraction multiplication and division, and helping me get a better understanding of the topic while I was researching it.

ix

Thank you to the researchers at the University of Michigan, especially Hyman Bass and Deborah Ball, for inviting me to their summer program to think about aspects of mathematical knowledge for teaching teachers. I appreciate all of the wonderful people that I met there, especially Deborah Zopf, Maggie Rathouz, and Rheta Rubenstein, for having wonderful discussions with me about my topic and supporting me in my work, even after the summer was over. Thank you to the professors and graduate students at the University of Delaware, especially Jim Hiebert, who helped me to think about learning goals as a way of looking at teacher knowledge, and Dawn Berk, for sitting down with me to talk about their teacher educator professional development and mathematics courses for prospective elementary teachers. Thank you also to the graduate students who talked with me and allowed me to observe their classes. A huge thank you goes out to Tom Walker, Stephanie Mitchell, and Karen Freeny, you know who you are, for allowing me to interview you and observe and record your classes. It may have been difficult to open up your classrooms and understandings to a stranger, but you all helped me immensely to understand all of the hard work involved in teaching prospective teachers about multiplication and division of fractions. I certainly could not have done it without you. I know that working with you has helped me become a better teacher educator, and I hope that I have been able to help each of you in some way in return. Thank you so much to my committee. To Jeff Meyer, thank you for agreeing to read a mathematics education dissertation, and for bringing the mathematical perspective to the work. Your questions helped me think about framing the work for someone who is

x

not familiar with all of the education background, and to make it understandable for a larger audience. To Sharon Dotger, I was so happy to be able to have you on my committee. Although you do not study mathematics education, I felt that you believed in and supported my idea from the beginning, and you also brought a unique perspective to the work. Thank you for helping me think about how to explain the different mathematical concepts to someone who started off unfamiliar with them, and thank you for always saying that you just felt like teacher educator knowledge was different, even if you could not put your finger on exactly why. Last, but certainly not least, thank you so much to my wonderful advisor, Joanna Masingila. I know that I could not have done this without you. Thank you for listening to me any time I had crisis of confidence and always believing in me. I always felt like you had faith in me, even when I did not necessarily believe in myself. Thank you for reading numerous drafts of each chapter, even if I did not always get them to you in a timely manner. Thank you for meeting with me each week to help keep me on task, and never actually breaking out the Kenyan porridge spoon, even though I may have needed it at times. I am truly thankful to have such a supportive and knowledgeable advisor to help me through this long and difficult process.

xi

List of Tables, Figures, and Appendix

Table 2.1 Summary of knowledge frameworks………………………………………………. 36

Figure 1.1 Ball et al.’s (2008) model of mathematical knowledge for teaching………………... 5 2.1 Ball et al.’s (2008) model of mathematical knowledge for teaching……………….. 31 2.2 Za slavsky and Leikin’s (2004) teacher educator teaching triad……………………. 52 2.3 Perks and Prestage’s (2008) teacher knowledge and teacher educator

knowledge tetrahedra………………………………………………………….…… 53 2.4 Jaworski’s (2008) Venn diagram modeling knowledge in

teacher education.…… ..55 4.1 A problem from Tom’s textbook dealing with the area model of multiplication…103

5.1 Stephanie’s representations of three -fourths times two- thirds…………………….146

5.2 Stephanie’s model of

…………………………………………………….149

5.3 Karen’s array representation of …………………………………………….150

5.4 An operator model representing of ………………………………………....154

5.5 A bad representation of ……………………………………………………155

5.6 An operator model and an area model of

…………………………………157

5.7 Tom’s complex fraction method of showing why one multiplies by the

reciprocal when dividing fractions………………………………………………..163

5.8 3 quarts divided into quart servings showing 7 ½ servings in total……………165

5.9 Pattern block model of …………………………………………………….167

5.10 Karen’s model showing by thinking of division as the inverse o f

multiplication……………………………………………………………………169

Appendix First Interview Questions……………………………………………………………. 219

xii

List of Relevant Abbreviations

CBMS: Conference Board of Mathematical Sciences CCK: Common Content Knowledge CGI: Cognitively Guided Instruction KCS: Knowledge of Content and Students KCT: Knowledge of Content and Teaching LMT: Learning Mathematics for Teaching Project MKT: Mathematical Knowledge for Teaching MKTT: Mathematical Knowledge for Teaching Teachers MT: Mathematics Teacher MTE: Mathematics Teacher Educators MTEE: Mathematics Teacher Educator Educator PCK: Pedagogical Content Knowledge PST: Pre-Service Teacher PUFM: Profound Understanding of Fundamental Mathematics SCK: Specialized Content Knowledge SMK: Subject Matter Knowledge

1

Chapter 1 — Introduction

In his 1985 Presidential address at the American Educational Research Association (AERA) annual meeting, Lee Shulman identified what he called the ―missing paradigm‖ in the research on teaching (Shulman, 1986). This missing paradi gm referred to the lack of a research focus on the content knowledge needed for teaching, and consequently in exams for teacher certification and evaluation. Specifically, Shulman introduced a type of knowledge which he referred to as pedagogical content knowledge , which linked knowledge of teaching pedagogy with knowledge of the specific content that was taught. In response to this idea, large numbers of studies (e.g., Ball, Hill, & Bass, 2005; Brophy, 1991; Grossman, 1990) have been undertaken in a variety of subjects to identify the kinds of knowledge that are required to teach well; specifically what types of knowledge do teachers have that set them apart from non-teachers? One of the areas where the study of the knowledge needed for teaching has flourished has been in mathematics. Recent studies (e.g., Ball, Lubienski, & Mewborn, 2001; Ma, 1999) indicated that the mathematics teachers need to know, even at the elementary level, is much more complex than was originally thought. Furthermore, research has shown that elementary teachers, both preservice and inservice, may not be well equipped with this deeper knowledge (e.g., Ball, 1990; Borko et al., 1992; Ma, 1999; Simon, 1993; Thanheiser, 2009). The growing ideas about the knowledge needed to teach elementary mathematics increases demands on teacher educators to help teachers acquire this knowledge. While a large amount of research has been and continues to be done on what mathematical knowledge is necessary for elementary teachers in mathematics, there has been little research on the knowledge demands placed on

2

mathematics teacher educators to support teacher learning. This project attempts to address this gap by looking at the mathematical knowledge needed for teaching teachers. Rationale

It has generally been assumed that more knowledgeable teachers lead to better student performance. However, it is not clear what types of knowledge produce better student outcomes. Mixed results on studies looking at factors such as the number of mathematics classes taken, the number of mathematics methods classes taken, and teachers‘ results on various exams have led researchers to question what it i s that teachers really need to know in order to teach mathematics (Begle, 1979). Recent studies (e.g., Hill, Rowan, & Ball, 2005; Hill, Schilling, & Ball, 2004), have worked to identify what specific types of teacher knowledge produce better student outcomes. The researchers in these studies have developed measures to test aspects of what they call ―mathematical knowledge for teaching,‖ and their studies have shown evidence that first and third grade teachers who performed better on these measures had higher achieving students. The researchers determined that ―students got an extra one -third to one- half of a month‘s worth of learning growth for every standard- deviation rise on their teacher‘s test scores‖ (Viadero, 2004, p. 8). Thus, we are making progress as a field in determining the nature of mathematical knowledge of teachers that produce better student outcomes. In a similar manner, we can assume that more knowledgeable mathematics teacher educators will produce better student outcomes where the students are prospective teachers. However, at this point, we are unaware of the nature of the knowledge required by teacher educators in order to improve teacher learning. It is commonly understood that teachers need to know more than their students. Thus the

3

knowledge demands placed on teacher educators must be more complex than those placed on their students (pre- and inservice teachers). This research study attempted to delve into this complexity and determine some of the aspects of mathematical knowledge needed by teacher educators. The benefits of clarifying the knowledge needed by mathematics teacher educators are many. Scholars in the field of mathematics education have been calling for the better preparation of mathematics teachers (e.g., Askey, 1999; Howe, 1999; Ma, 1999). Understanding the mathematical knowledge needed for teaching teachers will help the field of mathematics education in preparing doctoral students to become teacher educators, in writing textbooks and teachers‘ guides for content courses for prospective teachers, and in giving teacher educators the opportunity to reflect on the knowledge and skills necessary to help produce high quality teachers. In addition, the development of a knowledge base for teacher educators will help professionalize the field of education. Murray (1996) states, ―Without a sure sense of what constitutes educational malpractice, teaching and teacher education are behind other professions that have fairly well- articulated codes of good practice, which by extension define malpractice as the failure to follow good practice.‖ Thus, defining a knowledge base for teacher educators will give the field a framework for ―good practice,‖ which may help quiet the critics of the professionalism of teaching and teacher education. Historical Perspective In his presidential address, Shulman (1986) identified three different types of knowledge needed by teachers: content knowledge, pedagogical content knowledge, and curricular knowledge. By its name, content knowledge refers to knowledge of content.

4

However, Shulman says that it goes beyond this. ―The teacher need not only understand that something is so; the teacher must further understand why it is so, on what grounds its warrant can be asserted, and under what circumstances our belief in its justification can be weakened and even denied‖ (p. 9). Thus, teachers must understand the organizing structure of their discipline, how the concepts are related, how truth is established, and so on. Shulman (1986) defined pedagogical content knowledge as ―subject matter knowledge for teaching ‖ (p. 9). This entails making the subject accessible to students. It includes knowing the best representations of material, what makes the subject easy or difficult for students, and places where students commonly make mistakes. Pedagogical content knowledge provides a link between knowledge of teaching and knowledge of a subject, to give us a type of knowledge unique to the profession of teaching a specific content area. The third form of knowledge, curricular

knowledge , involves understanding the curriculum one is teaching. However, Shulman also included in this type of knowledge, knowing what students are studying in subjects other than the teacher‘s own content, as well as understanding what has come before and after the particular piece of the curriculum that one is teaching. In other words, teachers must know where their students have been and where they are going. Building on Shulman‘s three categories of knowledge, Ball and her colleag ues (e.g., Ball, Hill, & Bass, 2005; Ball, Thames, & Phelps, 2008; Hill & Ball, 2004; Hill, Ball, & Schilling, 2008; Hill, Schilling, & Ball, 2004) have expanded and defined ―the mathematical knowledge for teaching.‖ By researching the work that teachers do, they

5

have developed a framework that expanded Shulman‘s knowledge categories and applied them to elementary mathematics. This framework, illustrated below, breaks Shulman‘s categories of subject matter knowledge and pedagogical content knowledge into pieces. Included in subject matter knowledge are two main categories called Common Content Knowledge (CCK) and Specialized Content Knowledge (SCK) , as well as Knowledge at the Mathematical Horizon . Pedagogical content knowledge is broken down into Knowledge of Content and Teaching (KCT) , Knowledge of Content and Students (KCS) , and Knowledge of the Curriculum . While each of the categories of subject matter knowledge and pedagogical content knowledge contain three sub-categories, these researchers have thus far only developed two of the three sub-categories in each domain.

Figure 1.1 . Mathematical knowledge for teaching (Ball et al., 2008). Ball and her colleagues define CCK

as ―the mathematical knowledge and skill used in settings other than teaching‖ (Ball, Thames, & Phelps, 2008, p. 399). Thus it is the mathematical knowledge that anyone might know. Examples of common content knowledge include knowledge of algorithms and procedures for adding and subtracting, fi nding the area and perimeter of a given shape, or ordering a set of decimals.

6

Specialized content knowledge is defined as ―the mathematical knowledge and skill uniquely needed by teachers in the conduct of their work‖ (Ball, Thames, & Phelps, 2008, p. 400). In studying this knowledge, the researchers determined that this type of knowledge is unique to teaching, in that others would not need it in the course of their work. Examples of specialized content knowledge include being able to evaluate student algorithms to determine their validity, explaining why we invert and multiply when we divide fractions, and understanding and being able to correctly use mathematical vocabulary. While these types of knowledge may be found in people other than teachers, the researchers argue that this knowledge is a necessity for teachers, but is generally not needed by the typical learner of mathematics. ―The third domain, knowledge of content and students (KCS) , is knowledge that combines knowing about students and knowing about mathematics‖ (Ball, Thames, & Phelps, 2008, p. 401). This type of knowledge includes anticipating student difficulties, understanding students‘ reasoning, and knowing common errors and misconceptions that students will have with specific material. Ball, Thames, and Phelps (2008) explain how the first three types of knowledge work together in the classroom: Recognizing a wrong answer is common content knowledge (CCK), while sizing up the nature of the error, especially an unfamiliar error, typicall y requires nimbleness in thinking about numbers, attention to patterns, and flexible thinking about meaning in ways that are distinctive of specialized content knowledge (SCK). In contrast, familiarity with common errors and deciding which of several errors students are most likely to make are examples of knowledge of content and students. (p. 401)

7

The final domain that these researchers have expanded on in detail is knowledge of content and teaching (KCT) . This type of knowledge combines knowing about te aching with knowing about mathematics. It involves knowing how to sequence a particular set of topics and understanding the power and value of different mathematical representations. Ball and her colleagues (2008) describe one of the aspects of KCT in so me of the roles that the teacher has to play in helping students: ―During a classroom discussion, they have to decide when to pause for more clarification, when to use a student‘s remark to make a mathematical point, and when to ask a new question or pose a new task to further students‘ learning‖ (Ball, Thames, & Phelps, 2008, p. 401). These types of teaching tasks require that the teacher have both a deep understanding of the subject of mathematics, as well as understanding how their actions and decisions will affect how and what the students learn. Much of the current research on teacher knowledge uses Ball and her colleagues‘ framework as a starting point (Thanheiser et al., 2009). This project attempt ed to develop components of a framework for the mathematical knowledge required for teaching teachers. Since little is known on the learning trajectories of teacher educators, Stein, Smith, and Silver (1999) suggest that ―we might turn to what is known about the learning of teachers in the context of the c urrent reforms‖ (p. 243) in order to better understand how teacher educators might learn. Thus, we can use current frameworks for teacher knowledge as a basis for teacher educator knowledge.

While the Ball and colleagues‘ ―egg‖ (Figure 1 .1) framework provides a basis for looking at mathematics educator knowledge, it does seem incomplete, as teacher educators need to know more than what is known by their students; future teachers.

8

Questions that we might ask about teacher educator knowledge include: Are there other aspects of teacher educator knowledge not covered by this framework? Is there knowledge for teaching that is unique to teacher educators? What does this knowledge entail? Is there some sort of specialized , specialized content knowledge that is deeper than what Ball and colleagues have identified as specialized content knowledge? In this study I attempt ed to answer these questions by using a grounded theory study of experienced teacher educators in practice, to determine how teacher educator knowledge is qualitatively different than what others have identified as teacher knowledge.

Content Area Since the subject of mathematics is very broad, it would be difficult to look at elementary mathematics as a whole to determine aspects of a framework for teacher educator knowledge. I decided to narrow my content area by focusing on a domain that has been historically challenging for students and both pre and inservice teachers: multiplication and division of fractions (Ball, 1990a; Fischbein et al., 1985; Graeber & Tirosh, 1988; Ma, 1999). Much of the current research dealing with teachers‘ knowledge of fractions has focused on division of fractions. Researchers give justification for this focus such as the fact that ―division of fractions lies at th e intersection of two mathematical concepts that many teachers never have had the opportunity to learn conceptually — division and fractions‖ (Sowder, Phillip, Armstrong, & Schappelle, 1998, p. 51), and ―since division with fractions is most often taught algorithmically, it is a strategic site for examining the extent to which prospective teachers understand the meaning of division itself‖ (Ball, 1988, p. 61). As expected from these statements, researchers have found that both

9

students and teachers struggle with teaching and learning this topic. While the majority of teachers are able to perform the ―invert and multiply‖ division algorithm, researchers have found that teachers are unable to explain why the algorithm works (e.g., Borko et al., 1992; Eisenhart et al., 1993), or develop story problems that model division of fractions (e.g. Ball, 1988, 1990a; Ma, 1996, 1999). While there is less research on teachers‘ understanding of multiplication of fractions, many of the difficulties teachers have with division result from not having a deep understanding of multiplicative ideas in general. There are many reasons why multiplication and division of fractions are difficult for both students and teachers. First, unlike addition and subtraction, multiplication and division, both of fractions and whole numbers, are not unit-preserving operations. That is, when a person adds or subtracts, we can think of combining ―like terms,‖ and the result is also the same like thing. For example, 2 apples added to 3 apples results in 5 apples . However, when one multiplies or divides, the units sometimes change. We do not multiply one number of apples by another number of apples. Instead, we would multiply 2 apples by 10 children, giving us 20 apples total. If we divide, we can divide 20 apples by 10 children, and we get 2 apples per child. Alternatively, we can divide 20 apples among children so that each child receives 2 apples, and our quotient would give us the number of children who would receive apples. While this idea is not difficult to understand, it can become more complicated when talking about fractional pieces of a number; knowing what one‘s answer should even look like can be complicated.

Another reason why multiplication and division of fractions are difficult concepts for students and teachers is that many people have the misconception that multiplication

10

always makes bigger, division always makes smaller, and when we divide, we must divide a larger number by a smaller number (Greer, 1994). When students first learn multiplication and division (with whole numbers) these ideas are true, however, with the introduction of fractions, this is not always the case. Multiplying a value by a number between 0 and 1 will decrease the original value, while dividing by a number between 0 and 1 will result in an increase. These misconceptions cause problems when students and teachers are unable to identify the correct operation to use to solve a word problem. For example, when students are asked a question such as: Cheese costs $3.75 per pound. How much for 6 pounds of cheese? , the inclination is to multiply the two quantities together to result in a larger value. However, if the question read: Cheese costs $3.75 per pound. How much for pound of cheese? , many students will choose to divide $3.75 by , because they believe correctly that their answer should be less than $3.75, but incorrectly that division always makes the result smaller. This ―nonconservation of operations‖ ha s been found both in elementary students and prospective and practicing teachers (Fischbein et al., 1985; Graeber, Tirosh, & Glover, 1989; Greer, 1994; Harel & Behr, 1995). An added problem that researchers have identified regarding teachers‘ knowledge of rational numbers in general, is ―that one critical aspect of teachers‘ knowledge of rational numbers is that they do not realize that they lack the understanding of rational numbers necessary to teach this topic in a meaningful way‖ ( Sowder, Armstrong, et al., 1998, p. 145). Because many teachers know the procedures for multiplying and dividing fractions, they believe that they understand what they need to know in order to teach the topic. However, this procedural knowledge does not allow them to respond to student

11

questions about why the algorithms work, examine alternative student algorithms, or pose meaningful problems for their students. Thus the job of the teacher educator becomes more complicated around the ideas of multiplication and division of fractions. Not only do teacher educators need to help prospective teachers deepen their knowledge of these topics, they may also need to convince prospective teachers of the need for this in the first place. Research Questions In this research study I attempt ed 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? Specifically, using a qualitative study of teacher educators teaching mathematics content courses for prospective teachers, I attempt ed to determine some components of a framework for teacher educator knowledge as it relates to multiplication and division of fractions. Smith (2003) states that ―one of the aims of doctoral t heses for teacher educators should therefore be to build up a knowledge base for teacher educat ion‖ (p. 205). This thesis