# Middle school mathematics students' justification schemes for dividing fractions

TABLE OF CONTENTS Page LIST OF FIGURES : xiv CHAPTER 1 INTRODUCTION 1 Rationale for Studying Justification in Mathematics Classrooms 1 Statement of Research Questions 2 Motivation for the Study 2 Why Should We Study Justification? 3 Why Fractions? 3 Why Middle School Students? 4 Definition of Terms 4 Overview of the Study 4 2 REVIEW OF RELATED LITERATURE 7 Social Norms 7 Sociomathematical Norms 10 Teachers' Promotion of Sociomathematical Norms 11 Justifying as Part of Learning Mathematics 19 Fractions 19 Difficulties Teaching and Learning Fractions 20 Students' Conceptions of Fractions 20 Teacher Support and Mathematical Knowledge 23 Representation Modes 24 vii

CHAPTER Page Representations (Models) of Fractions 26 Interpretations (Constructs) of Fractions 28 Part-Whole (Part-Whole Relations) 28 Quotients (Division) 29 Measures (Measure Continuous Quantities) 29 Ratios (Comparison of Two Quantities) 30 Operations 30 Unifying Elements of Subconstructs 31 Constructive Mechanisms for Students 33 Division of Fractions 36 Opportunities to Learn Fractions with Conceptual Understanding..42 Suggestions for Assessment Opportunities 43 Textbook and Curricula 44 Selecting Appropriate Curricula 45 Summary 46 3 THEORETICAL FRAMEWORKS 47 Kazemi and Stipek's Sociomathematical Norm 47 Harel and Sowder's Proof Schemes 48 Observations and the Process of Proving 48 Harel and Sowder's Comprehensive Perspective of Proof 51 Harel and Sowder's Proof Schemes 55 Externally Based Proof Schemes 56 viii

CHAPTER Page Authoritarian Proof Scheme 56 Ritual Proof Scheme 58 Non-Referential Symbolic Proof Scheme 58 Empirical Proof Schemes 60 Perceptual Proof Scheme 60 Inductive (Examples-Based) Proof Scheme 61 Deductive Proof Schemes 62 Transformational Proof Scheme 62 Axiomatic Proof Scheme 66 Students' Difficulty Justifying with the Deductive Proof Schemes 67 Limitations of Studies on Justification 68 Kribs-Zaleta's Dividing Fractions 68 Summary 69 4 METHODOLOGY 70 Data Generation 70 Design 71 Setting 72 Participants 72 Process for Selection of Participants 73 Selecting Participants to Interview 74 Preliminary Surveys and Oral Interviews 76 ix

Page Taylor's Case 77 Description of the Intervention 79 Groups Worked on Problems 80 Groups Shared Multiple Strategies as a Class 80 Data Collection 81 Types of Data Collected 81 Preliminary Surveys 81 Oral Interview (Pre) 82 Video Recordings of Group Discussions 83 Field Notes 86 Teaching Logs 87 Students' Written Classwork 87 Oral Interviews (Post) 88 Data Organization 88 Folders 89 Coding Data to Keep Track of Them 90 Using the Computer to Organize Data 90 Data Analysis 91 Cobb and Whitenack's Methodological Approach 91 Timeline 94 Summary 94 5 RESULTS 96 x

CHAPTER Page Taylor Used Strategies that Made Sense because she was Aware of her Own Thinking and Learning 96 Taylor Made Sense of Problems and Thought about her Learning 97 Taylor and April Made Sense of the Lemonade Problem 99 Taylor Made Sense of the Orange Problem 102 Taylor Did Not Always Make the Unit as a Whole Explicit 108 Taylor Did Not Make the Unit Explicit on the Orange Problem ..109 Taylor Reflected on Not Making the Unit Explicit 122 Taylor and April Gave Solutions with Units that Represented Different Wholes 124 Taylor Asked Clarifying Questions when she Disagreed with Another Student's Justification 130 The Girls Overcame a Barrier to Ask Clarifying Questions 131 Taylor Asked Rosalee Clarifying Questions about the Ribbons Problem 133 Taylor Worked with her Group on the Pill Problem 138 Summary 139 6 CONCLUSIONS AND DISCUSSION 140 Conclusions and Discussion about Implications for Teaching 140 Claim 1 140 Claim 2 142 Claim 3 145 xi

CHAPTER Page Future Studies 146 Limitations of this Study 149 Reflection 150 REFERENCES 151 APPENDIX A CONSENT FORMS 155 B ASSENT FORMS 157 C PRELIMINARY SURVEYS 159 D INDEX CREATED FROM THE FRAMEWORK OF HAREL AND SOWDER'S PROOF SCHEMES 161 E CONTENT LOG/FIELD NOTES 163 F TEACHING LOGS 165 G TAYLOR'S ORAL INTERVIEW ABOUT MAKING SENSE OF PROBLEMS 167 H TAYLOR'S ORAL INTERVIEW ABOUT MAKING SENSE OF THE ORANGE PROBLEM 171 I TAYLOR'S ORAL INTERVIEW ABOUT EXPLAINING HER UNDERSTANDING TO OTHERS 175 J ORANGE PROBLEM DIALOGUE 177 K POSTER PROBLEM DIALOGUE 192 L PILL PROBLEM DIALOGUE 195 M RIBBONS PROBLEM DIALOGUE 199 xii

CHAPTER Page N LEMONADE PROBLEM DIALOGUE 204 O POST-INTERVIEW 212 xin

LIST OF FIGURES Page Krummheuer's Approach to Argumentation 14 Mathematical Differences and Sophisticated Solutions 17 Conceptual Model for Addition of Fractions 22 Bruner's 3 Stages to Conceptual Fraction Understanding 26 Representations (Models) of Fractions 27 Missing Pieces of a Part-Whole Model 29 Conceptual Modeling for Multiplication of Fractions 31 Measurement (Repeated Subtraction) Model for Division of Fractions 38 Partitioning Model for Division of Fractions 39 Measurement Division Fraction Problem with Oranges 41 3 Characteristics of an Observation 49 Process of Proving 50 Mathematical and Historical-Epstimological Factors 52 Instructional-Cognitive-Socio-Cultural Factors 53 Proof Schemes 56 Selecting Participants to be Interviewed 76 Initial Justification Schemes of Participants Interviewed 77 Coding by Data Type 90 Taylor's Solution on her Preliminary Written Survey 105 Taylor's Explanation on her Preliminary Written Survey 106 Group's Individual Strategies and Solutions 110 xiv

Figure Page 22. The Girls Using Oranges as Concrete Models 116 xv

Chapter 1 Introduction This chapter gives an overview of this dissertation. First, I provide a rationale for studying justification in mathematics classrooms. Second, I will state the research question. Third, I will describe the motivation the researcher had for conducting this study. Next, I will provide a definition of terms related to the study. Finally, I will present a general overview of the study. Rationale for Studying Justification in Mathematics Classrooms Reform-oriented teachers are focused on students gaining deeper understandings of mathematical ideas, relations, and concepts rather than focusing just on accuracy (Kazemi & Stipek, 2001; NCTM, 2000). Although many teachers want to promote goals of reform such as focusing on conceptual understanding, reform-oriented teachers don't always understand reform or are unsuccessful in implementing reform in their classrooms (Pang, 2000, 2001, 2003, 2005). A teacher in McClain and Cobb's (2001) study felt frustrated with the traditional textbook so she tried reform with centers, but whole-class discussions were unsuccessful. She had difficulty finding an instructional approach that met her students' needs and her pedagogical agenda. As a teacher I have also struggled to effectively implement suggestions consistent with reform. According to Pang (2000, 2001, 2003, 2005), reform doesn't mean changing social structures or adding techniques. Many teachers learned differently than reform efforts suggest as optimal so they try suggestions from the National Council of Teachers of Mathematics, but are not always able to stimulate deep conceptual understanding of mathematics on part of their students. Justifying in mathematics classes can help teachers understand their students'

understanding of mathematical concepts. It is important for mathematics teachers to understand and implement sociomathematical norms that promote students' engagement in conceptual mathematical thinking and conversation (Kazemi & Stipek, 2001; Yackel, 2001, 2002; Yackel & Cobb, 1996; Pang, 2000, 2001, 2003, 2005; McClain & Cobb, 1996, 2001; Hershkowitz & Schwarz, 1999; Wood, 1999). It is important to understand how students justify their mathematical thinking and ideas for themselves, their peers, and their teachers (Harel & Sowder, 1998, 2007; Sowder & Harel, 1998; Flores, 2002). Statement of Research Question Based on a desire for my students to build conceptual understandings of fractions, the following question was developed: What happens to my middle school mathematics students' justification schemes for dividing fractions when I implement a sociomathematical norm? • What happens when Taylor attempts to make sense of a problem for herself? • What happens when Taylor attempts to convince her peers? • What happens when Taylor attempts to make sense of other students' justifications? Motivation for the Study As a teacher it is frustrating to see your students learning to recall information they receive from the teacher rather than understand the concepts underlying the procedures. Beyond my own desire as a teacher, other researchers (Harel & Sowder, 1998, 2007; Sowder & Harel, 1998; Flores, 2002, 2006) have studied proof to understand students' mathematical conceptions. These researchers' findings were fascinating to me

3 and gave me the motivation to conduct a study to discover what would happen to my students' justification schemes when dividing fractions. Why Should We Study Justification/Proof? According to Harel and Sowder (1998), students have difficulties understanding, appreciating, and producing proofs. Oftentimes teachers take for granted what is evidence for students. Therefore, many students are not learning that proofs are convincing arguments, that proofs and theorems are a product of human activity, and that proofs are an essential part of doing mathematics. We need to shift students justifications from surface perceptions, symbol manipulation, and proof ritual to justifications based on intuitions, internal conviction, and necessity. Teachers need to focus instruction on proof understanding, proof appreciation, and proof production. In order to advance students' proof schemes we need to look at Harel and Sowder's (2007) comprehensive perspective to consider students' difficulties with proofs, roots of their difficulties, and the types of instructional interventions that are needed to advance students' conceptions of proof as well as their attitudes of proof. Proof schemes are a way to evaluate justification so we can plan for instruction to move students to more sophisticated ways of reasoning (Sowder & Harel, 1998). According to Flores (2002, 2006), when teachers understand the ways that students justify, they can build on those ways to help students develop their abilities to provide convincing arguments in mathematics. Students need to have the ability to develop methods to justify the correctness of a procedure or the truth of a fact. Otherwise students will not know when they are correct. Why Fractions?

4 Rational number knowledge is a rich part of mathematics. It is the idea of numbers beyond whole numbers. Rational number knowledge can be related to many aspects of mathematics and its applications (Kieren, 1992). Fractions are important for the development of proportional reasoning and future mathematics studies (Clarke, Roche, & Mitchell, 2008). Why Middle School Students? I chose to study how seventh graders justify their fraction understanding for two main reasons. One reason is that fractions are a large part of the middle school curriculum. Another reason was that I was currently teaching seventh grade and was curious about my students' understandings of fractions. Definition of Terms Since justification is the main focus of this study it is important to explain meanings of justification. Justification has many meanings. However, for the purpose of this study I am going to use Harel and Sowder's (1998, 2007) definition for justification. Harel and Sowder (2007) describe that proof is traditionally considered to be a relatively precise argument given by mathematicians. They look at proof in a different way. They describe proof as "the process employed by an individual (or community) to remove doubts about the truth of an assertion" (Harel & Sowder, 2007, p. 808). Proof is what establishes truth. They use the terms justification and proof interchangeably. Therefore, in this study, when I use the terms justification or proof I am referring to Harel and Sowder's (2007) definition. Overview of the Study

5 I conducted a qualitative intervention case study to understand how my students justified their understanding for dividing fractions. I purposefully implemented the sociomathematical norm that an explanation consists of a mathematical argument, not simply a procedural description or summary to foster justification when my students were learning fractions. Chapter 2 will describe the literature related to social norms and sociomathematical norms that were used to design the intervention for the case study. It will also describe literature about fraction concepts that students focused on during the intervention. Chapter 3 will describe the three theoretical frameworks used to guide this study: a sociomathematical norm described in Kazemi and Stipek's (2001) study; the proof schemes described in studies by Harel and Sowder (1998, 2007); and division of fractions problems created by Kribs-Zaleta (2008). These frameworks were the lenses I used to make sense of how students justified their understanding of fractions when I implemented a sociomathematical norm. Chapter 4 will provide a description of the methodology used for the study. I will explain how data were generated, collected, organized, and analyzed. This chapter will also describe the details of the intervention and the timeline used in the study. The chapter will end with a summary of the methods. Chapter 5 will describe the results of the intervention. Three guiding principles were expected of students to justify based on reasoning: Taylor is a case of a student who used strategies that made sense to her because she was aware of her own thinking

6 and learning; Taylor is a case of a student who was not always explicit about the unit as a whole when solving measurement division fraction problems; and Taylor is a case of a student who used clarifying questions when she disagreed with another student's justification. Data sets will be presented to reveal these claims in this chapter. Chapter 6 will first provide a conclusion and discussion about implications for teaching for each of the three claims presented in Chapter 5. Then possibilities of future studies will be discussed. Next, limitations of the study will be shared. I will conclude this chapter with a personal reflection of what I learned from conducting this study.

7 Chapter 2 Review of Related Literature Since I was purposefully implementing a sociomathematical norm to foster justification when my students were learning fractions, it is important to describe the literature related to social norms, sociomathematical norms, and fractions. Social Norms Social norms are the general ways that students participate in classroom activities (Kazemi & Stipek, 2001). Developing understandings of social norms requires both explicit and implicit negotiations from the teacher and students (Yackel, 2001). Yackel (2001) described a classroom norm as, "A norm is a sociological construct and refers to understandings and interpretations that become normative or taken-as-shared by the group" (Yackel, 2001, p. 6). Every classroom has social norms, but social norms vary from class to class. Social norms do not always favor deep learning. Pang (2000, 2001, 2003, 2005), Kazemi and Stipek (2001), Yackel (2001), and McClain and Cobb's (2001) studies described social norms found in classrooms that were consistent with mathematics reform. Some of the social norms included were an open/safe environment, meaning making, whole-class discussion, collaboration, and problem solving. Here we examine the tools that teachers employed to develop social norms in their classrooms. The teachers in the studies did the following to create an open/safe environment: • teachers and classmates accepted and supported students who made mistakes;

8 • teachers and students established a permissive and open atmosphere so that students' ideas and even their mistakes were welcomed; • errors were accepted as a normal part of learning; • when a student's contribution was judged to be invalid the teacher stated that the student did the right thing by attempting to explain their thinking; • teachers deemphasized performance of the "correct answer"; • teachers emphasized student effort; • active participation was expected of students; • teachers supported students' autonomy of thought; • students complied with the teacher's instruction; and • each lesson consistently consisted of the brief review of the previous lesson, the teacher's introduction of new mathematical contents of activities, students' activities, and whole-class discussion. The teachers in the studies did the following to encourage meaning making: • students were expected to develop personally-meaningful solutions to problems; • students were expected to listen to and attempt to make sense of each other's interpretations of and solutions to problems; • the teacher mentioned students by name if they appeared not to be listening; • students were expected to ask questions and raise challenges in situations of misunderstanding or disagreement; • students were expected to pose clarifying questions to the student explaining the problem when they did not understand;

9 • students rarely said they disagreed, but challenged the solutions instead; • students were expected to explain why they didn't accept explanations that they considered invalid; • teachers focused on students' learning and understanding; and • teachers commented on or redescribed students' contributions and often times notated their reasoning on the white board or overhead projector. The teachers in the studies did the following to enhance whole-class discussion: • the discussion pattern of social interaction predominated with a sequence of teacher-student turn taking; • the turn taking pattern broke down when students didn't understand explanations - they then justified their reasoning to the student directly; • teachers supported students' contributions to the discussion by providing praise and encouragement; • students solved problems for themselves and presented them to the whole class; • students were expected to explain and justify their thinking and solutions; • students were expected to speak loud enough for everyone to hear; • students usually listened carefully to their friend's explanations; and • different strategies were discussed. The teachers in the studies supported collaboration in the following ways: • students worked in small groups; • teachers circulated the room during small group work; • students collaborated with each other while working together; and

10 • teachers emphasized mathematical activity and utilized small group formats to encourage collaboration and discussion among students. The teachers in the studies encouraged problem solving in the following ways: • teachers encouraged students to describe how they solved problems; • teachers introduced mathematical contents in relation to real-life situations, and emphasized the process of problem solving; and • teachers encouraged students to find different solution methods for a given problem and to provide critiques of their peers' presentations. Sociomathematical Norms Pang (2000, 2001, 2003, 2005) and Kazemi and Stipek's (2001) studies found that although social norms and a positive social environment are consistent for mathematics reform, they are not enough to achieve conceptual understanding. All four of the teachers in Kazemi and Stipek's (2001) study shared the same four social norms consistent with mathematics reform. However they found that there were differences in the quality of mathematics discourse such as the kind of conversations and interactional patterns that promote conceptual thinking. Different sociomathematical norms were present in the successful teachers' classrooms. They propose that sociomathematical norms help to provide a framework for what teachers can do to promote the development of students' mathematical ideas. Researchers (McClain & Cobb, 1996: Hershkowitz & Schwarz, 1999; Kazemi & Stipek, 2001; Yackel, 2001) have studied sociomathematical norms of classrooms.

11 Sociomathematical norms are "normative aspects of interactions that are specific to mathematics" (Yackel, 2001, p. 6) and their activities (Kazemi & Stipek, 2001). It is the actual process of making a contribution that counts as an acceptable mathematical explanation and justification (McClain & Cobb, 1996; Hershkowitz & Schwarz, 1999). They involve a taken-as-shared sense of when it is appropriate to contribute to a discussion (Yackel & Cobb, 1996; McClain & Cobb, 2001). They are the social constructs that individuals negotiate in discussions to develop their personal understandings (Hershkowitz & Schwarz, 1999). A solution should be different, sophisticated, efficient, and elegant (Yackel, 2001). Teachers' Promotion of Sociomathematical Norms Sociomathematical norms provide learning opportunities for students and teachers (Pang, 2000, 2001, 2003, 2005) that regulate classroom discourse about mathematics (McClain & Cobb, 2001). Not all the sociomathematical norms that were created were beneficial for students' understanding of mathematics. While some of the teachers from Pang's (2000, 2001, 2003, 2005) studies created sociomathematical norms that encouraged mathematical insightfulness and difference, some teachers encouraged mathematical accuracy and automaticity. Hershkowitz and Schwarz (1999) completed a study that utilized an educational program for middle school students called the CompuMath Project. The project provided rich social interactions that supported high-level discursive activity. According to Hershkowitz and Schwarz (1999), the study was rich because the tasks were open-ended

12 problem situations, the tasks were designed with multiple phases (small group problem solving, reporting, reflection), and used computerized tools (graphic calculators or graphers, spreadsheets, dynamic geometry software) with actions within each (drawing, plotting, computing, scaling, dragging). The students worked in small groups on open-ended tasks. Students reported on the task by discussing their solution processes with others in their group and with the class. The following occurred during class time: • the students made an hypothesis without a computerized tool; • students were then given time for investigation; • the teacher then initiated whole class discussion on possible hypotheses; • students described the processes they went through not the computations they completed; • students then shared their agreement or disagreement and gave judgments on the nature of the hypotheses and learning strategies; • students then worked in small groups to test their hypothesis with computerized tools such as graphing calculators; and • lastly, the teacher led the class in summing up the mathematical ideas that were involved. Over the course of the study sociomathematical norms were developed and students learned within them. Students learned that their hypotheses were good because of the grounds they were based on, not if they were correct. Judging mathematical

13 hypotheses with tools in order to confirm or refute them became normative. Students learned that opposition or counter-examples were acceptable. Some students who felt uncomfortable sharing their misunderstandings later felt comfortable sharing when their intuitions were wrong. Students improved mathematically because they were able to concentrate on higher level activities since they were freed from technical burdens. Kazemi and Stipek (2001) found differences in the quality of discourse based on the sociomathematical norms that were developed. All the teachers had the social norm of expecting students to explain their thinking. However, some teachers pressed for conceptual thinking by promoting the sociomathematical norm that an explanation consists of a mathematical argument, not simply a procedural description or summary (Kazemi & Stipek, 2001). Wood's (1999) study looked at the interaction between the whole class and the teacher. Wood looked at how teachers create a context for argumentation. According to Wood, when argumentation takes place the following should happen in the mathematical discourse. The child gives an explanation. There is then a challenge from another learner. The explainer justifies their explanation. This continues until all members are satisfied and the disagreement is resolved. According to Yackel (2001), students and teachers give explanations to clarify aspects of their mathematical thinking that they think might not be apparent to others. Students and teachers then give justifications in response to challenges. Yackel (2001) described that Krummheuer's Approach to Argumentation (see Figure 1) provides a way to explain why emphasis on explanation and justification in

14 mathematics leads to mathematics learning that emphasizes reasoning. First, a student makes a conclusion in the form of a claim. This is a statement that is made as though it is true. Second, the student provides data. This is the support the student gives for the conclusion. Next, the student provides warrant which is the rationale that might be given to explain why the data are considered to provide support for the conclusion. Finally, the student provides backing which is further support for the warrant. It is why the warrant should be accepted as having authority. Yackel suggested that the students were more engaged in reasoning and "thinking" than they were in solving problems because "the discussion focused on data, warrants, and backing to support conclusions but not on the conclusions themselves" (Yackel, 2001, p. 13). Krummheuer's Conclusion 1 Data 1 Warrant 1 Backing Approach to Argumentation — • — • — • — • a statement that is made as though it is true (claim) the support that someone gives for the conclusion the rationale that might be given to explain why the data are considered to provide support for the conclusion further support for the warrant - why the warrant should be accepted as having authority Figure 1. Yackel described how this approach promotes reasoning in mathematics. Teachers would engage in sustained mathematical exchanges and pose questions that would challenge and extend students' mathematical thinking (Kazemi & Stipek, 2001; Pang, 2000, 2001, 2003, 2005; & Yackel, 2001). In Kazemi and Stipek's (2001)

15 study teachers would engage the whole class about a particular student's problem. They helped students to focus on concepts rather than procedures by expecting students to explain and justify their mathematical understanding by giving reasons for their actions. Students' actions were expected to be represented verbally, pictorially, and symbolically. Verification was a normal part of group activities. Teachers in Yackel's (2001) study repeatedly asked students for reasons for their claims. Students were required to attempt to make sense of other students' arguments. Teachers helped students realize that there might be more than one argument to support a given claim. They were required to compare and contrast their reasons with those of others (Yackel, 2001). Students were not allowed to just say whether or not they agreed or disagreed, but they were required to give their reasons for agreeing or disagreeing. (Kazemi & Stipek, 2001). Teachers would try to encourage mathematical arguments by not evaluating validity in students' claims, but by asking other students what they thought about their idea (Yackel, 2001). All the teachers had the social norm of expecting students to discuss different strategies. However, teachers can press for conceptual thinking by promoting the sociomathematical norm that mathematical thinking involves understanding relations among multiple strategies (Kazemi & Stipek, 2001). It was not effective for students' conceptual understanding when teachers gave the answers rather than allowed the students to have the opportunity to justify their own answers. Teachers would not accept only one way to get the correct answer. Students needed to prove how to solve problems in multiple ways. The teacher encouraged this