# Enhancing Primary Students' Mathematical Communication through Dyads

i Table of Contents Section 1: Introduction .........................................................................................................1 Problem Statement .........................................................................................................4 Nature of the Study ........................................................................................................8 Research Questions ........................................................................................................9 Research Objective ......................................................................................................10 Purpose of the Study ....................................................................................................11 Conceptual Framework ................................................................................................11 Operational Definitions ................................................................................................12 Assumptions .................................................................................................................14 Limitations ...................................................................................................................14 Scope and Delimitation ................................................................................................16 Significance..................................................................................................................16 Transition Statement ....................................................................................................18 Section 2: Literature Review .............................................................................................20 Introduction ..................................................................................................................20 Content of the Literature Review .......................................................................... 20 Organization of the Literature Review ................................................................. 21 Search for Literature ............................................................................................. 21 Conceptual Framework ................................................................................................22 Social Constructivism ........................................................................................... 22

ii Dialogic Teaching ................................................................................................. 24 Collaboration......................................................................................................... 25 Mathematic Problem-Solving ............................................................................... 28 Mathematics Promoting Democracy and Social Justice ..............................................29 Dyads………… ...........................................................................................................32 Potential Themes ..........................................................................................................32 Primary mathematics workshop structure. ............................................................ 33 Mathematics communication. ............................................................................... 35 Mathematics Communication and Diverse Learners ...................................................43 Students Acquiring English as a Second Language .............................................. 46 Students with Special Needs ................................................................................. 49 Promoting Mathematics Talk with Student Dyads ......................................................52 Multicase Studies in Education Research ....................................................................55 Teacher-focused Multicase Studies ...................................................................... 55 Student-focused Multicase Studies ....................................................................... 57 Need for Further Research and Rationale for Research Method .................................59 Conclusion ...................................................................................................................63 Section 3: Methods ............................................................................................................65 Introduction ..................................................................................................................65 Research Design...........................................................................................................68 Research Questions ......................................................................................................71 Context of the Study ....................................................................................................72

iii Ethical Considerations .................................................................................................73 Procedures for Gaining Access to Participants ..................................................... 74 Measures for Ethical Protection of Participants.................................................... 75 Role of the Researcher .................................................................................................76 Criteria for Participant Selection .................................................................................77 Data Collection ............................................................................................................79 Timeline ................................................................................................................ 79 Data Sources ......................................................................................................... 80 Organizing and Storing Data ................................................................................ 85 Data Analysis ...............................................................................................................85 Timeline ................................................................................................................ 85 Coding of Categories and Themes ........................................................................ 86 Data Analysis after Coding ................................................................................... 87 Discrepant Cases ................................................................................................... 88 Validity ........................................................................................................................88 Internal Validity .................................................................................................... 89 External Validity ................................................................................................... 90 Reliability .....................................................................................................................91 Case Study Data Base ........................................................................................... 92 Audit Trail ............................................................................................................. 92 Conclusion ...................................................................................................................92 Section 4: Results ...............................................................................................................94

iv Introduction ..................................................................................................................94 Participant Teacher Background, Classroom Context, and Math Lesson Structure ...........................................................................................................95 Gathering, Recording, and Keeping Track of Data and Emerging Understandings ................................................................................................99 Classroom Observations ....................................................................................... 99 Teacher Interviews .............................................................................................. 100 Analysis of Student Work Samples .................................................................... 101 Examples of Typical Math Lessons ...........................................................................103 My Third Observation of Ms. Garcia ................................................................. 103 My Third Observation of Ms. Torres .................................................................. 106 Findings......................................................................................................................109 Central Question 1 .............................................................................................. 109 Central Question 1 Summary .............................................................................. 140 Central Question 2 .............................................................................................. 144 Central Question 2 Summary .............................................................................. 155 Discrepant Cases and Nonconforming Data ..............................................................159 Evidence of Quality ...................................................................................................161 Internal Validity .................................................................................................. 161 External Validity ................................................................................................. 163 Reliability ............................................................................................................ 163 Conclusion .................................................................................................................164

v Section 5: Conclusions .....................................................................................................167 Overview ....................................................................................................................167 Interpretation of Findings ..........................................................................................170 Central Question 1 .............................................................................................. 170 Central Question 2 .............................................................................................. 181 Implications for Social Change ..................................................................................188 Equity in Math Education ................................................................................... 188 Students with Special Needs ............................................................................... 189 Recommendations for Action ....................................................................................191 Incorporation of Dyads ....................................................................................... 191 Professional Development .................................................................................. 192 Recommendations for Further Study .........................................................................193 Formal and Informal Dyads ................................................................................ 194 Dyad Focus ......................................................................................................... 195 Dyads and Bilingual Students ............................................................................. 195 Longitudinal Study.............................................................................................. 196 Reflections on the Research Process ..........................................................................197 Possible Personal Bias ........................................................................................ 197 Possible Effects of the Researcher on the Participants and Situation ................. 198 Changes in My Thinking as a Result of this Study ............................................. 199 Concluding Statement ................................................................................................200 References ........................................................................................................................202

vi Appendix A Letter of Cooperation from a Community Partner ......................................225 Appendix B: Data Use Agreement ..................................................................................226 Appendix C: Classroom Observation Protocol ................................................................229 Appendix D: Primary Classroom Mathematics Communication Interview Guide 1 ......230 Appendix E: Primary Classroom Mathematics Communication Interview Guide 2 .......234 Appendix F: Student Work Sample Analysis Protocol....................................................238 Appendix G: Qualitative Multicase Study Data Coding Matrix ......................................239 Appendix H : Audit Trail .................................................................................................240 Appendix I: Code List Used During Data Analysis.........................................................251 Appendix J: Portion Of Coded Observational Notes From Ms. Garcia’s Class Observation 3 .......................................................................................................253 Appendix K: Portion Of Coded Observational Notes From Ms. Torres’ Class Observation ..........................................................................................................256 Appendix L: Portion Of Coded Between-Case Analysis For Observation 3...................259 Curriculum Vitae .............................................................................................................264

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Section 1: Introduction The New Mexico Education Department has called for a high quality mathematics curriculum that focuses on the process standards (New Mexico Public Education Department, 2007, p. 14). One of the process standards vital to high quality mathematics instruction is communication. Math communication is a valuable learning tool because students’ metacognition about mathematics in enhanced when formulating their own words to communicate their thinking and when hearing the ideas of others (Chapin, O’Connor, & Anderson, 2003, p. 5). Valuing mathematics communication is essential to equitable teaching because discourse can make math more comprehensible and accessible for students (Burns, 2007; Coates, 2005; Varol & Farran, 2006). Mathematics can be challenging for all students, but those pupils acquiring English as a second language and students with special educational needs can have greater difficulty with math vocabulary and concepts (Burns, 2007; Cavanagh, 2005; Rubenstein & Thompson, 2002; Shellard, 2004). Developing language and discussing ideas with peers can promote math achievement for all students but especially those with special needs (Ernst-Slavit & Slavit 2007b; Shellard, 2004). Collaborative mathematics communication can also promote democracy and social justice. When students work together to solve a challenging problem, discuss theories, and justify their thinking they are doing the sort of work they will need to do later to fully participate in the democratic process (Ball & Bass, 2008). Therefore, fostering mathematics communication is essential not only as a means to increase math achievement for students but also to promote equity and democracy in the classroom.

2 Despite these benefits, incorporating mathematics discourse in meaningful ways can still be challenging for teachers. Many teachers, even when using a more reform- based curriculum, revert back to more traditional classroom talk structures such as lecturing and quizzing (Chapin et al., 2003, p. 5). Therefore, teachers need support as they try to implement change. This qualitative research study was formulated after I had worked several years at the school of interest exploring approaches to facilitate mathematical communication as a classroom teacher and teacher leader. The site is a Southwestern New Mexico primary school with students in grades preschool to sixth grade. I served as the site math goal team leader, helped plan math professional development, administered and analyzed teacher surveys, and guided the goal team in using student data to guide instructional practices. I was also a kindergarten teacher at the site and have studied issues related to early childhood mathematics through classroom action research. The educators at the campus have strived to take responsibility for their own professional development. They routinely participated in collaborative planning and lesson study. There was a Math Process Trainer on site to enhance professional development and provide teachers with individualized assistance. Although the school of interest had adopted a student-centered curriculum and had received ongoing teacher driven professional development, teachers still reported difficulties in facilitating mathematics communication in their classrooms. The students at the school were still lacking in the area of mathematics achievement. The school had made adequate yearly progress on the state standardized

3 test every year it has been given; however, the scores had been just meeting the state criteria for several years. For example, according to the New Mexico Public Education Department (2008), for the 2007-2008 school year 41.1% of the students scored proficient on the mathematics portion of the state standardized assessment; 58.9% of the students scored as not proficient. To make adequate yearly progress, 41% of students needed to score proficient. Therefore, the school was barely classified as making adequate progress for that school year. In the more recent 2008-2009 school year, an inequity was evident in the state testing data with regard to the achievement gap between the majority of the students at the school and the English language learners and students with disabilities. Overall, 53.3% of students at the site scored proficient in math. However, only 48.7 % of English language learners and only 21.1% of students with disabilities were ranked as proficient in mathematics (New Mexico Pubic Education Department, 2009). This achievement gap in mathematics must be investigated. Equally troubling, there was a lack of achievement of the students on mathematics constructed response problems. On the 2008-2009 New Mexico Standards Based Assessment, students in third through sixth grade scored dramatically lower in short answer and open ended questions compared to their achievement on multiple choice questions (New Mexico Pubic Education Department, 2009). For example, these data showed that in the third grade English curriculum classes students scored proficient on 63% of the multiple choice items but only 43% of the constructed response items. The bilingual third grade students answered 60% of the multiple choice questions proficiently

4 but only 36% of the constructed response questions were scored as proficient or advanced. Given the site’s emphasis on mathematical problem solving and site-based professional development, the poor achievement on open-ended math items was puzzling. On constructed response items, students must not only correctly solve problems, but also communicate their mathematical problem-solving processes and metacognition. Such a drastic achievement gap showed that students were having difficulty expressing their mathematical thinking. The campus had been exploring ways to promote math communication among students, such as using student math dyads. A dyad occurs when “two people (a dyad) take turns listening to each other for a fixed amount of time. In a dyad, the talker has the opportunity, indeed, the responsibility, to talk authentically” (Weissglass, 1990, p. 359). Due to the testing data it is unclear if these strategies were being used consistently if at all or if they were not appropriate for fostering mathematics communication and transferring that communication to independent student work. Problem Statement There is a problem with mathematics instruction at a southwestern New Mexico primary school (kindergarten through sixth grade). The problem specifically is that students are underperforming in the area of mathematics and are specifically lacking in the area of mathematics communication. Consequently, there is awareness of the need for new approaches, such as dyads, that purport to encourage students’ mathematical thinking and communication; however, there is a lack of (a) shared professional knowledge about how to implement dyads or other new practices, and (b) a lack of

5 research findings about the particular ways students’ mathematical communication is improved upon using them. This study will address that gap. Currently, the campus had adopted a mathematics curriculum focused on exploration and problem-solving, and the teachers have been trained on methods for enhancing mathematical communication. However, student achievement on open-ended math problems where students must describe their thinking was extremely low. The students may not have been taught meaningful ways to discuss their mathematical thinking or those methods may not be implemented consistently. Teachers might be committed to mathematics reform intellectually and believe this is reflected in their teaching, but reform-based ideals may not be evident in their actual classroom practice (Stigler & Hiebert, 2004, p. 13). Pratt (2006) explained that “the move from classrooms in which individualized written work was the dominant form of instruction, to ones in which…oral work is far more prominent, may not have been matched by appropriate changes in teachers’ underlying theoretical perspectives” (p. 232). Teachers and districts might adopt new curricula, which occurred at the school of interest, but this does not always lead to profound changes in the classroom. In particular, teachers working with reform curriculum often have difficulty facilitating student discourse in the classroom. While researching classroom communication, Schleppenbach, Perry, Miller, Sims, and Fang (2007) concluded that discourse can be very useful in classrooms; however, there is still a much to be done to help teachers use discourse meaningfully in the classroom (p. 393). Teachers need support exploring ways to promote meaningful student discussions. Teachers should

6 strive to increase student mathematics communication (Hyde, 2006; National Council of Teachers of Mathematics, 2007). The highest level of student communication is referred to as exploratory (Fisher, 1993). When students engage in exploratory conversations they share their ideas and reasoning and justify their thinking. Primary teachers build the foundation for students’ mathematical success (National Association for the Education of Young Children & National Council of Teachers of Mathematics, 2006, para. 1). The content and processes primary educators focus on during their mathematics lessons contribute to the future achievement of their students. Primary students may solve math problems efficiently, yet have great difficulty discussing the problem-solving strategies they employed in meaningful ways (Hartweg & Heisler, 2007). Teachers need support exploring ways to promote meaningful student discussions. The problem of underperformance on assessments of math communication impacts primary students dramatically. Without meaningful mathematics communication skills students will not be able to describe the higher level thinking they perform while solving mathematics problems, and they may not learn as deeply from the solutions and work of their peers. According to the New Mexico Public Education Department (n.d.), half of the weight of the New Mexico Standards Based Assessment (NMSBA) mathematics score is focused on open-ended response or short answer, also known as constructed response, problems. As described above, students at the school had low performance in this area for the past 2 years. Research needed to be conducted on the ways in which elementary students’ higher level thinking and communicating skills in

7 mathematics could be enhanced. This current research study was conducted to investigate and contrast the use of dyad groupings in two elementary classrooms as a means of enhancing students’ abilities to communicate mathematically. There were many possible factors contributing to this problem. Some of these included: lack of teacher professional development, the structure of mathematics lessons, and the student groupings incorporated during math instruction. Further investigation was needed to explore the problem of students’ underachievement in math communication. One grouping possibility for increasing student communication is the use of dyads. During dyads students are paired, and one student talks for a given amount of time while the other student actively listens without commenting. Then, students reverse roles and the speaker becomes the listener and vice versa. This type of communication could encourage exploratory talk in classrooms. Some researchers have had success with implementing student mathematics dyads to promote meaningful discourse in their classrooms (Fraser 1996; Krol, Janssen, Veenman, & van der Linden, 2004; Kumpulainen & Kaartinen 2003; Schmitz & Winskel, 2008; Wickett, 2002). Researchers have suggested looking more closely at student grouping and strategies for enhancing student mathematics discourse and learning (Bodovski & Farkas, 2007; Littleton et al., 2005). Thompson & Chappell (2007) explained: Mathematics classrooms should be learning communities in which students are expected to share their understanding with the teacher and with their peers. Such communities require that students be fluent with mathematical language. The

8 enhancement of this fluency is the responsibility of the mathematics teacher who must consciously work to encourage the use of language. (p. 195) Mathematics dyads could not only encourage students to use language, but also increase their mathematical understanding. As students discuss math problems they consider mathematical content and problem-solving methods (Hyde, 2006, p. 163). This increase in awareness could help students become better mathematics problem solvers overall. However, it is the responsibility of teachers to implement any communication strategy effectively. Jansen (2008) explained that small group discussions may encourage increased student participation and give students opportunities to learn about mathematics; however, these groupings may provide unexpected challenges as well (p. 95). Nature of the Study The study was a qualitative multicase study exploring how one kindergarten teacher and one third grade teacher introduced and facilitated math dyads in their classrooms. The research site was located in Southwestern New Mexico, and serviced pre-kindergarten through sixth grade students. The sample consisted of one kindergarten teacher and one third grade teacher. There were approximately 20 kindergarten students and 26 third grade students in the classrooms. Purposive sampling was used to provide maximum variation of the subjects. The participants were at the two extreme ends of the spectrum of primary grade teachers. The diversity in grade levels provided a variety of data and multiple perspectives to explore the issue of implementing math dyads to enhance student communication.

9 Teachers implemented student dyads during math lessons daily for 9 weeks. I collected three main sources of data during the implementation period. Video taped observations of math lessons using student dyads were conducted 5 times in each participating classroom. Participating teachers were interviewed about math curriculum, student communication, and dyads at the beginning and end of the study. Student work samples were collected for analysis at the beginning and end of the study as well. The data was analyzed using an open coding matrix to discover categories and themes related to using student dyads in primary classrooms. I used both within-case and between-case analysis to compare the coded data and find relevant themes and trends. More specific details about the nature of the study can be found in section 3. Research Questions A qualitative study was designed to explore the central research questions. The first central question focused on the teachers as they used dyads in their classrooms: What specific methods do two primary general education teachers (a kindergarten and third grade teacher) in a Southwestern elementary school use to introduce and facilitate student dyads to encourage communication during math lessons? This question was supported by the following subquestions: 1. How do teachers describe their experiences using math dyads to encourage student communication? 2. What are the similarities and differences between using math dyads to encourage student communication with a kindergarten and a third grade class?

10 3. What teacher choices while implementing dyads enhance mathematical communication by students? 4. What are the implications for best practices for primary teachers wanting to increase students’ communication about mathematics? The second central question focused on student communication about mathematics: What are the types and qualities of student communication that occur during the course of participating in dyads during math lessons? 5. How do students communicate about mathematics when just beginning to use dyads and when they have gained more experience with dyads? 6. What types of math communication between students (disputational, cumulative, or, exploratory) occurs while using mathematics dyads? 7. What types of communication do students use when completing open- middled exploratory math tasks while also participating in dyads during math lessons? Research Objective The objective of this study was to examine and explore primary teachers’ implementation and facilitation of a new student communication strategy, dyads, during mathematics lessons. I created a multifaceted description of how a kindergarten and third grade teacher introduce, modify, and utilize student mathematics dyads to support pupil learning and mathematics discourse. I also explored teacher opinions about using math dyads.

11 Purpose of the Study The purpose of this research option doctoral study was to explore the way primary teachers implemented a new cooperative communication strategy, student dyads, into mathematics lessons. In past research, dyads have been used to promote mathematical communication in the classroom (Fisher, 1996; Wickett, 2000). This study also observed the types of mathematics communication students used in dyads and discussed the process of using dyads with primary grade teachers. Through exploring the introduction and facilitation of math dyads in primary classrooms, this study added to the current body of educational research by: illuminating the process of using dyads, describing the types of student communication in dyads, examining the differences and similarities between using math dyads at different grade levels, and analyzing the influence of dyads on student math communication. The goal was to determine how the teachers introduce and sustain student mathematics dyads to promote communication. If they were successful, then their experiences could serve as a model to other educators wishing to promote equity and discourse in primary mathematics. If the dyads did not promote meaningful student talk, then the research may show why the strategy was not successful, what the educators may need in the area of professional development, and potential stumbling blocks that other educators may be able to avoid in the future. Conceptual Framework The use of student dyads is built upon the foundation of several key educational concepts. In social constructivism, social interactions such as those occurring in student

12 dyads are seen as essential for the promotion of student learning. The theory of dialogic teaching describes the way students learn through talk, and the student dyad is meant to promote such communication. While working in dyads, students must also collaborate with one another, so this research is also grounded in the theory and study of collaborative learning. Finally, the work students do in the math dyads will focus on authentic mathematics problem-solving, so the research in problem-solving mathematics instruction is important to this inquiry. These four concepts provide the foundation and rationale for the use of student math dyads to promote communication and are discussed below. These theories will be explored more thoroughly in the literature review. Operational Definitions Dialogic Teaching: Dialogic teaching embraces the idea that students can learn content area knowledge through extended and meaningful dialogue in the classroom. Dialogic talk is “characterized by the teacher rephrasing a student’s response for clarity, using student responses to generate new meaning, and using utterances as thinking devices” (McGuire & Harshman, 2002, p. 4). According to Alexander (2005, p. 4-5) dialogic teaching consists of five main elements. It is reciprocal, supportive, purposeful, collective, and additive. Teachers and students work together to extend their thinking and understanding through dialogue (McGuire & Harshman, 2002). Discourse: The National Council of Teachers of Mathematics (2009) described classroom discourse as “written and oral ways of representing, thinking, communicating, agreeing, and disagreeing that teachers and students use” (para. 1). This study will focus

13 on both the oral discourse with students and teachers as well as the written discourse in student work samples. Dyads: This educational grouping structure involves pairs of students actively listening to one another. As Wickett (2000) explained, in a dyad “students are paired, and each receives an equal amount of time to talk while the other listens without interruption or judgment” (para. 2). While engaging in dyads, the talkers speak about issues important to them and the listeners help clarify their thinking. Students may begin to learn this process by only speaking for five minutes, but with practice, the time each person spends talking can be increased gradually (Weisglass, 1990, p. 359). A dyad can also be more informal where pairs of students are simply asked to turn and talk to one another about a given topic (Chapin, Conner, & Anderson, 2003, p. 125). Mathematics instruction problem-solving approach: This study will use Burns’ (2000) description of problem-solving to guide the selection of challenging and worthwhile mathematical problems that foster student thinking and communication. “Problem-solving situations…demand that the child develop a plan for execution as well as execute the plan. Problem solving techniques do exist, but they are general approaches, not algorithms that can be routinely applied to specific problems” (Burns, 2000, p. 18). Therefore, problem-solving challenges students to apply the mathematics they know in unique and well-thought out methods. Open-middled problem: This is a problem presented to students that has only one correct answer. However, there are many potential methods for correctly solving the problem. This is opposed to both a closed task with only one correct answer and one

14 method for reaching that answer and an open-ended task with several appropriate answers and multiple methods for finding an answer (Fritz, 2002, p. 4). Types of student talk: Fisher (1993) identified three types of student talk: disputational, cumulative, and, exploratory (p. 255). In disputational talk students disagree and are unable to reach consensus. During cumulative talk students quickly reach agreement without clarifying and explaining their ideas. Exploratory talk involves sharing an initial idea which is debated, transformed, and clarified until consensus is reached. Assumptions It was assumed that the participants were honest in their replies. The participants knew that their identities would not be shared in this study; therefore, the anonymity should have helped them feel free to reply candidly to all questions. I also assumed that the theories grounding the theoretical framework of the study were the most appropriate ones to use given the purpose of the study. Finally, I assumed that to some extent the experiences of the sample teachers and students were representative of larger populations in education. Limitations This study was conducted in a Southwestern New Mexico elementary school that had adopted an investigatory math curriculum valuing teacher leadership and on-site professional development. The specific setting was unique as are the results of the qualitative multicase study. If conducted in another area, the outcomes would most likely differ due to the uniqueness of every school setting.

15 I worked as a classroom teacher in the school of interest and was the campus math goal team leader. This relationship with the teachers could have influenced their responses and instructional decisions. Only two teachers were observed, and observing more or different teachers could have lead to different data and conclusions. Although approximately half of the teachers at the research site teach bilingual curriculum only English language curriculum teachers participated in this study because I am not fluent in Spanish. I considered using a translator to allow participation of bilingual teachers. However, this idea was rejected due to the nature of data collection in qualitative case study research, where “the investigator is the primary instrument for gathering and analyzing data and, as such, can respond to the situation by maximizing opportunities for collecting and producing meaningful data” (Merriam, 1998, p. 20). The primary data source for this study was classroom observations, and it would be difficult for a translator to capture all the conversations that may be occurring in various dyads. Although the experiences of bilingual teachers are very valuable and should be considered, my inability to analyze Spanish data meant a lack of bilingual teachers participating in this current study. The qualitative analysis was open-ended and could be open to differing interpretations. The following strategies known for increasing qualitative validity were employed to promote the validity of study. By analyzing multiple data sources, the findings were triangulated and examined for common themes. I used a peer debriefer, our campus MPT, to review the research and question the findings. Finally, I clearly discussed possible bias to ensure that the nature of my role is clearly understood