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The use of questions and gestures in discourse during problem-based interviews

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Title:
The use of questions and gestures in discourse during problem-based interviews
Creator:
Lambert, Richard Kay ( author )
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
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Language:
English
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1 electronic file (54 pages). : ;

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Mathematics -- Study and teaching ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Review:
This research thesis examined the possible links between teacher questions and how students responded during seven problem-based interviews each consisting of a researcher and two 7th grade students. The interviews were conducted by another researcher at a mid-western middle school. This study focused on the types of questions the researcher asked (during the interviews), the resources (verbal and non-verbal) students used to answer the questions, and the manner in which students interacted with the researcher and their partner. Video recordings and transcripts of the interviews were coded for questions and gestures, revealing five types of questions, and five types of gestures. This study found that the type of questions asked of students during the interviews guided their thinking and explanation of the problem they were presented. Students used gestures to support their answers.
Thesis:
Thesis (M.S.E.)--University of Colorado Denver. Mathematics education
Bibliography:
Includes bibliographic references.
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System requirements: Adobe Reader.
General Note:
School of Education and Human Development
Statement of Responsibility:
by Richard Kay Lambert.

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University of Colorado Denver
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|Auraria Library
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All applicable rights reserved by the source institution and holding location.
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902872959 ( OCLC )
ocn902872959

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THE USE OF QUESTIONS AND GESTURES IN DISCOURSE DURI NG PROBLEMBASED INTERVIEWS by RICHARD KAY LAMBERT B.S., Strayer University, Woodbridge, Virginia, 199 5 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science in Education Mathematics Education 2014

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ii This thesis for the Master of Science in Education degree by Richard Kay Lambert has been approved for the Mathematics Education Program by Heather Johnson, Chair Ron Tzur Michael Ferrara 7/21/2014

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iii Lambert, Richard K. (M.S.Ed, Math Education) The Use of Questions and Gestures in Discourse Duri ng Problem-based Interviews Thesis directed by Professor Heather Johnson ABSTRACT This research thesis examined the possible links be tween teacher questions and how students responded during seven problem-based i nterviews each consisting of a researcher and two 7th grade students. The interviews were conducted by a nother researcher at a mid-western middle school. This stu dy focused on the types of questions the researcher asked (during the interviews), the r esources (verbal and non-verbal) students used to answer the questions, and the mann er in which students interacted with the researcher and their partner. Video recordings and transcripts of the interviews were coded for questions and gestures, revealing five ty pes of questions, and five types of gestures. This study found that the type of questio ns asked of students during the interviews guided their thinking and explanation of the problem they were presented. Students used gestures to support their answers. The form and content of this abstrac t are approved. I recommend its publication. App roved: Heather Johnson

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iv DEDICATION Historically, the United States has been a melting pot where people come seeking refuge, solace and freedom. Many of them come with very little knowledge of the English language and that puts their children at a disadvantage in school. Additionally, there are children growing up in the United States that are disadvantaged because they grow up in low socio-economic communities. This wor k is dedicated to all the teachers who have raised the bar in mathematics education by pushing their students to a higher level of understanding through discourse in their c lassrooms.

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v ACKNOWLEDGEMENTS First and foremost, I am eternally grateful to my wife, Yvonne, for her love, encouragement, and patience. Without her I would no t be completing this work. Thank you for the many hours spent reviewing my paper, be ing a sounding board, and encouraging me to keep going. I wish to thank my parents Kay and LouAnn Lambert. My father taught the value of hard work and attention to detail, while my moth er taught me to have confidence in myself. I wish to thank my grandmother Thora Lambert, who t aught me the value of education and instilled in me the desire to continu e learning throughout my life. I am deeply grateful to my faculty advisor, Dr. Hea ther Johnson, for her encouragement, honest feedback, and gentle prodding that kept me going throughout this process. Furthermore, it was a privilege to assist her with her research.

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vi TABLE OF CONTENTS CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . 4 Discourse and Reform in Mathematics Education . . . . . . . . . . . . 4 English Language Learners (ELLs) Participation in t he Mathematics Discourse . 6 Working on Challenging Math Problems Students Const ruct Their Own Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 StudentsÂ’ Mathematical Communication in Small Group s . . . . . . . . . 9 Types and Level of Teacher Questions Shape the Math ematics Classroom . . 10 Gestures Convey More than Words Alone Can . . . . . . . . . . . . . 12 III. METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 14 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 IV. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Questions Guide Student Participation . . . . . . . . . . . . . . . . 25 Student Use of Gestures . . . . . . . . . . . . . . . . . . . . . . 32 Student to Student vs. Student to Researcher Commun ication . . . . . . . . 34 V. DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 40 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . 42

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vii LIST OF TABLES TABLE 1. Examples of four question types . . . . . . . . . . . . . . . . . . . 17 2. Examples of type 5 questions . . . . . . . . . . . . . . . . . . . . 22 3. Examples of first three gestures types . . . . . . . . . . . . . . . . . 23 4. Examples of type d and e gestures . . . . . . . . . . . . . . . . . . 23 5. Number and percentage of question types asked ea ch student pair . . . . . . 25 6. Number and percentage of student replies to rese archerÂ’s questions . . . . . 30 7. Percentage of student responses vs. number of ty pe 3 questions researcher asked . . . . . . . . . . . . . . . . . . . . . . . . . 31 8. Student gestures by individuals . . . . . . . . . . . . . . . . . . . 33 9. Student-researcher vs. student-student discourse . . . . . . . . . . . . . 34

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viii LIST OF FIGURES FIGURE 1. Areas of filling a rectangle . . . . . . . . . . . . . . . . . . . . . 15 2. Areas of filling a triangle . . . . . . . . . . . . . . . . . . . . . . 16 3. Number of student responses vs. number of type 3 questions . . . . . . . . 32

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1 CHAPTER I INTRODUCTION According to the Programme for International Stude nt Assessment (PISA) 2012 report, the United States, while spending more per student than most countries, ranked 27th in mathematics among the 34 participating countrie s. Reportedly, students from the United States are weak in applying mathematical kno wledge when solving real-world problems (PISA, 2013). Furthermore, the student pop ulation in the United States is becoming more diverse, in particular the percentage of the Latino/a and Asian/Pacific Islander student population is increasing (Barbu & Beal, 2010; National Center for Education Statistics). Given the growing diversity of students, educators and policy makers are faced with the dilemma of meeting the ne eds of an increasing number of diverse learners (Walshaw & Anthony, 2008) while en hancing the cognitive knowledge of students. Teaching that is focused on mathematical concepts a nd allows students to wrestle with the mathematics improves student achievement ( Moschkovich, 2012). Moschkovich (2002) noted that learning mathematics is more than just “developing competence in completing procedures, solving word problems, and u sing mathematical reasoning” (p. 192); she further stated that it involves learning sociomathematical norms, such as how to present mathematical arguments, and participate in classroom discussions (Yackel & Cobb, 1996). Through classroom discourse, students can develop a “mathematical disposition” (Walshaw & Anthony, 2008, p. 520), learn ways of th inking about, reflecting on, and clarifying mathematical ideas. The more students ex press their ideas, the more visible

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2 their mathematical reasoning becomes and the easier it is for the teacher to understand what they know “and what they need to learn” (Walsh aw & Anthony, 2008, p. 526). Walshaw and Anthony (2008) suggested that discourse “centered on powerful ideas” (p. 517) can make a difference in students’ learning. H owever; such discourse, while effective, is often challenging to obtain. In parti cular, educators want students to learn the content, while allowing the students’ ideas to guid e the discussion (Sherin, 2002). In other words, teachers must strive to support the pr ocess of the classroom discourse while guiding the students toward an understanding of the content. Reform based teaching, includes discourse that cat ers to the needs of more students, may hold the key to achieving more equita ble results in our mathematics education (Boaler & Staples, 2008; Sherin, 2002; Wa lshaw & Anthony, 2008). However, the transition from a traditional teacher-directed classroom instruction to a reform type student-centered classroom does not come easy and i s demanding on teachers (Sherin, 2002). Teachers must be able to incorporate questio ns, in their instruction, that guide student learning of mathematical content, set expec tations for sociomathematical norms, and help students develop conceptual knowledge (Boa ler & Brodie, 2004; Franke, et al., 2009; Hiebert & Wearne, 1993; Purdum-Cassidy, Nesmi th, Meyer, & Cooper, 2014; Sherin, 2002).When participating in mathematical di scourse, students use a variety of resources; such as gestures, mathematical represent ations, objects, everyday experiences, and language; to help them communicate their unders tanding of mathematical concepts. Meeting the needs of a diverse student population i s a challenge faced by teachers of mathematics. For this study I examined students’ communication with each other and the researcher during problem-based interviews. In particular, I asked the following

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3 research questions: 1) What type of questions did t he researcher ask and how might the type of question afford or constrain how students e xplained their thinking? 2) What resources (such as gestures) or funds of knowledge (Moll, Amanti, Neff, & Gonzalez, 1992) did students use to explain their thinking an d how did they use those resources to communicate their understanding of the mathematics, support their thinking, and justify their responses? 3) In what patterns of communicati on did the researcher and students engage?

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4 CHAPTER II LITERATURE REVIEW Discourse and Reform in Mathematics Education Reform in mathematics education raises the expecta tion for discourse in the classroom (Hiebert & Wearne, 1993). Gee (1991), def ined discourse as “more than just language” (p. 142), it is the “socially accepted…wa ys of using language, of thinking, feeling, believing, valuing, and of acting that can be used to identify oneself as a member of a socially meaningful group” (p. 143). Furthermo re, discourse involves the use of “language and other symbols systems to talk, think, and participate in the practices that lead to…learning” (Moschkovich, 2007, p. 28). Mosch kovich (2002) emphasized that “gestures, artifacts, practices, beliefs, values an d communities” (p. 199) play an important role in the way students, in particular English Lan guage Learners (ELLs), communicate about mathematics. While there have been numerous studies on mathemati cs discourse in the classroom, there has been limited research on secon dary students’ discourse (Huang, Normandia, & Greer, 2005). Furthermore, Huang et al (2005) noted research at the secondary level is typically focused on communicati on as a means of acquiring mathematics content rather than gaining an understa nding of it. They suggested there is a correlation between communication and understanding and argued teachers should be explicit in the use of reasoning in the classroom b ecause students do not automatically pick up on the nuances of mathematical discourse fr om the teacher or classroom discussion. Therefore, systematic integration of ma thematical thinking and talking into

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5 the mathematics curriculum would greatly benefit ma thematics education (Huang et al., 2005). Ideally, an increase in discourse between students and teachers helps students “gain a greater understanding of…mathematics and be come better problem solvers” (Brenner, 1998, p. 154). In an analysis of several articles, Janzen (2008) found that teachers can help ELLs by paying attention to how t hey interact with each other and allowing students to talk about and explain their r easoning as they work through problems. When students participate in mathematical discourse, they are able to find alternative ways of solving problems that also help s the teacher understand what the students know and what gaps they have in their unde rstanding (Janzen, 2008). Mathematical discourse that supports students’ lear ning, is complex and presents new challenges for teachers and students, especiall y ELLs (Brenner, 1998; Sherin, 2002). Although it is challenging, developing this type of classroom contributes to the development and learning of students and teachers ( Sherin, 2002). Research suggests that starting students, particularly ELLs, in small grou ps, leads to increased student participation in large group discussions (Brenner, 1998). In a study of two classrooms using the same curriculum, Brenner (1998) found stu dents who were given explicit guidance on how to participate in small group discu ssions participated more actively in whole-class discussions. Perssinni and Knuth (1998) suggested discourse pla ys a vital role in the mathematics classroom and should be fostered so tha t students treat the speech of their classmates as “thinking mechanisms to be questioned and extended” (pp. 107-108). In a study of a fifth grade Japanese classroom, Wertsch and Toma (1995) found that discourse

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6 in the classroom was either used to deliver accurat e information between the speaker and audience or to invoke thought and discussion. Cobb and Bauersfeld (1995), defined dialogue that invokes thought and discussion as inq uiry mathematics. Knuth and Peressini (2001) described these functions of disco urse as being either univocal or dialogic. Univocal discourse is used to convey exac t meaning, such as when one participant speaks and the other one listens; and d ialogic discourse is defined as discourse that takes place between participants, where one pa rticipant initiates the discourse and a dialogue ensues (Knuth & Peressini, 2001). Reform-b ased classrooms tend to involve more dialogic discourse than univocal. English Language Learners (ELLs) Participation in t he Mathematics Discourse When referring to students as ELLs, I mean “studen ts who participate in multiplelanguage communities” (Moschkovich, 2002, p. 198). ELLs present their own set of challenges for teachers striving to develop more ma thematics based discourse in the classroom. However, when working with ELLs, teacher s should not limit their consideration to the obstacles these students face, they also need to consider the resources they bring to the classroom (Moll et al., 1992). In addition to supporting native English speaking students, classroom instruction needs to support “bilingual students’ engagement in conversations about mathematics that go beyond the translation of vocabulary and in volve students in communicating about mathematical concepts” (Moschkovich, 2002, p. 208). It is important for students to be afforded the opportunity of using mathematica l language, in particular “to pose mathematical questions, describe the solutions to p roblems, explain and justify their solutions, present arguments for or against conject ures, and defend their generalizations”

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7 (Moschkovich, 1999, p. 10). ELLs are supported in t his goal when they are provided opportunities to participate in mathematical discus sions that are rich in mathematical vocabulary and discourse (Moschkovich, 1999 & 2002) Teachers can support ELLs acquisition of English b y using instruction that is rich in mathematical vocabulary (Moschkovich, 2012). Lan guage experts suggest that acquisition of vocabulary in a second language is m ost successful when they are repeatedly exposed to the vocabulary and actively i nvolved in using the language, for example, consistently hearing and using mathematica l terms and vocabulary in a variety of ways throughout the year (Moschkovich, 2012). In their study of two elementary school classrooms, Khisty and Chval (2002) found st udents needed to hear appropriate mathematics before they could use it. They observed that in the beginning of the year students used few words when responding to the teac herÂ’s questions and/or each otherÂ’s comments; towards the end of the year they were spe aking in complete sentences and using correct mathematical vocabulary. This transfo rmation came about because their teacher, Ms. Martinez, modeled the mathematics voca bulary for them and demonstrated by her behavior that she expected the students to p articipate. This follows with SherinÂ’s (2002) observation that when there is an expectatio n for students to share their ideas they are more likely to do so. Working on Challenging Math Problems Students Const ruct Their Own Knowledge As classrooms become more diverse, the role of the mathematics teacher is changing. It is no longer sufficient for teachers t o stand in front of the classroom dispensing knowledge while students sit at their de sks listening, taking notes, and completing worksheets. In reform oriented classroom s teachers support the development

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8 of studentsÂ’ cognitive knowledge by presenting real -world problems that allow students to actively grapple with mathematical problems and construct their own knowledge (Stein, Engle, Smith, & Hughes, 2008). Stein et al. (2008) described this classroom setting as one that starts with the teacher present ing a real-life, challenging math problem, allowing the students to explore the problem while working on finding a solution, and concluding the lesson with a whole-class discussion that involves students presenting and justifying their solutions. During the process of e xploring and solving the problem, students are encouraged to work together, in pairs or small groups; afterwards the teacher guides the students in a whole-class discussions th at leads to an understanding of the mathematical concept being studied. However, this change comes with a price. Reform-or iented mathematics classrooms place new demands on teachers and studen ts, especially ELLs. For example, students work on open problems that can be solved i n different ways, and are expected to discuss their solutions in pairs, small groups, and /or with the whole class. Additionally, there is an expectation they will ask good question s, rephrase problems, provide good explanations for their solutions, be logical in the solutions, justify their work, and consider the answers of others (Boaler & Staples, 2 008). In a five year longitudinal study of three high schools, Boaler and Staples (2008) fo und that Railside, the most ethnically and linguistically diverse of the three schools, wa s the most successful in closing the achievement gap among the students. They found teac hers were successful because they valued the multi-dimensionality of mathematics. At Railside, the multidimensional nature of the mathematics classes led to an increased leve l of student success because students

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9 knew there was more than one way to be successful i n mathematics and they could excel at some of them (Boaler & Staples, 2008). Students’ Mathematical Communication in Small Group s Brenner (1998) suggested small group discussion ca n help ELLs build confidence in their mathematics discourse. Teachers should be aware of the fact that small group discussion needs to be developed among their studen ts. For ELLs whole-class discussion can be somewhat challenging if not daunting. In her observations of two mathematics classrooms, Brenner (1998) noted that the teacher w ho used small group discussions effectively had more student participation in whole -class discussions that were rich in mathematical content. Another benefit of small groups is that ELLs are a ble to enjoy and participate in social interactions with other students (Brenner, 1 998). ELLs should also be given the opportunity to participate in large group instructi on, because they are able to gain access to mathematical terms and vocabulary they may not g et working with their “peers who are also developing their second language skills” ( Brenner, 1998, p. 158). Student participation in group discussion benefits both the students and the teachers by providing students the opportunity to achieve new goals that are in line with reform mathematics and allowing teachers to assess students’ understan ding of the lesson content (Brenner, 1998). Perhaps, small group discussion provides the great est benefit to ELLs because it provides them the opportunity to discuss mathematic s in a smaller setting prior to discussing it in a larger group. Brenner (1998) sug gested language minority students often have “more sophisticated conversation” (p. 17 0) when working in small groups that

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10 can eventually lead to similar discussions in large r groups. When denied the opportunity to converse in small groups, ELLs are less likely t o participate in large group discourse. Boaler and Staples (2008) found that when students are held responsible for each otherÂ’s learning they tend to work more effectively in groups. Blunk (1998) described how one teacher, Lampert, worked with her students to establish effective group work habits. In the beginning of the year Lampert explic itly told her students what small groups were and why they would be working in groups Furthermore, she talked about her expectations; in particular, their responsibili ty for their own behavior, how they were to help each other, and how they could only ask for the teachers help when everyone had the same question (Blunk, 1998). Throughout the yea r, Lampert continued to support the studentsÂ’ work in small groups by recognizing them when they were working well as a group and correcting groups when they were not work ing well together. Students in LampertÂ’s class became successful working collabora tively in small groups because she taught them and they understood what was expected ( Blunk, 1998). Types and Level of Teacher Questions Shape the Math ematics Classroom The types and level of questions teachers ask shap es the mathematics classroom (Boaler & Brodie, 2004), impacts student engagement (Kazemi & Stipek, 2001), and influences student learning (Purdum-Cassidy, et al, 2014). Effective teacher questions lead to deeper mathematical thinking and reasoning within students (Purdum-Cassidy, et al, 2014) and when teachers ask important questions students ask important questions (Boaler & Brodie, 2004). Therefore, if teachers wan t their students to ask more conceptual, probing, or exploring type questions, t hen teachers need to ask more conceptual, probing, and exploring type questions; if they want students to extend their

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11 thinking, teachers need to ask questions that exten d students’ thinking (Boaler & Brodie, 2004). In a reform-oriented classroom, questions are used to elicit information about students’ thinking, encourage dialogue, and help st udents build conceptual knowledge (Purdum-Cassidy, et al, 2014). It is important for teachers of mathematics to develop a repertoire of questions that push students’ cogniti ve development of mathematics and hold them accountable for rigorous, disciplined way s of communicating their thinking (Smith & Stein, 2011). Questions that elicit high e xpectations of conceptual thinking in students; permits mathematics to drive student enga gement (Kazemi & Stipek, 2001), helps students discover how to reason mathematicall y, and realize they can make sense of mathematics (Smith & Stein, 2011). Effective teacher questions are powerful instruct ional tools and have been recognized as a “critical and challenging part of t eachers’ work” (Boaler & Brodie, 2004, p. 774). The questions teachers ask play an importa nt role in guiding students as they navigate “the mathematical terrain of lessons” (Boa ler & Brodie, 2004, p. 781). It is easy for teachers to ask students to describe their stra tegy for solving a problem; “it is more challenging…to engage students in genuine mathemati cal inquiry and push them to go beyond what might come easily for them” (Kazemi & S tipek, 2001, p. 123). In a study of three separate schools, Boaler and B rodie (2004) developed a method of coding teacher questions to inform their study. They developed nine categories of teacher questions that came from a study of diff erent examples of teacher instruction (Boaler & Brodie, 2004). The nine categories were: 1) gathering information, leading students through a method, 2) inserting terminology 3) exploring mathematical meanings

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12 and/or relationships, 4) probing, getting students to explain their thinking, 5) generating discussion, 6) linking and applying, 7) extending t hinking, 8) orienting and focusing, and 9) establishing context (Boaler & Brodie, 2004, p. 777). Teachers can use these types of questions during instruction to extend student thin king and deepen their conceptual knowledge of mathematics. Gestures Convey More than Words Alone Can Not only do gestures help convey the meaning of a speaker’s word, but gesturing reduces the demands placed on the “speaker’s cognit ive resources” (Goldin-Meadow, Nusbaum, Kelly, & Wagner, 2001). Gestures can be a reflection of a student’s problem solving strategies as well as completing the pictur e of their verbal explanations (Bjuland, Cestari, & Borgersen, 2008; Goldin-Meadow, et al, 2 001). According to Radford, Berdini and Sabena (2007), gestures provide students with a “visual and sensory-motor representation” (p. 526) of their explanations and helps them imagine it. Furthermore, gesturing helps students visualize what they are ta lking about, reduces the cognitive load of explaining, improves memory, and helps students to organize information (GoldinMeadow, et al, 2001; Radford, et al, 2007). Through the use of purposeful gestures, teachers c an give meaningful instruction (Yoon, Thomas, & Dreyfus, 2011). Goldin-Meadow, Kim and Singer (1999) suggested that gestures can guide the students’ attention to certain aspects of a problem or present another representation of a concept. For example, d uring a lesson on perimeters, a teacher could describe what the perimeter is while tracing the outside of a rectangle with her finger. Yoon, Thomas, and Dreyfus (2011) suggested that teachers use the following two strategies to help students in their study of mathe matics: first, provide students with the

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13 space necessary to use gestures, and second increas e studentsÂ’ awareness of how to use them to elaborate on their thinking. By using gestu res in their instruction, teachers can support studentsÂ’ learning and help them develop th e use of gestures in their own communication.

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14 CHAPTER III METHOD Data Collection The data for this study came from research conduct ed by Heather Johnson, a professor and researcher at University of Colorado Denver. As part of my own study, I assisted in the collection of data. The study was c onducted in an urban middle school in the western United States and consisted of six less ons taught to four sections of 7th grade students. The lessons used task situations involvin g a changing rectangle to support studentsÂ’ reasoning about covarying quantities (Joh nson, 2013). From the four sections, 14 students were chosen to participate in a follow up interview with Johnson. The students were chosen because they frequently commun icated their mathematical ideas during whole group lessons in the classroom. Students were paired with another student from thei r section. When more than two students were from the same section, Johnson co nsulted with the teacher to determine which students would work best together. The student pairs were Sergio & Rico (S&R), Tomas & Simon (T&S), Jorge & Tien (J&T) Navarro & Daria (N&D), Blanca & Crista (B&C), Myra & Elena (M&E), and Oliv ia & Sierra (O&S). The names of the students, used in this study, are pseudonyms Interviews were conducted with pairs of students at the school and were recorded. At this school, 46% of the student population had n ever exited the ELL program (i.e., they are not proficient in English), 18% had been r e-designated from one level to another (or exited ELL), and 36% have never been in ELL.

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During the interviews the students were give was a continuation of the relationship between the increasing height of a rec tangle and the volume of the rectangle (Figure 1). During the first task, s EFGH changed as the length of side EH increased. Fo r the second task students were asked to compare the relationship between the area of ABCD and the length of si (Figure 2). F or more details regarding the tasks see Johnson (20 13). Figure 1. Areas of a filling rectangle During the interviews the students were give n two separate tasks lessons taught during the sections, and examined the relationship between the increasing height of a rec tangle and the volume of the rectangle During the first task, s tudents were asked to explain how the area of recta ngle EFGH changed as the length of side EH increased. Fo r the second task students were asked to compare the relationship between the area of ABCD and the length of si or more details regarding the tasks see Johnson (20 13). filling rectangle 15 The first task examined the relationship between the increasing height of a rec tangle and the volume of the rectangle tudents were asked to explain how the area of recta ngle EFGH changed as the length of side EH increased. Fo r the second task students were asked to compare the relationship between the area of ABCD and the length of si de AD

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16 Figure 2. Areas of a filling triangle Data Analysis Following the interviews, the video recordings were transcribed by myself and the researcher. To ensure credibility and accuracy of t he transcriptions, each transcript was reviewed by an additional researcher (Patton, 1999) For the purposes of this study, I reviewed the vide o recordings and transcripts with the intent of examining the questions the rese archer asked and how the students responded to the researcher and each other. Additio nally, I made note of how the students used gestures, mathematical representations, manipu latives, and language to communicate about quantity and covariation, in part icular the relationship between the side length of AD and the area of ABCD, and how the y supported and justified their understanding. Even though I reviewed both the rect angle and triangle recordings and transcripts, I chose to focus on and use the result s of the triangle interviews because it was a new problem for the students.

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17 After an initial review of the transcripts and vid eos, I developed four types of questions that I felt were used in the interviews a nd discussed them with Johnson. The four question types were; 1) an overarching questio n meant to focus students on quantity and covariation, 2) a question related to the overa rching question, but more specific, 3) a question that engaged the other student in the conv ersation, and 4) a question that redirected the conversation. These types were chose n based on the anticipation that these would be the types of questions to come up during t he interviews. Table 1 shows examples of the four question types. In the discuss ion that follows I will describe how I coded questions as type 1, 2, 3, or 4. Table 1. Examples of four question types n r !!n"#$ %& n '()*+ (+,#-%+'n! '(+'()*+' (+!! ,#%. / 0! #'()*+ (+ ,12,#3n%1!!,"""'n """!!, #)%* 4 1n #'.'()*+5' '"6#.% 1n& ! n,#$%& 7 n2n'''()*+,82 n! ,#-%+

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18 After a quick review of the transcripts, Johnson an d I decided that I should start by reviewing and highlighting the rectangle transcr ipt of Myra & Elena, because not only did they talk with the researcher, but they talked with each other as well. The review consisted of highlighting each question the researc her asked and determining where each question group started and ended. In deciding what counted as a question, I followed a pattern similar to Boaler & Brodie (2004). In particular, I did not include utt erances in the form of a question (for example, “would you like to see the graph?”) as que stions. However, utterances that were meant to solicit an answer (for example “then talk to me about what changes and what stays the same”) were counted as a question. Initially, I looked at the questions as groups. A g roup consisted of questions that were related to each other either because they were related to the same overarching question, followed a students’ chain of thought, or refocused the students’ thinking. For example, in the following dialogue the question gro up starts with the researcher asking “what changes and what stays the same?” A new quest ion group starts with the researcher asking about a graph for the filling rectangle. Researcher: What changes and what stays the same? [Start of a question group] Elena: Okay, what changes is the length. Myra: Of EH and the area of EFGH and what stays the same is the length of EF. Elena: F, yeah the base. Researcher: Okay, what if we change the side length of EF, like make it really small. Elena: Then EF still says the same but the, um, are a [Sweeps palm of right hand in front of computer screen.] is smaller, because each of the side lengths, each of the, H and E side length gets bigger and EF stays the same. Researcher: And, what if I make EF really big?

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19 Myra: Um, the length of EF will change and how Elen a said, Researcher: Wait, the length of EF? Did you mean th e length of EH? Myra: EF would change [Moves pencil back and forth along EF shown on the computer screen.] I mean would not change, and the length of EH wil l [Emphasizes ‘will’ when speaking] change, and also the area will change. It may get bigger or smaller. Start of new question group Researcher: And if I were to press the show graph s o that graphs would appear, and if you would compare the graph of when EF is really lo ng to when EF is really short, do you have any sense of how the show graphs would compare? Like in the example above, new groups often started when Johnson asked the students an overarching (type 1) question. Occasion ally, a new group started when the students were asked a question that redirected the conversation (type 4). After highlighting the questions and identifying qu estion groups, I started coding the questions by type. In deciding how to classify a question, first I asked myself if it was an overarching question, a derivative thereof, or r elated to an overarching question, but more specific. The overarching questions (type 1) were based on th e researcher’s interview questions. Type 1 questions were one of the followi ng or a derivative thereof; a) what do you anticipate will happen when you click animate p oint, b) how is the area of ABCD changing as the length of AD is increasing, or c) s uppose you created a graph relating area ABCD and the length of AD, what do you expect the graph to be like. For example, one of the questions asked of Olivia & Sierra, “as the length of AD gets larger, how does the area of ABCD change?” was coded a type 1, becau se it was a derivative of the

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20 overarching question “how is the area of ABCD chang ing as the length of AD is increasing?” Questions related to the overarching question, but more specific (type 2), were also based on Johnson’s interview questions, but co ntained more detail. For example, she asked Myra (referring to the aforementioned overarc hing question), “Could you just describe how the area gets bigger? Like, does it ge t bigger just like the area of the rectangle got bigger?” was coded type 2 because the researcher was more specific by relating it to the rectangle problem. Questions were coded type 3 when the researcher ask ed a question of a specific student and the dialogue prior to the question show ed that the student had not been participating in the discussion. For example, at th e start of her interview with Jorge & Tien, Jorge answered the questions and Tien made on ly a small remark until the researcher asked what she is thinking. Researcher: Instead of a filling rectangle, it's g oing to be a filling triangle. And so what I'd like one of you to do is just go ahead and press animate point, and then talk to me about what changes and what stays t he same. [ Jorge presses touchpad, ] And then anywhere on the touch pad. Tien: Oh Jorge: I wanna say that these two points are getti ng smaller as it goes up to like an actual triangle. Researcher: And which points are these two again? You can use the letters for them, just so I'm sure I know which ones you mean. Jorge: I can’t see it. [ Researcher moves computer closer. ] D and C Researcher: Okay Jorge: They're going to get like, well, I can see as you animate the point, they're going to get smaller and smaller. It gets smaller a s it gets to the point.

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21 Researcher: What do you notice Tien about what changes and what stays the same? I coded the last question in this example as a type 3, because the researcher specifically asked Tien a question and she had not been participating in the dialogue. A type 4 question was different from a type 3 in th at generally, either one or both of, the students’ attention had drifted from the cu rrent objective and the researcher asked a question to bring the discussion back to the inte nded topic. For example, in the excerpt below Crista and Blanca are trying to calculate the area of ABCD, then the researcher attempts to redirect their attention with the final question. Crista: [continues to calculate the area of ABCD] Blanca: So see how that's [points to AB] like the same length as right there [points to DC] ? Um, I think it stays the same, cuz no matter what you're going to have that same length, but, you're gon – you're g onna' divide it by two. Researcher: Ok. So is shape ABCD, is that a triangle? [Type 4 question] As I analyzed the transcripts I kept a record of th e question type in a table, which included a column for each student pair, the type o f question asked, follow up questions and notes. Following my review of Myra & Elena’s rectangle tra nscript, I reviewed, highlighted, grouped, and coded questions in the re ctangle transcripts of Tomas & Simon, Jorge & Tien, Blanca & Crista, and Navarro & Daria respectively. At this point I left my work with the rectangle transcripts and began highl ighting and coding the triangle transcripts. After completing the triangle transcri pts, I went back to highlighting and coding the rectangle transcripts of Olivia & Sierra and Sergio & Rico. While working with the triangle transcripts, I foun d several instances where the questions did not fit any of the four types. In par ticular, the researcher was probing

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22 students for an explanation of their responses and thinking (Boaler & Brodie, 2004; Smith & Stein, 2011). Therefore, I added a fifth ty pe (table 2), a question that was meant to push the students thinking and explanation. Noti ce in the following example how the researcher follows SierraÂ’s response with a questio n that is pushing her to elaborate on her thinking (type 5). Researcher: What do you think Sierra? How does the area increase? Is it like the rectangle? If you had to describe it, what's the i ncrease like? Sierra: I don't think it's going to be like the re ctangle. Researcher: Okay. Why not? [Type 5 question] Table 2. Examples of type 5 questions n r 9 1n,#)%* nn ,*n! ,#3n%0!,*n ,#.% During my analysis of the transcripts, I noticed st udents using gestures, such as moving their hands and pointing to the computer scr een or handout, as they responded to questions, and explained their answers. This led me to ask questions about the gestures, such as how often were students using gestures, wer e all the students using them, how were they being used, etc. (Corbin & Strauss, 2008) Based on my analysis of the video recordings and transcripts, I identified three type s of gestures (Table 3). In particular; a) pointing at the screen, paper, or etc.; b) illustra ting a motion such as pinching fingers together to represent the filling of a triangle, ra ising arm at an angle, or etc.; and c) nodding in agreement.

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23 Table 3. Examples of first three gestures types :&n r ; <*! <$!! n=82'(+<&' '()*+ <&' '!n!' 3! < > <-= r= n! r'
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24 After coding the transcripts, I began examining the data for connections between the researcher’s questions and student responses. I used constant comparison analysis (Corbin & Strauss, 2008) to look for similarities a nd differences. For example, I noticed when Johnson asked the overarching question “how is the area of ABCD changing as the length of Ad is increasing?” each of the student pa irs in the same way (a similarity), they said it was increasing. Then when she followed up w ith “how does the area increase?” their responses became diverse (a difference). Oliv ia, for example, did not know how to respond while Sergio said it will go up “by a small amount.” To address the three research questions, interview recordings and transcripts were analyzed by comparing the types of questions asked by the researcher to the student participation, and gestures. Additionally, I examin ed how often students communicated directly with the researcher (student-to-researcher ) and compared it to their communication with each other (student-to-student). My intent in analyzing the studentto-researcher (S-R) and student-to-student (S-S) di scourse was to consider whether or not S-S discourse served as a resource in student disco urse. The data from the interviews and transcripts were reviewed multiple times to find co mmon themes and patterns in the discourse between the students and researcher. The results of my analysis follow in the next chapter.

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25 CHAPTER IV RESULTS Questions Guide Student Participation In the interviews, Johnson began with overarching q uestions then guided studentsÂ’ participation with effective follow up qu estions. Table 5 shows the five types of questions, how many of each type were asked of each student pair, and the percentage of each. The totals show how many questions student pa irs were asked and how many questions of each type Johnson asked overall. Table 5. Number and percentage of question types as ked each student pair In the interviews Johnson asked 102 type 5 question s (table 5). That equates to 41% of the questions (table 5) being higher order q uestions that encourage students to think more deeply and explain their thinking in gre ater detail. The type 1 and type 2 questions guided student thinking and participation and accounted for another 40% of the questions. Lastly, type 3 and type 4 questions dire cted the student discourse and accounted for 19% of the questions. &n&n/&n4&n7&n9& @A@A@A@A@A@A )%*7B"CAD44"4A7"/A9/>"DAB/9">A/7E"CA$%&7D"EA>//"/AB4"4A9"A/>77"7A79D"A3n%4B"9A>/"CA/7"4AB4">A/997"4A7BD"9A-%+4>"4A/7"7A4>"4A74"DAC/7"A/E" CA %.7C"9A/4E"BA7C"9A"EA/474"7A94/"7A.%77"DA7>"CA77"DA/C"7AB//"/A/C>"EA&%.7B"CA/D"4A4/"9A>>">A9B/"9A/7E"CA &/B>"9AC7/E"DA/4E"4A/4E"4A>/7"A/7D

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26 Using type 5 questions, not only was Johnson able t o push students to elaborate on their thinking but she was also able to help stu dents be specific in their responses. Notice in the following example how Johnson uses a question to get the student to specify what “it” is. Researcher: And now let’s imagine how the area incr eases as the length of AD increases. So maybe let’s just start with a questio n here. If you press animate point, can you tell me how the area of ABCD changes as AD increases? Simon: It increases, but as it gets bigger and bigg er then it starts increasing more slowly because there’s not as much um, room I guess Tomas: Like that [Points to the white space near the top of the tria ngle] it goes like that and then it starts slowing down. Researcher: What does it mean to increase more slowly? Simon: Um, that it has a smaller rate of change, th e rectthe go up, ah—as it gets bigger because uh, when you press animate point it goes like, wait for it still. Tomas: It goes slowly right there. [Points to the triangle as it is reaching the fille d amount.] Simon: It goes one, two, three, four, five, six, se ven-, eight— nine— ten—, and then it just starts to increase, increasing by less each time. Researcher: And when you’re saying it’s the it—it goes slowly. Can you tell me what you’re looking at? What’s it? [type 5 question] Simon: The area. In the above dialogue, Johnson used the type 5 ques tion to push Simon to be specific and use mathematical language in sharing h is thinking about the increase in area ABCD. Even if she understood what he meant by “it,” she expected him to use mathematical language to specify that “it” meant “t he area.” This example demonstrates

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27 one of the ways teachers can use questions to push students to use mathematical language and be specific in the thinking. Throughout the interviews Johnson used student resp onses to guide her use of follow on questions during the interview and adjust her questions for the following interviews. For example, in the dialogue below John son asked Sergio & Rico, the first pair to be interviewed, what they expected to happe n when she pressed animate point (type 1). Rico responded first by trying to remembe r how to find the area of a triangle, then he identified the shape as a trapezoid. Follow ing his response, the researcher rephrased the question and asked, “as the length of AD changed, what do you think would happen to the area?” Researcher: If I click on the show rectangle, you can see how we are just filling this part of the rectangle, rather than the whole rectan gle like we were filling before. So if I press animate point, what do you ex pect is going to happen? Rico: Um, as I know with a triangle, you have to do um wait don’t you have to do, I’m not sure, base times height divided by two. And that looks like a trapezoid and if that was increasing, you’d have to do base one plus base two divided by two or times two I’m not sure, so th is would be more complicated. Researcher So if we were actually finding the amoun t of area…it would be more complicated. And I'm thinking, let’s not worry so m uch how we would actually find it using a formula. But if I press a nimate point…that’s going to change the length of AD, and so as the length of AD changed, what do you think would happen to the area? Sergio: You know, it will increase, but, well, by a small amount. In the rest of the interviews, the researcher asked each pair about what would change and what would stay the same, as can be seen in the following excerpt from the second pair to be interviewed, Tomas & Simon.

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28 Researcher: Okay, I’m going to have us look at a s ituation that is no longer a filling rectangle. So instead of a filling rectangle we are going to have a filling triangle. And so what I want one of you to do is ju st press animate point. And, let’s talk about what changes, what stays the same Simon: Um, the area changes and the length of AD ch anges and uh, but uh, I guess nothing technically stays the same because AB also changes and I guess because the right triangle so does the hypote nuse. Researcher: What do you think Tomas? Tomas: I notice that line segment DC changes. It be comes more narrower, more narrow, as it gets to the top. The area has to chan ge, the area because it’s not constant, it’s not one equal shape, so that wou ld change the area. By changing the wording of this question the studen ts’ responses became less procedural in nature and more conceptual. For examp le, in the excerpts above Rico’s initial response was procedural (he wanted to calcu late the area of ABCD); whereas, Simon and Tomas answered by noting the changes in t he area and side lengths (conceptual). Johnson used questions to redirect students’ thinki ng (type 4) from procedural to conceptual thinking. In the following example, noti ce how Myra & Elena begin by explaining the relationship between the length AD a nd the area ABCD procedurally. Johnson uses a combination of type 2 and 4 question s that guide the students to delve further into a conceptual understanding of covariat ion. Researcher: So, yeah, so if I press animate point, what's the increase in the area of ABCD like? [type 2 question] Myra: Um— Elena: What is the increase like? Researcher: Yeah. Elena: Well, you just multiply the length of AD [ points to Length of AD ] and the length [ pointing toward Length of AB ].

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29 Researcher: Mhm. And if we— Elena: Wait, doesn't it just increase by [ laughs ]— Researcher: Well, and if we don't worry so much ab out, I purposely picked a shape that is really hard to calculate area, and I did that pu rposely, and if we don't worry so much about how you actually find the area, if you just think about— [researcher uses an utterance to redirect the stude nts’ thinking] Myra: Base one plus base two times height divided by two. Researcher: Mhm. and so if we just think about how does this area [ motions fingers over shaded region of filling triangle ] change? How does it increase as AD gets bigger? What's the increase like? [researcher uses type 4 question first to redirect students’ thinking, then a type 2 followed by another type 2 quesiton] Myra: Oh, I get you now! Uh [ laughs ], the, how does it increase? Like, it's getting smaller and smaller every time [ motions hands upward creating the top of a triangle ]. In the above example, Johnson was able to help the students’ move from thinking procedurally about the problem to thinking about it in a deeper conceptual way. As Johnson continues to push Myra & Elena to explain t heir thinking, both students demonstrate attention to quantity and covariation a s can be seen in the following excerpt. Researcher: So when you say it's getting smaller a nd smaller every time, do you mean there's less— Myra: The area. [ said over Researcher ] Researcher: The area's getting smaller? Myra: The area's getting…it's getting biggerElena: Bigger, I think it's getting bigger. [ Said as Myra was talking ] Myra: Yes, the area's getting bigger, but how much it increases, it's getting smaller.

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30 Another consideration was how often each student re sponded to the researcherÂ’s questions and how the questions may have affected s tudent participation. Table 6 shows the number of times each student replied to the res earcherÂ’s questions. It also shows the percentage of the studentsÂ’ replies based on the nu mber of questions Johnson asked each pair. Students are listed by pairs, the pair with t he least responses is listed first. Table 6. Number and percentage of student replies t o researcherÂ’s questions ;' & ; / 4"BA 4D . /B BD"7A 4D /7 B>">A 7> & & B 7>">A 7> + 4> 97"9A 99 /9 79"9A 99 + $ 44 9/"7A B4 & & 4> 7C"BA B4 $ 49 9>">A C> . 49 9>">A C> ) 9 9/"BA EC * 7B 7C"7A EC ) 9E 9/"/A 4 3n 3n 97 7C"DA 4 Interestingly, except for two pairs, Sergio & Rico, and Tomas & Simon, there was less than a 10% difference in the number of times s tudent pairs replied to the researcherÂ’s questions. On average, one student, from each pair, responded 56.7% of the time and the other student responded to 43.3% of the questions. The pairs with the lowest percentage of questions t o engage another student (type 3), also had the smallest difference in their parti cipation. On the other hand the pair with the largest difference in participation, Sergio & R ico, also had the highest percentage of type 3 questions (table 7). This led me to believe that there was a correlation between the

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31 level of participation and the number of type 3 que stions the researcher asked. Upon further examination, I found that in five of the in terviews the researcher asked more type 3 questions of the student who participated the mos t. Table 7 shows the percentage of student participation (percent of responses) and th e number of type 3 questions that researcher asked the pair (percent of type 3 questi ons). The last column of the table shows how many type 3 questions the student was ask ed. The students are listed by pairs, the pair with largest level of difference in percen tage of responses is listed first. Table 7. Percentage of student responses vs. number of type 3 questions researcher asked Further analysis of the type 3 questions and studen t responses shows a correlation between how many type 3 questions students were ask ed and how often they responded. Johnson used type 3 questions to directly engage a student in conversation. Students asked three or four type 3 questions responded more frequently (a mean of 55 responses), than did students asked zero, one or two type 3 que stions (a mean of 46 responses). ; ;' ;' n 4 n4 BDA 7"4A 4 4/A 7"4A & B>A /">A 4 & 7>A /">A > $ & 9/A 4">A 7 & $ 7CA 4">A / + 79A >">A > + 99A >">A 4 ) 94A 7">A ) 7CA 7">A > 3n 7DA 7"4A / 3n 9/A 7"4A > 9>A C"7A 9>A C"7A 4

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32 Figure 3 shows the number of type 3 questions stude nts were asked as it related to the number of times the student responded during th e interview. Students who were asked zero type 3 questions responded an average of 46 ti mes. Figure 3. Number of student responses vs. number of type 3 questions I found that JohnsonÂ’s use of questions, in particu lar type 1, 2, and 5 questions, effectively assisted students as they navigated thr ough a mathematical task that was new to them. She used type 1 and 2 questions to set the stage and help students explore this new concept. Furthermore, type 5 questions were uti lized to probe studentsÂ’ conceptual knowledge and solidify their understanding. Through the effective use of questioning Johnson was able to guide student responses, help t hem stay focused on the objective, and elicit a deeper level of thinking from them. Student Use of Gestures In the interviews, Johnson encouraged students to u se manipulatives and gestures to explain their reasoning. For example, when Tien explained that the area of ABCD will nnnrn nnnnrn

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33 grow just a little bit at a time, Johnson suggested that Tien use the computer animation to explain what she meant. Each of the students in this study used gestures in some form. Overall Elena, Olivia, Tien, and Crista had the highest use of ges tures, respectively. Table 8 shows how often each student used gestures during their inter view. The total column shows how many gestures were recorded for each student and to tal row shows how often each gesture type was used. Students are listed from the fewest number of gestures used to the greatest. Table 8. Student gestures by individuals B > 7 9 B B D / > 7 /> ) 9 4 C 7 /> 9 > 9 / 7 /B + B 4 > E > /D 3n / 9 > C D 4/ D 9 4 B 44 $ > 7 > C 4 47 & E / > D 4> E 7 / 9 B 7B /E / B 9 94 & / /7 4 C 4 9D /B /B 7 C / B9 4 /4 C 4 C9 &E74E/4>CCE97/ Elena had the highest (75) recorded use of gestures while Simon had the lowest with 16. Overall, the most frequently used gestures were pointing (type a), illustrating a motion such as pinching fingers or raising an arm a t an angle (type b) were second, and using manipulatives, tracing an object, sketching, or drawing (type d) were third; nodding (type c) and others (type e) were used the least.

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34 Student-to-Student vs. Student-to-Researcher Commun ication I analyzed the amount of student-to-student (S-S) v s student-to-researcher (S-R) discourse. Table 9 shows the number of times studen ts directed their conversation to each other (S-S) and how often they spoke to the researc her (S-R). The total column shows how often the students spoke during the interview, while the total row shows the number of times remarks were directed at the researcher (S -R) and how often they were directed to the other student (S-S). Students are listed in pairs. Table 9. Student-researcher vs. student-student dis course .F .F. & ) 7B 7C 7/ 74 $ 4 / 44 & /C > /C 3n 7> 7 99 / 9C 47 4 4C + // 7 /B 49 > 49 49 > 49 /B > /B B > B & 9 / C /7 /9 & 77D C 7B9 While Navarro & Daria had the most student to stude nt discourse (11.1%), 5 of their S-S exchanges occurred in the first minute of the interview, they were arguing over who would press the “animate” button (Daria finally “won” by picking up Navarro’s hand and placing his finger on the touchpad). Durin g their subsequent S-S communication they were seeking assistance from eac h other. In the first one, Navarro asked Daria “what happens?” and during the second one Daria turned to Navarro for

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35 reassurance that she had the correct formula for ca lculating the area of ABCD. Other, SS exchanges typically involved one student explaini ng something to the other one or completing sentences. For example, when Myra was st ruggling to understand how the area of ABCD was changing, Elena compared it to pre -algebra when they “found the area of the whole shape” and then continued to explain f inding the area for a triangle. The vast majority or the discourse, 96.3%, was betw een a student and the researcher. Although it may not be the only reason, it is probable that the design of Johnson’s study caused students to direct most of t heir conversation to the researcher. In particular, Johnson had designed the interviews to solicit individual students’ reasoning related to quantity and covariation, therefore her questions elicited more S-R communication than S-S communication.

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36 CHAPTER V DISCUSSION To promote students’ mathematical reasoning, teac hers are being encouraged to move classroom practices away from teaching computa tional accuracy to focusing on understanding “mathematical ideas, relations, and c oncepts” (Kazemi & Stipek, 2001, p. 59). This focus is challenging pedagogically and de mands that teachers engage their students in mathematical inquiry that pushes studen ts to “go beyond what might come easily to them” (Kazemi & Stipek, 2001, p. 59). Ask ing good questions is one of the ways teachers expand students reasoning and thinkin g, because it affords students the opportunity to take ownership of their own learning (Purdum-Cassidy, et al, 2014). The questions teachers ask not only shape the mathe matical terrain of the instruction but, guides students as they navigate t hrough it (Boaler & Brodie, 2004). In comparing the question types used in this study to Boaler and Brodie’s (2004) nine types of questions, I found a correlation between four of the types. In particular, the overarching questions (type 1) is closely related t o Boaler and Brodie’s (2004) type 3, “Exploring mathematical meanings and/or relationshi ps” (p. 777). In the interviews, Johnson used type 1 questions to support students’ exploration of covarying quantities involved in the task. This same thing is accomplish ed in the mathematics classroom when teachers design and use questions that cause studen ts to explore the mathematical relationships and meanings of concepts being taught (Boaler & Brodie, 2004). The type 2 questions in this study correlated to Bo aler and Brodie’s (2004) type 8 question, “Orienting and focusing” (p. 777), in tha t they were more specific than type 1, because they were more specific they helped student s focus on key elements. For

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37 example, if students did not say anything about the area of ABCD or the length of AD changing, when asked the first overarching question “what changed and what stayed the same?”, Johnson was able to orient their focus by s pecifically asking about the area of ABCD or the length of AD. By focusing their attenti on on the specifics, students were able to make connections between the mathematical r epresentation (Boaler & Brodie, 2004) and the concept of covariation. Questions that engaged another student (type 3) rel ated directly with Boaler and Brodie’s (2004) type 5 question, “generating discus sion” (p. 777), in that it solicited contributions from the other student. Stein and Smi th (2011) recommended using these types of questions in conjunction with other teache r moves (such as revoicing, restating another’s reasoning, or comparing their own thinkin g to someone else’s) to guide mathematical discourse and student understanding. B ased on the results of this study, this type of question is more effective when used more t han once with a student and suggests two to three questions to be most effective in gene rating student participation. Type 5 questions pushed students to elaborate on th eir thinking through probing, and correlated to Boaler and Brodie’s (2004) type 4 question, “probing, getting students to explain their thinking” (p. 777). In this study Johnson used type 5 questions to get students to be explicit in their explanations, expl ain their thinking, and delve deeper into their understanding of covariation. For example, wh en Sergio used “it” for the area Johnson asked what “it” was and when students told Johnson that the area of ABCD was increasing more slowly, she asked them to explain w hat more slowly meant. Incorporating questions that press students to elab orate their thinking can be challenging,

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38 because a teacher usually has many options at this point and it is uncertain where the students will go with the question (Boaler & Brodie 2004). I did not find a correlation between the type 4 (a question that redirected the conversation) in this study to Boaler and BrodieÂ’s (2004) question types. The lack of correlation suggests that type 4 questions represen t another type of question teacherÂ’s use in the classroom. One way Johnson utilized type 4 q uestions was to redirect the students from thinking procedurally (for instance, calculati ng the area of ABCD) to thinking conceptually (noticing the relationship between the increase in the length AD and the area of ABCD). As suggested by JohnsonÂ’s use, type 4 questions may support students shift from procedural to conceptual thinking. Limitations My initial intent for this study was to examine th e affect ELL status had on the level of student participation. However, the ELL st atus of each student was not available and limited the scope of this study to examining th e student participation in general. The United States is becoming more diverse and so are i ts classrooms. More students are learning English at the same time they are studying mathematics. Therefore, understanding how ELLs communicate their comprehens ion through verbal and nonverbal means is becoming increasingly important to teachers. Due to the nature of the research conducted for thi s study, I was unable to contrast the nature of student to researcher (S-R) vs. stude nt to student (S-S) communication. As noted in the previous chapter the original research was designed to solicit more S-R communication; therefore, there was very little S-S communication. When the students did engage in S-S communication, it was to solicit support and/or reassurance.

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39 Another limitation was the small sample size. Whil e I was able to examine the discourse within this small group it may not be rep resentative of the larger student population. Further study of student interactions a mong themselves and with teachers, how they use gestures, and the impact of teacher qu estions on student participation will benefit teachers and educators. Implications An issue for further research would be to study th e correlation between a studentÂ’s ELL status, teacher questions, and how st udents communicate in response to the questions. Research of this nature would serve to p repare teachers for the continuing increase in the diversity of the student population Furthermore, as classrooms move towards reform-oriented teaching there will be a gr eater need to understand how questions affect ELLs learning. Additionally, research on how gestures are affecte d by gender and culture would benefit educators. Questions for further research i nclude, do students of one culture use gestures more than another? Is there a difference i n how girls and boys use gestures? Does the way students are grouped change how they u se gestures (i.e. girl-girl, girl-boy, or boy-boy)? In JohnsonÂ’s study the students tended to communic ate more with her than with each other. Was this a result of the nature and des ign of her study? In this study, the students were sitting next to each other, facing th e researcher, with the laptop between the students and Johnson. Does the way a teacher sets u p a classroom impact the amount of student-to-teacher (S-T) vs. S-S discourse? Also, t he questions were designed to elicit individual student responses, hence did this affect the amount of S-T vs. S-S discourse?

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40 Does the nature and design of teacher questions cha nge the way students interact with each other and the teacher? Lastly, the potential link between the types of qu estions asked of students (e.g., questions to engage another student (type 3)) and t heir level of participation (e.g., number of responses), merits further study. Students in th is study who were asked three or four type 3 questions responded more than students who w ere asked zero, one, or two type 3 questions. Continued study of the correlation betwe en the number and type of questions asked to students and their subsequent participatio n could benefit mathematics teachers of mathematics. This study was conducted in a small group setting with two students. Might a similar correlation be present within a lar ger group of three or more students or with students participating in a whole-class discus sion? Concluding Remarks The landscape of the mathematics classroom in the United States is changing, and a greater emphasis is being placed on students obta ining a conceptual understanding of mathematics (National Governors Association for Bes t Practices Council of Chief State School Officers, 2010). This transition, from simpl y learning procedures to exploring mathematical concepts, is challenging for both teac hers and students. Questions and gestures play an important role in student interact ion during classroom discourse, whether it be whole-class, small groups, or a pair of stude nts. In a small way this study captured how teacher que stions can guide studentÂ’s navigation of mathematical terrain that is new and unfamiliar. It showed how gestures can play an important role as students communicate their understanding of mathematics.

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41 However, additional research is needed to understan d how this reform in mathematics education is affecting the increasing number of div erse learners in the United States.

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