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Petkova, Mariana M.
Classroom discourse and Teacher talk influences on English language learner students' mathematics experiences
h [electronic resource] /
by Mariana M. Petkova.
[Tampa, Fla] :
b University of South Florida,
Title from PDF of title page.
Document formatted into pages; contains 242 pages.
Dissertation (Ph.D.)--University of South Florida, 2009.
Includes bibliographical references.
Text (Electronic dissertation) in PDF format.
ABSTRACT: This study examined the features of the classroom discourse in eight Algebra I classes from two urban high schools with diverse student populations. In particular, by using the discursive analysis perspective, the type of communication between teachers and students was examined. The study investigated to what extent teachers' patterns of discourse change as a result of the number of ELLs present or their particular teaching experiences and ESOL endorsement. Furthermore, the impact of teachers' cultural and linguistic backgrounds upon ELLs' mathematics experiences was explored, particularly the teachers' patterns of discourse and adjustments to their teacher talk, or modifications of instructions that contributed to ELLs' engagement in the mathematics classroom.Data analysis from various sources (observations, video-recordings, frequency counts, interviews, the teachers' self-evaluations, and the researcher's and the ELLs' evaluations) indicated that to some extent all teachers changed their patterns of discourse simply due to the presence of ELLs, regardless of the total number in the class. Teachers with more teaching experience and with ESOL training had a smaller number of ELLs in their classes, whereas in both schools the novice teachers were assigned to teach classes with the highest number of ELLs. The novice teachers frequently used almost the same strategies as their more experienced colleagues did.Yet the qualitative analysis of the type of modifications to their speech they made, the type of questions they asked, and the provision of information of higher cognitive demand according to Bloom's Taxonomy indicated that even though all teachers needed improvement in using these strategies, the more experienced teachers with ESOL training applied those strategies to a fuller extent. They more often used slower and simpler speech and different questioning techniques sensitive to the ELLs' level of English language acquisition (i.e., pre-production, early-production, and speech emergence) and provided the students with content specific, enriched information. However, they still did not ask enough questions that could provide the ELLs with opportunities to justify and explain their opinions, and rarely led the discussions to a point which could move the ELLs to the highest level of the subject-specific literacy intermediate speech and fluency in mathematics in English.
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Advisor: Gladis Kersaint, Ph.D.
Second Language Acquisition (SLA)
English Speakers of Other Languages (ESOL)
x Mathematics Education
t USF Electronic Theses and Dissertations.
Classroom Discourse and Teacher Talk Influences on English Language Learner Students' Mathematics Experiences by Mariana M. Petkova A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics Education College of Education University of South Florida Major Professor: Glad is Kersaint, Ph.D. Denisse Thompson, Ph.D. Jeffrey Kromrey, Ph.D. Anthony Erben, Ph.D. Date of Approval: April 3, 2009 Keywords: Mathematics education, discourse analysis, mathematical discourse, Second Language Acquisition (SLA), English Sp eakers of Other Languages (ESOL) Copyright 2009 Mariana M. Petkova
ACKNOWLEDGEMENTS Foremost, I would like to thank my majo r professor Dr. Gladis Kersaint and the committee members, Dr. Denisse Thomps on, Dr. Anthony Erben, and Dr. Jeffrey Kromrey. It is due to their invaluable assi stance throughout the study that I was able to complete my dissertation. I am grateful to the principals and the school personnel who allowed me access to their schools. A special thank you is due to the mathematics teachers and the ELL students who participated in testing the st udyÂ’s instruments and i ndicated interest to participate in the main study. I would also like to acknowledge the assistance of Roger Scruggs in making copies of the videotapes, Mamie Ashby and Sandra Rizzi in helping with the Spanish translation of the instruments, and Amy Ri ffe and my students for their assistance in achieving accuracy of the transcriptions of the video recordings. Many thanks go to Dr. Lori-Sue Grieb Severino, a great friend whose particular expertise c ontributed greatly in preliminary testing of my instruments, and later in checking the accuracy of the transcriptionsÂ’ analysis and description. Lastly, I express my gratitude to my fa mily without whom none of this would have been accomplished. My daughter and son, for proofreading countless drafts before this final paper emerged; my husband and pa rents, for their love and support and their understanding when I could not spend enough time with them.
i TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES vi ABSTRACT viii CHAPTER I: INTRODUCTION, PURP OSE OF THE STUDY, RESEARCH QUESTIONS, DEFINITIONS A ND TERMINOLOGY CLARIFICATION 1 Purpose of the Study 4 Research Questions 4 Definitions and Terminology Clarification 5 ESOL Requirements 6 Teacher Talk (TT) 8 TeachersÂ’ Patterns of Discourse 8 Discourse 8 Mathematical Discourse and Mathematical Discourse Communities 8 CHAPTER II: LITERATURE REVIEW 10 Difficulties that ELLs Face in the Mathematics Classroom 11 StudentsÂ’ Learning of Math ematics from the Discourse Perspective 11 Mathematics Classroom Discourse when ELLs are Present 13 Second Language Acquisition (SLA ) and Literacy Development in Connection with Mathematics 14 English Language Profic iency Influences on ELLsÂ’ Experiences in Mathematics 14 Cultural Influences on ELLsÂ’ Experiences in Mathematics 15 Discourse and SLA: Genesis 17 Classroom Discourse, SLA and Learning in Mathematics 19 The Nature of the Mathematics Clas sroom Environment when Discourse is a Feature 20 Tasks and Discourse 21 Small-Group Work and Discourse 23 The Role of the Teacher in Promo ting Discourse as a Means of Negotiating Meaning in Mathematics 25
ii The Multiple Roles of the Teach er in Mathematics Classroom Discourse 25 TeachersÂ’ Beliefs and Perceptions About Discourse 25 TeachersÂ’ Expectations and Methods of Teaching and Their Effect on ELLsÂ’ Mathematics Experiences 28 Teacher Talk and Voices Used in Discourse 29 Teacher Questioning 31 Error Treatment and Feedback 32 The Role of Students in Mathematics Classroom Discourse 33 Relationships between Interventions and ELLsÂ’ Mathematics Achievement 34 Mathematics Instruction in Bili ngual Programs 35 Alternative Mathematics Programs for Migrant Students 37 Teacher Education and In-Service Programs 38 Culturally Relevant Education 39 TeachersÂ’ Instructional Pr actices that Promote Discourse 42 Methods of Assessment and their Effect on ELLsÂ’ Mathematics Experience 43 Research Methods for Examining Classroom Discourse 45 Chapter Summary 54 CHAPTER III: METHODOLOGY AND PROCEDURES 55 The study 56 Context 56 Participants 58 Instruments 60 Â“Teacher Talk TestÂ” (TTT) Forms 1 and 2 60 Pre-observation Teacher Questi onnaire 62 Student Questionnaire 63 Data Collection Procedures 64 TeachersÂ’ Demographic Data 64 ELL StudentsÂ’ Demographic Data 65 Classroom Observations 65 Videotaped Observations 66 Field Notes 66 Interviews 66 Teacher Interviews 66 Student Interviews 67 Data Analysis 68 Data from TTT Forms 1, 2, and 3 68 Characteristics of Krussel et al.Â’ s (2004) Framework 69 Purpose 70 Setting 70 Form 71 Consequences 71
iii Method of Analytic Induction 72 Validity, Reliability, and Objectivity Check of the Analysis Process 73 Credibility 74 Transferability 76 Dependability and Confirmability 77 CHAPTER IV: RESULTS AND INTERPRETA TIONS 78 Characteristics of the Sample 79 Years of Teaching Experience 81 ESOL Endorsement 81 Number of ELL Students Present 81 Years of Teaching Experience, ESOL Endorsement, and Number of ELL Students Combined 82 TeachersÂ’ and ELL StudentsÂ’ Linguistic Backgrounds Combined 82 Case Study Analysis 83 Green Bay High School 84 Mr. Able 84 Typical classroom discourse. 85 Krussel et al. framework. 89 Perceptions of classroom discourse. 92 Summary of the frequenc y count of the teacherÂ’s discursive strategies. 96 Ms. Barrera 98 Typical classroom discourse. 99 Krussel et al. framework. 102 Perceptions of classroo m discourse. 106 Summary of the freque ncy count of the teacherÂ’s discursive strategies. 108 Ms. Chandler 111 Typical classroom discourse. 112 Krussel et al. framework. 116 Perceptions of classroo m discourse. 120 Summary of the frequenc y count of the teacherÂ’s discursive strategies. 123 Mr. Davison 123 Typical classroom discourse. 124 Krussel et al. framework. 127 Perceptions of classroom discourse 131 Summary of the frequenc y count of the teacherÂ’s discursive strategies. 134 Lincoln High School 136 Ms. Andersen 136 Typical classroom discourse. 137 Krussel et al. framework. 142 Perceptions of classroo m discourse. 146
iv Summary of the frequenc y count of the teacherÂ’s discursive strategies. 148 Ms. Brown 151 Typical classroom discourse. 152 Krussel et al. framework. 157 Perceptions of classroo m discourse. 161 Summary of the freque ncy count of the teacherÂ’s discursive strategies. 165 Ms. Cortez 167 Typical classroom discourse. 168 Krussel et al. framework. 173 Perceptions of classroo m discourse. 177 Summary of the freque ncy count of the teacherÂ’s discursive strategies. 180 Mr. Daniels 183 Typical classroom discourse. 184 Krussel et al. framework. 188 Perceptions of classroo m discourse. 191 Summary of the freque ncy count of the teacherÂ’s discursive strategies. 194 Summary of Results 196 Question 1 201 Question 2 203 Question 3 206 Question 4 207 CHAPTER V: DISCUSSION AND CONCLUSIONS, LIMITATIONS, AND RECOMMENDATIONS 210 Discussion and Conclusions 210 Limitations 218 Recommendations for Further Research 220 REFERENCES 222 APPENDICES 231 Appendix A: Teacher Talk Test (TTT) Form 1 232 Appendix B: Pre-observation Teache r Questionnaire 235 Appendix C: Post-observation Teach er Questionnaire 236 Appendix D: Questionnaire for ELL Students 239 Appendix E: Post-observation Teache r Questionnaire 240 ABOUT THE AUTHOR End Page
v LIST OF TABLES Table 1 SchoolsÂ’ Demographics 57 Table 2 Overall Sample Information 80 Table 3 Pearson Product-Moment Correlati on Coefficients 197
vi LIST OF FIGURES Figure 1. TeacherÂ’s, researcherÂ’s, a nd ELLsÂ’ evaluations of Mr. AbleÂ’s frequency of use of various discur sive strategies. 93 Figure 2. Frequency count of Mr. Ab leÂ’s use of various discursive strategies during the three 20-minute video-recorded session s. 97 Figure 3. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evaluations of Ms. BarreraÂ’s frequency of use of various disc ursive strategies. 107 Figure 4. Frequency count of Ms. Barre raÂ’s use of vari ous discursive strategies during the th ree 20-minute video-recorded sessions. 110 Figure 5. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evaluations of Ms. ChandlerÂ’s frequency of use of various disc ursive strategies. 121 Figure 6. Frequency count of Ms. Cha ndlerÂ’s use of various discursive strategies during the thre e 20-minute video-recorded sessions. 124 Figure 7. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evaluations of Mr. DavisonÂ’s frequency of use of various disc ursive strategies. 133 Figure 8. Frequency count of Mr. Davi sonÂ’s use of various discursive strategies during the th ree 20-minute video-recorded sessions. 135 Figure 9. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evaluations of Ms. AndersenÂ’s frequency of use of various disc ursive strategies. 147 Figure 10. Frequency count of Ms. Ande rsenÂ’s use of various discursive strategies during the th ree 20-minute video-recorded sessions. 149 Figure 11. TeacherÂ’s, researcherÂ’s, a nd ELLsÂ’ evaluations of Ms. BrownÂ’s frequency of use of various disc ursive strategies. 162 Figure 12. Frequency count of Ms. Br ownÂ’s use of vari ous discursive strategies during the thre e 20-minute video-recorded sessions. 166
vii Figure 13. TeacherÂ’s, researcherÂ’s, a nd ELLsÂ’ evaluations of Ms. CortezÂ’ frequency of use of various disc ursive strategies. 179 Figure 14. Frequency count of Ms. Co rtezÂ’ use of various discursive strategies during the thre e 20-minute video-recorded sessions. 182 Figure 15. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evaluations of Mr. DanielsÂ’s frequency of use of various discursive stra tegies. 192 Figure 16. Frequency count of Mr. Dani elsÂ’ use of vari ous discursive strategies during the th ree 20-minute video-recorded sessions. 195 Figure 17. TeachersÂ’ frequencies of used strategies during the three 20-minute video-recorded sessions. 200
viii Classroom Discourse and Teacher Talk Influences on English Language Learner StudentsÂ’ Mathematics Experiences Mariana M. Petkova ABSTRACT This study examined the features of the classroom discourse in eight Algebra I classes from two urban high schools with dive rse student populations In particular, by using the discursive analysis perspective, the type of communication between teachers and students was examined. The study investigat ed to what extent teachersÂ’ patterns of discourse change as a result of the number of ELLs present or thei r particular teaching experiences and ESOL endorsement. Furthermor e, the impact of teachersÂ’ cultural and linguistic backgrounds upon ELLsÂ’ mathematics experiences was explored, particularly the teachersÂ’ patterns of discour se and adjustments to their teacher talk, or modifications of instructions that contributed to ELLsÂ’ engagement in the mathematics classroom. Data analysis from various sources ( observations, video-recordings, frequency counts, interviews, the teacher sÂ’ self-evaluations, and th e researcherÂ’s and the ELLsÂ’ evaluations) indicated that to so me extent all teachers change d their patterns of discourse simply due to the presence of ELLs, regardless of the total number in the class. Teachers with more teaching experience and with ES OL training had a smaller number of ELLs in
ix their classes, whereas in both schools the novi ce teachers were assigned to teach classes with the highest number of ELLs. The novice te achers frequently used almost the same strategies as their more experienced colleag ues did. Yet the qualitative analysis of the type of modifications to their speech they made the type of questions they asked and the provision of information of higher cognitive demand acco rding to BloomÂ’s Taxonomy indicated that even though all teachers needed improvement in using these strategies, the more experienced teachers with ESOL training applied those strategies to a fuller extent. They more often used slower and simpler speech and different questioning techniques sensitive to the ELLsÂ’ level of English language acquisition (i.e., pre-production, earlyproduction, and speech emergence ) and provided the students with content specific, enriched information. However, they still did not ask enough questions that could provide the ELLs with opportunities to justify and e xplain their opinions, and rarely led the discussions to a point which could move th e ELLs to the highest level of the subjectspecific literacy Â– intermediate speech and fluency in mathematics in English.
1 CHAPTER I: INTRODUCTION, PURP OSE OF THE STUDY, RESEARCH QUESTIONS, DEFINITIONS AND TERMINOLOGY CLARIFICATION The impending changes that accompany the United StatesÂ’ continued transition into the highly technological and specialized twenty-first century pose a unique challenge to its current educational system. The challe nge arises from the need to provide equal access to a high quality education to a c onstantly increasing and diverse student population. According to the U. S. Census Bu reau (2004), the number of people (five or more years old) in the United States who speak a language other than English at home increased from nearly 47 million in the ye ar 2000 to 49.6 million in 2004. This accounts for nearly 18.5% of the total U. S. populat ion. The changes in th e general population of the country are inevitably refl ected in the schools, with an increasing number of students categorized as English Language Learners ( ELLs). The need to address the demographic shift in the schooling populati on poses serious questions fo r those involved with the educational system, including teachers, admi nistrators, teacher educators, publishers, curriculum developers, politicians, researcher s, as well as parents, and the students themselves. The dilemma in the field of mathematics education, in essence, is that of providing each student with a quality and challenging mathematics education independent of his/her initial level of proficiency in the English language. At the same time, it is not sufficient for such students to attend the same schools, have the same
2 teachers, same textbooks, and be exposed to the same curriculum as their fluently English speaking peers. ELLs are not provided with equal education and oppor tunities if they do not understand the material becau se of a lack of fluency in the language in which this material is presented (Lau versus Nichols, 1974). Thus, the ultimate goal must be to develop these studentsÂ’ knowledge both in ma thematics and in the English language. To do so, schools must still provide all students with the knowledge and skills necessary to develop their abilities to creatively appl y mathematics, to analyze problems and determine the most appropriate ways to solve them (Glenn Commission, 2000; U. S. Department of Education, 2001). It is important to consider the stat ement by the Mathematics Learning Study Committee of the National Research Council (2002), Proficiency [in mathematics] is much more likely to develop when a mathematics classroom is a community of learners rather than a collection of isolated individuals. In such a classroom, student s are encouraged to generate and share solution methods, mistakes are valued as opportunities for ev eryone to learn, and correctness is determined by the logic and structure of the problem, rather than by the teacher. (p. 26) This notion is consistent with the Nationa l Council of Teachers of Mathematics (NCTM, 1991) recommendation that Â“the discourse of a classroomÂ—ways of representing, thinking, agreeing and disagreei ngÂ—is central to what student s learn about mathematicsÂ” (p. 34). Several studies have investigated the influences of classroom discourse on studentsÂ’ learning in mathematics (Ben -Yehuda, Levy, Linchevski, & Sfard, 2005;
3 McNair, 2000; Sfard, 2002). Other studies have examined the specific nature of the mathematics classroom (Jacobson, & Lehr er, 2000; McClain, & Cobb, 1998; McNair, 1998, 2000) or small-group work (Blunk, 1998; Leonard, 2000; Zack, 1999) environment when discourse is a feature. Fu rthermore, a group of studies ha ve investigated the role of mathematics teachersÂ’ beliefs and perceptions about discourse (Blanton, 2002; Blanton, Berenson, & Norwood, 2001; Branderfur, & Frykholm, 2000; Nathan, & Knuth, 2003; Renne, 1996). Another group of studies investigat ed teachersÂ’ instructional practices and employed strategies in promoting discourse (Patrick, Turner, Meyer, & Midgley, 2003; Sherin, 2002; Turner, Meyer, Midley, & Pa trick, 2003; Wood, 1999). Some studies have focused specifically on mathematics teachersÂ’ interaction patterns (Forman, & Ansell, 2001, 2002; Kovalainen, Kumpulainen, & Va sama, 2001; Rittenhouse, 1998), teacher questioning techniques (Steele, 1999-2000), a nd error treatment and feedback (Weingrad, 1998). Other studies have examined the role of individual studentsÂ’ communication in relation to the mathematics classroom cu lture and discourse (Bills, 1999; Davidenko, 2000; Manouchehri, & Enderson, 1999). Only a relatively small group of rese archers have focused their efforts on investigating the natu re of classroom discourse when ELL students with linguistically and culturally diverse backgrounds are pr esent (Brenner, 1994, 1998; Davidenko, 2000; Moschkovich, 1999, 2002). However, questions su ch as whether mathematics teachersÂ’ patterns of discourse relate to the number of ELL students present in the classroom, how a mathematics teacherÂ’s experience and E nglish Speakers of Other Languages (ESOL) endorsement relate to his or her patterns of discourse, and how teachersÂ’ own linguistic and cultural backgrounds affect their patterns of discourse when teaching mathematics in
4 English, and especially to classes with ELL students present, still remain open for investigation. Purpose of the Study It is important that preand in se rvice teacher education programs provide information to assist teachers to reach all students and improve their instructional practices so Â“that all students have the opportunity to develop their mathematical potential, regardless of a lack of proficie ncy in the language of instructionÂ” (NCTM, 1989, p. 142). Recent Standards documents reveal that current reform efforts Â“[demand] that reasonable and appropriate accommodations be made as needed to promote access and attainment for all studentsÂ” (NCTM, 2000, p. 12). To illustrate this need NCTM points out that ELLs Â“may need special attent ion to allow them to participate fully in classroom discussionsÂ” (p. 13). The aim of this study is to examine features of mathematics classroom discourse that may contribute to ELLsÂ’ engagement in the mathematics classroom. In particular, the st udy examines the impact of studentsÂ’ and teachersÂ’ cultural and lingui stic backgrounds on studentsÂ’ experiences in mathematics. Research Questions Specifically, the study examines the type of communication that occurs between teachers and students in mathematics clas srooms when ELLs are present. The study addresses the following research questions: 1. To what extent do teachersÂ’ patterns of discourse in the mathematics classroom change as a result of the number of ELL student(s) present?
5 2. To what extent do mathematics teache rsÂ’ experiences and teachersÂ’ ESOL endorsement (i.e., training) relate to their patterns of discourse when teaching mathematics to classes with ELLs present? 3. How do teachersÂ’ own linguistic and cultu ral backgrounds affect their patterns of discourse when teaching mathema tics in English to classes with ELL students present? 4. What patterns of discourse do teacher s use when ELLs are present in the mathematics classroom? What adjustments to teacher talk or modifications of instructions are observed? Definitions and Terminology Clarification Extant literature uses varying term inology to classify children who learn mathematics in their non-native tongue. As a result, it becomes necessary to clarify the intent of the terminology used in this manus cript. For example, Bradby (1992) provided the following definition for Language Minority (LM) and Limited English Proficient (LEP) students: Language Minority refers to children who come from homes in which a nonEnglish language is spoken. The Englis h language skills of language minority children range from not being able to speak English at all to being very fluent in English. Since [ sic ] those who study language acquisi tion are still debating about definitions, Limited English Proficient has several definitions; conceptually, however, LEP means that the children have sufficient difficulty with English that they are at a disadvantage in classe s taught entirely in English. (p.1)
6 In the literature, LEP is used synonymously with English Speakers of Other Languages (ESOL) and English Language Learners (ELLs). In this manuscript, the term ELLs will be used to emphasize the process of acquiring language skills while learning the content of mathematics. The emphasis is then placed on teaching mathematics content while teaching language. However, because the use of the phrase ELLs is in transition in the setting in which this study takes place, the te rm ESOL will also be used to represent the normative educational practic es. In Florida, the terminol ogy LEP or ESOL students is still used to be consistent with the language used in the Florida Consent Decree. ESOL Requirements In 1990, as a result of a lawsuit filed by the League of United Latin American Citizens (LULAC), Farm WorkersÂ’ Associat ion of Central Florid a, Haitian Refugee Center, and similar organizations against the Florida State Board of Education, The Florida Consent Decree was signed. The Consen t Decree addresses issues regarding the right of access to all educational programs by students whose primary language is not English. The settlement agreement was develo ped in compliance with Â“federal and state law and regulations including the federal Equity Educational Opportunity Act, Title VI of the Federal Civil Rights Act, of 19964, and Flor ida Educational Equity Act, and related federal and state provisions regarding compen satory, migrant, and special educationÂ” (Florida Consent Decree, 1990). As a result, the United States Di strict Court for the Southern District of FloridaÂ—Miami Division issued a ruling by which ESOL endorsement became a requirement for any teacher who is a primary provider of instruction or services to ELLs. Category I (Primary Language Arts/English) Teachers in
7 the State of Florida are requi red to complete 15 credit hour s or 300 in-service credit points in courses specifically designed to help ELLs in the mainstream classroom. The courses should address areas such as: (a) Methods of teaching English to speakers of other languages (ESOL); (b) ES OL curriculum and materials development; (c) Cross-cultural communication and understa nding; (d) Applied linguistics; and (e) Testing and evaluation of ESOL. Six years or more are allowed for the completion of the ESOL Endorsement, or 3 years for K-12 ESOL Coverage obtained by a passing score on an ESOL Subject Area Test. To meet state ESOL requirements, t eachers of Basic Subject Areas (Social Studies, Mathematics, Science, and Computer Literacy) are grouped in Category II, with its own set of specific timelines and requirements. For example, a mathematics teacher who provides instruction to any ELL student is required to complete 3 credit hours or 60 in-service credit points of ESOL quality training and instruction. The courses should address methods of teaching the subject matter paralleled with: (a) Methods of teaching ESOL; (b) ESOL curriculum and materials de velopment; and (c) Te sting and evaluation of ESOL. The timeline for a beginning teacher to complete these requirements is two years, while for an experienced teacher the timeline is a year However, the completion of these courses only grants compliance with the Florida Consent DecreeÂ’s minimum requirements for subject area teachers, while for an ESOL Endorsement 15 credit hours are still needed. Recently, many colleges and universities offering degrees in education for Category I teachers include ESOL E ndorsement as part of their graduation requirements.
8 Teacher Talk (TT) In the literature teacher talk refers to the language used by teachers in classrooms as opposed to their use of language in other set tings (at home, at the store, at the doctorÂ’s office, etc.). In this study, I will use EllisÂ’ (1994) definition of teacher talk as the process through which Â“teachers address classroom la nguage learners differently from the way they address other kinds of cl assroom learners. They make adjustments to both language form and language function in order to fac ilitate communication. These adjustments are referred to as Â‘teacher talkÂ’ (Ellis, 1994, p. 726). TeachersÂ’ Patterns of Discourse The phrase, patterns of discourse, will refer to the different types of communication a teacher used with his or her students. Krussel et al. (2004) referred to patterns of discourse as Â“teachersÂ’ discourse movesÂ” (deliberate actions taken by teachers) to facilitate the discour se in the mathematics classroom. Discourse Â“The term Â‘DiscourseÂ’ with a capital Â‘D ,Â’ [refers to] ways of combining and integrating language, actions, interactions, wa ys of thinking, believi ng, valuing, and using various symbols, tools, and objects to enact a particular sort of socially recognizable identityÂ” (Gee, 2005, p. 21). Mathematical Discourse and Mathematical Discourse Communities Before discussing research regarding disc ourse in mathematics classrooms, some clarification of the concept of such discourse must be provided. The definition for mathematical discourse, and mathematical discourse communities provided in Sherin (2002) is certainly applicable:
9 the process of mathematical discourse refers to the way that the teacher and students participate in class discussion s. This involves how questions and comments are elicited, and through what m eans the class comes to consensus. In contrast, the content of mathematical discourse refers to the mathematical substance of the comments, questions, and responses that arise. (p. 206) By extension, the term mathematical discourse communities refer to classroom environments where Â“students are expected to state and explain ideas and to respond to the ideas of their classmates. Teachers are asked to facilita te these conversations and to elicit studentsÂ’ ideas (p. 207). Thus, Â“becomi ng a member of a mathematical discourse community involves learning to talk about math ematics in ways that are mathematically productiveÂ” (p. 208).
10 CHAPTER II: LITERATURE REVIEW This chapter will present the collected research knowledge and leading views about the influences of classroom discourse on studentsÂ’ learning of mathematics. Of particular interest are studies from cl assroom-based and second language acquisition (SLA) research that provide insights regarding the influen ces of classroom discourse on what and how ELL students learn in mathematics classrooms. A related area of interest is research that provides information about the process of developing ELLsÂ’ literacy and reading skills in conjuncti on with the development of conceptual understandings and skills in mathematics. Specifically, theoreti cal and empirical work that addresses the following will be discussed: 1. Difficulties that ELLs face in the mathematics classroom. 2. The nature of the mathematics classroom when discourse is a feature. 3. The role of the teacher in promoti ng discourse as a means of negotiating meaning in mathematics. 4. The relationship between intervention s and ELLsÂ’ mathematics achievement. 5. Methods of assessment and their effect on ELLsÂ’ mathematics experiences. Examination of findings from these studies wi ll provide insights on the current state of knowledge regarding influence of cla ssroom discourse on ELLsÂ’ learning of mathematics. In addition, research approaches used in these studies guided the formation of approaches utilized in the study that will be described later.
11 Difficulties that ELLs Face in the Mathematics Classroom StudentsÂ’ Learning of Mathematics from the Discourse Perspective Socio-cognitive theories pr ovide an invaluable persp ective on studentsÂ’ learning of mathematics in an environment that fosters classroom discourse. According to Vygotsky (1978), learning occurs when the in dividual internalizes external knowledge to supplement his/her knowledge. Such exte rnal knowledge can be accessible from interactions with ot her individuals possessing different or more knowledge in the domain under discussion. Bakhtin (1981) supports this notion and suggests th at the process of learning is intrinsically bot h social and individual. Based on these perspectives, one can conc lude that discourse between a learner (the student) and an expert (the teacher or peer) contributes to th e learnerÂ’s cognitive development (and learning of mathematics in particular). As such, Â“the shift in perspective from looking at mathematics learni ng as an internal reasoning process to looking at what is interactively accomplished through talk is a critical on e to noteÂ” (Hicks 1998, p. 243). Research reveals how the ex ternal knowledge accessed in interactions with the teacher and peers is internalized, and how new meaning and understanding of mathematics is appropriated in order to form new knowledge. For example, Sfard (2002) reported on how students devel op their thinking and learn ne w knowledge in mathematics by becoming skillful in the discursive use of new symbolic toolsÂ— specifically, a bar diagram and a dot plot (a visual representation of data simila r to the bar diagram, where a series of dots are used instead of bars). The flow of di scourse as students solved two word problems was examined. In particular the study examined how the discourse was
12 mediated by the graphic display of data, how a specific graphic display was accessed by the learners and used in initiating specif ic discourse, and finally, how much learning occurred as a result of such discourse (i .e., how skillful the students had become in participating in the mathematical discourse). Distinguishing between pronounced (a specific question), attended (the procedure involved in answering the question) and intended focus (the answer), the anal ysis of the learning epis odes revealed that the process by which students move from a pronounced focus to an attended and an intended focus is not straightforward, but is rather complex and happens gradually or in cycles. In order to solve the pr oblem, students use intimations (Â“an association of the present situation with an experience of the past that enables a new discursi ve decisionÂ” (p. 331)) and implications (examining the applicability of thei r decisions). For example, if the inducement element (the element encouraging discussions) of the present situation asks for Â“betterÂ” batteries, the word Â“bette rÂ” induces the asso ciation with the source Â“longer lastingÂ” in the studentÂ’s mind, which is translated into the target of Â“longer barÂ” on the graph, and then the decision is made to draw the upper li mit line through the tip of the longest bar. In this tendency to draw presumed inferences Sfard noted that sometimes discursive decisions are influenced by studentsÂ’ inherent assumptions about the discursive mechanism, or presumptions about the expected solution (metalevel intimations). Nevertheless, this study indi cated that students in discourse-rich environments significantly improved their ab ilities to participate in mathematical discourse (thus learning occurred). McNair (2000) reported on the charact eristics of mathematics classroom discussions that result in better student learning. Two sm all group discussions were
13 compared to investigate the factors that contribute to maximum or minimum learning outcomes. The research findings indicated that discussions that have three main characteristics Â— mathematical subject, pur pose, and frame Â– provide maximum learning opportunities for students. The mathematical subjects are usually numbers, shapes, spaces, variables and the patterns and rela tionships between them; a mathematical purpose could be to solve a mathematical problem and must Â“add structure and understanding to mathematical systems of r easoningÂ” (p. 206), and a mathematical frame is the system of organization of studen tsÂ’ experience in searching for patterns, generalizing and formalizing procedures, ma king connections, logical reasoning, proofs, and communicating their ideas. Ben-Yehuda, Lavy, Linchevski, and Sfard (2005) provided additional information regarding how studentsÂ’ learni ng of mathematics can be maximized. The unique aspect of this study is the methods used to investigate the mechanism of failure in mathematics of two students with learning difficulties. The resu lts revealed that some studentsÂ’ failure in mathematics is due to the instructorÂ’s inab ility to use discourse to improve studentsÂ’ comprehension and problem-solving skills in math ematics. The result is an inability to provide each individual student with a choice of tools to approach mathematics problems without the fear of exclusion. The researcher s noted that each studentÂ’s learning potential can be significantly maximized with improveme nts in discourse that recognize individual needs and abilities and that explo it each studentÂ’s strengths. Mathematics Classroom Discourse when ELLs are Present The research discussed above reveals the mechanism by which external knowledge accessed in interactio ns between the teacher and th e class is internalized and
14 how new meanings and understa ndings of mathematics are a ppropriated in order to form new knowledge. However, there is a pa ucity of research (Brenner, 1994, 1998; Moschkovich 1999, 2002; Secada, 1996) that specifically focuses on the nature of the discourse in mathematics classrooms when th e students present have linguistically and culturally diverse backgrounds. Although the recent reform agenda in education is oriented toward trying to involve all students in meaningful communication in the mathematics classroom (NCTM 1989, 1991; U.S. Department of Education, 2001), very little is known about what occurs in the classroom wh en a large population of ELL students is trying to learn mathematics at the same time as they are learning the language in which the subject is taught. Second Language Acquisition (SLA) and Litera cy Development in Connection with Mathematics The process of learning mathematics cannot be examined in isolation from other aspects of learning such as cognitive deve lopment, general literacy development, language learning, writing and r eading development not only for ELLs but for the student population in general. To that end, in this se ction I will discuss rese arch that focuses on these issues. Specifically, research will be discussed that examines development of studentsÂ’ general literacy (listening, speaking, writing and reading abilities) simultaneously with the development of thei r mathematics conceptu al understanding. English Language Proficiency Influences on ELLsÂ’ Experiences in Mathematics Several studies reveal a positive correlation between English proficiency and achievement in mathematics (Abedi and Lord, 2001; Bradby, 1992). Bradby (1992) examined a variety of factors to determ ine which of them were predominantly
15 influencing the performance on achievement tests of ELL students from Asian and Hispanic backgrounds. The data from the National Educational Longitudinal Study of 1988 were analyzed and findings revealed a direct relationship between Hispanic ELL studentsÂ’ achievement both in mathematic s and reading and their English language proficiency and familyÂ’s socio economic status (SES) characteristics. In contrast, he found that for Asian ELL students, their SES was a more influential factor on their achievement in mathematics than their English language proficiency. Abedi and Lord (2001) reported the results for 1,174 eighth-grade students from 11 schools in Los Angeles with diverse linguist ic, ethnic, and SES backgrounds after they had been tested on two mathematics tests (one with and one without linguistic modifications) created with the use of the National Assessment of Educational ProgressÂ’s (NAEP) released items. The purpose of this study was to determine the influence of studentsÂ’ language proficienc y on their achievement on mathematics tests in which there was an emphasis on word problems. Results indicated lower performance for students who were in the process of learning English as a second language as compared to language proficient students. Results also in dicated higher improvement in scores for the language-deficient, as well as language-proficien t, students on test items with linguistic modifications. Cultural Influences on ELLsÂ’ Experiences in Mathematics The literature that examines culture a nd mathematics education focuses on two primary areas: studentsÂ’ view s on learning mathematics and culturally relevant teaching. More recently, because of the diversity of the student population in classrooms, research has focused on the need to provide culturally responsive or releva nt instruction to
16 students. Regarding the former, several chapte rs were written in the late 1980s that discuss the influence of stude ntsÂ’ cultural background on th eir views about mathematics learning. For example, several studies indicate that there is a re lationship between the studentsÂ’ cultural background and their math ematical achievement (Cocking & Chipman, 1988; MacCorquodale, 1988; Leap, 1988; Tsang, 1988). Primarily, these studies suggest that studentsÂ’ culture provides a lens with which they may examine their experiences in mathematics. In some cases, the importance the community and the family places on mathematics can enhance studentsÂ’ experi ences in learning the subject by providing additional motivation (Tsang, 1988), whereas in other cases it may limit the studentsÂ’ (Cocking & Chipman, 1988) and parentsÂ’ (MacCorquodale, 1988; Leap, 1988) involvement in education, particularly in mathematics. For example, Cocking and Chipman reported that Hispanic women in their study perceived mathematicians as sloppy, remote, obsessive, and calculating, and th us, because of thei r perceptions they and their children tend to shy away from ma thematics. Other research indicates that Native Americans tend to view education as something reflective, visual, and more holistic/global, and they learn better when they work in c ooperative small group settings (Eieife, 2002; Reyhner, Lee, & Gabbard, 1993). Thus, if teachers are not familiar with individual studentsÂ’ ways of learning and cultural values, th e typical competitive western education with its use of traditional a uditory teacher centered setting may lead Native American students to underachievement in math ematics. The possibility of such reactions is an important consideration for the math ematics education community, particularly when the emphasis is placed on Â“mathematics for all.Â”
17 Discourse and SLA: Genesis The first traces of the idea that s econd language acquisition develops in interaction with others and not in individuals in isola tion can be seen in Wagner-Gough and Hatch (1975). In investigating the inte ractions between child ren from different backgrounds (Chinese and Ira nian) throughout thei r process of learning English, the researchers presented the fo llowing picture of second la nguage acquisition. First, language learners produce forms (i.e., words indicating verbs, nouns) without understanding their functions. Th en, they start understanding the functions of these forms by referring to their native language knowledge, and by tr ying to group frequently occurring forms without yet knowing the target language rules. Thus, they move to a stage where they start to understand the seman tic difference between variations of forms of a verb for example, and start to incor porate them in their sp eech. However, if the process of rule formation is placed in a so cial discourse where th e language learners are provided with an input from na tive speakers (NS) of the targ et language, then the process of second language acquisition is more eff ective. Thus, Â“we should not neglect the relationship between language and communication if we are looking for explanations for the learning processÂ” (p. 307). Another significant contributor in the de velopment of the inte raction perspective (later to lead to the discursive pers pective in research) is Long (1981, 1983, 1985). According to Long (1980), Â“ Input refers to the linguistic forms usedÂ” (morphemes, words, utterances) and studies concentrati ng on input usually consider the forms the learner is exposed to; while Â“by interaction is meant the functions served by those forms, such as expansion, repetition and clarificat ionÂ” (p. 259) and studying interactions must
18 concentrate on Â“describing the functions of those forms in (conversational) discourseÂ” (Long, 1983, p. 127). Thus, to investigate the inte ractions of students and teachers in the classroom the researcher must take into acc ount the participation in conversation of both the native speakers (NS) and nonnative speak ers (NNS), taking turns, negotiation of meaning, etc. Long (1981, 1983) analyzed the interactio ns between NS-NS and NS-NNS, where the NNS in his study were from various li nguistic backgrounds. Long found that NS do use more modifications to the input when they interact with NNS in comparison to interaction with NS. These m odifications include more freq uent use of selfand otherrepetitions, lower type-token ratio (i.e., sl ower speech patterns), comprehension and confirmation checks, and expansions. Modifications appeared to be used in order to avoid conversational difficulties or to repair the discourse when difficulties in conversation already had occurred. Late r, Long (1996) formulated an updated version of the Interaction Hypothesis which relates th e factors of importance in SLA: Â“ negotiation for meaning and especially negotiation work that triggers interactional adjustment by the NS or more competent interlocutor, facili tates acquisition because it connects input, internal learner capacities, pa rticularly selectiv e attention, and output in productive waysÂ” (pp. 451-452). Pica, Young, and DoughtyÂ’s (1987) study also revealed a positive impact of interactions and negotiati on of meaning on comprehension. This study revealed that interactional modificatio ns of input lead to more signifi cant levels of comprehension than conventional ways of linguistically s implifying input. In Mackey (1995), ELLs participated in communicative ta sks; some learners received a pre-modified input with no opportunities to interact, while other learners co uld use interaction in the process of input.
19 The study indicated that the le arners who participated in interactions progressed more quickly in their SLA development. Classroom discourse, SLA and learning in mathematics Many researchers focused th eir attention on the interactions that occur in a classroom setting because a significant part of second language learning takes place in such an environment. Parallel to such research in the field of SLA, in the past decade, a similar trend has developed in classroom-orie nted research by subj ect area specialists. This line of research adopts the discourse perspective in investigating the mechanisms by which ELL students learn in diffe rent content areas while faced with the obstacles of adjusting to a new culture and learning the la nguage in which the different content areas are taught in school. Regarding mathematics, Brenner (1 998) and Moschkovich (1999, 2002) have investigated the nature of classroom discourse when the students involved have linguistically and culturally diverse backgrounds. Moschkovich (1999) observed discussions in a computer-based dynamic instructional environment, noting which teaching techniques improved ELLsÂ’ participation in the mathematics discussions about the geometric shapes and figures in a tangr am puzzle. She found that teachers improved ELLsÂ’ participation in discussions by empl oying techniques such as utilizing objects to encourage students to talk about their propertie s and characteristics, giving sufficient time for group discussions (student-t o-student discussions), asking students to repeat their statements using different expressions in or der to clarify their statements, and using Â“revoicingÂ” (reformulating the studentsÂ’ statem ents using formal mathematical terms) in order to show acceptance of the ELLsÂ’ response s and thus encourage their participation
20 in discussions. Instead of correcting ELLs Â’ linguistic mistakes and concentrating on language development, the teacher focused on whether students demonstrated conceptual understanding. Moschovich (2002) examined b ilingual studentsÂ’ learning of mathematics with English as a second language (L2). Findi ngs indicated that bi lingual students may be capable of communicating meaning and compet ence in mathematics without learning the correct vocabularyÂ— by using gestures, objects, or everyday examples as resources, or simply by using their first language (L1). Findi ngs also indicated that involving bilingual students in classroom discourse provided them with practice leading not only to their L2 development, but also their mathematical development. An examination of the mathematical co mmunication in two algebra classes with large populations of ELLs (predominantly Hisp anics) revealed that in classrooms in which small-group discussions were encour aged and computers were employed to stimulate discussions, more successful mathematical communi cation was exhibited, which later was spread to a large-group setti ng (Brenner, 1998). In contrast, in the class in which the teacher employed mostly whol e-classroom instruction, the ELL students were more reluctant to speak aloud in front of a large group. The Nature of the Mathematics Classroom Environment when Discourse is a Feature Many researchers have studied the na ture of the mathematics classroom environment when discourse is a feature. A lthough they did not spec ifically look at the effects of such an environment on ELL student sÂ’ learning of mathematics, the findings that are reported are valid for all learners of mathematics and thus for ELLs as well (Blunk, 1998; Jacobson, & Lehrer, 2000; Leonard, 2000; McClain, & Cobb, 1998; McNair, 1998, 2000; Zack, 1999).
21 Tasks and Discourse A characteristic of classroom discourse that fosters and maximizes studentsÂ’ learning potential of mathematics is that it must have a mathematical subject (McNair 1998, 2000). This usually requires that stud ents be involved in a meaningful mathematical task. However, the completion of the same task by different students does not guarantee that they will absorb the lesson equally as well. Jacobson and LehrerÂ’s (2000) research provides evidence that the difference among studentsÂ’ discourse while solving the same task chiefly determines what these students will learn and retain. They examined the difference in classroom discourse and st udentsÂ’ learning in four 2nd grade mathematics classrooms. The students were to design a quilt by performing transformations (slides, flips, reflections turns, and rotations) of Â“core squaresÂ” composed of different shapes of triangles using computer software. The researchers attended to the actions of the teachers and th e nature of the discourse promoted and how this affected studentsÂ’ learning of the geom etry involved in the project. The findings indicated that the patterns of discourse promoted by the teache rs were related to differences in teacher understanding and knowledge of studentsÂ’ reasoning and learning process regarding geometry and space. For example, Teacher A emphasized the new terminology focused on the core squaresÂ’ tr ansformations. The students were asked to clarify and elaborate their ideas about space and transformations in geometry. Teacher BÂ’s focus was on Â“whyÂ” and Â“howÂ” questions, which required students to explain the process of making a quilt: Â“And how did that change the design?Â” Thus, she encouraged the students to reflect on their thinking pr ocess of why and how they used geometric motions to make the quilt design. Teacher C also encouraged discussions, but her focus
22 was on the content and not the proces s of making the design. Her use of revoicing was mostly in the form of repeti tion and clarification of what the students said and rarely asking them to further elaborate on their ideas. Similarly, Teacher D did not ask the students to reflect on the transformations used to make the quilt. Even though the students were involved in disc ussions, they were only as ked to recognize shapes and colors. Further investigation was performed on the studentsÂ’ learning and retaining of the knowledge in each of these classes to meas ure the effectiveness of the classroom discourse. The results indicated that the mo re knowledgeable the teachers were about studentsÂ’ thought and learning process of ge ometry (in classes A and B), the more students learned and retained the gained know ledge about geometry transformations and their applications. Thus, these studies show that the teacher plays an important role in the creation of an environment that facilitates di scussions in mathematics by trying to involve all students in meaningful tasks. McClain and Cobb (1998) reported on the role of imagery and discourse in studentsÂ’ conceptual understa nding of mathematics. The st udents were involved in a year-long teaching experiment using the instructional theory Realistic Mathematics Education (RME). According to this theory, the in struction should start with the teacherÂ’s description of a problem situ ation using statements in a way that Â“students can evoke imagery of the situations described in the problem statements when solving tasksÂ” (p. 61). For example, the teacher in the study us ed a narrative about a pumpkin seller whose pumpkins were carried in crates of ten; the goal was for students to use this imagery to make sense of the taskÂ—working with tens. Â“In this way, the studentsÂ’ construction of situation-specific imagery allows them to engage in personally meaningful mathematical
23 activity and constitutes a basis for the student sÂ’ subsequent mathematization of activityÂ” (p. 61). The teacher then encouraged students to participate in subsequent activities by using this imagery to explain their thinki ng in terms of relati onships rather than numerical patterns. McClain and Cobb introduced the two terms folding back and dropping back of discourse to describe observe d changes in the discourse when difficulties in communication were encounter ed. In the first situation, communication was based on the prior discourse activity in which the imagery was Â“taken-as-sharedÂ” from all students since they we re familiar with it. In contra st, in the second situation, the teacher needed to introduce new information because of the lack of shared imagery. The teacher played a central role in directing disc ourse to use both strategies when difficulties in discourse were encountered. Small-Group Work and Discourse While most research investigates the natu re of discourse in the classroom, some researchers have examined the characteristic s and nature of discour se elicited in smallgroup work (Blunk, 1998; Leonard, 2000; Zack, 1999). For example, Leonard examined the discourse in the small-group work of three diverse sixth grade mathematics classrooms during a lesson on making a hydr ometer and measuring humidity. He investigated the effects of different disc ourse patterns on stud entsÂ’ (and teachersÂ’) learning. The results of the study indicated that the studentsÂ’ personali ties affected their behavior in small-group discourse much more than other factors such as gender. Usually, assertive students (male and female) were more involved in discourse. Also, during the whole-classroom discussions the teache rs had more control and used more institutionalized discourse (g iving clues to students of the changes in the lesson and
24 expectations from them), in small-group di scussions the discourse was more emergent and natural in nature. However, planning the task in advance, giving hands-on activities, and applying group pairing based on gender di d not always guarantee that the emergent discourse was mathematical in nature. Furt hermore, observed patterns revealed that students often used previous knowledge to initiate discussions. The teachers who took advantage of this and used both emergent and institutionalized discourse were more successful in facilitating discussions. The re sults also indicated that throughout their discourse, students exhibited improved knowle dge about relative humidity. They also exhibited an understanding and application of important vocabulary in context. Zack (1999) reported on an the argumenta tion of proofs of three members of a small group with the goal Â“to convey the sounds of mathematical talk in a classroom and school culture in which the child ren have been encouraged sin ce their entry to the school (for many, at 6 years of age) to engage in conversation about ideasÂ” (p. 134). More specifically, the focus was on the students Â’ use of logical c onnectivesÂ—Â“culturally grounded elements of language.Â” The results revealed that the ch ildren used logical connectives such as because, but, and if Â… then Â… in order to create a strong argument and connect their ideas. They also used para llel logical and syntactic structures such as ifÂ… then Â…, but it doesnÂ’t so you canÂ’t, which also contributed to the logical coherence of their argument. This also demonstrated the childrenÂ’s development toward use of more formal mathematical language. However, while Leonald (2000) and Zack (1999) highlighted studentsÂ’ talk in small-group discussions, Blunk (1998) focuse d on the communication of the teacher involved in creating and maintaining small-gr oup discussions. The subject of this study
25 was another researcher, Magdalene Lampert, and her fifth-grade mathematics class. Findings indicated that in order to creat e and maintain small-group discussions, the teacher viewed her role not as a transmitter of information or merely assigning students to groups, but rather as facilita ting studentsÂ’ social and cogni tive skills of communication about mathematics. For example, early in the year, the teacher talked about the characteristics and the nature of the small groups and why the students will work in such groups. Later in the year, the teacher made e xplicit her expectations for student behavior during small group interactions and explained how she woul d evaluate their group work. This case study suggests that allowing student s to engage in sophisticated, complex discussions about mathematics, creating meani ngful tasks for students to discuss within group interaction, and maintaining a climate in which the studentsÂ’ sp irit of curiosity is encouraged, are more important than finding the Â“right Â” answer (p. 210). The Role of the Teacher in Promoti ng Discourse as a Means of Negotiating Meaning in Mathematics The Multiple Roles of the Teacher in Mathematics Classroom Discourse TeachersÂ’ Beliefs and Perceptions about Discourse Many researchers have concen trated their efforts speci fically on the effects of teachersÂ’ beliefs about discourse and th eir impact on the classroom environment (Blanton, Berenson, & Norwood, 2001; Brenderfur, & Frukholm, 2000; Nathan, & Knuth, 2003; Renne, 1996). For example, Renne i nvestigated the factor s that influenced a teacherÂ’s attempts to incorporate students Â’ questions and initiatives in classroom discourse. Although the teacher attempted to shift to a more student-centered instructional approach and to incorporate studentsÂ’ questions and initiatives, some
26 deviations from this pattern were observed. Of ten studentsÂ’ initiativ es were converted by the teacher into teacher initiatives. That is, some of their questions were not directly answered or were ignored. Consequently, co mmunication in the classroom was in the traditional initiation-reply-evaluation (IRE) sequence (Mehan, 1979) where the teacher initiates (with a question or statement), a st udent responds, and the teacher evaluates the studentsÂ’ response (verbally or by a gesture). Further invest igations indicated that the teacherÂ’s detours to such teach er-centered instructions were influenced mainly by cultural beliefs and assumptions about teaching, lear ning and knowledge. Additionally, a lack of details about how to implement the reform, tim e constraints to complete the course, the number of students in the cla ss, and struggle for control were also found to be influential factors in the observed teacherÂ’s behavior. Brendefur and Frukholm (2000) reported on an investigation of two preservice teachersÂ’ beliefs and perceptions about disc ourse in relation to their mathematical understanding and internship practices. The findings revealed that even though both teachers were similar in age, attended the sa me mathematics methods class that promoted the reform-based perspective of discourse, and were assigned to intern in the same school with similar teachers, each teacher employed di fferent instructional practices. One of the teachers encouraged communication and facilita ted students in sharing ideas, while the other used a teacher-centered ap proach. Further investigation indicated that the observed differences in teaching practices were in acco rdance with the teachersÂ’ initial beliefs and dispositions toward mathematics and its teaching and learning. Nathan and Knuth (2003) investigated th e effects of a sixth-grade teacherÂ’s beliefs on her instructional practices when di scourse and interactions were promoted over
27 a period of two school years. They reported th e Â“pivotal roleÂ” of th e teacherÂ’s goals and beliefs in shaping her classroom practic es (p. 178). Specifically, by analyzing the flow of information they found that even though the teache r believed that students learn from their peers when they actively share ideas, little student-to-stude nt (S-to-S) talk occurred during the first year when she attempted to apply a reformed curriculum to promote discourse. The vast majority of communicati on was vertical, teacher -to-class talk (T-toC), which is very similar to the traditiona l IRE sequence. However, during the second year, S-to-S talk increa sed to 33%. By analyzing the nature of scaffolding they found that while during the first year the T-to-C communication was predominantly analytic (addressed mathematical conten t), during the second year it dropped to 50% analytical and 50% social in nature. The S-to-S analytical and social talk also showed a similar pattern. Furthermore, by analyzing the patterns of in teraction at a global level they found that while the teacher had a central role in interactions during the first year, during the second year Â“a star patternÂ” emerged with a less evident teacher authority (p. 198). Similar findings were reported in Bl anton (2002) and Blan ton et al. (2001). Blanton et al. thoroughly examined one preservi ce teacherÂ’s perceptions of discourse and her teaching approach in a seventh grade math ematics classroom. They noted a change of pattern in her methodology. Initially she prim arily used the IRE pattern of classroom discourse and perceived the teacher as Â“a te ller.Â” Later, her peda gogy shifted to using questions that explored student solutions and strategies. At this point the student was perceived as Â“a teller.Â” This shift in instru ctional approach was a crucial step in the teacher using a dialogue-based form of discour se and perceiving the student as an active participant in mathematics disc ourse. This study thus contri buted to the notion that Â“a
28 teacherÂ’s developing practice is inherently linked to the social dynamics of the classroomÂ” (p. 228). Blanton (2002) found that pre-service teach ersÂ’ initial beliefs, despite being very influential in the beginning of the teachi ng practice, could be changed by a reflective study of their classroomsÂ’ discou rse. Discursive reflections could provide teachers with information not only about st udentsÂ’ learning in mathema tics, but also about how teachers themselves could learn how to teach mathematics more successfully. TeachersÂ’ Expectations and Methods of Teac hing and Their Effect on ELLsÂ’ Mathematics Experiences Several studies link teachers Â’ expectations of ELLsÂ’ performance in mathematics, and understanding (or lack thereof) of their le arning process of mathematics, to the way the teachers teach mathematics (Davidenko, 2000; Rhine, 1995a, 1995b, 1999). For example, Rhine analyzed the tutoring sessions of intermediate grad e teachers of classes that include ELL students and examined th e teachersÂ’ expectations of the ELLsÂ’ performance in mathematics. Rhine videotaped the interactions between the teachers and students during these tutoring sessions, used recall interviews with teachers, and performed quantitative and qualitative ev aluations of their assessments. From the videotaped sessions, Rhine found that the teach ers tended to teach differently when ELLs were present in a group. He reported that the teachersÂ’ limited understanding of ELLsÂ’ mathematics learning became apparent duri ng the interview process. Teachers often linked the lack of English prof iciency to a similar lack of mathematical knowledge or understanding. When asked to make predictions about studentsÂ’ achie vement on tests, the
29 teachers usually underestimated the ELLsÂ’ perf ormance in comparison to their English speaking peers. Other studies suggest that teachers limit their instru ctional approaches when teaching classes with ELLs. For exampl e, Davidenko (2000) investigated the instructional practices and communication used in two algebra classes that included ELLs in order to evaluate the effects of teachi ng methods on studentsÂ’ learning practices. The data collected by classroom obs ervations, videotaped interactions, and interviews with the algebra and English as Second Language (ESL) teachers, and with 9 students (both ELLs and English speakers) from the two algebra classes were analyzed. Because mathematics teachers were aware that ELLs we re present in the classroom, they often reinforced computational skills and Â“ins trumental learningÂ” (learning experiences involving reinforcement of good behavior). Additionally, they usually assumed that ELLs could not handle higher-level mathematic s involving word problems, mathematics communication, and discussions in English a bout mathematics concepts. Consequently, students taught in such a manner received only a limited conceptual understanding of mathematics and their knowledge was only at the procedural and computational level. Davidenko concluded that the ELLsÂ’ proficiency in English was not th e sole factor that influenced their performance in mathematic s. Other very influential factors are the teachersÂ’ expectations and methods of teachi ng which also contribute to the studentsÂ’ learning process. Teacher Talk and Voices Used in Discourse Some scholars focus their attention speci fically on studying the nature of the teacherÂ’s communication and on finding patterns that provide insight into how teachers
30 facilitate classroom interactions (F orman, & Ansell, 2001, 2002; Kovalainen, Kumpulainen, & Vasama, 2001; Rittenhouse, 1998). For example, Rittenhouse investigated how Magdalene Lampert (a te acher/researcher) enacted discursive norms and routines in the first month of the school year with her fifth-grade mathematics class. Rittenhouse noted that the teacher facili tated discourse by skillfully employing techniques described as stepping in and stepping out of discussions. When the teacher stepped into discussion her talk was rather convers ational in nature, and she mainly participated in discussions by asking questions, providing additional information from her knowledge base of mathematics and thus contributing to the predominantly studentcommunication by presenting new ideas and in troducing and explaining new vocabulary. In contrast, when stepping out of discussion, the teacher talked in a more didactic manner. Here, she commented on discussions (Â“talk about the di scussionÂ”) or was formally teaching the rules and norms the stud ents should employ in order to participate in a polite argument. Thus, the study demonstr ated how the teacherÂ’s talk and Â“her dual role as participant and commentator provide us insight into one teacherÂ’s vision of what fostering studentsÂ’ understanding of mathematics looks likeÂ” (p. 187). Kovalainen et al. (2001) examined how t eachersÂ’ use of scaf folding strategies (involving the students in building on one anotherÂ’s ideas) faci litated classroom interactions. The investigation identified f our complementary and partially overlapping strategies of scaffolding: evocativ e (asking stimulating questions), facilitative (relating culturally established knowledge revoicing, modeling, monitoring) collective (enforcing the rules of discussions) and appreciative (expressing support, interest, pacing the
31 tempo) The study demonstrated how, by using these four strategies, the teacher orchestrates classroom interactions into productive discourse about mathematics. Forman and Ansell (2001, 2002) investigated the nature of the teacherÂ’s communication in relation to his/her personal experience in mathematics. They found that teachers often use different voices (different types of teacherÂ’s talk) when discussing theoretical versus standard strategies in ma thematics. For example, one teacher used one voice when orchestrating discussions of stude nt-invented strategies and used another when a standard algorithm was de monstrated. When using the first voice the teacher usually emphasized studentsÂ’ persistence as cr itical thinkers and ri sk-takers. She often used revoicing and encouraged studentsÂ’ th inking. However, when using the second voice the teacher often talked a bout her own past mathematic s experience, or about the experience of the studentsÂ’ parents or older si blings. Then, she talked about the confusing nature of standard algorithms and their limited use. Revoicing was rarely used and the algorithms were not explicitly explained. Teacher Questioning Some research reports on the particular effects of some specific parts of teacher talk such as questioning tec hniques, error treatment and provision of (or lack of) feedback (informing the student if his/her responses or remarks are correct or are accepted) on ELL studentsÂ’ experiences in the mathematics classroom. For example, Steele (1999-2000) investigated how one t eacher employed discourse and questioning techniques to develop student sÂ’ algebraic reasoning while finding patterns. The activity of finding size, color, shape, and number patte rns in a calendar was used as a tool to develop reasoning skills and vo cabulary building in context. Steele found that the teacher
32 was able to create an atmosphere of pr oductive discourse in which students were facilitated in the development of their alge braic thinking. The teacher achieved that by employing challenging questioning techniques which stimulated studentsÂ’ high-level thinking. For example, she asked students not only to make predictions for possible patterns, but also to support them with logical reasoning. Additionally, the teacher skillfully involved students to provide logical arguments and to correct themselves when necessary. Writing in mathematical logs was us ed in order for students to organize their thoughts in anticipating the teach erÂ’s questions and their pos sible answers. The teacher not only asked students questions, she was also an active listener. She was always open to change her initial plan based on students Â’ predictions and ideas. Thus, this study demonstrated how the teacher successfully Â“used questions to probe, stimulate, and initiate studentsÂ’ alge braic thinkingÂ” (p. 96). Error Treatment and Feedback Weingrad (1998) investigated what type of error trea tment and feedback provided to students from the teacher cultivated polit e mathematical argumentation. The study also provided insights into how teachers encourage st udents to take risks and participate in discussions about mathematics by overcoming the Â“face-threatening actsÂ” (FTAs) when voicing their opinions or making public stat ements. Weingrad found that the teacher achieved this by balancing between politely requesting for all stude nts to participate (requesting for bids) and nominating particular students to do so. The teacher also used polite criticism (when students violate the no rms and rules of interaction) or provided challenges to elicit further elaboration of ideas. Moreover, without simplifying the request or without repeating it, the teacher used a Â“second nomination of challengeÂ” and
33 thus implied to a student that his or her re sponse does not need fixing, but rather more elaboration. Weingrad also demonstrated how the teacher used politeness strategies to repair Â“breakdownsÂ” in discourse (a situation when a student offered an incorrect answer or idea) and to return the discourse to its usual patt ern. Furthermore, Weingrad demonstrated how the teacher may let students know that he/she is interested in their ideas. However, the study was limited in scope because it did not provide information on how the students perceive the teacherÂ’s politeness and how they respond to it personally. The Role of Students in Math ematics Classroom Discourse Bills (1999) examined the role of in dividual studentsÂ’ communication in relation to the classroom culture and discourse. He applied linguistic comparative analysis to study the speech patterns of two high school boy s involved in one-to-one interaction with the teacher. As Â“a useful lens through which to review th e relationship between social positioning and mathematical enculturation in teacher-pupil relationshipsÂ” (p. 162), Bills used modality markers One example of such modality markers was the speakerÂ’s use of propositions in attachment to Â“privateÂ” verbs (verbs whose value cannot be measured and is known only in relation to its subject) such as think, believe, suspect The way the students used these particular markers was c onsidered to demonstrate their commitment or detachment from the mathematics classr oom/community. A similar modality marker involved examining the use of we and you in mathematics talk. Moreover, the speakerÂ’s addition of such adverbs as obviously actually frankly and the use of tag questions such as isnÂ’t it? was also considered an indicator of commitment. Bills found that one studentÂ’s communication mode was more impersonal in nature. The student often used we and you to show commitment to the mathematic s community. The fist-person singular
34 pronoun I was mostly used to express mathematical fact or action rather than personal opinion or statement. That speaker also exhibited confidence of his knowledge of mathematics and use of the technical termi nology involved by frequent use of adverbs such as just and obviously For example: Â“So obviously if the gradient of the normal is 1/2 the gradient of the tangent will be 2Â” ( p. 166). In contrast, the other student exhibited a more personal aspect of mathematics learni ng. He often demonstrat ed insecurities, and used questions in order to receive affirmation from the teacher of his ideas or actions. The significance of this study is in its modeli ng of how researchers might use linguistic analysis to examine the role of individuals (students or the teac her) in the social environment of classroom discourse. Manouchehri and Enderson (1999) also reported on the examination of mathematics classroom discourse to illumina te the role of students in shaping the ambiance of such discourse. Their findings indicated that the students mutually influenced each othersÂ’ lear ning of mathematics by engaging in small-group or wholeclassroom discussions about mathematics. The mechanism by which this learning occurred involved the compilation of ar gumentation, collaboration, negotiation of meaning, and refinement of conclusions. By ex tension, the students were also involved in systematic group inquiry, where they were actively involved in idea sharing, finding patterns, and collaboration. Relationships between Interventions and ELLsÂ’ Mathematics Achievement Several studies revealed improvement in ELLsÂ’ mathematics achievement versus just positive differences in ELLsÂ’ experiences in mathematics, by examining standardized test scores. Some linked this to the adopti on of bilingual programs (Liberty, 1998), or
35 summer training programs for both teachers and students (Lara-Alecio, Cmajdalka, Parker, Cuellar, and Irby, 1996). Other stud ies specifically examined what effect teachersÂ’ instructional practices that promote discourse (i.e ., specific teacher Â“discursive movesÂ”) have on ELLsÂ’ mathematics achieveme nt (Patrick, Turner, Meyer, & Midgley, 2003; Sherin, 2002; Turner, Meyer, Midle y, & Patrick, 2003; Wood, 1999). However, several studies indicated that standardized tests are usuall y based on an English speaking population and thus are inhere ntly biased against ELLs. They suggested that new assessment instruments need to be developed in order to more accurately measure ELLsÂ’ achievement in mathematics (Abedi, Lord, Hofstetter, & Baker, 2000; Gronna, ChinChance, & Abedi 2000; Liu, Anderson, & Thurlow 2000). Mathematics Instruction in Bilingual Programs Educators have considered different outle ts to enhance the academic experience of ELLs. Regarding mathematics, for example, students may receive mathematics instruction in their native tongue as part of a bilingual education program. These classes allow students to develop their mathema tical understanding while developing their literacy skills in their native language a nd English. Results from several programs revealed that ELL students were able to make achievement gains in mathematics while engaged in these programs. For example, Liberty (1998) conducted a study to examine the effects of a 2-year program in a sc hool that employed an English-as-a-SecondLanguage (ESL)/Transitional Bilingual Educat ion Program. The program addressed staff development, material adoption, and parent al education. On a c ontent knowledge of mathematics test written in Spanish, ELL stude nts showed achievemen t levels near the national average. These positive results were at tributed to rigorous teacher professional
36 development programs, ESL certification, the ac quisition of better materials, and parent education. As another example, the study conducted by Lara-Alecio, Cmajdalka, Parker, Cuellar, and Irby (1996) reveal s the influence of a summer program on both teachers and students. They conducted a 3 year-long study (from 1993 to 1995) of 200 fifth-grade ELL students (mostly Hispanic) from an ur ban Houston public school. Students, eight bilingual teachers, and eight bilingual aids participated in a 6-week voluntary summer program to improve studentsÂ’ English proficiency. The math ematics content was used as a means for teaching English. The researchers analyzed the results from preand posttests (students were given the choice of the language on each test) and investigated four main indices of pedagogy: Â“(a) Activity struct ures, (b) Language content, (c) Language of instruction, and (d) Communi cation modeÂ” (p. 4). The results of the assessments indicated that ELL students gained mathematical knowledge in four targeted areasÂ— fractions, charts and graphs, measurement a nd geometry, and problem solving. Most of the gain was observed during the first year of the program. Additionally, data collected from interviewing teachers revealed genera l satisfaction with the program, curriculum materials, real world problems orientati on, and instructional st rategies learned for teaching mathematics concepts in both language s. TeachersÂ’ aides and small class size were also pointed out as positive facets of the program. Another program that reported positive resu lts in the mathematics achievement of ELL students is the QUASAR (Qualitativ e Understanding Amplifying Student Achievement and Reasoning) project (Lane, Silver, and Wang, 1995). Results from that project indicated that gains were evident in all groups of students, including bilingually
37 educated Latino students. All students benef ited almost equally from this reformed mathematics education and sufficiently develo ped their reasoning skills and critical thinking in mathematics. Alternative Mathematics Pr ograms for Migrant Students In a descriptive report, Celedon-Pattichis (2004) di scussed various programs designed to assist migrant st udents in learning mathematicsÂ—The University of Texas Migrant Student Program, Project SMART, ESTRELLA, and The Portable Assisted Study Sequence. Each program was designed in order to incorporate migrant studentsÂ’ linguistic and cultural considera tions in delivering the mathem atics content. Some of the programs used distance learning forms for de livering instruction and assessment via emails, interactive discussions, and lessons on video or TV, while others delivered information face-to-face by providing tutoring sessions for the students during a convenient timeafter school or during the summer. The programs were developed using accumulated knowledge about second language acquisition and incorporating CumminsÂ’ (1992) distinctions between studentsÂ’ exhi bition of basic interp ersonal communication skills (BICS) and cognitive/academic language proficiency (CALP) needed in academic subjects such as mathematics. The author discussed the source of diffi culties and challenges of mathematics word problems for migrant students. From think-aloud protocols teachers can understand what words help or hinder studentsÂ’ understanding of the problem. Celedon-Pattichis found that, often, the language used in ma thematics word problems could confuse students because it contains a mixture of everyday social language with academic language. Thus, students are often confused with words having a double meaning in
38 different discourse. Additionally, some word problems might be difficult to understand due to a cultural problem because student sÂ’ personal experiences do not match the linguistic expression used for st ating the problem. For exampl e, students might not have a schema of Â“HallÂ’s planetariumÂ” because they have never heard of or visited one. Thus, the programs used word problem s and activities in mathematics that incorporate migrant studentsÂ’ experiences. For example, an activity is suggested that can be used to teach migrant students the concept of distance ( d = r t ). In this activity, the students can be asked to plan a trip, find the best route, how long each route is, and plan the budget for the trip. The success rate of the programs in the recent years has made them widely used to meet the needs not only of migrant, but also of any alternative, nontraditional, culturally, and linguistica lly diverse students. Teacher Education and In-Service Programs Educators have considered different appr oaches for helping teachers enhance the mathematics learning experiences of ELL student s. In this section, I highlight several projects that reform traditional education in mathematics in an attempt to accommodate ELL students. For example, Cahnmann and Ho rnberger (2000) implemented a 3-day summer institute to address Â“language-base d mathematics learning of ESOL students from low-income urban contextsÂ” (p. 42). During the workshop they presented teachers, administrators, and resource specialists with samples of student work to raise their awareness about the link between mathema tics, language, and assessment practices. Based on the analysis of studentsÂ’ work, the educators reflected on the complexities that are associated with the use of content specific vocabulary and grammar involved in mathematics.
39 Other researchers have examined the effects of innovative programs on the mathematical performance of ELLs in a classroom setting. For example, Halpen, Patkowski, and Brooks (1996) examined the eff ects of a pilot program with a class at the City University of New York Brooklyn Colle ge, which combined the teaching of ESL with Calculus I. The results of the progr am demonstrated how the studentsÂ’ language developed through their study of the concepts, reading, and vocabulary of Calculus. Their lingual development was thus dually enhanced, as they employed a glossary of English terms commonly used in Calculus and mathema tics in general. Similarly, Brenner (1998) investigated the use of an innovative mathematics programÂ—College Preparatory MathematicsÂ—with Hispanic ELL student s in order to evaluate classroom communication. The results revealed that stude nts in the classroom in which techniques such as small-group discussions were enc ouraged and computers were employed to stimulate discussions, showed more successf ul mathematical communication, which later spread to a large-group setting. Culturally Relevant Education During the past decade, much attention has been given to the need for culturally relevant or responsive instru ction. This notion was highlight ed by Ladson-Billings (1994) who examined the effective te aching of African American St udents. Culturally relevant instruction refers to pedagogy that recognizes the importan ce of studentsÂ’ culture and empowers students by using cu ltural references as a sp ringboard for student learning. Currently, culturally relevant in struction is encouraged as a means to address the needs of a diverse student population (Cahnmann & Remillard, 2002; Gay 2000; Gustein, Lipman, Hernandez & de los Reyes, 1997; Mattews, 2003). To provide instruction that is
40 culturally relevant, teachers need to unders tand how studentsÂ’ culture (i.e., values, beliefs, customs, social norms, and language) influences their exp ectation for learning, their preferred learning styl es (e.g., independent vs. coll aborative), their preferred communication, and preferred problem solv ing style. For example, Cahnmann and Remillard (2002) studied the issues and challenges two teachers experienced while teaching mathematics in culturally, linguist ically, and socio-economically diverse classrooms. They focused on the role of the teachers in providing equal mathematical experiences to all students while exercising culturally relevant teaching. Cahnmann and Remillard found that even though both teach ers were deeply committed to foster mathematics understanding in their students, they implemented the ideas of culturally relevant teaching in different manners. The t eachers had different in terpretations of the reformed ideas, received different support and professional development in their educational communities, and had differe nt comfort levels and knowledge in mathematics. The researchers also indicated that even though it might be beneficial for the cultural and the linguistic background of the teacher to be similar to that of the students, all mathematics teachers could use some ideas from research and incorporate culturally relevant instruc tion in mathematics to di verse student populations. Mattews (2003) also studie d how teachers utilized st udentsÂ’ prior knowledge in mathematics, along with their diverse cultural background, in th eir instruction in order to develop studentsÂ’ critical th inking and empower them with experiences in mathematics related to their cu lture. Research results revealed that mere Â“good intentionsÂ” for teaching mathematics by using a culturally relevant approach are not enough. Deeper changes in teaching methods, practices, beliefs, values, a nd expectations are needed. He provided an
41 illuminating example of a successful applicati on of culturally relevant teaching when he discussed a teacher who was able to involve students in critical discussions about the real-life application of large numbers and scientific notation. The teacher skillfully fostered the studentsÂ’ critical thinking and engaged them in an activity to compare the areas of parks in the community, thus rela ting studentsÂ’ informal knowledge about large numbers and cultural background. Mattews (2003) also provided examples of the challenges te achers encountered when they tried to incorp orate culturally relevant t eaching in their mathematics instruction. They sometimes failed to involve their students in critical discussion because the discussion of social issues often interfered w ith teachersÂ’ comfort level about such issues or their knowledge in mathematics. So metimes, the mathema tics experiences that teachers provided to their students were base d on the teachersÂ’ ideas of what might be relevant to studentsÂ’ culture, but often such relation was artif icial and peripheral, because the teacher failed to understand the studentsÂ’ deep cultural and i ndividual experiences, and thus failed to build an activity based on gl obal cultural characteristics still related to the particular group of students. Teachers tend ed to build Â“toÂ” instead of Â“onÂ” studentsÂ’ cultural background and informal knowledge of mathematics. Another factor impeding on teachersÂ’ use of discussion as a critical source of knowledge that needs to be incorporated into their mathematics lesson is that some teachers viewed conversations about studentsÂ’ personal experiences and culture as a devi ation from the lesson focus, and as providing too much Â“extraÂ” information.
42 TeachersÂ’ Instructional Practic es that Promote Discourse Some scholars focused their attention on th e successful instruc tional practices and techniques teachers use in order to promote discourse more effectively (Patrick, Turner, Meyer, & Midgley, 2003; Sherin, 2002; Turn er, Meyer, Midley, & Patrick, 2003; Wood, 1999). For example, Sherin observed 78 eighthgrade classes, scrutinizing the dilemma teachers faced when they tried to promote a student-centered instruction approach that involves facilitation of the discussions, while at the same time ensuring that discussions are mathematical in nature and that learni ng occurs. Research results indicated that teachers who used instructional practices balanced in process and content of discourse, and shifted focus between both aspects, were actually more successful in the facilitation of classroom discourse. Wood (1999) also investigated the role of the teacher in crea ting an environment in which students were engaged in mathem atical discourse. He found that teachers who were able to first establish discursive norms and patterns of behavior in their students actually lifted the cognitive attention and focus off the social organization of the interactions (such as turn-t aking and the use of courteous language) and shifted it to discourse about mathematical ideas. Thus, the students were able to follow each otherÂ’s logic and ideas, and focused on the mathematical context rather than on its social form. Turner et al. (2003) examined the effect of the teacherÂ’s discourse and instructional practices on studentsÂ’ motiva tion and performance in mathematics in two sixth grade classrooms with similar highmastery/high-performance students. The researchers found that practices in the teacher Â’s organizational and motivational discourse which were consistently positive and supportive resulted in academic self-regulation and
43 higher mastery and performance in the math ematics classroom. On the other hand, the classroom in which the teacher was less c onsistent and sometimes used non-supportive discourse had a negative result that often consisted of failu re in mastery and performance by the students. Similar findings were reporte d by Patrick et al. (2003). Patrick et al. observed analogous patterns in 6th grade teacher behavior elicited from the eight 6th grade classes observed. Thus, both studies indicate the decisive role of the teacher in providing a positive and supportive discursive environmen t in which students are encouraged to take risks, make mistakes, try out ideas, and collaborate with others. In such an environment, students become more motivated to pursue goals toward higher mastery and performance in mathematics. Methods of Assessment and their Eff ect on ELLsÂ’ Mathematics Experience Another area of research concentrat es on studying the influences of the assessment structure on ELL studentsÂ’ achieve ment in mathematics. Several studies suggest that standardized te sts are usually based on an E nglish speaking population and thus are inherently biased against ELL students (Gronna, Chin-Chance, & Abedi 2000; Liu, Anderson, & Thurlow 2000). For example, Gronna, Chin-Chance, and Abedi (2000) investigated the performance of a large 3rd, 5th, and 7th grade Hawaii public school population on the Stanford Achievement Test (9th edition), administ ered during the 19981999 school year, in order to study the relatio nship between ELL studentsÂ’ achievement on such tests and their English language prof iciency. They found that performance varied significantly between ELL and English speaki ng students. The scores of ELL students indicated a higher level of achievement in mathematics as opposed to reading when compared to the scores of non-ELL students. This higher level of achievement was
44 specifically evident on calculation-type ma thematics problems as opposed to word-type problems. Kiplinger, Haug, and Abedi (2000) evalua ted the effects of studentsÂ’ English reading ability on their success on mathematics tests. The chief goal of this study was to provide research data to gui de future development and construction of new assessments with linguistic accommodations for ELLs and students with special needs. The researchers administered three versions of a mathematical test to 1,198 fourth-grade students. The first version of the test wa s written in English with no accommodations. The second version consisted of simplifie d phrasing in the pr oblemsÂ’ descriptive language. The third version was written in E nglish without simplifications, but students were provided with a glossary to use duri ng the exam. The researchers found that ELLs performed better when they used accommoda tions, most notably on the linguistically simplified version of the test. Furthermore, ELLsÂ’ performance on mathematics tests with a higher number of word problems was strongly related to their Engl ish reading abilities. In a similar study, Abedi, Lord, Hofstett er, and Baker (2000) investigated the effects of specific accommodation strategies and aimed to determine their impact on ELLsÂ’, as well as on English proficient studentsÂ’, performance on mathematics tests containing word problems. They collected da ta from 946 eighth-grad e students from five California middle schools who had been tested on five different tests: (a) with original NAEP items for comparison, (b) with linguisti cally modified items (simplified English version), (c) with a glossary, (d) with extra time provided, (e) with a combination of extra time and a glossary. They found that a major ity of the ELLs improved their scores when extra time was given and when they had access to a glossary or were aided in
45 understanding the mathematical concepts pres ented to them through the use of simpler language. Particular combinations of these methods increased performance even more. Research Methods for Examining Classroom Discourse In order for researchers to examine more closely and accurately ELLsÂ’ mathematics experiences and achievement in classrooms in which the dynamics involve teachers and students in rigorous discussions, new paradigms of research need to be adopted. This is supported by the fact that, as research already ha s implied, Â“there is a need of more interdis ciplinary collaboration in resear ch design, data collection, and analyses requiring close attention to ta lkÂ” (Adler, 2001, p. 513). The new methodologies need to be able to properly evaluate th e process and the content of mathematical discourse and assess the predominant factors that contribute to ELLs Â’ membership in the mathematical discursive communities, in th e sense that Sherin (2002) defines such membership. In order to organize and classify the main research traditions (main research approaches and adopted methods of research ) of past and present research, and draw attention to the current trends in exam ining classroom discourse, the framework developed by Ellis (1994) is used in the present study. He applied ChaudronÂ’s four main categories to describe the traditions in resear ch in the field of second language acquisition (SLA): (1) psychometric tradition, (2) interac tion tradition, (3) discourse analysis, and (4) ethnographic tradition. An assumption is made here that research on discourse in the mathematics classroom has similar characteri stics to research on discourse in the language classroom. (1) The psychometric tradition usually examines mathematics achievement as an end product of the applica tion of different methods of t eaching, curricula, and use of
46 materials. Experimental methods of res earch are employed under this tradition and, usually, data of preand post-tests between control and experimental groups undergoing a specific treatment is analyzed. The impedime nt to the application of such methods of research is that usually little or no account is taken of the vari ety of the contributing factors to studentsÂ’ performance on post-tests. Usually, more positive results of the experimental group over the control group are automatically attribut ed solely to the treatment applied (Ellis, 1994). Due to these inherent flaws, no studies in the field of discourse in the mathematics classroom were identified as generally employing such a research methodology. (2) Studies that investigate the relati onship between stude ntsÂ’ behavior and performance as well as the teacherÂ’s in teraction and methods usually fall under the interaction analysis tradition Such research methods t ypically include counting the frequency of occurrences of events during in teractions. Then, using coding schemes, the classroom interactions are categorized and analyzed. However, a problem with this method might be that in concentrating on differe nt utterances in isol ation (i.e., what the teacher said, what question was asked, and how often), the global picture of the discourse could be missed (i.e., why this was said, what were the teacherÂ’s intentions, what was the sequential flaw of the conversation), thus Â“casting doubt on the reliability and validity of the measurementsÂ” (Ellis, 1994, p. 567). In the early 1980s Allen, Frohlich, and Spada developed a means of examining the interactions taking place in a classroom setting called the Communicative Orientation of Language Teaching (COLT) Observation Scheme The COLT Observation Scheme was used to investigate the effects of communicative language teaching on second
47 language acquisition in program s developing bilingual proficie ncy, or in second language (French, English, etc.) immersi on programs. Most studies using the COLT observation scheme Â“provided evidence that a combination of form and meaning worked better than exclusive focus on either meaning or form Â” when adopting comm unicative methods of instruction (Spada, & Frohlich, 1995, p. 7). In addition, rationales for each of the categories used to code communication in a classroom are provided. However, the system is divided into two parts. Part A is used in real-time coding and emphasizes seven main categories to describe the events that take place in the language classroom: time, activities and episodes (drill, game, discussi on), participant organi zation (whole class, individual or group work), content (topics -language, subject matter, management), content control (teacher/text, teacher/text/st udent, student), student modality (listening, speaking, reading, writing, other), and materials (text, audio, vi sual). As it is described, this part of the COLT coding scheme focuses on the instructional practices adopted in the classroom, while Part B is focused on coding the communicative features in the classroom. The coding is done from the audio (and/or video) record ings obtained during the classroom observations and focused on the verbal interactions that took place between students and teachers within activities a nd episodes. Seven main communicative categories are identified: use of target la nguage, information gap (giving or requesting information), sustained speech (minimal or not), reaction to form/message, incorporation of utterances in discourse (correction, repetition, paraphrase, comment, etc.), and discourse initiation and form restriction (coded only for the students). However, there are important caveats to be considered before one uses the COLT observational scheme in the classroom:
48 First, it is important to emphasize that the COLT scheme offers one way of looking at instructional pract ices and procedures in L2 classrooms and, depending on the userÂ’s purpose and needs, it may be more appropriately used in some contexts than in others. For example, if one is interested in un dertaking a detailed discourse analysis of the conversatio nal interactions between teachers and students, another method of coding analys is of classroom data would be more appropriate. Similarly, if one is interested in carrying out ethn ographic research in classrooms, the COLT scheme (or any othe r scheme with a set of predetermined categories) would not be suitable gi ven the difference in theoretical and methodological perspectives between ethnogr aphic and interaction approaches to classroom observation. (Spada, & Frohlich, 1995, p. 10) To avoid such impediments, some res earchers investigating the discourse in mathematics classrooms have proposed the use of linguistic tools (Bills, 1999; Rowland, 2002) that aid in examining utterances not in isolation, but in relati on to the global picture of discourse. For example, Rowland proposes th e use of linguistic tools such as Â“hedgesÂ” ( maybe probably possibly ), Â“attribution shieldsÂ” (so-and-so says Â…), and Â“shieldapproximatorsÂ” ( about around basically ) that focus on the Â“pragmatic meaningÂ” of the mathematical discourse. The term Â“pragma tic meaningÂ” is defined by Rowland as the means frequently (though not necessa ry consciously) used by speakers to convey affective messages to do with social relations, attitudes and beliefs, or to associate or distance themselves from the propositions they articulate. That is to say, pragmatic meaning is an important tool in fulfilling the interactional function of language. (p. 2)
49 This correlates to the already-addressed notion of BillsÂ’ modality markers that are used as linguistic tools to examine the role of individuals (stude nts or teachers) in classroom discourse by measuring the status, re liability, and truth value of a statement. (3) A more systematic description of the interactions that occur in mathematics classrooms may be gleaned from the discourse analysis tradition Methods used in the discourse analysis tradition include analyzing classroom tr anscripts where account is taken of the nature of the mathematics clas sroom environment as a whole in addition to the role of both the teacher and students in contributing to the interactions in order to negotiate meaning and understanding in the process of teaching and learning mathematics. In such analysis, the functions of individual utterances are combined in a larger discourse unit (Brenner, 1994; Lobat o, Clark, & Ellis, 2005; Knuth, & Peressini, 2001; Krussel, Edwards, & Springer, 2004; Sfard, 2002). For example, Knuth and PeressiniÂ’s (2001) framework for examining and classifying the teachersÂ’ discourse in mathem atics classrooms used two such larger units for classificationÂ— univocal and dialogic Univocal discourse refers to the teacher being the authority and, usually, any discrepancie s in student answers from the teacherÂ’s expected responses are evaluated as mistakes. Dialogic discourse is when such discrepancies are used as Â“t hinking devicesÂ” to generate further discussions and thus generate new mathema tical understanding. On the other hand Krussel et al. (200 4) proposed their own framework for categorizing Â“teachersÂ’ movesÂ” (d eliberate actions they take) as facilitators of discourse in the mathematics classroom. According to this framework, the teacherÂ’s discourse: (1) has an intended purpose (to move the activity to reflections, justifications), (2) takes
50 place in a setting (small-group or whole-class discourse), (3) has a particular form (verbalÂ—questions, directions statements, clarifications challenge; or non-verbalÂ— gestures, face expressions), and (4) results in consequences (immediate or long-term). Enriching the discourse analysis tradition, Brenner (1994) developed a Communication Framework for Mathematics which he found very useful in classifying the communication in mathematics classrooms with a predominantly ELL population (for the application of this framework see Brenner, 1998) According to this framework, communication in mathematics classrooms or in small group-discussions falls into three main categories: 1) communication about mathematicsÂ—which reflects on cognition, reasoning, and metacognition; 2) communication in mathematicsÂ—math register, special vocabulary, symbolism, and repr esentations; 3) communication with mathematicsÂ— problem-solving tools, investigations, alterna tive solutions, etc. The framework aids in finding patterns in the differ ent types of mathematical co mmunication and the languages (English, Spanish, or others) us ed in such interactions. Lobato et al. (2005) categorized comm unicative actions, taken by teachers to facilitate students in th eir conceptual understanding of mathematics, in an initiationeliciting framework By analyzing three teaching episode s, the authors demonstrated how teachers used Â“tellingÂ” in a reformed way to promote conceptual understanding: (1) by focusing their communicative acts on function rather than on form, (2) by presenting new information in a conceptual rather than a procedural manner, and (3) by presenting each action in relation to other act ions. Thus, the authors made a reformulation of teachersÂ’ Â“tellingÂ” and demonstrated how they can use Â“initiationÂ” (introducing new mathematical ideas that stimulate studentsÂ’ thought in constructing conceptual understanding in
51 mathematics) and Â“elicitingÂ” (when the teacher uses an idea originating from a student in order for students to make further conject ures and employ new ways in viewing and conceptualizing mathematical situations) to foster studentsÂ’ learni ng of mathematics. (4) The ethnographic tradition in research involves na turalistic Â“uncontrolledÂ” observations and detailed descriptions of th e classroom discourse. Thus, it could provide more insight into teacher and student cogni tion, and a deeper account could be taken of the uncontrolled additional variables that a ffect the process of teaching and learning mathematics. However, one of the problems in this tradition of research is that its methods require a very experienced and well-trained ethnographer, independent of whether he/she is an active or non-active participant in the cl assroom discourse. Yet another problematic area with this type of res earch is the fact that it is time consuming to collect and analyze the data, and difficult to make generalizations and warrant the transferability of the study in other conditions and situations However, despite the listed difficulties in applying such a line of research, naturalistic studies provide more insight into what and how things happen in the classroom (Ellis, 1994). Being mostly descriptive in nature, they contribute to the bank of knowledge in research, and as Lampert (1998) asserted Â“the purpose of such interpretive re search is not to determine whether general propositions about learning and teaching ar e true or false, but to further our understanding of these particul ar kinds of human activity in the contexts where they occurÂ” (p. 160) Following this idea, Lamper t, along with teaching mathematics for seven years (from 1982 until 1989), conducted resear ch on her own teaching. Later on, she built a research team that used ethnographic re search methodology Â“in order to examine the practical dynamics elements of establishing and maintain ing a culture, developing and
52 using tools for mathematical communication, an d creating a curriculum in the context of work on problemsÂ” (p. 159). (Findings from this research teamÂ’s investigations on discourse have been listed throughout this paper.) Many studies cannot be simply classified into one distinctive category because they often employ a combination of research methods and span across several categories. Often, data from quantitative and qualitative studies of classroom discourse is used in order to create a better depiction of the natu re of classroom disc ourse (Ellis, 1994). And as Adler (2001) stated in The Handbook of Discourse Analysis even though discourse analysis and qualitative methods of resear ch Â“are not widely accepted even within the educational establishmentÂ” (p. 513), if expe rts from different domains (i.e., first and second language acquisition, linguists, child renÂ’s cognitive development specialists, psychologists, mathematics educators, and ma ny more) work in collaboration in studying discourse, better result s could be achieved. As an example of such collaboration, Atweh, Bleicher, and Cooper (1998) used Hallidey and HasanÂ’s (1989) socio-linguist ic model to propose a framework for investigating the mathematics discourse in two ninth grade classes in two different schoolsÂ—one a boysÂ’ school from high SES and th e other a girlsÂ’ school from low SES. The researchers focused on examining how ge nder and studentsÂ’ socioeconomic status (SES) affect the teacherÂ’s perception of thei r mathematics abilities and needs, and shape the discourse in the classroom. By the propos ed framework, the comparison concentrated on finding different patterns in thr ee areas: (1) differences in the field (social actions; what is happening in discour se), (2) differences in the tenor (the relations and role of the
53 participants in discourse), and (3) differences in the mode (what part the language is playingÂ—rhetorical mode; symbols in context). Findings of differences in the field indicate that even though the discourse in both classes was teacher-centered, th e two teachers used different approaches to present the curriculum. The teacher from the boysÂ’ school used more rigorous formal math language and related the topic under inve stigation to other s ubjects and real world practices. This was related to the teacherÂ’s perception that most of the students in his class are collegebound and need advanced mathematics skills. In contrast, the teacher from the girlsÂ’ school perceived his students as consumers a nd thus needing skills in consumer math. He used less formal language and defined only us eful concepts applying Â“rules-of-thumb.Â” These differences in tenor revealed that in accordance to his perceptions that his students need to be self-regulated, independent lear ners, the teacher from the boysÂ’ school created a general atmosphere of competition and often used sarcasm to challenge his students. In contrast, the teacher from the gi rlsÂ’ school assisted his studen ts with their difficulties and corrected their mistakes in a very polite and courteous manner. Nevertheless, his language was personal and differed from im personal mathematics formal talk. The comparison of the mode of discourse between the two cla sses revealed that the teacher from the boysÂ’ school used a voice of aut hority and stressed important terminology. At the same time, he often encouraged argumenta tion and applied sarcasm. In contrast, the teacher in the girlsÂ’ school stressed little on key words and little or no argumentation was used. Thus, the authors conclude that Â“clas sroom interactions, being consistent with teacher perceptions, tend to have a self-ful filling role for teacher expectationsÂ” (p. 82).
54 Chapter Summary In this chapter I have provided insights about the theoretical frameworks, research approaches, main findings and discussions that have influenced my studyÂ’s ideas. I have highlighted what is currently known about ELL studentsÂ’ learning of mathematics in classrooms in which discourse is a featur e. Research has al ready indicated one encompassing idea: that involving ELLs in m eaningful discussions about mathematics is a giant step in reaching the goal of providing membership in the mathematical discourse community and providing a quality mathematical education for all students.
55 CHAPTER III: METHODOLOGY AND PROCEDURES Based on knowledge collected from the litera ture review, the goal of this research is to investigate the nature of the di scourse in mathematics classrooms adopting approaches from the discourse analysis trad ition. Â“In discourse an alysis, the units of analysis are variable and may range from words, phrases, and sentences to paragraphs or even larger unitsÂ” (Wood & Kroger, 2000, p. 28). Mathematics education researchers who have adopted the discourse analysis trad ition use methods that consist of analyzing classroom transcripts in which account is taken of the nature of the mathematics classroom environment as a whole in additi on to the role of both the teacher and the student. Thus, using discourse analysis, I examined the influences of teacher talk on the inclusion or exclusion of Eng lish Language Learners (ELLs) in classroom interactions. I investigated how teachers negotiate mean ing and understanding in the process of teaching mathematics to classes with a ve ry diverse linguistic and cultural student population. This includes an examination of whether teachers adjust or modify their patterns of discourse depending on the number of ELL students present. Furthermore, I investigated whether differences exist in teachersÂ’ discourse methods based on their experience in teaching mathematics and their ESOL endorsement. The following research questions are addressed: 1. To what extent do teachersÂ’ patterns of discourse in the mathematics classroom change as a result of the numb er of ELL student(s) present?
56 2. To what extent do mathematics teache rsÂ’ experiences and teachersÂ’ ESOL endorsement relate to thei r patterns of discourse when teaching mathematics to classes with ELL students present? 3. How do teachersÂ’ own linguistic and cultu ral backgrounds affect their patterns of discourse when teaching mathematics in English, and especially to classes with ELL students present? 4. What patterns of discourse do teachers use when ELL students are present in the mathematics classroom? What adjustments to teacher talk or modifications of instructions are observed? The Study The study was conducted during the Fall Se mester of the 2007-2008 school year and explored the patterns of discourse between high school mathematics teachers and their students, especially wh en ELL students are present. Context The participants were teachers and their mathematics classes fr om two urban U.S. public high schools in the Sout heast with diverse student populations. The schools were deliberately chosen because they have a la rge population of ELL students from a variety of backgrounds (See Table 1). Table 1 illustrates that th e two schoolsÂ’ student populati ons are comparable in size and diversity. However, there are some discer nible differences in the percentages with respect to their raci al and ethnic groups. More speci fically, while both schools have comparable percentages of Non-Hispanic Bl ack, American Indian/Alaskan Native, and Multi-Racial students, there are differences in the propor tion of Hispanic, White, and
57 Table 1 SchoolsÂ’ Demographics Demographic characteristics Green Bay High School1 Lincoln High School Student population 1939 students 1872 students American Indian / Alaskan Native 0.31% (6 students) 0.27% (5 students) Asian / Pacific Islander 1.44% ( 28 students) 10.47% (196 students) Non-Hispanic Black (African American) 34.61% (671 students) 39.58% (741 students) Hispanic 46.73% (906 students) 14.64% (274 students) Multi-Racial 3.51% (68 stude nts) 4.97% (93 students) Non-Hispanic White (Caucasian) 13.41% (260 students) 30.07% (563 students) ELLs 10.93% (212 students) 6.20% (116 students) 1 Pseudonyms are used for schoolsÂ’ names.
58 Asian/Pacific Islander students. Also, while in Green Bay High School the majority of the students are Hispanics, followed by Bl acks and Whites, in Lincoln High School the majority are Blacks, followed by Whites, Hispanics, and Asians/Pacific Islanders. Furthermore, while in both schools the pe rcentage of students from economically disadvantaged families constitutes approximate ly half of the population, the schools have different percentages of ELL students. Table 1 indicates that the percentage of ELLs in Green Bay High School is almost tw ice the one in Lincoln High School. Both schools offer programs to aid the ELLs in their subject area classes. In addition to their core courses, most of the ELLs are encouraged to take intensive2 reading and mathematics elective classes or are prov ided with after school tutoring programs. In some of the Algebra I or Intensive Mathematic s classes, bilingual te acher aides Â– fluent in Spanish in Green Bay High School and fl uent in Spanish, French, and Arabic in Lincoln High School Â– were avai lable to assist ELLs. Participants The scope of the study was limited to eight teachers (four teachers per school) to allow focused attention on the discour se and to allow the researcher to examine the type of communication that occurs between teacher s and students in mathematics classrooms when ELLs are present and provide answers to the studyÂ’s resear ch questions. Eight teachers participated in the study there were two female teachers and two male teachers from Green Bay High School, and three female teachers and one male teacher from Lincoln High School. The teachers also va ried by their linguistic and cultural 2 Such intensive courses are Intensive Reading I, II, and III (wherein guided instructions are provided to improve studentsÂ’ vocabulary, comprehension, and criti cal reading skills) and Intensive Math I, II, and III (wherein instructions focus on helping students acqui re the competencies necessary to pass the StateÂ’s Comprehensive Test).
59 backgrounds Â– in each school there was one African American, one Hispanic, and two non-Hispanic White (one male and one female ) teachers. Teachers were selected based on the following criteria: 1. Years of teaching experienceÂ—two teach ers in each school with many years of teaching experience and two teacher s in the beginning of their teaching careers were chosen to participate. 2. Teachers with/without ESOL endorsementÂ—two of the teachers in each school with many years of teaching e xperience have an ESOL endorsement, and the inexperienced teachers did not ye t have, or were currently working on, their ESOL endorsement. (Theoreticall y, a teacher who does not have their ESOL endorsement cannot teach an ESOL child, unless he or she is perhaps in compliance with the timelin e set for a beginning teacher to complete these requirements in the first two years of teaching mathematics to ELLs; for an experienced teacher the timeline is a year.) 3. The teachers teach mathematics courses at the same level Algebra 1 (with zero or a small number of ELLs or with a larger number of ELLs depending on the population of the classes). Algebra I classes were sele cted because this is a core subject that is a graduation requirement for all students. Because of this, ELLs must also take and succeed in this course to graduate. The study required finding teachers with cer tain years of experience and ESOL endorsement who are teaching Algebra I to classes with a varied number of ELL students, and who are willing to participate in the study. In both schools, three of the
60 participants were easily identified, however the fourth participants were teaching a similar curricular course such as Intensiv e Mathematics in one school and Liberal Arts Mathematics in the other that include some al gebra content. To be consistent across the curricula, I observed all teachers when they were teaching topics identical with the Algebra I curricula Â– linear e quations. Additionally, in both sc hools, some of the Algebra I classes were taught using a comput er-based instructional program, I Can Learn Lab. Instruments Several tools were used to collect data relating to the communication that occurs between teachers and students in mathematics classrooms when ELLs are present. A description of each instrument is provided below. Â“Teacher Talk TestÂ” (TTT) Forms 1 and 2 Each teacher was observed using the TTT protocol. They were observed while teaching similar topics to their mathema tics class (with or without ELL students) Â– Coordinates and Scatter Plots, Graphing Line ar Equations, The Slope of a Line, Quick Graphs Using Slope-Intercept Form, Functions and Relations, Writing Linear Equations in Different Forms, F itting a Line to Data, and Predicting with Linear Models. The TTT Form 1 was used to obtain information about the teach erÂ’s patterns of discourse and teacher talk, measuring the teachersÂ’ frequenc y of the below mentioned teaching techniques (see Appendix A). The TTT Form 1 is partially based on the ELL Strategies Verification Form provided to teachers by the local school dist rict and used by the State Department of Education to perform Â“walk throughÂ” ELL Compliance Audits. The version used in this study, TTT Form 1 includes not only items that rela te to second language acquisition
61 (SLA), but also items that relate to teaching mathematics as a content area, thus reflecting the idea that content area teachers should encourage ELL students to participate in classroom discourse and thus help such students develop thei r abilities in both mathematics and the English language (Brenner, 1994, 1998; Moschkovich, 1999, 2002). Guided by these ideas, items were included th at account for teachers asking inferential and higher order questions according to BloomÂ’s Taxonomy of Six Cognitive Levels and ELL studentsÂ’ four stages of language developmentÂ— pre-production, early production, speech emergence, and intermediate fluency The instrument also includes items that indicate the extent to which the teachers use modifications, or accommodations of their speech, when ELL students are present. For ex ample, do they use synonyms for difficult mathematics terms, or any potentially difficu lt words in English? Additionally, guided by GeeÂ’s (2005) definition of Â“Dis course,Â” with a capital Â“DÂ”, and the notion that Â“people build identities and activities not just through language but using language together with other Â‘stuffÂ’ that isnÂ’t languageÂ” (p. 20 ), I have also included items in the TTT Form 1 that reflect not only teachersÂ’ talk and interactions with their students, but also teachersÂ’ actions and behaviors in general such as gest ures, models or visual images, Â“hands onÂ” activities and the like, that formulate communication with ELLs while they teach mathematics. The TTT Form 1 is comprised of items that are deemed to be among the best practices by educational research, especi ally with regards to ELL students. The list does not suggest that all strategies should be used in each lesson. It rather encapsulates the most widely used strategies for teach ing mathematics to ELL students according to the research found in th e literature review (see Chapter Two).
62 After classroom observations, teachers were asked to complete a different version of the TTT Â– Form 2 (see Appendix C). This instrument includes the same items as Form 1 however the teachers were asked to complete a checklist (yes/no/needs improvement) to evaluate their own patterns of discourse. A dditionally, teachers were asked to rate their use of each strategy on a frequency scale from 1 to 5, with 5 as the most frequent. This provided an opportunity to collec t data about each teacherÂ’s pe rceptions of his or her own teaching and on the classroom experiences they provide. Pre-observation Teacher Questionnaire To collect data about teachersÂ’ Â“way of thinking, believing, valuingÂ” (Gee, 2005) not only about the subject they teachÂ—mathema tics, but also about the way they teach it to a linguistically and cultura lly diverse student population, a Teacher Questionnaire (see Appendix B) was developed. This instrument includes questions about teachersÂ’ years of teaching experience, ESOL certification, a nd their cultural and linguistic background. To gather data about their perceptions about their ELL student s, teachers were asked to identify the ELL students (if any) in the observed class and to comment on their perceived stage in SLA. (Each teacher was provi ded with a list of the definitions for each of the stages in SLA Â– pre-production, early produc tion, speech emergence, and intermediate fluency .) The goal was to determine what understanding teachers have about their ELL students, linguistic and cultural differences, and whether they use this knowledge to modify their mathematic s instruction. Teachers were provided opportunities to comment on their experien ces with teaching mathematics to ELL students, and/or indicate if they have concerns or recommendations for improvement related to these experiences. To avoid the Hawthorne effect (the fact that the teachers
63 might say what they want me to hear), data obtained from the teachers were compared to official data about the teac hers and their ELL students collected from the schoolÂ’s personnel and guidance departments. In order to ensure the studyÂ’s trustwor thiness and to eliminate any biases from influencing teachersÂ’ and ELL studentsÂ’ answer s to the post-observation questionnaires, the studyÂ’s participants were not provided access to any of the que stionnaires ahead of time. Student Questionnaire This Questionnaire (see Appendix D) was used to collect data about how students perceive their own ab ilities and experiences in SLA and mathematics; thus, this allowed an opportunity to discern similar ities or differences between studentsÂ’ and teachersÂ’ perceptions of their participati on in classroom discourse. This instrument includes questions that addre ss the studentsÂ’ ELL categorization according to their level of English language proficiency (each student was provided with a list of the definitions for each of the stages in SLA), and ma thematics experience backgroundÂ—previous mathematics courses taken, and grades. Additio nally, the instrument includes questions about studentsÂ’ family and personal attitudes about mathematics. Furthermore, during the interviews, students were given an opportuni ty to comment on their experiences with learning mathematics in English and to provide a self-evaluation about their participation in the observed lessons. All students were read the same ex act questions directly from the Student Questionnaire (see Appendix D for the specific questions) in order to limit the possibilities of asking a biased question and to minimize threats to the studyÂ’s
64 trustworthiness. To ensure that the questions were comprehensible to the ELL students of various levels of SLA, the complexity of the language used in the questionnaire was modified and simplified. The readability test of the Student Questionnaire indicated a reading ease of 83.4%, correspond ing to a fourth-grade read ing level according to the Flesh-Kincaid Grade Level Scale. The questi ons were translated for ELL students at the initial stages of Englis h language acquisition. The person translating and assisting in negotiating the meaning of each question and answer in the dialogue between the researchers and the students read the questi ons directly from the Spanish-translated version of the questionnaire (see Appendix E), which was checked for simplicity by a language professional who speaks Spanish, or was asked to use simplified studentfriendly language when translating in langua ges other than English and Spanish. Data Collection Procedures TeachersÂ’ Demographic Data Demographic data about each teacher pa rticipant were obtained from their personnel files. This data was compared to the data they provided on the Teacher Questionnaire. This comparison allowed the detecti on of similarities or differences between teachersÂ’ perceptions of their ELL students and the information on file. ELL StudentsÂ’ Demographic Data Demographic data about the ELL students in Algebra I classes were obtained from the schoolsÂ’ guidance departments to ve rify studentsÂ’ ELL level and placement in the ELL program. This information was used to supplement and verify the information provided by ELL students during the post-observational interviews.
65 Classroom Observations Each classroom was observed when the teacher introduced new material or reviewed an already taught t opic (e.g., similar topics at th e same level of mathematics Â– Coordinates and Scatter Plots, Graphing Line ar Equations, The Slope of a Line, Quick Graphs Using Slope-Intercept Form, Functions and Relations, Writing Linear Equations in Different Forms, F itting a Line to Data, and Predicting with Linear Models) In order to ensure that teachers taught similar topi cs of the Algebra I curriculum in different schools, all observations were conducted w ithin a three week period. Each teacherÂ’s Algebra I class (in one case Intensive Mathem atics class and in the other Liberal Arts math class) were observed on at least three o ccasions in order to better detect teachersÂ’ instructional patterns of talk or behavior (Â“teacher discour se movesÂ”). Each class was observed for approximately 20 minutes. The TTT Form 1 Observational Protocol was used to document the frequency of differe nt types of pattern s of discourse and teacher talk and interactions (i f any) with the ELL students. Multiple observations of the same teachers also provided evidence about the extent to which the observed patterns were robust and/or whether there were real differences in patterns based on content. Videotaped Observations Each observed instructional session was also videotap ed. According to Wood and Kroger (2000), Â“a videotape is clearly required if one is concerned with the coordination of discourse with other activiti es, for example, with the perf ormance of a (nonverbal) task or with features that are only available on video (e.g., facial expr ession)Â” (p. 70). The videotaped sessions were useful during the da ta transcription phase and when analyzing the teacher-student inte ractions, which included nonverbal communication.
66 Field Notes During each observation, field notes were ta ken to capture classroom interactions emerging in the process of teaching/learni ng mathematics throughout the entire class period (i.e., for approximately 45 minutes). This allowed the researcher to capture additional information about th e interactions between the teacher and the students and some specific characteristics of the nature of the classroom discourse including any unusual strategy that might not have been reflected in the TTT Form 1 instrument. Interviews After the classroom observations, the t eacher and group of ELL students in the class were interviewed. All interviews were conducted and videotaped on the day of the last observation to ensure that the discursive strategies used by the teacher are still vivid in the ELLsÂ’ and teachersÂ’ minds. The videotap ed interviews allowed for a reliable and accurate account of participant comments. Teacher Interviews The teachers were asked to comment on their already completed pre-observation Questionnaire for Teachers Teachers were also asked to self-evaluate the talk they had employed during the observed classroom se ssion (with various numbers of ELL students present) using the TTT Form 2 They were asked to reflect and comment on their experience not only during the classroom sessi ons under investigati on, but also on their general experiences in teaching mathematics to classes of a linguist ically and culturally diverse student population.
67 Student Interviews ELL students were asked to complete the Student Questionnaire during interviews. When needed, a translator (a t eacher, a staff member, or student who spoke the same language) was present to ensure th at the ELL student unders tood the nature of the interview. ELL students were asked to reflect on their part icipation in the particular lessons taught during the classroom sessions under observation. Emphasis was placed on what, in their opinion, was causing any problem s in their participation in classroom discourse and comprehension of the mathem atics lessons. Then, they were asked to comment on the talk their mathematics teacher had employed during the observed classroom sessions using the TTT Form 3 They were asked to reflect and comment on their experience not only during the classroom sessions under investigation, but also on their general experiences in learning mathem atics in English as their non-native language. As mentioned before in the description of the Student Questionnaire instrument, in TTT Form 3 the possible threats of the studyÂ’s trustw orthiness posed by the possibility of the researcher and/or the translator asking a bi ased question were also addressed. For this reason the questions were read directly from both the Student Questionnaire and TTT Form 3 (see Appendices D and E for the specific questions asked). The person translating and assisting the researcher either read the que stions directly from the versions translated in Spanish or used a simplified student-fri endly language when translating in languages other than English and Spanish. Data Analysis The data analysis will be discussed in two sections. First, the frequency count of the used discursive st rategies by using the TTT Forms instrument will be discussed.
68 Second, Krussel, Edwards, and SpringerÂ’s ( 2004) framework and the method of analytic induction (i.e., Â“from the ground upÂ”; Davidenko, 2000, p. 39) for analyzing the teacherÂ’s discourse will be described. Data from TTT Forms 1, 2, and 3 During each 20-minute observation, tally marks were used to record each observed use of particular discursive strategy on the TTT Form 1 Then, I counted the frequencies for each box. Re-playing the vi deo recordings of each observed session and checking its transcription permitted further re finement of the frequency count of the discursive strategies used by the teacher and allowed for additional qua litative analysis of the data. Additionally, two in ter-rater reliability (e.g., dependability ) tests were conducted. Initially, the researcher and a research associate, w ho is an expert in working with ELL students, both observed a classr oom session of exactly 20 minutes of a mathematics teacher outside of the study sample and filled in TTT Form 1 Then, each separately re-played the vide o recording and read the tr anscription of the observed session and counted the total frequency for each box in TTT Form 1. The inter-rater reliability score was .75. Afte r this, a training session was carried out, which permitted the researcher and research associate to clar ify and reach consensus regarding the nature and meaning of the codes of the various di scursive strategies. Then, a lesson that involved a teacher from the study sample wa s observed, coded and analyzed individually and the frequency counts from TTT Form 1 were compared again. The inter-rater reliability score was .87. Additionally, the data collected in TTT Forms 1, 2 and 3 was illustrated using various visual displays to permit further analysis.
69 Characteristics of Krussel et al.Â’s (2004) Framework The Krussel et al.Â’s framework was used to analyze the Â“teachersÂ’ discourse movesÂ” (deliberate actions they take) as f acilitators of discourse in the mathematics classroom. This framework provided a quali tative method of identifying patterns and themes to explain phenomena (i.e., to see if a trend existed in th e teachersÂ’ discourse individually and across the sample). Categor ies were formed from the data collected during the observations and the video-recordings of the clas sroom sessions and this data was then matched with the data from the teach ersÂ’ and ELL studentsÂ’ in terviews, to see if a trend existed. The transcript s of the observations and interviews were read and coded for categories that were prevalent from the teachersÂ’ observed discourse moves and teachersÂ’ and ELLsÂ’ answers and were relate d to the research goals and questions. The process of analyzing the Â“teachersÂ’ discourse movesÂ” involved three phases: (a) an initial reading of the transcribed data for an overall sense of meaning, (b) detection of Â“meaning unitsÂ” within the text, and (c ) the formation of themes by grouping key phrases or actions for each teacher. These them es were then compared to establish which ones were more customary for each teacher and to determine when they could be considered a teacherÂ’s emergent pattern of discourse. Each theme was related to a category of teacher discourse as described in the Krussel et al. (2004) framework. According to this framework, the teach erÂ’s discourse: (a) has an intended purpose (to direct the activity to reflectio ns, justifications, small-group or whole-class discourse); (b) takes place in a certain setting (assigned roles and norms in discourse); (c) has a particular form (verbalÂ—questions, directions, statem ents, clarifications, challenge; or non-verbalÂ—gestures, facial expr essions), and (d) results in consequences (intended or
70 unintended, immediate or long term). Each of these categories of teacher discourse is expounded upon below. Purpose The purpose of the teacherÂ’s discourse may be, for example, to shift discourse from a whole-class discussion to small-group wo rk so as to initia te participation in activities requiring justifications and reflections, or simply to deal with discipline issues. The teacherÂ’s purpose can only be perceived by a researcher if he/she pays considerable attention to the flow of the di scourseÂ’s text and the shift in its meaning. For example, if a teacher regularly asks the students questions such as, Â“And how did that change the problem? Correct. We switched the starting point,Â” it is evident that the teacher is trying to direct the students toward re flections on their thinking, explan ations, and jus tifications. Setting In observing the teacherÂ’s actions toward establishing a setting for classroom discourse, the researcher might infer not only from the presen t discourse, but also from previously-set norms of discourse. The rese archer may become aware of such norms by looking at the classroom layout, or by observing that the stude nts or the teacher already have established roles in discourse which na turally occur, without the teacher assigning them in front of the researcher. For example, if a teacher specifically states: Â“What is the slope in this equation? Please, raise your ha ndsÂ…,Â” it is evident that the teacher is trying to establish certain norms for students in taki ng turns to participate in the mathematical discourse. And if a teacher simply asks Â“What is m?Â” and many students raise their hands to answer the question, it is then clear that the teacher ha s already estab lished the norms of behavior for his/her student s in participating in classroom discourse. Furthermore, if
71 many students answer aloud the aforementioned questions posed by their teacher without raising their hands and being in dividually called upon, a resear cher can conclude that the students comply with more liberal and less explicitly stated classroom behavior. Form The form of the teacherÂ’s discourse includes actual teacher talk (verbal) and actions (non verbal) For example a question: Â“How do you know that is true?Â” indicates that the teacherÂ’s discourse takes the form of a chal lenge. If the teacher says Â“IÂ’m not sure I understandÂ…,Â” he/she is reques ting clarification. (For more details see Krussel et al., 2004, p. 309). The teacherÂ’s non-verbal discourse also may be displayed in different formsÂ—gestures, facial expressions, spatia l proximity between the teacher and student, pausing after having posed a quest ion, or the use of silence. For example, after collecting studentsÂ’ bell-work, if a teacher walks to th e board in silence and writes the title of a lesson topic, he/she is switching the classroom discourse to instructional mode. In another situation, if a teacher walks ar ound the studentsÂ’ seats and assist s them in their individual or group work, a researcher can conclude that in this class, the teacher uses spatial proximity with his students as part of his instructional techniques. Consequences The teacherÂ’s discourse always has some consequences which may be intended (to shift the cognitive level of the task performed) or unint ended (for example, lowering his/her expectations of ELLs), more immedi ate (shifting the dialogue from univocal to dialogic) or long-term (setting norms of politeness and turn-taking during classroom discussions, which may consequently accelerate future discourse toward meaning rather than form).
72 In conclusion, examine the classroom discourse by applying Krussel at al.Â’s framework provided valuable information about the studied sample. More specifically, the process of analyzing the teachersÂ’ discursive moves he lped in answering the first research question (i.e., what patterns of discourse the teachers used when ELLs are present in the mathematics classroom and if there were any observed adjustments to teacher talk or modifications of instructions). Th e previously described frequency count of each teacherÂ’s use of different discursive strategies also permitted the detection of which strategies were most prevalent for a sp ecific teacher. Furtherm ore, Krussel et al.Â’s framework permitted a qualitative view and co nsideration of not only prevalent themes (i.e., main patterns of discourse) typical fo r each teacher, but the consideration of secondary or less prevalent themes. These less prevalent themes were also important for this research because they provided informa tion needed to answer the other research questions. The gathered information revealed the extent to which teachersÂ’ patterns of discourse changed as a result of the number of ELL students present and the extent to which teachersÂ’ experience, ESOL endorseme nt, and their own linguistic and cultural background, affected their pattern of discourse. Method of Analytic Induction To analyze teachersÂ’ discourse, the qual itative method of analytic induction was applied to the data. Lincoln and Guba (1985) st ated that Â“inductive data analysis may be defined most simply as a process for Â“making senseÂ” of field dataÂ” ( p. 202). As quoted in Lincoln and Guba (1985), Reese asserted: The widespread distinction between i nduction as an inference moving from specific facts to general conclusions, and deduction as moving from general
73 premises to specific conclusions is no longer respectable philosophically. This distinction distinguished one kind of induction from one kind of deduction. It is much more satisfactory to think of induction as probable inference and deduction as necessary inference. (p. 251) According to Davidenko (2000), another re searcher who adopted the method of analytic induction in her dissert ation study: When we analyze data from a qualitative study, we look for codes that represent instances of a concept th at we are yet to define The larger categories emerging from the codes become our new concepts. We may use predefined categories to represent the concepts. In this case, thr ough the analysis of the data, we attach new meanings to them. After I began to think inductively, I found the qualitative research process to be ex citing and creative. (p. 40) However, whereas Davidenko studied Â“how [ELL] students learn mathematics in English-taught mathematics classesÂ” ( p. 30), this study focuses on the teachers. Specifically, this study examines teachersÂ’ Â“d iscursive movesÂ” and the influences of teacher talk on the inclusion or exclusion of ELL students in classroom interactions. Validity, Reliability, and Objectivity Check of the Analysis Process The framework of Lincoln and Guba ( 1985) dictates that when carrying out qualitative naturalistic methods of inquiry, rather than using the conventional terms internal and external validity reliability and objectivity four other terms are used instead: credibility transferability dependability and confirmability (p. 300). Below I describe how this studyÂ’s trustworthiness was attained by addressing each of the above criteria.
74 Credibility For satisfying the credibility criteria of trustworthiness of the instruments some of the techniques suggested by Lincoln a nd Guba (1985) were used (i.e., prolonged engagement and persistent observation). The te achers were observed for the entire class period (typically 45 minutes) on three different occasions. The teacher-led instruction portion of the lesson (typical ly 20 minutes) was videotaped. This videotaped segment allowed the researcher to obtain frequency counts of the used discursive strategies. Additionally, I collected data from the teacher s via preand post-observations interviews, and from post-observation interviews of thei r ELLs. Furthermore, triangulation of the data collection and analysis procedures wa s employed using different data collection models Â— observations, field not es, video-recordings, questio nnaires and interviews, and school recordsÂ’ files. This data was anal yzed according to Krussel et al.Â’s (2004) framework and frequency counts of the teachersÂ’ use of different discursive strategies were determined. The triangulation of sources data, and methods facilitated the creation of a more holistic view of the discur sive practices adopted by each teacher. Furthermore, I engaged in peer debriefing to ensure the studyÂ’s trustworthiness. In several of stages of the study, debriefing se ssions were performe d with two colleaguesresearchers who are experts in the fields of teaching ELLs and research in mathematics education, and written records of these sessions were mainta ined. These experts helped me to improve the instruments and include teacher-appropriate and student-appropriate language and ensure the proper r eadability levels as per my audience. With the assistance of these sessions, the achieve d readability levels as follo ws: 71.7% (corresponding to a 4.6-grade reading level) on the Teacher Questionnaire and TTT Form 2 and 80.3%
75 (corresponding to a 4.3-grade reading level) on the Student Questionnaire and TTT Form 3 which were respectively classified as Fa irly Easy and Easy Readability Scores, according to the Flesh-Kincaid Grade Level Scale (for more information refer to www.plainlanguage.gov or www.plainlanguagenetwork.org ). As described before, debriefing with a Ph. D. professional in th e field of educating ELLs facilitated the achievement of an inter-rater compatibility rating of 87% in the fr equency count of the teachersÂ’ use of the discur sive strategies as per TTT Form 1 instrument (see Data from TTT Forms 1, 2, and 3, pp. 65-66 in this manuscript) After the analysis of the data from the video record ings, the transcriptions, TTT Forms 1, 2, 3 the questionnaires, and a more detaile d description of the findings, another debriefing session was convened with the expe rts mentioned. In order to maximize the studyÂ’s credibility, the research associate participated in a training session to learn to employ Krussel at al.Â’s framework and was give n ample opportunity to code individually a sample of data using the same coding them es. The 22 discourse strategies reflected in TTT Form 1 (refer to Appendix A) were used as comparative constants. Then, the constant-comparison of the data analysis be tween the researcher and research associate showed a reliability rating of .83 (e.g., 83%). This process gave the researcher an opportunity to compare findings to determine the consistency of in terpretations and to resolve any discrepancies that were found. To confirm drawn conclusions, the research associate used the data from TTT Forms 1, 2, and 3 and compared his findings with the ones made by the researcher to determine the researcherÂ’s consistency of interpretations. In cases of disagreement, the researcher and research associate discussed any discrepancies until consensus could be reached. Furthermore, after the frequency count of
76 the teacherÂ’s used discursive strategies and the qualitative analysis, any deficiency in used strategies was indicated (e.g., negative case analysis was performed which Â“requires the researcher to look for di sconfirming data in both pa st and future observationsÂ” (Lincoln & Guba, 1985, p. 310) in or der to Â“refine a hypothesis until it accounts for all known cases without exception Â” (Lincoln & Guba, 1985, p. 309). Moreover, member checking was carried out with th e participating teach ers to ensure that Â“data, analytic categories, interpretations, and conclusionsÂ” correspond to an Â“adequate representationÂ” of reality, and to provide continuous po ssibilities for them to react to such representations. A summary of some of the cas e studiesÂ’ descriptions were also given to the studyÂ’s participants to read and commen t on, as part of this member checking. Transferability According to Lincoln and Guba, it is not the researcherÂ’s Â“task to provide an index of transferabilityÂ”; however, it is the rese archerÂ’s task to Â“provide only the thick description necessary to enab le someone interested in making the transfer to reach a conclusion about whether transfer can be contemplated as a possibilityÂ” (p. 316). Thus in establishing the studyÂ’s transferability, a thick description of how different stages of the study were carried out is provide d. In order to ensure a high st andard of transferability, a thick description of the observations and th e subsequent processes of transcribing and analyzing the data, as well as its various representational formats (e.g., tables, charts, histograms) and analyses thereof is provided. Additionally, the instruments used in the study are provided in the appendi ces. This ensures that a Â“ data base [has been provided] that makes transferability judgments possible on the part of potential appliersÂ” (Lincoln, & Guba, 1985, p. 316).
77 Dependability and Confirmability According to Lincoln and Guba, Â“a single au dit, properly managed, can be used to determine the dependability and confirmability simultaneouslyÂ” (p. 318): The auditor should see himor herself as acting on behalf of the general readership of the inquiry report, a read ership that may not have the time or inclination (or accessibility to the data) to undertak e a detailed assessment of trustworthiness. (p. 326) To comply with these criteria of trustworth iness, an auditor (an expert, who is not a member of my dissertation committee) in spected, verified, and examined drawn conclusions by examining the supporting docum entation for accuracy and fairness from the onset of the study. The auditor was introduced to the study at its inception, as well as the development and testing of its various inst ruments, raw data collection, data reduction and analysis, study findings and final repor t, and further methodological notes and trustworthiness notes. Audit tr ial notes followed the suggestions made by Lincoln and Guba (1985). (For further deta ils regarding this process re fer to the Appendices A and B in Lincoln and Guba (1985), pp. 382-392).
78 CHAPTER IV: RESULT S AND INTERPRETATIONS The results of the study are pr esented in two sections. Th e first section reports the results of the data analysis of the teacher questionnaires and interviews in relation to years of teaching experience, number of ELL st udents present, and the teachersÂ’ and ELL studentsÂ’ linguistic backgrounds. The second s ection presents each te acher and his or her classes as cases to be examined. The descri ption of each case includes sample classroom excerpts and the results from analysis that applied the framework developed by Krussel et al. (2004). Furthermore, each case study provides the results of comparing the data from the researcherÂ’s preliminary evaluation (i.e., before an actual count of the frequencies with which the teachers use different discourse strategies), the teachersÂ’ self-evaluations, and the evaluations of the ELL students. A dditionally, the frequencies with which different discursive strategi es were used by each teacher are provided. Bar graphs are used to visually represent the findings and pr ovide the reader with Â“a quick glance [of] an overall pattern of resultsÂ” as prescribed by the American Psychological Association (APA), 2001, p. 176. In sum, section two will repo rt the results of the data analysis in relation to research question one and will prov ide the results of the data analysis in relation to the research questions that are i llustrated with detailed examples and specific evidence.
79 Characteristics of the Sample Table 2 presents the demographic informa tion about the studyÂ’s participants that reflect the criteria used to id entify the studyÂ’s participants: 1. Years of teaching experienceÂ—at leas t two teachers in each school have many years of teaching experience a nd two teachers are in the beginning of their teaching careers. 2. Teachers with/without ESOL endorsement Â—at least two of the teachers in each school have many years of teac hing experience and have an ESOL endorsement, and the inexperienced teach ers either did not yet have their ESOL endorsements, or had just obtained them. 3. The teachers teach mathematics courses at the same level Algebra I (with zero or a small number of ELLs or with a larger number of ELLs, depending on the population of the classes). Information about the third criteria will be summarized and visually represented using graphs. In both schools, three of the participan ts taught Algebra I, however the fourth participants of both schools taught a similar curriculum course such as Intensive Mathematics (Ms. Brown in Lincoln High Sc hool) and Liberal Arts Mathematics (Mr. Davison in Green Bay High School). In order to be consistent acr oss the curricula, I observed all teachers when they were t eaching a common topic Â– linear equations. Additionally, in both schools, some of the Algebra I classes were taught using an individualized computer assi sted learning program called I Can Learn Lab. Furthermore, from the eight teachers who participated in the study, there were two female teachers and
80 Table 2 Overall Sample Information Name Years of teaching experience (teaching Algebra) Languages spoken by teacher ESOL endorsement (years) Number of ELLs in class Languages spoken by students Green Bay High School Mr. Able3 34 (34) English Yes (3) 2 Spanish Ms. Barrera 2 (2) English, Spanish Just obtained (0.5) 9 Spanish Ms. Chandler 9 (8) English Yes (5) 8 Spanish Mr. Davison 16 (8) English Yes (7) 4 Spanish Lincoln High School Ms. Andersen 23 (23) English, French Yes (11) 4 Spanish, French, Creole, Arabic Ms. Brown 0 (0) English, Yoruba, limited French Just obtained (0.5) 5 Spanish, Arabic Ms. Cortez 10 (5) English, Spanish No (0) 4 Spanish, French, Creole, Arabic Mr. Daniels 12 (12) English Yes (9) 3 Spanish 3 Pseudonyms are used for teachersÂ’ names.
81 two male teachers from Green Bay High School and three female teachers and one male teacher from Lincoln High School The teachers also varied in their linguistic and cultural backgrounds Â– in each school there was one African American, one Hispanic, and two non-Hispanic White (one male and one female) teachers (see Table 2). Years of Teaching Experience The teachers in the sample varied greatly in their years of experience in teaching (see Table 2). In both schools, there were at least two teachers w ith many years of experience Â– Mr. Able and Mr. Davison in Green Bay High School, and Ms. Andersen and Mr. Daniels in Lincoln High School. In both schools, th ere also was at least one teacher who had recently begun his/her teaching ca reer or at least was in the beginning of teaching Algebra I to classes with ELLs pres ent Â– for example, Ms. Barrera in Green Bay High School and Ms. Brown in Lincoln High School had just started their teaching careers, and Ms. Chandler (Green Bay High School) and Ms. Cortez (Lincoln High School) had taught Algebra 1 to ELLs for about five to seven years. ESOL Endorsement Only one teacher in the sample Â– Ms. Cortez from Lincoln High School, had not fulfilled the requirement for content area t eachers of 60 hours of training toward ESOL endorsement. Two other teachers from each school, Ms. Barrera and Ms. Brown, had recently completed their training and had littl e experience as teacher s of Algebra I to ELL students (see Table 2).
82 Number of ELL Students Present The number of ELL students in each classr oom is more evenly distributed in Lincoln High School, whereas it is very unequally spread in Green Bay High School (see Table 2). This may be due to a trend in Green Bay High School, wherein most of the ELLs are assigned to Algebra I classes that employ computer labs, tutorials, tests and quizzes (as was indicated by the guidance department chair in the school). Years of Teaching Experience, ESOL E ndorsement, and Number of ELL Students Combined An interesting observation is that the mo re years of teaching experience a teacher has, the smaller the numbers of ELLs present in his/her class. This is the case with Mr. Able and Mr. Davison in Green Bay High Sc hool; the situation is similar with Ms. Andersen and Mr. Daniels in Lincoln High School. In both schools, the teachers just beginning their teaching careers (Ms. Barrera in Green Bay High School and Ms. Brown in Lincoln High School) are assigned to teach classes with the highest number of ELLs. Even though these two teachers had recentl y completed their ESOL endorsementÂ’s requirement, they lack the practical experien ce of teaching Algebra I to classes with diverse student populations involving a high number of ELLs present. TeachersÂ’ and ELL StudentsÂ’ Li nguistic Backgrounds Combined Table 2 depicts the number of languages spoken by each teacher, the number of ELLs in his/her class, and how many ELLs spoke the same language(s) as the teacher. Only three of the teachers spoke the same language as their ELL studentsÂ—Ms. Barrera in Green Bay High School, Ms. Andersen a nd Ms. Cortez in Linc oln High School. Of these three teachers, only two of them Â— Ms. Barrera and Ms. Cortez also had similar
83 cultural backgrounds as some of their ELL stude nts. Ms. Barrera is from the Dominican Republic and Ms. Cortez is from Puerto Rico. In addition, four of the teachers in the sample Â– Mr. Able, Ms. Chandler, Mr. Davison, and Mr. Daniels Â– do not speak any languages other than English. One teacher, Ms. Brown, speaks three languages (English, Yo ruba, and limited French), yet none of her ELLs is from the same linguistic bac kground (4 of them speak Spanish, and one speaks Arabic). However, even though thes e teachers did not sp eak their ELLsÂ’ native languages, an analysis of th e interview data, as well as that collected from the observations and videotaped sessions, revealed the various ways in which these teachers dealt with the issue. Case Study Analysis In this section, a detailed descripti on of the eight case studies will provide information about how each teacher exhibited some strategic modifications of his/her Â“discursive movesÂ” in order to accommodate the ELLs present in his/her mathematics classroom. Each teacher will be discussed as a separate case as a member of a faculty of each school. Each case represents the synthesis of data obtained from video-recorded, transcribed and analyzed observati on field notes, responses to the Teacher and Student Questionnaires (see Appendices B and D) and the TTT Forms 1, 2, and 3 (see Appendices A, C, and E). Treating each teac her as a case allows for the examination of normal practices that constitute the classroom atmosphere and the mathematics classroom discourse. Finally, some of the similarities and differences in the teachersÂ’ observed discursive patterns wi ll be presented.
84 Green Bay High School Mr. Able4 Mr. Able, an African American in his la te fifties, has over 30 years of teaching experience. He has a MasterÂ’s Degree in Ma thematics Education, is certified to teach secondary mathematics, and has completed th e required 60 hours of training toward his ESOL endorsement three years prior to his involvement in this study. In addition to teaching Algebra I, he also teaches College Pr eparation classes and is one of the coaches for the schoolÂ’s track team. Mr. Able only speaks English. His Algebra I class consisted of 21 students, with only tw o officially indicated on his roster as ELL students of Hispanic background. However, the class appeared diverse, with an almost equal number of Hispanic, African-American and Caucasian students. Mr. AbleÂ’s classroom was very organized, clean, and laid out in traditional rows. The two ELL students were seated beside each other on the right side of the classroom. Bilingual students who spoke both English and Spanish were seated nearby to provide translation assistance if needed. During th e interview, Mr. Able confirmed this observation by stating that he often, Â“teams a person who speak[s] Spanish and English with the ELL student to help him.Â” He also pointed out that he is conscious of the presence of ELL students in his classroom a nd often Â“try[ies] to speak slow[ly] when lecturing.Â” Mr. AbleÂ’s classroom was well-decorated. On the rear wall were colored posters in creative shapes. The posters contained his studentsÂ’ answers to some autobiographical questions and information on their hobbies and interests. The rest of the walls were 4 Pseudonyms are used for both teachersÂ’ and studentsÂ’ names.
85 decorated with posters on which the stude nts wrote Â“Math is likeÂ…., because it Â…Â”, where each student had completed the senten ce in his/her own original way. Around the white board Mr. Able had posted mathematics vocabulary words in Spanish (which he said he did in order to assist his ELL students). Prior to his first period class, students would come by to ask for help with the homework, to ask for recommendations, or to ask for his signature on a field trip form. The student/teacher interactions were all impr essively carried out with mutual respect. The students seemed to respect him not only as their Algebra I teacher but also as a coach and as a person. Typical classroom discourse. During each of the three observed classroom sessions, Mr. Able used the same basic cla ss sequence that will be described in the examples that follow. First, he began the le sson with five-minutes of bell-work. He used an alarm clock to time the bell-work and placed a prepared transparency form that consisted of mathematics questions that th e students had studied a couple of lessons previously. For example: Solve the following equations for the corresponding variable: 1. 20 = 6x + 8 2. -10 Â– k = -3 3. 2y Â– 7 = 15 4. 15 Â– 4g = -33 5. 2.1 = 0.8 Â– z Mr. AbleÂ’s students complied with the fo llowing rules of behavior. All students worked on the assignment. After the allotted five minutes were over and they heard the sound of the alarm, they quietly got up (inc luding the ELL students) and placed their bellwork on his desk in the left corner of the room.
86 Then, Mr. Able explained the bell-work and demonstrated the correct solutions, which were already written on another tran sparency. Before starting a new lesson, he usually informed the class of any upcoming events. For example, during the first observed session, he informed the students that there would be a test on Friday and that he would check their notebooks while they completed the test. Following this initial stage of the class, Mr. Able wrote the title of the new lesson onto the white board: Lesson 4.6 Quick Graphs Using Slope-Intercept Form. He also wrote the equation: y = mx + b and asked: Â“Can anybody tell me what m represents?Â” As I analyzed all of the video-recorded sessi ons, a similar pattern of questioning emerged. Most of the questions thr oughout the lessons had a sim ilar purpose of involving the students in the mathematical discussions, a nd usually required one-w ord responses or a list of words. Thus, to this first question, a c ouple of students answered aloud: Â“the slopeÂ” and the teacher, nodding, said Â“GoodÂ” and simply continued: And, whenever you see the b it is the y-intercept. What we mean by the yintercept is the way it crosses the y-axis. When we talk about m today, the top is either you go up or down. When you go up, it is positive, when it is down it is negative. The bottom, you can go to the left or to the right, when you go to the left it isÂ…? Here Mr. Able changed the pitch of his voice, to indicate th at he is asking a question, and looked toward the class, nodding to ward a student (not an ELL) who raised his hand. The student (along with a couple of others) answered: Â“negative.Â” Then, Mr. Able continued, Â“when you go to the right it is?Â” and a few stude nts answered aloud, Â“positive.Â”
87 Mr. Able also drew a small diagram, m = = and continued explaining the graphing of lines by providing examples. The examples started with Slope-intercept form but later on moved to examples in which the students were expected to perform some algebraic oper ations in order to tr ansform the equations into y=mx+b form and then graph them. With each example, Mr. Able asked the students questions such as, Â“Someone tell me what is m? Â” and if they answered with just one number Â“threeÂ”, he would continue, Â“ it is always the number before the x ; donÂ’t write [it] as an integerÂ” and thus the students who an swered corrected themselves by saying Â“three over one.Â” Then Mr. Able continued asking the whole class: Â“what is b Â” and, nodding to a student, wrote the answer Â“(0, 5).Â” After this, he simply continued: Â“then you graph this one. Start with (0,5). Always start with the y-intercept, go up 5, put your dot on 0, go up 5. Now, so meone tell me how you do 3 over 1?Â” These excerpts from this lesson exemplify Mr. AbleÂ’s classroom discourse and climate. In all three observed twenty-minut e sessions and video-recordings, Mr. Able always began his lessons with bell-work a nd then explained the lessons on the board, oftentimes using the overhead projector to place pre-prepared grids with the coordinate system on them, which he used to demons trate plotting points and graphing linear equations by using the slope-interce pt form, or point-slope form. For example, for the second lesson, Lesson 4.8 Functions and Relations, he used various examples to review previously studied concepts. In this case, he used multiple representations to discuss functions (i.e., graphs, tabl es, Venn Diagrams, etc.):
88 1. 2. 3. 4. 5. Input Output -3 1 5 0 8 1 -2 0 6. Input Output 7. Input Output Later in the lesson, linear f unctions were represented using Point-slope equations of lines. In the third lesson, 5.2 Writing Linear Equations Given the Slope and a Point, Mr. Able made connections to the first less on by asking the students to find the slope ( m ) between two points and then demonstrated how this slope and either of the points could be used to write a Point-slope equation of the line containing them. InputOutput 10 1 0 3 -10 5 0 4 3 -1 0 5 3 4 1 1 7 8 2 5 -1
89 Krussel et al. framework. Applying the Krussel et al. (2004) framework to analyze Mr. AbleÂ’s Â“discourse movesÂ” as a facilitator of discourse in his Algebra I classroom, the following was determined. The purpose of Mr. AbleÂ’s discourse was to encourage participation in whol e-class discussions and activities. For example, he catered to his diverse student mix, including the ELLs, by using frequent questions such as, Â“What to do now, m or b?,Â” Â“what is m ?,Â” and Â“what is b?Â” (e. g., either/or and one-wordresponse questions). Thus, he purposefully used simpler talk, synonyms and various visual representations to ensure that hi s students attained a better grasp of the mathematical concept being presented. For example, in order to better explain the representation of slope as m = he used the words Â“the topÂ” and Â“the bottomÂ” when referring to the numerator and the denominato r of the fraction representing the slope. At the same time, he drew arrows to indicate that each positive or negative integer represents the number of units they have to move up or down the grid. However, Mr. Able did not try to dire ct the students toward reflecting on their thinking, or to provide explanat ions and justifications. Most of the questions were of the type Â“what isÂ…Â” or Â“tell meÂ…Â” and did not move the students to higher levels of cognitive demand according to BloomÂ’s Taxonomy: knowledge, comprehension, application, and analysis He asked only one Â“howÂ…Â” question: Â“How do we now graph this equation?Â” which, had he waited, would ha ve initiated a longer response than the one garnered. Moreover, had Mr. Able called upon an ELL student for the question, it would have encouraged that student to a higher level of SLA such as speech emergence or intermediate speech However, Mr. Able missed this opportunity by immediately
90 following that initial question w ith another, Â“What is this point here?Â” an d by assisting the students by pointing to the y-inte rcept in the equation. The setting for classroom discourse a ppeared to be established early in the school year as it was possible to dis cern certain norms of classroom behavior in each of the observed lessons. These normative practices we re exemplified via the classroom layout and the well-established roles of the teacher and student in the classroom discourse that required no clarification during any of the observed lessons. For example, after the alarm clock went off to indicate the end of the be ll work, all the studen ts stopped writing and turned in their bell work according to demonstrated pre-established practices. Additionally, when Mr. Able aske d general questions, despite the fact that a few students answered aloud, only one student took the next turn talking (by rece iving an encouraging nod from Mr. Able), and then the same student continued talking, extending or correcting his/her answer if such co rrections were needed. The form of the teacherÂ’s discourse includes both teacher talk (verbal) and actions (non verbal) For example, Mr. AbleÂ’s questions, Â“What would you have to do to get b by itself?Â” or Â“How do we now graph this equation?Â” indicat ed that Mr. AbleÂ’s discourse took the form of a challenge. However, he was satisfied with short responses and easily provided assistance in subsequent steps. He did not move the discourse to the higher levels of cognitive demand such as synthesis and evaluation according to BloomÂ’s Taxonomy. He did not ask students to categori ze, justify, or perform more critical analyses or to further explain some st eps of their problem-solving process. Close examination of Mr. AbleÂ’s less ons revealed the various forms through which Mr. AbleÂ’s non-verbal discourse was displayed. He often used gestures to
91 demonstrate the slope of the line as first go ing up or down (rise) a nd the left or right (run); or he faced the class a nd used eye contact after posi ng a question, and with just a nod indicated which student may answer. He also walked between the rows when students performed individual wo rk and assisted them or an swered questions (if asked), thus establishing spatial proximity between teache r and student. According to this Krussell et al.Â’s (2004) framework, the teacherÂ’s discourse always has some consequences, which may be intended or unintended immediate or long term For example, Mr. Able intended to shift the cognitive le vel of the task performed (graphing a line) by asking the students to explain how to do this, but he unintentionally assisted them in this task, t hus lowering his expectations of their abilities to complete the task on their own. More important, whenev er he asked ELL students to answer a question, after the ELLs provided a one-word re sponse, Mr. Able did not challenge them to further explain the steps needed, but instead pointed out each consecutive step. Consequently, he unintentionally demonstrated lower expecta tions for ELLs and did not provide them with opportunities to practice their mathematics vocabulary in English. Some of the immediate or long-term consequences of the teache r discourse were apparent when Mr. Able wanted to shift the dialogue from univocal to dialogic and to involve the class. He usually faced the students and asked questions such as, Â“can someone tell meÂ…,Â” thus indicating that each student could participate in the discourse. Mr. Able also had set long-term norms of formality, such as taking turns speaking, during classroom discussions. This, in turn, allowed the student s (and the ELLs in particular) to focus their attention on the meaning of the mathematical discourse rather than on its form, and set norms for general comm unication in English.
92 Perceptions of classroom discourse. On the graph below, I have represented the results of comparing the data from three sources of evaluation (i.e. TTT Form 1, 2, and 3 ) of teacher talk (see Figure 1). Figure 1 represents the researcherÂ’s preliminary evaluation of the teacher talk (i.e., before analyzing th e teacher talk and coun ting of the frequency with which each discursive category is used), Mr. AbleÂ’s self evaluation of the frequency with which he used the pre-determined categories of teacher talk and discourse characteristics, and his ELLsÂ’ evaluations of his use of the pre-determined teaching strategies and categories of teacher talk. An average ELL student score was determined by adding the evaluations of his ELLs for each discursive category and then dividing this by the number of ELL students. The numbers 1 to 22 on the y-axis correspond to the predetermined major categories of the teacher ta lk that are described in greater detail in TTT Forms 1, 2, and 3 (see Appendices A, C, and E). On the x-axis is the frequency that each evaluator attributed to Mr. AbleÂ’s talk or strategies, on a scale from 1 to 5 (with 5 representing the hi ghest frequency). The general level of agreement amongst th e teacher, the researcher, and the ELLs in their evaluations of the strategies used by the teacher are presented by the computed pair-wise correlations (Pearson product-mome nt correlation coefficients), which show whether the teacherÂ’s perceptions of his ow n use of strategies matches those of the researcher and the ELLs. For this particular case, the correlation be tween the teacher and researcher is .62; between the teacher and ELLs it is .25, and between the researcher and ELLs it is .65. As Figure 1 indicates, there are four strategies where th ere is complete consensus between the evaluations of th e researcher, the teacher se lf-evaluation, and the ELLsÂ’
93 Figure 1. TeacherÂ’s, reasearcherÂ’s and ELLsÂ’ evaluations of Mr. Ab leÂ’s frequency of use of va rious discursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
94 evaluationsÂ— Use of a variety of visual or audito ry stimuli: transparencies, pictures, flashcards, models, etc (strategy 17), Use of technology to enrich a concept presentation (18), Use of repetitions (4), and Use of gestures, facial expressions, eye contact or demonstrations to enhance comprehension (15). The first two strategies (17 and 18) were evaluated as most frequently used. The next tw o strategies (4 and 15) were also evaluated as ones traditionally employed by Mr. Able, but with a slightly lesser frequency. The video-recordings of Mr. AbleÂ’s cl assroom sessions also reveal that he consistently used an overhead projector, calcu lators, or pre-prepared spreadsheets with data so as to enhance his presentation of concepts. Mr. Able varied his presentation modes between transparencies containing preprepared bell work, lesson outlines, and the use of grids to graph linear equations. Mr. Ab le also often repeated or paraphrased his statements or asked students to repeat or restate them, especially when important mathematical concepts were formulated. Fo r example, he used the phrase Â“always start with the y-interceptÂ” and repeated it three times in three consecutive examples which the students were given to graph. At the same time, he asked them to recognize that b in the equation y = mx+b represents the y-intercept and to wr ite it as a coordinate pair such as (0, 5) and then to plot that point on the y-ax is. Mr. Able also used facial expressions, gestures, and eye contact th at exhibited awareness of culture-specific acumen. Figure 1 also reveals some differences be tween the evaluations of the researcher, the teacher self-evaluation, and the ELLsÂ’ evaluations of the teacherÂ’s use of change of tone, pitch, and modality (strategy 5) and providing opportunities for students to share experiences and build up on personal or cu ltural-specific knowledge while problem solving in mathematics (21). I evaluated Mr. Able as having used strategy 5 least
95 frequently, whereas he judged that he used th is strategy one to two times a month and the ELL students evaluated that Mr. Able used this strategy at least one to two times a week. Strategy 21, on the other hand, was evaluated by the ELL students as almost never having been used by Mr. Able, used rarely accordi ng to the researcher, and used three to four times a week according to the teacherÂ’s self-evaluation. Such anomalies in the results could be attributed to the fact that the conclusions of the researcher were elicited only from the observations of the three classroom sessions and the interviews with the teacher and students. However, since the focus of this investigation is on the classroom discourse and teacher talk influences on ELL studentsÂ’ mathem atics experiences, these studentsÂ’ opinions were placed un der special scrutiny. Thus, with regards to how ELLs feel in Mr. AbleÂ’s classroom, it is evident from Figure 1 that some strategies are not as often incorporated in his teaching style (look at the lowest bars from amongst the ELLsÂ’ evaluations of Mr. AbleÂ’s discourse)Â— Conclude a lesson with a summary of key concepts (strategy 10), Provide feedback (14) and Provide opportunities for st udents to share experiences and build up on personal or cultural-specific knowledge while problem solving in mathematics (21). The ELLsÂ’ evaluations reveal that Mr. Ab le did not encourage the studen ts to reflect on the concepts they had just learned by conc luding lessons with a summar y of important points. The ELL students in Mr. AbleÂ’s class shared that they were not immediately provided with clear feedback if they had answered a question correc tly or not. They also indicated that Mr. Able did not ask them to talk or give examples from th eir country or family when solving mathematical problems
96 Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 2 provides the frequency with which Mr. Able im plemented each of the strategies found on the TTT Form 1 on the observed lessons. The strategies most frequently employed are: Use of different questioning techniqu es sensitive to the level of SLA (strategy 12), Use of gestures, facial expressions, eye contact, or demonstrations to enhance comprehension (15), and Use drawing of charts and graphics organizers to enhance comprehension (16) This information provides additional eviden ce to support conclusions drawn from the previous graph (Figure 1), where most of the ev aluators also indicated that Mr. Able used strategies 12, 15 and 16 fairly frequently. Ho wever, additional analys is of the types of questions posed by Mr. Able re veals that they usually elic ited only one-word responses, or were general and thus elicited only a short list of words in response. This suggests that although Mr. Able is aware of the level of his ELLs Â– early production Â–he did not challenge them with questions th at could lead them to move to the next levels of subjectspecific literacy Â– speech emergence and intermediate speech in mathematics and in English. In order for ELLs (and all student s in general) to become more active participants in the mathematics classroom di scourse, they need to be given opportunities to share their opinions, to expl ore different methods of solvin g mathematical problems, or simply to be encouraged to participate more a nd thus to reinforce thei r learning of what is for them new and unfamiliar mathematics terminology. The lack of valuations in category 2 ( Use of idioms and slang words from the mathematics vocabulary which if used, ar e accompanied by a proper explanation) is likely due to the fact that Mr Able used shorter sentences, gestures, or drawings to provide visual representation of idioms or slang words from the mathematics vocabulary.
97 Figure 2. Frequency count of Mr. AbleÂ’s use of various discursive strategies during the three 20-minute video-recorded sessions. 0510152025 12. Use different questioning techniques 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 1. Use of a slower and simpler speech 6. Use of clarification of directions 11. Math discussions and problem solving 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 7. Comprehension checks 13. Use wait-time techniques 22. Content specific, enriched information 8. Identify vocabulary, pictures, or models 17. Use of visual or auditory stimuli 18. Use of technology 3. Use of synonyms 9. Review of related concepts 14. Provide feedback 20. Alternative forms of assessment 2. Use of fewer idioms and slang words 10. Summary of the key concepts 19. Using cooperative groups 21. Use of cultural-specific knowledge StrategyFrequency
98 For example, he used the drawing in order to assist his ELLs (and all students for that matter) in associating the words rise and run in the formula for slope m = rise runwith the directions of movement to which th ey refer. Figure 2 also reveals that Mr. Able utilized only one styl e of teaching and did not expose his students to different classroom work arrangements, such as cooperative groups or partner discussions (i.e., lack of use of strategy 19). Additionally, the two ELL stude nts indicated in the inte rview that they found the observed lessons easy and that this was the reason they did not have any difficulties understanding the concepts. Because of that th ey did not feel the need to ask questions and thus participated just once, if at all. They were able to simply listen to the teacherÂ’s explanations and could, without many difficulties, execute the task of graphing equations. Ms. Barrera Ms. Barrera, a woman in her mid 40s origina lly from Central America, is fluent in both Spanish and English. She worked in Gr een Bay High School for five years: three years as a Teacher Assistant (aiding the Ma thematics teachers with translations in Spanish), and the last two years as a teacher of Algebra I and Intensive Math. She earned an engineering degree in her country. Ms. Ba rrera initially obtaine d a temporary teaching certificate in Physics, and is currently working on her certific ation in Secondary Mathematics. She recently completed the requ irement for ESOL trai ning for subject area teachers by attending night classes. Ms. Barrera taught her Algebra I clas s with the aid of a Computer Aided Instructional Program, I Can Learn Lab, which has tutorials, tests, and quizzes aligned
99 with the Algebra I curriculum. The class c onsisted of 24 students, of which 9 were officially listed on the rost er as ELL students, all of th em with Hispanic backgrounds. Thirteen of the 24 students were also of Hisp anic origin, five were African-American and 6 students were White. Considering the teach erÂ’s bilingual abilities and educational background, the schoolÂ’s guidance department enrolled more Hispanic ELLs in Ms. BarreraÂ’s Algebra I class so th at they could be better assi sted with both mathematics and language issues. During an interview, Ms. Barrera shared th at because she speaks English with an accent, she often uses comprehension checks in order to ensure that she is understood by her students. Additionally, thes e checks allowed her to determine whether she needs to modify her talk or use synonyms in order to negotiate with her students the meaning of the particular idea she was trying to convey. Typical classroom discourse. The following excerpts provide specific examples of these tactics. Each line (a sent ence or fragments with the same idea) is numbered so as to allow for clearer references to it thereafter: Ms. Barrera :  If you remember yesterday I wrot e the steps how you ca n determine if two lines are parallel or perpendicular.  I am going to remember to you step one is: write the equations in slopeintercept form, which is y = mx + b  Do you remember guys, what is m and what is b ?  Raise your hands if you remember.  WhoÂ’s m ? Here the teacher, being ELL herself, used the words Â“rememberÂ” (line 2) instead of Â“remindÂ” and Â“whoÂ” (line 5) instead of Â“whatÂ”. The words were pronounced with a Spanish accent, but it appeared the students had no difficulties understanding, and if they did, they were negotiating the meaning th at the teacher was trying to convey. The
100 strategies used were numbers 9 ( Start with a review of related concepts ) (lines 1 and 2) and 12b ( Questioning technique sensitive to ELL level of early production and requiring a one-word response) (lines 3 and 5). ELL Student :  point-slope Ms. Barrera :  Just, uh, very good, slope.  That is goi ng to be that coefficient together with x.  And what is b?  You remember? The answer was not exact (line 6), but the teacher used strategies number 14 ( Provide feedback ) (line 7), strategy 3 ( Use of synonyms ) for Â“ slopeÂ” as Â“that coefficient together with xÂ”, and strategy 12b (another Use of questions requiring a one-word response ) to encourage and help the student better unders tand where the slope is located in the equation. ELL Student:  y Ms. Barrera:  y-intercept.  Correct, yintercept is b. Correct.  Then now, I want to ask you guys.  I am going to call equation one and equation two.  Uh, Melissa, is equation one written in slope-intercept form? Here again the answer was not exact (line 11) but the teacher used strategies number 14 ( Provide feedback ) (line 13) and 4 ( Use of repetitions or paraphrasing) to paraphrase the studentÂ’s answer into the more precise, correct answer (line 12), thus emphasizing subject-specific lesson vocabulary (strategy 8) Â– Â“slopeÂ” (lin e 7) and Â“y-interceptÂ” (line 13). Furthermore, by calling on an ELL for part icipation (line 16), the teacher tried to involve ELL students in mathematical discussions (strategy 11) by using a yes/no question (strategy 12 b) (line 16) which showed se nsitivity to the studentÂ’s level of SLA ( early production ).
101 ELL Student:  No Ms. Barrera:  Why? With this higher level question (strategy 12c) (line 18), th e teacher initiated a more thorough response and encouraged the ELL student to try to evolve to the next stages of SLA speech emergence or intermediate fluency. Furthermore, Ms. Barrera thus tried to involve the ELL student in justifications and explanations of answers and thus exhibited her higher expectations of th e ELL students that they could handle discussions requiring higher levels of cognitive demand ( evaluations, justification and explanations ), as per BloomÂ’s Taxonomy (strategy 22). At the same time, she encouraged the development of their linguistic abilities by involvi ng them in mathematics discussions. ELL Student:  Because y is not isolated. Ms. Barrera:  Correct.  B ecause y is not isolated.  You ha ve to write or leave y alone.  Then I am going to write equation one in slope-intercept form.  That means I have to isolate y.  What do I have to do in order to isolate my variable y? Here the teacher provided feedback (strategy 14) (line 20), then used repetition (strategy 4) (line 21) to emphasize the importance of is olating y, and finally used simple language to explain that y must be writt en or left alone (strategy 1cuse of slower and simpler speech ) (lines 21-24), yet at the same time she fo cused her talk on key concepts. With the last question (strategy 12 d) (line 25) she moved the discus sion to higher linguistic and cognitive levels by asking the student to reco mmend the next step, thus encouraging the development of intermediate speech level of SLA in her ELL students. Melissa:  You can subtract. Ms. Barrera:  You have to subtract 5x.  Okay, you ha ve to move this term.  In order to move this term, you have to do the opposite.
102  That is adding, you having to subtract minus 5x and minus 5x.  I have to do the sa me thing in both sides of my equation in order to keep the balance.  Okay guys? Here the teacher used repetitions and paraphrasing of the studentÂ’s an swer (strategy 4) (line 27) and thus provided feedback (strategy 14) (lines 27-29) by extending upon the studentÂ’s answer and ex planations. She then used various synonymous expressions (such as Â“move this termÂ” (line 28); Â“do the opposite Â” (line 29); Â“minus 5xÂ” (line 30), etc., in order to help the students better grasp the concept of opposite operations in transforming the equation into Slope-Intercept form and ma intaining the balance in the equation (line 31). Thus, she demonstrated use of synonymous expressions to teach the mathematical concept of performing Â“opposite operationsÂ” to transform equations (strategy 3). At the end, she performed a comprehension check (strategy 7), by asking the question Â“Okay guys?Â” (line 32). In the intervie w Ms. Barrera indicated that, by often asking her students this question, she was actually reassuring he rself that her students were following her explanations and understood the consequences of the performed steps. As she said Â“Okay guysÂ”, she faced her students, and was i ndicating that she expected a response by providing some wait time. Observing the students Â’ facial expressions or looking to see if they were shaking their heads (in agreement or disagreement) was an indication to her if they understood the operation she was perfor ming on the board. Thus, Ms. Barrera was actually performing a comprehension check (strategy 7). Krussel et al. framework. An application of Krusse l et al. (2004) framework revealed that the purpose of Ms. BarreraÂ’s discourse was to involve more ELLs in mathematical discussions and to encourage justifications and e xplanations of their
103 answers by moving from Â“What is thisÂ…Â” type s of questions, initiating usually one-word or short responses, to higher order ques tions (Â“Why?Â” or Â“BecauseÂ…?Â” [changing the pitch of her voice at the end to indicate that the student must provide a justification]). Ms. Barrera thus guided her student s (both ELLs and fluent Englis h speakers) to reflect on their thinking and to provide explanations and justifications. Mo reover, she regularly called upon an almost equal number of ELL and non-ELL students. On average, throughout the three observed sessions, Mrs. Ba rrera called on at le ast three to four different ELLs and the same number of non-ELL s, and then individually helped at least three ELLs and any other students (one or two) if they requested help. What was interesting with regards to Ms. BarreraÂ’s teaching style was that even though she taught this Algebra I class in a com puter lab, she often varied her instruction strategies. From a class-wide lecture usi ng the overhead projector and involving the students in discussions, she often switched to individual work on the computers and provided individual help to particular students. Ms. BarreraÂ’s actions towards establishing a setting for classroom discourse can be inferred from conversations with other teachers and from my own observations that she had strict rules in order to ensure that the computers in the room are used properly. For example, she had explicitly written on the board her rules stating in essence that no food or drinks are permitted and that student s should always bring their materials to school. Additionally, the student s had their binders on a shel f and neat note-taking was encouraged. The form of Ms. BarreraÂ’s discourse also included both verbal ( teacher talk ) and non-verbal (actions) forms Even though Ms. Barrera tried to emphasize both the
104 meaning and form of mathematical senten ces, being an ELL herself, she sometimes struggled with the pronunciation or proper usage of the mathematics terminology in English. In some instances, students were confused by her accent, improper sentence organization, or improper use of English grammar. For example, while reading a particular problem from the computer sc reen, Ms. Barrera was actually stating the following to a non-Spanish speaker who asked fo r her help: Â“The problem is determined the equation of the line conte nds that giving points and write answered in the standard form.Â” Obviously she was trying to direct the student by paraphrasing what the problem asks for (i.e., to determine the equation of th e line that contains the given points (to which she points), and to write the an swer in standard form), but she was using a long sentence in improper English. Next however, Ms. Barre ra negotiated the meaning of the text she just read, by breaking it into simpler sentences, using gest ures to point to the given information, and explaining what steps s hould be taken to solve the problem: Okay, the first thing they are giving to you is two points, this and this. (She points to the points on the computer screen.) The first thing you have to find a slope. Okay, after that solve for b. And after that so lve that equation. (She wrote the equation y=mx+b in the studentÂ’s notebook.) And then write that equation in standard form. This short excerpt illustrates some instances of Ms. BarreraÂ’s verbal form of discourse ( teacher talk), as well as the fact that she is a teacher who is an ELL herself. By being aware of that, she often tried to negotiate the meaning of whatever she was saying by using simpler and shorter sentences and by using non-verbal forms of discourse. Ms. BarreraÂ’s non-verbal discourse was displayed in different formsÂ—she often used gestures to demonstrate the slope of the line and also used colored markers when
105 writing on the overhead: she marked important formulas in red, each consecutive step in blue, and all other information in black. She also often sat next to a student to help him/her on the computer and once encourag ingly patted on the s houlder an ELL student who exhibited improvement in both his ma thematics and Englis h language skills. The consequences of Ms. BarreraÂ’s discourse fell into the following categories intended or unintended immediate or long term as described in the Kr ussel et al. (2004) framework. Ms. Barrera intentionally shifted the cognitive leve l of the task performed by asking the students to explain how to perform certain steps. She demonstrated equally high expectations of ELL and non ELL students. However, she unintentionally made occasional mistakes in her proper use of the mathematical terminology and tried to compensate for this by further explanati ons and use of synonyms (e.g., she asked her students to Â“moveÂ” a term from one side of the equation to the othe r, and further added Â“do the oppositeÂ…when addition, subtract th e term on both sidesÂ” (lines 28-31)). Nevertheless, some of the positive immediate or long-term consequences of the teacher discourse were also transparent. For exampl e, when Ms. Barrera wanted to encourage ELL students (and all students for that matter) to start using proper reasoning techniques and justifications of the an swers they proposed, she would sometimes ask a question: Â“can you add these two terms?Â” and would point to 3 +5x for example. After the student answered Â“NoÂ”, she would encourage him/her to properly state the reasons for that by simply stating Â“becauseÂ…Â” and changing the pitch of her voice at the end. Thus she indicated that she expected a complete stat ement of the sort Â“because they are not like terms,Â” immediately indicating to the student that in mathematics justifications, proper responses are needed. She thus set long-term norms of classroom discussions. This in
106 turn, ensured that the stude nts (and the ELLs in particular) focused on both the mathematical discourseÂ’s meaning (to justif y) and its form (using Â“becauseÂ…Â”), and set the norm for the proper usage of conten t-specific vocabulary in English. Perceptions of classroom discourse. Figure 3 represents in the form of bar graphs the researcherÂ’s preliminary evaluation (i.e ., before counting of the frequencies with which Ms. Barrera uses different discourse strategies), Ms. BarreraÂ’s self-evaluation, and the evaluations of her ELL students who voluntee red to participate in the study. The pairwise correlations (Pearson product-moment correlation coefficients ) for Ms. BarreraÂ’s case study are as follows: the correlation betw een the teacher and re searcher is -.07; between the teacher and ELLs it is -.23, and between the researcher and ELLs it is .27. The negative results could be attributed to an unrealistic self-evaluation and a lack of understanding of the ELL students, which is due to less years of teac hing experience and a lack of ESOL training. There is agreement on the use of strategies 7 ( Use of Comprehension Checks ) and 14 ( Provide Feedback ) as the most frequently used, followed by (slightly less frequently) strategies 3 ( Use of Synonyms ) and 18 ( Use of Technology ). Ms. Barrera used various strategies to ensure that she is understood by her students and encouraged their further participation in the discourse by often providing feedback. Throughout the observed lessons, she frequently asked her students whether they understood the content of her talk. If further clarifica tion was necessary, she often modified her talk by using synonyms, explained ideas more thoroughly, a nd helped students visualize the concept under discussion by using the overhead pr ojector or the computer screen.
107 Figure 3. TeacherÂ’s researcherÂ’s, an d ELLsÂ’ evaluations of Ms. BarreraÂ’s frequency of use of various discursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
108 Figure 3 illustrates interesting differences between the researcherÂ’s evaluation, the teacherÂ’s self-evaluation, and the ELLsÂ’ evalua tions of Ms. BarreraÂ’s frequency of use in the following strategiesÂ—strategies 1 ( Use of a slower and simpler speech) 2 ( Use of fewer idioms and slang words ), 5 ( Use of change of tone, pitch, and modality ), 15 ( Use of gestures, facial expressions, and eye contact ), and 22 ( Provides students with content specific, enriched information, thus exhibiti ng equally high expectations from both ELL and English speaking students) With respect to strategy 22, while the ELL students and the researcher indicated th at Ms. Barrera frequently used this strategy, her own reflections on her teaching styl e indicated that she thinks sh e is not applying the strategy frequently enough. Even though in the interview she revealed that this is something she strongly believes to help her ELLs, Â“For ESOL st udents, I like to use ex amples of the real life, and also do different activi ties that they can utilize the topic that I am teaching,Â” she clearly thinks that she needs to improve upon her use of the strategy. On the other hand, strategies 1, 2, 5, and 15 were evaluated by th e ELL students as rarely being used by Ms. Barrera (one to two times a month). Ms. Barrera and the researcher both felt that she used these strategies more frequently. Summary of the frequency count of the teacherÂ’s discursive strategies. The actual frequency count of Ms. BarreraÂ’s use of each strategy, shown in Figure 4 below, reveals the frequency with which each category was used during the three observed lessons, and is in better consensus with the ELLsÂ’ evaluations rather than with those of the researcher and the teacherÂ’s self-evaluation. Figure 4 indicates the frequency with which Ms. Barrera implements the strategies from TTT Form 1 At first glance, we can immediately notice that the most
109 typical strategies for Ms Barrera are strategy 12 ( Use of Different Questioning Techniques ) and strategy 14 ( Provide Feedback ) Next in use are strategies 7 ( Use of comprehension checks ), 4 (U se of Repetitions ) and 6 ( Use of Clarifications of Directions). This data is further supported by the interview with an ELL student who confirms Ms. BarreraÂ’s use of the above strategies: Uh, I like my mathematics class because the teachers know how to explain, you know, like for all to understand. [If] she got to explain like twenty times for you to understand, she will do, and she always a good person. SheÂ…, if you donÂ’t understand English she will talk Spanish. She make[s] it easy to like mathematics that way. And she uh, no the easy way of th e class is where you are the focus, if she see you. If you fail some quiz, she will help you for next time [so that] you can pass it, and sheÂ’s a great teacher. However, Figure 4 also reveals that Ms. Barrera less frequently applied strategies 15 ( Use of gestures, facial expressions or eye contact ), 2 ( Use of fewer idioms and slang words from the mathematics vocabulary, or if used a proper explanation was provided ), 13 ( Use of wait-time technique s after posing a question ), 16 ( Use drawings of charts and visual organizers ) and 20 ( Providing the students with al ternative forms of assessment ). As the excerpts demonstrate, Ms. Barrera of ten asked the students questions to check their knowledge of previously-presented con cepts, to finish subsequent steps of a problem, or to check their comprehension. Ho wever, she expected immediate responses and did not provide the student s with enough time to think and process information and subsequently provided the correct answers he rself. When studentsÂ’ answers were only partially correct, Ms. Barrera usually provi ded the correct statement or paraphrased
110 Figure 4. Frequency count of Ms. BarreraÂ’s use of various discursive strategies during th e three 20-minute video-recorded sessions. 0102030405060 12. Use different questioning techniques 14. Provide feedback 7. Comprehension checks 4. Use of repetitions or paraphrasing 6. Use of clarification of directions 22. Content specific, enriched information 8. Identify vocabulary, pictures, or models 3. Use of synonyms 11. Math discussions and problem solving 1. Use of a slower and simpler speech 18. Use of technology 5. Use of changes of tone, pitch, and modality 17. Use of visual or auditory stimuli 19. Using cooperative groups 9. Review of related concepts 21. Use of cultural-specific knowledge 10. Summary of the key concepts 2. Use of fewer idioms and slang words 13. Use wait-time techniques 16. Use of charts, graphic organizers 20. Alternative forms of assessment 15. Use of gestures, expressions, eye contactStrategyFrequency
111 studentsÂ’ responses so that th ey were correct without asking students to do so themselves. Although she exhibited sensitivity to her ELLsÂ’ levels of English fluency and encouraged them to talk using mathematical terms, she did not allow time for them to think after she posed questions. When ELL students were asked to ela borate on their expe riences in their mathematics classroom, they expressed their preferences: Â“Â…like maybe work in groups, that would be better Â‘cause other people would know how to do it too.Â” The ELL students were aware that Ms. Barrera was limited by the setting of their Algebra I class in a computer lab. They all indicated that she usua lly assisted them by explaining things to the class as a whole by demonstrating examples on the overhead projector. They also indicated in the interviews that Ms. Barrera helped them also individually by circulating between the desks and the computers and often sat next to them a nd provided additional assistance if needed. However, they still felt th at Ms. Barrera could allow them to work in cooperative groups or with partne rs (strategy 20), because this would also be beneficial if they needed help at a particular moment wh en she was assisting another student. Figure 3 also demonstrates that, according to the ELLs, strategy 20 is not utilized frequently enough by Ms. Barrera, and the frequency count of the use of strategy 20 throughout the three observed sessions confirms th at (Refer to Figures 3 and 4). Ms. Chandler Ms. Chandler is Caucasian, in her ea rly 40s, and speaks only English. She is certified to teach secondary mathematics a nd has previous higher level mathematics and technology application experience because she worked as an analyst for a software company prior to becoming a high school mathematics teacher. She has taught in her
112 current school for eight years, and feels very comfortable teaching Algebra I in a computer lab. She completed her ESOL endor sement requirement of 60 hours through course work six years ago. Ms. ChandlerÂ’s Algebra I class consists of 26 students, eight of whom are ELLs of a Hispanic background. In th e interview, she shared that even though Â“it is challenging to teach ESOL students,Â” she finds them Â“to be mostly motivated and capable mathematically.Â” She also added that she ha d equally high expectations in mathematics from her ELL and fluent E nglish-speaking students. Typical classroom discourse. The following pattern emerged from the three observed lessons Â— Ms. Chandler never us ed whole-class lect uring as a teaching technique, but continually circulated am ongst her students while they worked on the computers and assisted them if they asked for help or if she decide d they needed any. The excerpt below demonstrates which strategies in particular Ms. Chandl er utilized when a female ELL student-Narissa asked her for help: Ms. Chandler:  So when youÂ’re writing the notes when youÂ’re writing the notes, it wasnÂ’t making sense? Narissa:  No because... Ms. Chandler:  I seeÂ… go on. [Ms. Chandler ac knowledged that the student had a problem, and encour aged her to continue.] Narissa:  It was number five on the um sheet. Ms. Chandler:  Yeah? [Again: Encouragement to continue ] Narissa:  It was one of them where you had the fractions. Ms. Chandler:  Oh right, the fractions are more difficult, but, so hereÂ’s the deal.  You got y, so you were getting that problem because you have to know the slope and you also have to know b  So youÂ’re doing m and now you put the two in there.  That Â’s exactly right. In line 7, strategy 14 (provide feedback) is used, as Ms. Chandler acknowledged that fractions are more complicate d. Furthermore, she provided assistance and the
113 clarification of the steps that the student needs to execute next in order to solve the problem (strategy 6) Â– that is, to write th e equation in slope-intercept form when two points are given (lines 8 and 9). The t eacher also provided the student with feedback that she had found the slope correctly (strategy 14 again in line 10). During the interview, Narissa indicated that only after Ms. Chandler provided her with the feedback (strategy 14) that she had found the slope correctly and assisted her in the next steps (strategy 6) was she able to complete the problem she was struggling with. The remainder of the excerpt will demonstrate other specific strate gies (besides strategy 14 and 6) that Ms. Chandler used, as well as the examples that she provided, in order to guide the student to the taskÂ’s completion and complete the rest of the examples from that lesson on her own: Ms. Chandler:  So the only other thing you have to do isÂ… you have to use one of the points.  So they are your points, right? ( x, y ) ( x, y ).  So just you can pick either point.  It doesnÂ’t matter which.  So use this one because it has the one, so it seems easier.  So we put that in place in y  HereÂ’s your m you already did.  Put that in for x because your whole what youÂ’re trying to do is find b Here the teacher pointed to the given points (-1, 1) and ( 1, 5) (strategy 15 Use of gesture and demonstrations to enhance comprehension, lines 11 and 12), demonstrated why and how the student should substitute one of the points (lines 13, 14, and 15), and wrote out: 1=2(-1) + b. The language the teacher used to explain the solution was simplified (i.e., she used simple commands and shorter sentences when explaini ng concepts Â– strate gy 1a) and thus was adapted to the ELL studentÂ’s level of SLApre-production. However, the use of Â“put that
114 inÂ” instead of Â“substituteÂ” (lines 17 and 19) indicates that even though Ms. Chandler demonstrated for the ELL student the soluti on in writing, she did not model the correct vocabulary in order for the ELL to move to the next level of English development Â– early production (i.e., lack of use of strategy 1b, wherein teachers need to model/demonstrate correct responses both in mathematics and in English). The next excerpt further demonstrates Ms. ChandlerÂ’s use of strategy 1a ( Use of a slower and simpler speech ) when demonstrating to the ELL student how to solve the linear equation for b. However, Ms. Chandler continued to use Â“put inÂ” in stead of Â“substituteÂ”, and thus missed the opportunity to expose the ELL to the appr opriate mathematics terminology for the performed mathematical operation (i .e., lack of use of strategy 1b): Ms. Chandler:  So now youÂ’re just gonna solve this for b  So you multiply of course, trying to get b alone so you add two.  ItÂ’s just like solving an equation.  So now you know m and you know b and so then you can write the equation y equals Â…and you put in the m two x plus b  So youÂ’re half done with all of these.  You already got th e 2 but you just have to find the b  So for every one.  Go ahead. Narissa:  Oh, I t hought that you was supposed to stop atÂ… see? Mrs. Chandler:  Â…yeah, so you had one more step.  So like here.  Put in zero one.  So zero equals one plus b Â…and then you just solve it. Here Ms. Chandler provided the student with feedback to indicate that she understood where she was experiencing difficulty (strate gy 14, line 29). Initia lly she performed a comprehension check by encouraging the student to c ontinue on her own (strategy 7, line 27) but, upon observing that the student coul d not complete the necessary operations on her own, Ms. Chandler d ecided to demonstrate how to find b with one more example
115 (lines 30 to 32). However, while she continued using simplified speech in her demonstration (strategy 1a), she again used Â“put in Â” instead of Â“substitute (lines 23 and 31), and Â“get b aloneÂ” instead of Â“isolate b on one side of the equationÂ” (line 21) when explaining how to solve for or find b (i.e., lack of strategy 1b). Next, Ms. Chandler decided to show yet another example in or der to make sure the ELL student would be able to complete the exercise se t from this lesson on her own: Ms. Chandler:  IÂ’ll do one more so you can see it.  Zero equals two plus b Â…  So minus two.  b equals negative two.  So you just had one more step.  So y equals negative two x minus two.  Skyla, be quiet. Here the teacher interrupted her instructions and made a remark to a (non-ELL) student causing a disciplinary problem by changing her tone, pitch and mode of talking (strategy 5, line 39). This demonstrated that the teach er was able to handle a discipline problem and, as the following excerpt will demonstrate, continued to mainta in the instructional Â“momentumÂ” without much disturbance: Ms. Chandler :  Minus two.  So what you s hould do is take this one.  I mean you can fi nish this one in class today.  BecauseÂ…just finish out all the bÂ’s and write all the equations.  And youÂ’re done with that one. [Then, Ms. Chandler continued to assist the student with one mo re example (a fourth example) and clarified her directions (strategy 6) in order to make sure the student understood which tasks to complete in class a nd which to complete at home in order to learn the lesson]:  This one is actually even easier because what I can do, you can write it on the paper if you want or I can put you on the quiz on the computer.
116  Because on these, they tell you the slope so you donÂ’t even have to figure it out.  They tell you the slope and they tell you the point.  And so then you have to write y equals mx plus b. [She writes y=mx+b ].  You put the poin t, you put the slope, and you find b.  ThatÂ’s all you have to do.  So then youÂ’ll have those two done and youÂ’ll be totally caught up.  So this one let me know after you finish that one if you want to do it on the computer or if you want to take it home and do it.  You donÂ’t even have to write the notes because theyÂ’re so much like those notes.  And then you gotta ge t both of those done, and these IÂ’ll take back and you can do both of those later. In this longer excerpt, Ms. Chandler modele d for the ELL student (Narissa) the solutions of four sample problems. Then, she summarized what Narissa needed to do in order to complete the assignment on her own (strate gy 10, lines 41 to 50). Ms. Chandler also provided her with choices between alternative forms of assessment Â– a quiz on the computer or take-home completion of the task by modifying (i.e., shor tening) the notes at parts (strategy 20, lines 52 to 54). The excerpts above also demonstrate the atmosphere Â“typicalÂ” of Ms. ChandlerÂ’s classroom Â– i.e., how the teacher managed the discipline in her mathematics classroom and facilitated students in their individual wo rk on computers by providing assistance (to both ELLs and students fluent in English) when needed. Krussel et al. framework. The purpose of Ms. ChandlerÂ’s discourse was to individually assist her studen ts and attend to their individua l needs. She regularly called upon and helped an almost equal number of ELL and non-ELL students. On average throughout the three observed sessions, Mrs. Ch andler helped at le ast four to five
117 different students. By freque ntly circulating around the ro om, she not only helped in mathematics, but she was helping them fix problems with their computers, and managed discipline problems (as was demonstrated in the excerpts above). As the excerpts above also illustrated, she often gave ELL students the option of choosing between writing all the work in their notebooks and then taking th e quiz again, or finish ing their notes on two lessons at home and taking more quizzes the following day. Ms. Chandler also graded their notebooks and thus eval uated their progress in writing and reading in mathematics (i.e., utilization of strategy 20 Â— providing the ELL students with a lternative forms of assessments). In the interviews, Ms. Chandler indicated that she was grading all studentsÂ’ mathematics notebooks in the middle and the end of each semester. However, with the ELL students she carried out this check daily or weekly, which gave he r a better idea of their progress in her class and thus allowed her to determine which of them needed her immediate assistance. To classify Ms. ChandlerÂ’s actions towards establishing a setting for mathematics classroom discourse, inferences were made from the interviews with Ms. Chandler and with her ELLs, and also from the observati ons. For example, duri ng the interview, Ms. Chandler indicated that she enjoys teach ing Algebra I in a Computer Lab setting. However, she indicated that being unable to speak Spanish, which was the native language of eight of her ELL st udents, increased her challe nges. On the other hand, her experience with them thus fa r into the school year (alr eady 3 months have passed) indicated that they generally had good b ackground knowledge in mathematics and were very motivated to learn more. In my interv iews with them, her ELL students indicated that they also enjoyed the classÂ’ computer la b setting. They also said that they liked the
118 class because they worked at their own pace and knew that they could always ask Ms. Chandler to assist them if they encounte red difficulties while solving a problem. The observations confirmed that such a classroom setting facilitated Â“stress-freeÂ” studentteacher interactions in mathematics. Howeve r, in such a setting, the ELL students lacked exposure to group or partner di scussions about mathematics. As was demonstrated in the previous case study, Ms. Barrera, who also taught Algebra I in a computer lab, often switched between a class-wide lecture a nd individual work. She was involving her students in discussion even if it was just to explain something from the bell work or in order to present a concept to the whole class before assigning them to individual work on the computers. The form of Ms. ChandlerÂ’s discourse also included both verbal ( teacher talk ) and non verbal (actions) forms As was demonstrated in the above excerpt of her interaction with an ELL student (Narissa), Ms. ChandlerÂ’s teacher talk included shorter sentences and simple commands (i.e., she uti lized strategy 1a). Thus, she demonstrated awareness that she was explaining mathema tics to an ELL in a very early stage of English language acquisition Â– pre-production Â– and as a result used slower and simpler speech. She also used non-verbal actions such as gestures and demons trations to enhance her ELL studentsÂ’ comprehension of her explana tions (i.e., she utilized strategy 15). For example, she moved her hand up to show that a line with a positive slope goes up; then she moved her hand down, horizontally, and ve rtically to demonstrate lines with negative, zero, and undefined sl opes, respectively. She also demonstrated to her ELLs how to show their work in their notebooks. For example, during her explanations to Narissa in the excerpt above, Ms. Chandl er first wrote the equation of a line ( y=mx+b ),
119 then she pointed to the given points (-1, 1) and (1, 5) and demonstrated how the student should substitute one of the points into the equation, and then she wrote out: 1=2(-1) + b. However, even though Ms. Chandler demonstr ated for Narissa the correct solution in writing, the use of Â“put that in Â” instead of Â“substituteÂ” indica tes that she did not model for the ELL student the correct mathematics vocabul ary in English. This indicates omission of utilization of strategy 1b, wherein t eachers need to model/demonstrate correct responses both in mathematics and in English in order for the ELLs to be able to move to the next level of English development Â– early production. The consequences of Ms. ChandlerÂ’s discourse fe ll into the following categories intended or unintended immediate or long term according to Krussel et al.Â’s (2004) framework. For example, Ms. Chandler intentionally simplified her talk when talking to ELL students, as was demonstrated in the exce rpts of her discussi on with Narissa (the ELL from a pre-production stage of SLA) above. Thus, she demonstrated awareness of the level of SLA of her ELLs and their ne ed to still develop their conceptual understanding of mathematics in English. However, even though she stated in the interview that she holds e qually high expectations of ELL and non-ELL students, she unintentionally often Â“took the floorÂ” and was the main speaker, as exhibited in the above excerpts. Thus, she was not providing the ELL students with many chances to be equal participants in mathematics discussions. On a different note, some of the immediate or long-term consequences of her discourse were al so apparent. For example, Ms. Chandler often checked ELL studentsÂ’ comprehension of her explanations by asking them to complete the assignment on their own after he r assistance, but mainly she was doing this by just watching them quietly if they were writing the solution of subsequent problems
120 correctly or, as she indicated in the interview, by checking their written homework on the following day. As was demonstrated in the ex cerpts above, she encouraged her students to show all the work that they performed in solving a mathematics problem, but she did not encourage them to reason, analyze, or simp ly discuss the solution with her. Thus she set long-term norms of writing in mathematics in English, but did not target the development of her ELLsÂ’ or al linguistic abilities. Perceptions of classroom discourse. Figure 5 represents in the form of bar graphs the researcherÂ’s preliminary evaluation (i.e., before an actual count of the frequencies with which Ms. Chandler uses different discursive strate gies), Ms. ChandlerÂ’s selfevaluation, and the evaluations by her ELL student s that volunteered to participate in the study. The pair-wise correlations for Ms. Ch andlerÂ’s case study are as follows: the correlation between the teacher and researcher is .77; between the teacher and ELLs it is .53, and between the resear cher and ELLs it is .70. As Figure 5 indicates, accor ding to ELL students, Ms. Chandler most frequently employed the following strategies: 14 ( Provide feedback) 6 ( Use of clarification of directions) 13 ( Use of wait-time after posing a question) and 18 ( Use of technology to enrich a concept presentation) (see Figure 5). The students in Ms. ChandlerÂ’s class learn Algebra I using the I Can Learn Lab (i.e., computers were utilized as an inherent part of instruction), but the video-recorded sessions also reveal that the teacher assisted her students (and the ELLs in particular) in uti lizing the technology in their problem solving processes (strategy 18). As one could in fer from the excerpts provided above, Ms. Chandler certainly utilized strategies 14 and 6. Additionally, her Â“laid backÂ” style of explaining concepts and also telling students she would come to check on their work
121 Figure 5. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ efvaluat ions of Ms. ChandlerÂ’s freqeuency of use of various disc ursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
122 again ensured that she was providing them with enough time to grasp the new or difficult for them concept (strategy 13). However, Figure 5 also reveals that while the teacher evaluated herself as most frequently using strategies 2 ( Use of fewer idioms and slang words) 3 ( Use of synonyms) and 5 ( Use of change of tone, pitch, and modality) according to her ELLs (and the researcher), she used these strategies only on ce or twice a week. Such disparities between the teacher, ELL and researcher evaluations coul d be attributed to the difference between the teacherÂ’s self-perception of how frequently she used the same strategies and the results elicited from the three observed sessi ons and the interviews. From Figure 5, it is evident that Ms. Chandler evaluated that sh e frequently used idioms and slang words from the mathematics vocabulary when explai ning concepts to ELL students. She also reflected in her self-evaluation of her ta lk that she used more synonyms and often changed her tone, pitch, and modality (strategie s 2, 3, and 5 respectively) so as to better present the concepts behind the mathemati cal terms used. The ELL students indicated that their teacher frequently was giving directions and providing assistance when a specific task was posed to them (strategy 6) They also indicate d that Ms. Chandler provided to them feedback and extra wait-time (strategies 14 and 13, respectively) on a regular basis. However, from Figure 5, it is apparent that all evaluators agree that Ms. Chandler did not provide many opportunities for students to share cultural background experiences when solving mathematical problems (strategy 21). She also did not conclude the lesson by summarizing the key concepts (strategy 10).
123 Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 6 below indicates the frequency with which Ms. Chandler implemented the strategies identified in TTT Form 1 Figure 6 indicates that Ms. Ch andler most frequently used strategies 14 ( Provide feedback ) and 1 ( Use of slower and simpler speech ), followed by strategies 5 ( Use of change of tone, pitch, and modality ) and 6 ( Use of clarification of directions and assistance ). However, Ms. Chandler did not utilize strategies 11 ( Involve students in mathematical di scussions and problem solving ) and 19 ( Expose students to different classroom work arrangements ). These results further confirm the resu lts found by analyzing Ms. ChandlerÂ’s discourse by applying Krussel et al.Â’s (2004) framework and the researcher, the teacher, and the ELLs evaluations of Mrs. Chandl ersÂ’s discourse reported in Figure 5. Mr. Davison Mr. Davison is a 40 year ol d Caucasian, and speaks only English. He has been a teacher for 16 years and has taught Algebra I a nd Liberal Arts for eight of those years. He completed his 60 hours ESOL endorsement re quirement through in-service points and by taking additional evening courses. During the in terview, he shared that he often uses bilingual students to peer-tutor ELLs. He also said that he is aware of the presence of ELL students in his class and the fact th at he does not sp eak their language: I try to slow my teaching to give students a chance to ask questions. It also gives me a chance to read the expressions of th e students. I can usually tell if they understand or not. Also it gives me a ch ance to change the way I present the material. The class was diverse, with an almost equal number of Hispanic (9), African
124 Figure 6. Frequency count of Ms. ChandlerÂ’s use of various discursive strategies during the three 20-minute video-recorded sessions. 05101520253035 14. Provide feedback 1. Use of a slower and simpler speech 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 18. Use of technology 4. Use of repetitions or paraphrasing 7. Comprehension checks 12. Use different questioning techniques 8. Identify vocabulary, pictures, or models 15. Use of gestures, expressions, eye contact 17. Use of visual or auditory stimuli 20. Alternative forms of assessment 22. Content specific, enriched information 3. Use of synonyms 2. Use of fewer idioms and slang words 9. Review of related concepts 10. Summary of the key concepts 13. Use wait-time techniques 16. Use of charts, graphic organizers 21. Use of cultural-specific knowledge 19. Using cooperative groups 11. Math discussions and problem solvingStrategyFrequency
125 American (8), and Caucasian (5) students. Mr. DavisonÂ’s classroom had student desks lined on both side-walls, facing the center. His desk was in the back and, after taking attendance on the computer, he usually walked to the center of the room, where the overhead projector was located, facing the white board on the front wall. Mr. Davison usually taught by using both the white board and the overhead projec tor, often switching between the two. Typical classroom discourse. During all three observed classroom sessions, Mr. Davison used either the overhead projector or the white board to write solutions to problems the students were solving collectivel y under his guidance. During this time, the students referred to the book to read the lesson-specific vocabulary and examples. He created a Â“stress freeÂ” environment where st udents freely asked questions or readily provided answers to individual or general questions. From th e discussions that took place and the responses students provided during ea ch observed lesson, it was evident that the students were becoming more active in the classroom mathematics discourse, as the following excerpts will demonstrate. For exam ple, the third sessi on was a review of previously taught concepts, and when appro aching an application problem, the teacher asked the students to recall the difference between exact interest and ordinary interest After posing the question twice and seeing that the students still did not respond, Mr. Davison began leading the cla ss in the following dialogue: Mr. Davison:  How many days are in the year exactly? Ashley (not an ELL):  365 Mr. Davison:  365. How many days are in a bankerÂ’s year? Joshua (not an ELL):  360. Mr. Davison:  ThatÂ’s the difference.  Exact interest and thatÂ’s the way to remember it, exact interest is exactly 365 days, OK?
126  Ordinary interest or bankerÂ’ s interest is 360 days.  Because 360 is an easier number to work with.  ThatÂ’s what they tell you.  The real reas on isÂ…which will give the bankers more money? Here the teacher, even though he was ta lking to the whole class, was applying questioning techniques sensitive to the fact that ther e were ELL students present in his mathematics classroom (strategy 12b, lines 1, 3, and 10). More specifically, Mr. Davison was asking questions usually eliciting a one-wor d response which are appropriate for ELLs in the early production stage of SLA. From the follo wing two excerpts, it becomes evident that some ELL students began to par ticipate, and that Mr. Davison called on them by name if they did not raise their hands: Maria (an ELL):  360. Mr. Davison:  360, because look, letÂ’s say you have a six months loan.  That is 180 days. Mr. Davison:  Right?  So, 180 over 365 isÂ…s omebody with a calculatorÂ…?  Trevor, what is it? The teacher used a specific example from the real world (line 12, strategy 22c) and initiated participation of students in the calculations (s trategy 11, line 15) leading them to understanding the difference in value of both interests. Here the teacher involved another ELL student in the mathematical discussion by calling on him by name (strategy 11, line 16) and asking him to provide the answer by using a calculator (strategy 18, line 15). Trevor (an ELL):  0.4893149506. Mr. Davison:  OK, now, whatÂ’s 180 over 360? It Â’s just 0.5, itÂ’s just half. [ 19] So what Â‘s gonna ma ke them more money?  It is not much of a difference but what is bigger?  The 0.5.  ThatÂ’s the bankerÂ’s interest. Maria (the previous ELL):  But couldnÂ’t you round that up? Mr. Davison:  Well, but that Â’s the thing!  They donÂ’t round it up.
127  I mean, if I am paying interest, I ainÂ’ t rounding up, IÂ…, ItÂ’s gonna cost me money.  You see what I am saying? Maria again:  That is why the bankers want 360. Mr. Davison:  Right, thatÂ’s why they want 360.  ItÂ’s easier to work with because 360 is an even number, but they also use it because itÂ’s a little bit more. This excerpt illustrates how Mr. DavisonÂ’s attempts to involve all his students by particularly calling on some ELLs when notic ing that they do not participate in the classroom mathematics di scourse (strategy 11). Krussel et al. framework. The purpose of Mr. DavisonÂ’s discourse was to involve his students in discussions (strategies 11 and 12; lines 1, 3, 10, 15, 18, 19, 20 Â– involvement of all students, and 16 Â– a speci fic ELL student is called) which in this example makes them realize for themselves th e difference between the two definitions of interest (exact and ordinary). By using synonymous words Â“bankerÂ’s interestÂ” for Â“ordinary interestÂ” (lines 7a nd 22) and Â“more,Â” Â“not much of a difference butÂ…biggerÂ” for representing the difference between the tw o types of interest (exact and ordinary interest) in simplified sentences : Â“So whatÂ’s gonna make them more money? It is not much of a difference but what is bigger?Â” (lines 19 and 20) Mr. Davison tried to aid his ELL students (and all students for that matte r) in understanding the concepts behind the terminology Â“exact interestÂ” and Â“ordinary in terestÂ” (and its synonym Â“bankerÂ’s interest, line 22) (strategies 1a and 3). After an analysis of the three observed classroom sessions, a pattern unique to Mr. Davison emerged. He never directly corr ected his students when their answers were incorrect (strategy 1b). He used the strategy of repeating the question (strategy 4),
128 whereby the students seemed to perceive the Â“ unspoken feedback Â” that the answer is incorrect and they should try again (strate gy 14) until the right answer were provided. Whenever the right answer was provided, Mr Davison usually re-s tated and elaborated the response (strategy 4 agai n). For example, when first asking: Â“Â…quarterly is how many times a year?Â” and receiving the answer Â“1.25,Â” he repeated Â“how many times a year?Â” and when another student said Â“3Â” he asked again until someone answered Â“fourÂ”. Then, Mr. Davison indicated that this is the correct answer by repe ating: Â“four times a year which is three months.Â” This demonstrated that he was satisfied with short responses and easily provided the explanations as to w hy this is the correct response. Here the researcher is not stating that this is an appropriate stra tegy for use with ELLs, but is instead reporting on the observe d pattern in Mr. DavisonÂ’s mathematical discourse. The reader is thus provided with a glimpse into the actual discourse that took place during the classroom observations. However, the exam ple demonstrates how Mr. Davison usually missed opportunities to move the discourse to higher levels of cognitive demand ( synthesis and evaluation) as per BloomÂ’s Taxonomy (strat egy 22 e and f). Furthermore, by not asking the ELL students in particular (s trategy 12d) to further explain some steps while problem solving (strategy 12 c and d), he did not provide them with opportunities to expand their level of Eng lish language acquisition to th e more advanced levels of speech emergence and intermediate fluency. Mr. Davison had created a relaxed classroom setting Regardless of the classroom work arrangement Â– lectures, cooperative gr oups or whole-classroom discussions, his students (ELLs and non-ELLs) naturally partic ipated by asking questions or readily providing answers. Even though some of the calculations they provided were incorrect,
129 the students demonstrated active interest and involvement in classroom activities. Furthermore, they exhibited mutual respect toward each other. The form of the teacherÂ’s discourse included both teacher talk (verbal) and actions (non verbal) During the observations, Mr. Davison exhibited focus on the mathematical concept discussed (and e xplaining the concept by using more informal/conversational English), rather than on the form of presenting it (i.e., stating a formal definition of the concept in English). For example, this is demonstrated in the dialogue that took place when an ELL student asked, Â“But couldnÂ’t you round that up?Â” (line 23 in the excerpt provided above). Mr. Davison answered: Â“Well, but thatÂ’s the thing! They donÂ’t round it up. I mean, if I am paying interest, I ainÂ’t rounding up, it is gonna cost me money. You see what I am saying? Â” (lines 24 to 27), which indicates that he tried not to simply provide the answer to the question. He asked his students to critically think and realize the difference th at occurred if rounding was indeed performed (here we do see an attempt of applying strate gy 22 Â– providing the students with content specific information). However, even though Mr. Davison demonstrated that he was trying to provide opportunities for his st udents (both ELLs and non-ELLs) to build upon their prior knowledge an d be able to solve real world pr oblems, he did no t ask them to further explain, criticize, or justify their thinking while problem solving that could expand on ELLsÂ’ language skills and all stude ntsÂ’ critical thin king skills. Mr. DavisonÂ’s non-verbal discourse was displayed in different formsÂ—he often used his hands to gesture when talking, or used eye contact afte r posing a question, and with a nod or calling on a student indicated w ho may speak. He also walked between the
130 rows while students worked in pairs or groups and assisted them or answered questions (when called). According to Krussel et al.Â’s (2004) framework, Mr. Davison demonstrated intended efforts to make all his students (both ELLs and non-ELLs) feel as equal partners in the discourse that took place in his classroom. However, he unintentionally neglected to shift the cognitive level of the tasks performed (simple interests, deposits, etc.) by not asking the students to further explain how to carry out the particular steps, or to critically evaluate their answers or further check and expand upon them. He assisted them by asking mostly questions requiring oneÂ–word or short responses and thus demonstrated his lower expectations that they would not be ab le to complete the task on their own. Only on a few occasions throughout all three observed sessions did Mr. Davi son ask questions of the type Â“Well, if weÂ’re depositing all this but sheÂ’s getting cash backÂ…so what would we do now?Â” which encouraged the ELL student answering to reply: Â“So we subtract.Â” Â“Right, so weÂ’re going to s ubtract,Â” Mr. Davison reassu red him by providing feedback and re-stating the sentence, upon which the student completed th e sentence Â“thirty 20Â’sÂ”. Such instances of challenging the students to explain the strategies used to completely solve problems were rare and Mr. Davison usually assisted them by asking questions leading to the next step. Additionally, some of the immediate and long-term consequences of Mr. DavisonÂ’s discourse were apparent. For exampl e, when he wanted to shift the dialogue from univocal to dialogic and involve the class in figur ing out how to find the annual interest rate, he faced the students and said Â“Â…well, then you gotta figure out how much per year. LetÂ’s put it like this : If I paid hundred dollars worth of interest in 6 months, how
131 much would I pay in a whole year,Â” it was not surprising that ma ny students immediately answered Â“200.Â” Then, Mr. Davison continue d giving further examples such as the following: Â“What if I paid 50 dollars of interest in 3 months?Â” and thus was involving all students to participate in the discourse. However, in assisting them with more and more specific questions, which as an immediate consequence involved many of the students, Mr. Davidson inadvertently prevented th em from reasoning a nd justifying their responses. The apparent protocol was to si mply supply a short response and move on. An immediate consequence of Mr. DavisonÂ’s talk and questioning techniques was thus the encouragement of studentsÂ’ partic ipation in classroom discourse. The long-term consequences were student involvement in tasks requiring mental acuity up to the fourth level of BloomÂ’s Taxonomy (knowledge, compre hension, application, and analysis), and also the encouragement of ELLs to attain the speech emergence level of SLA. This, in turn, indicated that while Mr. Davison provide d the students (and the ELLs in particular) with equal opportunities to fo cus on the mathematical discourseÂ’s meaning rather than on its form, he was not directing them to critica lly reflect on the result s and to explain their thinking, and he was not provi ding the ELLs with opportunities to develop to the next stage of SLA Â– intermediate speech Perceptions of cl assroom discourse. On Figure 7 below are represented the results of comparing the data from th ree sources of evaluation (i.e., TTT Form 1, 2, and 3) of teacher talk. The pair-wise correlatio ns (Pearson product-moment correlation coefficients) for Mr. DavisonÂ’s case study are as follows: the correlation between the teacher and researcher is .68; between the teacher and ELLs it is .17, and between the researcher and ELLs it is .43.
132 As Figure 7 indicates, there are three st rategies where there is almost complete consensus between the evaluations of the rese archer, the teacher se lf-evaluation, and the ELLsÂ’ evaluationsÂ— Use of a slower and simpler speech (strategy 1), Provide feedback (14), and Use of gestures, facial expressions, eye contact, or demonstrations to enhance comprehension (15). Another strategy that all agreed th at Mr. Able used in a consistent manner but in a smaller frequency is: Provides students with content specific, enriched information, thus exhibiti ng equally high expectations from ELL and non-ELL students (22). The observations, as well as the vide o recordings, provide evidence that Mr. Davison employed the above-listed strategies very often. He habitually used shorter sentences and adapted his speech to his audi ence (a diverse group of students, including ELLs). To foster his ELLsÂ’ development of mathematics communication in English, he focused his teacher-talk during whole-classroom discussi ons on key concepts and then provided opportunities for his students to en gage in small group-work and partner discussions, where they could apply these c oncepts in problem solving. (Refer to the excerpt at the beginning of th e description of Mr. DavisonÂ’ s classroom discourse for an example.) However, in reference to which strategi es were least frequently used by Mr. Davison according to his ELLs, from Figure 7 it is evident that thes e are strategies 2 ( Use of fewer idioms and slang words) and 20 ( Provide opportunities for students to share experiences and build up on personal or cu ltural-specific knowledge while problem solving in mathematics ). During the interviews, Mr. DavisonÂ’s ELL students (three students) shared a similar opinion that the obs erved lessons were ra ther easy for them
133 Figure 7. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evalua tions of Mr. DavisonÂ’s fr eqeuncy of use of various discursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
134 because, as one of the ELL students said: Â“the teacher gave a very good explanation of [them]Â”[referring to the lessons]. Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 8 indicates the frequency with which Mr. Davison implemented each category of teacher talk and teaching strategies found on the TTT Form 1 As Figure 8 indicates, the strategies most frequently employed by Mr. Davison are: Use of different questioning techniques sensitive to the ELLsÂ’ l evel of SLA (strategy 12) and providing feedback (strategy 14). This pattern additionally supports the da ta reflected in th e previous figure (see Figure 7), which included the researcher Â’s evaluation, the teacher self-evaluation, and the ELLsÂ’ evaluation of Mr. DavisonÂ’s styl e of teaching mathematics to classes with ELLs. However, further examinations of the questions with which Mr. Davison addressed his ELLs indicate that most of th e questions required only a one-word response or a short list of words. Thus, the data in dicates that by using que stions appropriate for ELLs from initial stages of ELL (English) language development (i.e., questions that initiate simple responses), Mr. Davison was aware that his ELLs were in the stage of production of English. However, as the excerp ts above also demonstrated, he was satisfied with his ELLsÂ’ short responses and di d not challenge them with questions that could lead them to move to the highest levels of the subject-specific literacy Â– intermediate speech and fluency in mathematics in English. Moving the mathematics discussions to higher levels of cognitive demand (i.e., analysis, synthesis, and evaluation ) on BloomÂ’s taxonomy creates more opportunities fo r all students (and ELLs in particular) to become critical mathematics thinkers.
135 01020304050607080 14. Provide feedback 12. Use different questioning techniques 7. Comprehension checks 4. Use of repetitions or paraphrasing 6. Use of clarification of directions 1. Use of a slower and simpler speech 22. Content specific, enriched information 3. Use of synonyms 18. Use of technology 8. Identify vocabulary, pictures, or models 10. Summary of the key concepts 11. Math discussions and problem solving 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 19. Using cooperative groups 15. Use of gestures, expressions, eye contact 9. Review of related concepts 13. Use wait-time techniques 21. Use of cultural-specific knowledge 2. Use of fewer idioms and slang words 5. Use of changes of tone, pitch, and modality 20. Alternative forms of assessmentStrategyFrequency Figure 8. Frequency count of Mr. DavisonÂ’s use of various discursiv e strategies during the three 20-minute video-recorded sessions.
136 Despite the fact that Mr Davison created opportuniti es for his ELL students to participate in the mathematics discourse, he did not ask enough higher order questions such as Â“Why?Â”, Â“How?, Â“What is your opini on?Â”, or Â“Compare/contrast ideas,Â” and did not provide them with opportunities to jus tify and explain, to dr awn conclusions and, consequently, expand on their learning of math ematics and English. Figure 8 also reveals the aforementioned omission of teaching strategies 2 (U sing of fewer idioms and slang words) and 21 ( Providing opportunities for students to share experiences and build up on personal or cultural-specific knowledge while problem solv ing in mathematics and thus building cross-cultural knowledge). Lincoln High School Ms. Andersen Ms. Andersen is Caucasian and in her mid 40s. She has a professional teaching certificate in secondary mathematics and is presently working on her National Board Certification (NBC). She has 23 years of t eaching experience, the entire duration of which she has taught Algebra I, Geometry, and Intensive Mathematics classes. She completed her requirement of 60 hours of ESOL training through coursework 10 years ago. However, during the interview, she shared : Â“I am still learning to incorporate more strategies that I find via classes Â– CRISS Â…or NBC [Nati onal Board Certification] classes as well.Â” Ms. AndersenÂ’s Algebra I class was very small; it consisted of only 11 students, eight of whom were Black or African Am erican, two were Hispanic, and two were White. Four of the students in the class were ELLs. Two of them (one male and one female) were Haitians and spoke Creole, French, and very limited English. They were
137 repeating their Algebra I class. During the in terview, Ms. Andersen indicated that she has a minor in French and can speak it fluen tly, even though her read ing and writing skills were becoming more limited from not prac ticing the language more often. She also indicated her Â“love [to] pract ice French with Haitian students.Â” One of the other ELL students in her class (a male student) was of Hispanic background. Even though the teacher was aware that at home his family sp eaks mostly in Spanish, she indicated that he was the most comfortable with English of th e ELLs in her class. She classified him as having an intermediate level of fluency in English. Th e other ELL student spoke Arabic and was in pre-production to early production level in his English proficiency. He was repeating his Algebra I class as well. In connection to this, Ms. Andersen said: Since we are receiving an influx of Bo snians, Palestinians, and Muslims [from other countries], they are trying to le arn English with mostly poor American school habits. Because of their limited schooling due to political and religious issues, it is of utmost importance they are screened, tested, and placed in small learning communities with pairing/sharing grouping. Typical classroom discourse. From the three observed sessions in Ms. AndersenÂ’s Algebra I class, the following pattern of cl assroom organization became evident. First, she usually assigned bell work which she eith er wrote directly on the overhead projector, or had pre-written on transpar encies. Then, the students were expected to take notes while she presented the new lesson by again using the overhead pr ojector. Often, she asked the students review questions, thus i nvolving them in classroom discourse and building upon their prio r knowledge in her explanation of the new mathematical concept, evidence of which will be provided throughout the excerpts below. The class often used
138 the McDougal Littell Publishing CompanyÂ’s Algebra 1 textbook by Larson, Boswell, Kanold, and Stiff (2004). Ms. Andersen usua lly pointed out which information the students needed to outline in their notes or which exercises they mu st do by specifically asking them to Â“dog earÂ” (a n idiom she used in reference to marking the pages by bending) the corner of the page containing the information she wanted them to pay specific attention to. Her efforts to teach her students good note-taking skills are demonstrated in the following excerpt: Ms. Andersen:  I want you to Â“dog earÂ” two page s that we keep looking at in this book all the time. [Here in order for her ELLs to also understand what she meant by Â“dog earÂ”, Ms. Andersen demonstrated the be nt corners of the pages from her book to which she was referring].  I wa nt you to just memorize perhaps one of the graphs, because you make mistakes like you did last year.  Just keep in mind an equation thatÂ’s just x equals, just goes like thisÂ… [While repeating this, Ms. Andersen was also writing on the overhear projector x=a, and drew out a coordinate system and with her ma rker showed the direc tion of the graph of the equation x=a (parallel to the y-axes). She also pa used so as to provide her students with enough time to start note-taking a nd draw the same graph that she was demonstrating, and which was also drawn on th e page in the book she was pointing to her students]. Thus, Ms. Andersen utilized strategies 17 ( Use of variety of visual stimuli : transparencies and pictures from the book) and 20 ( Provide opportunities for students to read and write in mathematics, lines 1 to 3). Ms. Andersen:  OK? [Here the teacher looked up and checked if her students were done writing the equation x=a and sketching the graph of the equati on, and whether or not they were nodding confirmation that they understood the direction in which the graph goes].  You know what that means and then, hopefully, your y you know goes the other way. [Here the teacher wrote on the overhead the equation y=b and drew its graph, and then pointed for the students to see how the graph of this equation is in a different direction from the first graph. And, again, after providi ng sufficient time for her students to take notes, she continued]:
139  And IÂ’ve pointed out numerous times I want you to Â“dog earÂ” your bookÂ… Here the teacher checked if her students we re following her and comprehending her drawings and explanations (strategy 7, lines 3 and 4). And because this observation was a review lesson, she wanted them to take notes of the important sections of the already completed chapter and she was also checking their previous knowledge (strategy 9, line 5) by observing if they nodded in agreement that the graph of y=b should go in the opposite direction. Then, Ms. Andersen repeated her directions (strategy 4, line 6). At this moment, a student (the ELL student of Hispanic background who was most comfortable in English in co mparison to the other ELLs in her class) interrupted: Jennifer:  IÂ’ve got a question.  You said that the lines should be parallel, but should the axes and the lines should be parallel if you have x equals it would be, or y equals? Ms. Andersen:  The x -equation is parallel to the y -axis. Jennifer:  Right. Ms. Andersen:  The y-equation would be parallel to the x -axis. If you would like to write that down as your personal note. Jennifer:  Right, and also if itÂ’s on the x -axis there is no way itÂ’s going to be on horizontal? Ms. Andersen:  Exactly. GoodÂ…good perception and g ood information for the rest of the students to pick up on. [Then the teacher continued by facing the whole class.]  The two pages I want us to go back to, and when I mean Â“dog earÂ”, fold it over, this might be used next year. [Here, Ms. Andersen demonstrated again wh at she meant by using the idiom Â“dog earÂ” by holding the corner of the page, thus clarifying to all students (especially to her ELLs) that she meant for them to mark the page as important].  LetÂ’s go back to page 213Â… On page 213 in the book, in a box entitled Equations of horizontal and vertical lines, are presented the graphs of two general equations: y=b and x=a. On the same page, the graphs and the solutions of two model examples Â– example 5, which asks the students
140 to Â“graph the equation y=2,Â” and example 6, asking them to Â“graph the equation x= -3 Â” Â– are provided. The teacher now explained the ex amples in detail and then continued: Ms. Andersen: Â…The other page that I can rea lly continue to push that IÂ’d like you to Â“dog earÂ” is on pa ge 2Â…, I think itÂ’s page 228: c lassification of lines by slope. On this page, in the box are presente d four graphs of lines with slope m>0 m<0 m=0 and m as undefined. Here (line 15), Ms. Andersen utilized strategy 17 ( showing charts and graphic organizers to enhance teacher talk ) and thus directed th e students to pages of the chapter with important information for them to mark and revi ew in detail. Then, after some outside interrupti ons and once the students opened their books to this page, she continued explaining the visual repr esentations provided on page 288 of the book. The excerpt that follows provides a more detailed glimpse at Ms. AndersenÂ’s teacher talk: Ms. Andersen:  You can see how the line is going down based on the slope being negative.  Okay? We did those problems.  When itÂ’s going up, is in the case where it is undefined, which is the fourth one.  And then if I wanted to include another line, this is crossing through the y -axis at negative four, theref ore it would have a level slope.  A level flat line which w ould mean it equals zero.  Keep in mind zero, and does not exist or undefined, are not the same things. Here, Ms. Andersen utilized strategy 3 (lines 19 to 21) Â— use of synonyms a level slope and a level flat line in the descriptio n of the mathematical term for slope m=0, and strategy 17Â— showing charts and graphic organiz ers to enhance teacher talk, and thus helped her students (and her ELL students in particular) better unde rstand the underlying concepts in that particular mathematic vocabulary. She also utilized strategy 1c ( teacher
141 talk focused on key concepts lines 16-21) and strategy 1d ( teacher talk fostering conceptual understanding through content line 21). Ms. Andersen:  Make sure you are looking at the last two lines on page 228.  Know we donÂ’t always go ahead.  We go back as a reference.  HavenÂ’t you read a novel and forgot something in a chapter and you wanted to go back a few pages? Students (a few students said together):  Uhhuh. Ms. Andersen:  ThatÂ’s what we do in this book.  We condense the notes.  WeÂ’re already on page 244.  And if youÂ’ve take n notes by me every day, youÂ’ve made your own personalized Algebra 1 st udy guide, bell work, examples. This segment of the teacherÂ’s discourse (lin es 30, and 33 below) displays not only how Ms. Andersen guided her students in not e taking, but also that she was setting expectations for all of them (including her ELL st udents) to practice writing in mathematics in English and thus create their Â“own personalized Algebra IÂ” journals with definitions, visual representations (graphs), and examples from class work, bell work, and homework: Ms. Andersen:  Alright, so I mean we have not written 244 pages of notes, have we?  Every thing has been condensed.  You know what you need to know exam wise and Algebra, so it takes you to the next class.  Okay, before we go to th e last section, which is dealing with functions and identifying func tions, some of you have your homework out.  If you have any questions on page 244, letÂ’s go over it at this time.  13 to 45 and list every fourth problem. The assigned homework from page 244 in cluded practice and application problems asking the students to find the slope and the y-intercept given the e quation of a line, and
142 then to graph the line. Here, Ms. Andersen utilized strategy 22 ( application of the content specific information the students were exposed to wh ile learning this chapter). Ms. Andersen:  Okay?  Any ques tions from last night?  I prefer to start th at before we do the next section. This excerpt from Ms. Anders enÂ’s discourse illustrates the teacher talk pattern that she naturally adhered to by utilizing the above mentioned strategies, and will be analyzed in the following paragraph usi ng the Krussel et al. (2004) framework. Krussel et al. framework. The purpose of Ms. AndersenÂ’s discourse was to foster conceptual understanding and expanded literacy through content (strategy 1dÂ— Use of pattern of speech appropriate to students with Intermediate fluency in SL, lines 16, 18, 19, and 20; Refer to Appendix A and Figur e 9). The fact that she regularly used a variety of visual stimuli (strategies 16 and 17) Â– transparen cy sheets on the overhead projector when giving the students bell work or teac hing a new lesson, pictur es from the textbook (line 22, 29, 35, and 36), or modeling solutions on the white board using different colors Â– indicated that Ms. Andersen catered to th e needs of the ELLs present in her Algebra 1 classroom. Her focused use of slower and simpler speech (strategy 1, lines 16 to 22) and various visual representations were done in order to aid her ELL st udents to better grasp mathematical concepts. For example, she strategically pointed out to students visual representations (strategy 17) of the special cases of equations of vertical (line 18) or horizontal lines (lines 19 and 20), and she explicitly taught them Â“condensedÂ” notetaking skills (strategy 16, lines 28 to 32) by poi nting out which parts of the book are important to mark (Â“dog earÂ”) and read for future reference (strategy 20, lines 22 to 25).
143 However, Ms. Andersen did not involv e her students or her ELL students in particular, toward reflections on their thinking, further expl anations, or justifications (strategy 22e and f). Most of her questions were of the type Â“wha t isÂ…Â”, Â“whichÂ…Â” or Â“do you rememberÂ…Â” (strategy 12a and b) wh ich only tested studentsÂ’ knowledge and perhaps comprehension and app lication (strategy 22a, b, and c) but did not force them to perform analysis, synthesis, and evaluation (strategy 22d, e, and f). There were only a few isolated instances of such lines of que stioning in each of the observed sessions. The setting for classroom discourse appeared to have been established early in the school year, as the students conformed to ce rtain pre-established norms of behavior during the observations. For example, in tu rn-taking, usually once a student provided an answer to a general question, the same student continued talking (lines 6, 9, and 11) until the small task was completed (in the partic ular example, the teacher answered and explained questions to this particular student) or anothe r student would be asked to continue. However, during the classroom obs ervations, Ms. Andersen did not provide different classroom settings for her students. She usually started with bell work on an overhead projector and then presented the new lesson (again on an overhead projector) while the students were expected to take notes. In this classr oom setting, the students freely asked questions, as the ex cerpt above illustrated (lines 6, 9, and 11), but they were not exposed to classroom a rrangements that fostered cooperative group work, partner discussions, or games (i.e., lack of utilizing strategy 19). The form of Ms. AndersenÂ’s discourse included both actual teacher talk (verbal) and actions (non verbal) Even though the teacherÂ’s natural talk in her native English was not very simple, and she often included phrases such as Â“I believe firmly,Â” Â“you may be
144 assured,Â” Â“increments of 1,Â” she often accomp anied this talk with actions such as gestures (strategy 15), drawings, or the use of colored markers while writing on the overhead or the white board (strategies 16 and 17). Thus, by enhancing her teacher talk via strategies such as gestures and visu al stimuli, Ms. Andersen demonstrated attentiveness to the presence of ELL students in her mathem atics classroom, and used demonstrations to enhance their comprehe nsion of her talk. For example, while explaining the signs of numbers representing th e coordinates of ordere d pairs in the four different quadrants, Ms. Andersen said: Â…and what is so neat about this, I want you to see. Look at I and III, arenÂ’t they total opposites of each otherÂ” Two plus es and two minuses. And look at the reverse, of quadrants II a nd IV. Look at the signs, becau se if you could fold, there would be symmetry. Fold and see. The previous excerpt illustrates that wh ile Ms. AndersenÂ’s talk included words with precise meanings such as Â“reverse,Â” Â“total opposites,Â” Â“sym metry,Â” her discourse often took the form of a challe nge; yet at the same time she modeled, or asked the students to model, the si tuation she was explaining (strategy 1b). Hence through her verbal and non verbal discourse, she demonstrated correct responses both in mathematics and English that were appropriate for ELLs from the stage of early production of English language (also part of strategy 1b). Howe ver, she was usually satisfied with ELL studentsÂ’ short responses and did not challenge them to further experiment with the English language in explaining their thinking while problem solving (i.e., she did not apply strategies 1d, 12c or d, and 22f).
145 Some of the consequences ( intended or unintended immediate or long term ) of Ms. AndersenÂ’s classroom discourse became clea r after an analysis of the three observed sessions. For example, she intended to include her ELL students from lower levels of English language acquisition in various task s (plotting points or graphing lines) by asking for their participation in the task; however, she unintentionally assisted them in the taskÂ’s execution by asking them what the next step of the task would be and by posing questions that required only single-word answers or sh ort responses. Thus, sh e exhibited her lower expectations that her ELL students would not be able to complete the task on their own. Consequently, she unintentionally did not provide her ELLs with enough opportunities to practice their mathematics vocabulary in English. Despite the fact that the students coul d freely ask Ms. Andersen questions while she was explaining a lesson (as the excerpts above demonstrat ed), the ELL students with lower levels English language acquisition exhibited limited immediate participation in the mathematics discourse. However, in the long-term, Ms. Andersen still facilitated them in learning Algebra in English by expecting them to write in their mathematics notebooks in English. Under her directions, the ELL st udents were creating their Â“personalized Algebra I study guidesÂ” as were the rest of the students in the class, and later they could refer to their own notes and study from them (see the excerpts above, lines 30 and 33). Furthermore, by explicitly showing work, and often using a red colored marker to show important steps while solving mathematical problems, Ms. Andersen was modeling for her ELLs how to approach mathematical prob lems and how to describe their own work by using proper math notation and vocabulary. This, in turn, provided all her students (and the ELLs in particular) with opport unities to study from their self-created
146 mathematics journals and in the long-term improve their knowledge of the proper mathematical notation and terminology used in the mathematical discourse. Perceptions of classroom discourse Figure 9 represents in the form of bar graphs the researcherÂ’s preliminary evaluation (i.e., before an actual count of the frequencies with which Ms. Andersen uses different di scourse strategies), Ms. AndersenÂ’s selfevaluation, and the evaluations by her ELL stude nts who volunteered to participate in the study. The pair-wise correlations for Ms. A ndersenÂ’s case study are as follows: the correlation between the teacher and researcher is .61; between the teacher and ELLs it is .12, and between the resear cher and ELLs it is .46. Figure 9 displays complete agreement be tween the evaluations of the researcher, the teacher self-evaluation, and the ELLsÂ’ eval uations that Ms. Andersen most frequently Provided feedback (strategy 14). The other strategies for which there was almost complete agreement over their frequent and consistent use by Ms. Andersen wereÂ— Use of a slower and simpler speech (1),Use of wa it-time techniques after posing a question (13), and Providing students with co ntent specific, enriched in formation, thus exhibiting equally high expectations from ELL and non-ELL students (22) The excerpts from the video-recordings of Ms. AndersenÂ’s classroom also revealed that she readily provided her ELL students with feedback to th eir answers as she did, for example, in line 12: Â“Exactly. GoodÂ…good perception and good information for the rest of the students to pick up onÂ” (strategy 14). In the excerpts above, it was also demonstrat ed that Ms. Andersen talked slowly and often paused between sentences (strategy 1, lines 3 and 4), thus providing enough time for her students to take notes (including her ELL students). In the interviews, the ELL students indicated that Ms. Andersen provided enough
147 Figure 9. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evalua tions of Ms. AndersenÂ’s frequency of use of various disc ursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
148 wait-time after asking a question (strategy 13), which gave them the opportunity to organize their answer to the question in English better. The video-recordings also confirmed that usually there was provided ad equate wait-time after a question was posed, thus giving equal chance s for both ELL and non-ELL students to participate in the classroom discourse. The excerpts above (lines 16 to 33), as well as from the rest of the video-recordings, also demons trated that Ms. Andersen provided her students with content specific, enriched information, thus exh ibiting equally high expectations from ELL and non-ELL students (strategy 22). Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 10 represents the frequency count of the strategies employe d by Ms. Andersen during the three observed classroom se ssions. Figure 10 indicates th at Ms. Andersen employed strategies 12 ( Use of different questioning techniques sensitive to the ELLsÂ’ level of SLA), 14 ( Providing feedback ), 13 ( Use of wait-time techniques after posing a question) and 1 ( Use of a slower and simpler speech ) most frequently. For example, Ms. Andersen often used expressions such as Â“Correct!Â” and Â“Very good!Â” which indicated frequent utilization of strategy 14 ( providing feedback) not only to inform her students of the correctness of their answers, but also to en courage their participation in discourse. She even used expressions in French such as Â“Tres bien!Â” or Â“Bon!Â” when addressing her Haitian ELL students who also spoke French, wh ich also indicated frequent use of the technique providing feedback (14) when addressing ELL students too, and as a result also encouraged their par ticipation in discourse. Both charts (Figures 9 and 10) indica te that Ms. Andersen frequently used wait
149 Figure 10. Frequency count of Ms. AndersenÂ’s use of various discursive strategies during the three 20-minute video-recorded sessions. 0102030405060 12. Use different questioning techniques 14. Provide feedback 13. Use wait-time techniques 1. Use of a slower and simpler speech 7. Comprehension checks 4. Use of repetitions or paraphrasing 3. Use of synonyms 17. Use of visual or auditory stimuli 19. Using cooperative groups 22. Content specific, enriched information 6. Use of clarification of directions 8. Identify vocabulary, pictures, or models 9. Review of related concepts 5. Use of changes of tone, pitch, and modality 11. Math discussions and problem solving 15. Use of gestures, expressions, eye contact 2. Use of fewer idioms and slang words 20. Alternative forms of assessment 16. Use of charts, graphic organizers 21. Use of cultural-specific knowledge 18. Use of technology 10. Summary of the key conceptsStrategyFrequency
150 time techniques (strategy 13). By re-playing th e video-recordings and measuring the waittime after a question was posed, it was found that Ms. Andersen provided different waittimes after posing questions in relation to each questionÂ’s difficulty and to whom the question was addressed (recordings revealed that she provided at least from three to five seconds of wait-time, and often more, with repetition of the questions, when addressing an ELL from a lower level of English language acquisition). She also used simpler commands and shorter sentences when explaining concepts (s trategy 1). For example, in the excerpts above, Ms. Andersen used the simpler command Â“LetÂ’s go back to page 213Â…Â” (line 14), or modified her talk usi ng shorter sentences when explaining the differences in graphs of the equations x=a and y=b to an ELL student: Â“The x -equation is parallel to the y -axis.Â” (line 8), and Â“the y-equation w ould be parallel to the x-axisÂ” (line 10). Such adaptation of her speech to her a udience exhibited Mr. AndersenÂ’s awareness of the presence of ELL students at different stages of SLA. Further analysis of the teacher talk revealed the means by which she fostered her ELLsÂ’ early production of correct responses both in mathematics and E nglish. For example, she demonstrated her solutions to a mathematical problem on the overhead projector by using different colors for definitions (green), solutions (blue) and important facts (red). Then, she focused her teacher-talk on key concepts and enc ouraged her ELL students to apply these concepts while explaining the steps of the problemÂ’ s solution. However, she assisted them by asking leading questions which required only short responses, and often provided teacher-directed instructions and explained thin gs, rather than expecting the students to finish more problems on their own Â– her disc ourse was usually similar to that in the sample excerpts at the beginning of this section on her discourse.
151 However, from both graphs (see Figure s 9 and 10), it becomes evident that strategies 2 ( Use of fewer idioms and slang words from the mathematics vocabulary) and 21 ( Provide opportunities for students to share experiences and build up on personal or cultural-specific knowledge while problem solving in ma thematics and thus building cross-cultural knowledge) were the most lacking in Ms. AndersenÂ’s discourse methods. Furthermore, the lack of use of strategy 2 indi cates that by rarely using idioms such as Â“if and only ifÂ”, Â“right-angleÂ” etc., Ms. Ande rsen was exhibiting awareness that an ELL student in an early stage of English language acquisition might think that ther e is a Â“leftangleÂ” or simply might misunderstand her. In the excerpts above, Ms. Andersen used the idiom Â“dog earÂ”, but each time she used it, she also demonstrated and explained that she expected the students to mark the page by fold ing the corner of the page she was referring to. At the same time, by not providing opport unities for her ELL students to share their previous experiences while problem solving and also by not enhancing her instructions by building up on studentsÂ’ personal or cultur al-specific knowledge (strategy 21), Ms. Andersen demonstrated a lack of awareness of the benefits of applying this strategy Â– benefits which are indicated in research in the field of teaching ELLs and were discussed in the review section of this manuscript. Ms. Brown Ms. Brown is in her early twenties. She has just gra duated from college and is teaching at the high school from which she graduated. She was not a mathematics education major and did not ta ke any courses on methods of teaching mathematics. She has a temporary teaching certificate and is cu rrently working on her ESOL certification.
152 She is fluent in Yoruba, her native language English, and has limited fluency in French. This is her first year of teaching Algebra I and Intensive Mathematics. Ms. BrownÂ’s class consisted of 17 student s, five of whom were ELLs. Four of these ELLs are Hispanics and one is Arabic. Th ree of the Hispanic ELLs are in their early stages of English language acquisitionÂ—between early-production and speechemergence. The fourth of the Hispanic students, and the Arabic student, are in the intermediate stage of fluency in English. The class is diverse with three White, eight Hispanic, and nine Black student s. In the interview, Ms. Brow n indicated that she tries to aid her ELL students by using coope rative groups as part of he r instructional techniques and always allows discussions in which stud ents could share their previous personal and cultural-specific experiences in mathematic s. When asked to comment on any concerns she has about teaching mathematics to ELL stude nts, she said: Â“My greatest concern is the language barrier. Oftentimes [I] have to have students translate processes into Spanish. My students who are bilingual enab le me to bridge the language gap.Â” She nonetheless further commented that, Â“My e xperience with [ELL students] has been extremely positive. The students who are strong in English oftentimes translate for the students who are not as strong. This greatly helps me in th e classroom.Â” In the Interview, Ms. Brown also shared that for the Arabic student she occasionally used the help of a teacher assistant (TA) who knew Arabic. Howe ver, because the Arabic ELL student was relatively more comfortable in English and only occasionally needed help (usually in word problems), this TA was often busy he lping in other classes. During the three observations in Ms. BrownÂ’s class, the TA was on other assignments, and the Arabic student was working well in the class on his own.
153 Typical classroom discourse. To better visualize the atmosphere in Ms. BrownÂ’s mathematics classroom, a detailed excerpt from an activity Ms. Brown used to teach her students Scatter Plots and the concept of data correla tion will be provided. The following excerpt will demonstrate both the teacher-student interactions and sometimes student-tostudent interactions that arose in Ms. Brow nÂ’s classroom during that activity due to the particular blend of students (ELL and nonELL students), as well as the teacherÂ’s discursive moves with the students: Ms. Brown:  Alright, can you guys get in height or der for me at the front of the room?  ( After a while) Alright, Okay.  So, can you guys line up in reverseÂ—IÂ’m sorry, fromÂ—come up here, so all switch, so it would beÂ…( the students line up shortest to tallest from left to right): Jasmine, Rosita, George, and Bryan (here fictitious names are used for easier presen tation; Jasmine, Rosita, and George were ELL students, and Bryan was a non-ELL).  Alright, so we have our class member s organized from tallest, I mean, from shor test to tallest, right?  So in part D itÂ’s asking us if Â“whether our height is correlated with the number of siblings we haveÂ” ( the teacher read this from a prepared worksheet)  So, if it is so, Jasmine would have the least amount of siblings, right?  And Bryan would have the most amounts of siblings.  So letÂ’s see if it works.  Jasmine how many siblings do you have? Jasmine:  Five-four, five-four. Ms. Brown:  Five? Jasmine:  Four. Ms. Brown:  Five? Jasmine:  Five-four. Class:  What? H uh? What is she talking about? Jasmine:  Me, I am five-four. Ms. Brown:  No, no, no, how many siblings do you have, not how tall you are. Lines 8 to 17 reveal that Jasmine (an ELL student from the early-production stage of English language acquisition) encountered di fficulties in understand ing the meaning of the word Â“siblings.Â” However, rather that using a synonym for the word Â“siblings,Â” Ms.
154 Brown corrected Jasmine by saying she didnÂ’t m ean to ask her how tall she is (i.e., Ms. Brown exhibited lack of use of strategy 3use of synonyms ). Additionally, the incident revealed that even though Ms. Brown was aw are that she had ELLs and tried to involve them in the activity, she was not aware that this particular ELL student was from the early production of English language and is just beginning to experiment with the language. In this case, the teacher needs to model/demonstrate correct responses for her both in mathematics and English or use synonyms in order to negotia te the meaning of her instructions and questions (i.e., n eeds to apply strategy 1b or 3). Rosita ( next to Jasmine ):  She got nine all to gether, she said five-four. Jasmine:  No, no, no ( thinks and holds up three fingers ), three. Ms. Brown:  Three? Jasmine:  ( nods ) Student ( in background ):  She got a twin sisterÂ… Rosita ( to Jasmine ):  I got more than you. Ms. Brown:  Okay, you have three.  Okay, Ro sita, so for us to have a positive correlation, Rosita should have more.  Rosita, how many do you have? Rosita:  Thirteen. Lines 19 to 27 reveal that Rosita (an ELL transitioning between the speech-emergence and intermediate-fluency level of English language acquisi tion) had initially assumed that Jasmine was claiming to have 9 siblings ( line 18). Then, because she was turning around to grab her other classmatesÂ’ attention, sh e did not take note of JasmineÂ’s gesture (holding up three fingers, line 19 ), and Ms. BrownÂ’s repetitio ns that Jasmine has three siblings (lines 20 and 24). Rosita even teas ed Jasmine (interrupting her dialogue with Ms. Brown) that she has more siblings than her (line 23). Then, she simply heard the teacherÂ’s explanation of a positive correlation (line 25) and the question toward her (line 26), and thus answered with her pre-pr epared number of 13 (line 27). That answer reveals that
155 Rosita had not understood that the point of the exercise was not to provide fictitious data that meets the criteria of positive correlation, but rather to answer with real data so that the class could determine whether there was in fact a correlation between height and number of siblings. This further reve als how some ELLs, despite developing good basic interpersonal communication skills (BISC), still need time to transition to the more advanced level of cognitive academic language proficiency (CALP) to fully understand the true context of the academic task (Cummings, 1983; Ellis, 2000). Although Ms. Brown questioned the number of siblings Ros ita reported (in line 28 next), she did not voice any doubts (if she had any such doubts) in RositaÂ’s understanding of the cognitive demands of the task: Ms. Brown:  Thirteen? ( Rosita nods), perfect.  Alri ght, Rosita has thirteen.  So for our correlation to work if this is a pos itive correlation, Jose should have more siblings than Rosita.  Jose, how many siblings do you have? Jose:  You m ean like brothers and sisters? Ms. Brown:  Yeah. Here it was the third ELLÂ—Jose (in the speech-emergence stage of English language acquisition) that finally initiated for Ms. Br own to negotiate the meaning of the term Â“siblingsÂ” and used the synonymous words Â“like brothers and sistersÂ” (line 32). When he received feedback (strategy 14, line 33) that his a ssumption of the word Â“siblingsÂ” meaning is correct, he answered: Jose:  I only got a sister. Ms. Brown:  You only got one, hmmÂ…alright. Bryan:  ( mumbling a big number in the vicinity of 13 ). Jose:  Oh, twenty Ms. Brown:  No, uh-uh, thatÂ’s fine, donÂ’t lie.  You donÂ’t have to lie.  ItÂ’s al right, we are prov ing our point.
156 Here Ms. Brown realized that the students need assistance in order to clarify their misunderstanding of what answers are expected from them and clarified the directions (strategy 6) by stating that in order for the students to understand the point of the activity they need to provide tr uthful (real life) data: Ms. Brown :  Bryan, how many brothe rs and sisters do you have? Bryan:  Five. Thus, Bryan who was not an ELL could clear hi s confusion of what answer to provide (his confusion was exhibited in line 36), and truthfully reporte d the actual number of his siblings (in line 42). Ms. Brown:  Five?  Okay.  So we go from 3 to 13, to 1, to 5, right?  Hmm, doe s that look like it has a re lationship with the height? Class:  No. Ms. Brown:  Â‘Cause we donÂ’t have a positive.  Is that, do we have a negative correlation there? Class:  No. Ms. Brown:  No, so we donÂ’t have any correlation, alright.  So, that was a ll I was trying to prove with that.  Thank you guys. As is evident, the point of the activity had b ecome apparent to the class, and by analyzing the data provided from the students participa ting in the activity, they were able to reach the conclusion that there was no correlation be tween the studentsÂ’ heights and the number of siblings they have (lines 45 to 53). Here Ms. Brown summarized the reported data (i.e., utilized strategy 10), and asked the students to analyze it (strategy 22d) in order to complete the activity.
157 Krussel et al. framework. According to Krussel et al.Â’s (2004) framework, the purpose of Ms. BrownÂ’s discourse was to initiate participation for all her students, including ELLs, in whole-class or group activit ies, leading to better understanding of the new concepts she is teaching (in the above ex cerpt Â– scatter plots a nd data correlation). Throughout the observations it became clear that Ms. Brown frequently called on her ELL students (i.e., utilized strategy 11) to co mplete problems she had just modeled for them how to solve, and tried to involve them in discussions. For example, three of the four students (Jasmine, Rosita and Jose) who pa rticipated in the activity from the excerpt above were ELLs. Ms. Brown also oftentimes used Â“hands onÂ” activities or games such as the one described in the excerpt above, or group work so that all her students (and ELLs in particular) could better grasp a math ematics concept. This indicates that she utilized strategy 19 and, as a result, her ELL students were exposed to different classroom work arrangements such as group work, partner and whole-class discussions. During the observed lessons, she attended to the fact th at she had a diverse student population and ELLs present in her mathematics classroom and often called on them, thus involving them in the problem solving and math discussions (strategy 11) or asked them questions (strategy 12). However, as the excerpt above exhib its, Ms. Brown had a limited understanding that ELLs from different stages of English language acquisition have different needs and that she needed to adjust her talk in order to accommodate them in the classroom discourse. Thus, initially Ms. Brown omitted to start the activity with clear directions and did not utilize strategy 6 until later in the activ ity (lines 38-40), when it became apparent that the students need assistance in order to provide realistic data and to complete the
158 activity. As the excerpt also demonstrated, in itially Ms. Brown did not realize that using the word Â“siblingsÂ” was not familiar to her ELLs from the early-production to speechemergence stages of English language acquisition. Thus, she did not utilize strategy 6 of using the synonymous words Â“brothers and sistersÂ” until an ELL (Jose) was unable to negotiate the meaning of the word (lines 3133). Thus, the example above demonstrates that even though Ms. Brown involved her ELLs in the classroom discourse, she did not adjust her talk to their level of English language development (i.e., lack of use of strategy 1b). The setting for classroom discourse was evidently established early in the school year, as the students exhibited a familiarity with certain expectations and norms of classroom behavior in each of the observed lessons. For example, Ms. Brown regularly started her lessons with bell-work. Then, she us ually collected the stud entsÂ’ bell-work, as well as their homework from the previous ni ght. However, during the observed and video recorded sessions of classroom discourse, Ms. BrownÂ’s students often did not follow the norms of turn-taking she was trying to esta blish. For example, when Ms. Brown asked questions (general or specific ), even though a specific studen t might be asked to answer, many students either answered aloud or cont inued interrupting each other. But, as was demonstrated in the excerpt above, usually Ms. Brown repeated her questions to the student they were addressed to and kept most of her st udents focused on completing the mathematics task at hand. Furthermore, as illustrated in the exce rpt above, Ms. Brown established a classroom environment that encouraged active learning by often asking students to perform Â“hands onÂ” activ ities or to work in groups, thus exposing her ELL students to different clas sroom work arrangements (strategy 19).
159 The form of Ms. BrownÂ’s discourse included both teacher talk (verbal) and actions (non-verbal discourse) For example, the use of questi ons of the type Â“how,Â” Â“tell me about,Â” Â“compare/contrast,Â” as the fo llowing excerpts illustra te: Â“How does this question look like that equation?Â” and Â“As x is going up, can anyone tell whatÂ’s happening to y ?Â”, reveal Ms. BrownÂ’s efforts to encourage her ELL students to expand their literacy in both the Eng lish language and mathematics (i .e., utilization of strategies 12 c and d). In order to provide answers to such questions, the st udents (including ELLs) need to move to operations involving higher cognitive demand according to BloomÂ’s taxonomy Â– analyzing, distinguishing, and explaining content-specific, enriched information (strategy 22d, e, and f). Ms. BrownÂ’s non-verbal discourse was disp layed in different forms. For example, she moved her hand up or down to help her ELL students understand positive or negative scatterplotsÂ’ correlations, or she moved her hand up or down and left or right to demonstrate the slopes of lines as going up or down (rise) and the left or right (run). Thus, these examples demonstrate th at Ms. Brown utilized strategy 15 ( use of gestures, facial expressions, eye contact or demonstrations ) so that her ELL students could better understand the concepts of scatter plot and slope of a line. Anot her display of Ms. BrownÂ’s non-verbal discourse is her use of he r index finder in front of her lips whenever she wanted to indicate to the class that th ey need to quiet down and listen. Additionally, when switching between activities, she usuall y would raise her arm up and say Â“Alright, so today weÂ’re gonna be discus sing scatter plotsÂ” or Â‘Alr ight, so, example oneÂ…Â” She also circulated between the rows when stude nts performed group work and assisted them or answered questions.
160 The consequences of Ms. BrownÂ’s discourse fell into the following categories Â– intended or unintended immediate or long term as described in the Krussel et al. (2004) framework. Ms. Brown demonstrated intentions to shift the cognitive level of the task performed (strategy 22 d and e) for bot h ELL and non ELL students. For example, by intentionally choosing three ELLs from the four stude nts to participate in the activity described in the excerpt above, she involved the ELLs in the classroom discourses (strategy 11). Most notably, by frequently pos ing questions to ELL st udents (strategy 12), she was demonstrating an intentional goal to provide them with opportunities to practice their mathematics vocabulary in English a nd was demonstrating hi gh expectations. She provided them with equal opportunities not only to share personal data (strategy 21), but also to use that data to help them better understand the concept of scatter plots (strategy 22c) and to identify whether they saw any correlation (strategy 22d). In the excerpt above, it was also demonstrated that Ms. Brown unintentionally caused confusion by using certain words (siblings) and by not clar ifying the directions of the activity. Later, negotiation of the meaning of the word Â“sib lingsÂ” was initiated by her ELL student Jose, who asked Ms. Brown for feedback (strategy 14) whether Â“s iblingsÂ” is synonymous to Â“brothers and sistersÂ” (lines 31-33). After this, Ms. Brown clarified the directions by explaining that there is a need to provide real istic data in order to discover whether there is indeed a correlation between studentsÂ’ hei ght and the number of siblings they have (lines 38-40). Some of the immediate or long-term consequences of Ms. BrownÂ’s discourse are also of interest. For example, sh e often shifted the dialogue from univocal to dialogic and exposed her students to differe nt classroom work arrangements (strategy 19), such as
161 using cooperative groups or pa rtner discussions. This was de monstrated not only in the excerpt above, but also in the other obse rvations. However, she had problems setting long-term norms of turn-taking and politeness duri ng the discussions (as illustrated by the instance involving Rosita in the excerpt a bove (lines 18 and 23)). Thus, even though she provided her ELL students (and all students for that matter) with opportunities to participate in the classroom discourse by i nvolving them in the activity of the above excerpt (strategy 11), setting norms of polite and orderly communication in English could further enrich the studentsÂ’ possibilities to b ecome better team-players and equal partners in future work collectives. Perceptions of classroom discourse. Figure 11 represents the researcher evaluation, the teacherÂ’s self-evaluation, and her ELL studentsÂ’ evaluations of Ms. BrownÂ’s teacher talk and discourse characteristics. Th e pair-wise correlations (Pearson product-moment correlation coefficients) for Ms. BrownÂ’s case study are as follows: the correlation between the teacher a nd researcher is -.14; betwee n the teacher and ELLs it is -.26, and between the researcher and ELLs it is .59. This negative result is due to an unrealistic self-evaluation and perhaps also due to a lack of unde rstanding of the ELLs, since Ms. Brown has no teaching experience and lacks ESOL certification. Figure 11 reveals that all evaluators agr eed that Ms. Brown uses two strategies frequently Â– Use of wait-time techniques after posing a question (strategy 13) and Provide feedback (14). For example, the above excerpt al so demonstrates that Ms. Brown consistently provided her stude nts with feedback (strategy 14) to indicate whether their responses were correct or needed further modi fication, by the use of such expressions as
162 Figure 11. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ eval uations of Ms. BrownÂ’s frequency of us e of various discursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
163 Â“alrightÂ” (see lines 2, 4, 35), Â“perfectÂ” (lin e 28) Â“okayÂ” (lines 2, 24, 25, 29, 44) or Â“hmmÂ” whenever she wanted them to reconsider their answers. During the observations (and measuring the time after lines 45, 46, and 49 in the excerpt above ) and in the postobservation interview, Ms. Brown indicated that after asking a question she often paused and thus indicated to the students: Â“Think a bout it!Â” to provide them enough wait-time to rethink and/or correct their answers (strate gy 13). Figure 11 also reveals that Ms. Brown consistently used the followi ng strategies in her discourse: Identify subject specific and important lesson vocabulary and provide co ntext embedded exampl es, pictures, or models ( strategy 8) and Use of gestures, facial expressions, eye contact, or demonstrations to enhance comprehension (15). For example, when introducing the lesson about Scatter Plots, Ms. Brown began as follows: Today weÂ’re gonna be discussing scatter plotsÂ…So what I have on the overhead, you also have on the bottom of your note shee t. But first we have a little vocab. A scatter plot is a graph that shows th e relationship between two separate data, okay? And the way that the data can have a relationship. They can have a positive relationship, a negative rela tionship, or no correlation. Furthermore, as we saw in the excerp t at the beginning describing Ms. BrownÂ’s classroom discourse, she did not simply form ally introduce the vocabulary to her students but rather used an activity to embed the defi nitions in a real-life context by looking for a correlation between the studentsÂ’ heights and nu mber of sibling (i.e., utilized strategy 8). ELL students also felt that they understood th e vocabulary by the manner it was taught to them, and noted that Ms. Brown used this stra tegy very frequently (strategy 8). Figure 11 also makes it apparent that all agr eed that Ms. Brown frequently used gestures, facial
164 expressions, eye contact, or demonstrations to enhance comprehension (strategy 15) Â– a facet described and analyzed in the section regarding the form of Ms. BrownÂ’s discourse as per Krussel et al.Â’s (2004) framework. However, with regards to how ELL student s feel in Ms. BrownÂ’s classroom and how they evaluated the teacherÂ’s discourse, Figure 11 reveals some wide disparities between the teacher and ELL student evalua tions. For example, for a few of the strategies, Ms. Brown and the ELL students di sagreed on the frequency with which the strategies are used. The part icular strategies where there is disagreement between Ms. Brown and her ELL students are: Use of synonyms (strategy 3), Use of change of tone, pitch, and modality (5) and Provide opportunities for students to share experiences and build on personal or cultural-specific know ledge while problem-solving and thus build cross-cultural knowledge (21). As revealed in the provi ded excerpt, some of Ms. BrownÂ’s ELL students who participated in the activity encountered difficulties in understanding the meaning of the word Â“siblings.Â” Only later in the dialogue when Jose asked Â“You mean like brothers and sisters ?Â” (line 32) did Ms. Brown real ize that the student needed help with vocabulary, and in turning to Bryan she asked Â“B ryan, how many brothers and sisters do you have?Â” (line 41). Thus, as i ndicated by the ELLsÂ’ evaluations and verified by the excerpt provided, Ms. Br own did not make frequent use of synonyms or other expressions that could help he r students better understand th e concepts (strategy 4). The ELLs also indicated, contrary to Ms. BrownÂ’s opinion, that she does not change her pitch or modality of talk (strategy 5) and thus they are comp letely unaware if certain words or phases in her talk carry a greater degree of importance. According to ELL students, Ms. Brown did not differentiate her speaking tone and grammatical or in structional tone (e.g.
165 lack of use of strategy 5). As a result, ELLs could not discern whether something was important as part of the lesson instruction or if it was just said in a conversational mode. The ELL students also indicated that Ms. Brown does not ask them to give examples from their country or family when solving mathematical problems (strategy 21). Even though the initial excerpt about the scatter pl ot activity is an instance of this strategy, it was an isolated one which occurred infrequently, as the ELLs indicated in the interview. The observed classroom sessions did not reveal another in stance of the use of this strategy. However, Ms. Brown evalua ted that she utiliz ed her ELLsÂ’ cultural perspectives and backgrounds as insights to further modify and eventually improve her instructional approaches. Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 12 indicates the frequency with which Ms. Br own implemented each of the discursive strategies found in the TTT Form 1 The strategies most frequently employed by Ms. Brown areÂ— Provide feedback (strategy 14) Check for comprehension (7), and Use of different questioni ng techniques (12). This provides additional evidence to support conclusions drawn from the previous graph (see Figure 11). Furthermore, in the text above were demonstrated many examples of Ms. BrownÂ’s implementation of strategies 12 (see the examples provided under the discussion of the form of Ms. BrownÂ’s discourse) and strategy 14 (see lines 2, 4, 25, 28, 29, 35, and 44 in the excerpt above). Examples of her use of strategy 7 Â– Use of comprehension checks throughout the lesson are the observed instances when she often stated definitions by finishing them with the question Â“Â…right?Â” thus eliciti ng the studentsÂ’ reactions (o r nods) of agreement or
166 Figure 12. Frequency count of Ms. BrownÂ’s use of various discursive strategies during the three 20-minute video-recorded sessions. 0102030405060 14. Provide feedback 7. Comprehension checks 12. Use different questioning techniques 1. Use of a slower and simpler speech 11. Math discussions and problem solving 19. Using cooperative groups 3. Use of synonyms 4. Use of repetitions or paraphrasing 15. Use of gestures, expressions, eye contact 20. Alternative forms of assessment 6. Use of clarification of directions 8. Identify vocabulary, pictures, or models 2. Use of fewer idioms and slang words 22. Content specific, enriched information 13. Use wait-time techniques 9. Review of related concepts 18. Use of technology 5. Use of changes of tone, pitch, and modality 10. Summary of the key concepts 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 21. Use of cultural-specific knowledge StrategyFrequency
167 disagreement with, or unders tanding or not, her statements (see lines 4, 6, 45, 51 in the excerpt with the scatter plot activity). Figure 12 also reveals the previously discussed omission by Ms. Brown of implementing the following characteristics of t eacher talk and other di scursive strategies: Conclude a lesson with a summary of the key concepts (strategy 10), Use of charts, graphic organizers Â—Venn diagrams, tree diagr ams, time lines, semantic maps, outlines, etc. (16), Use of a variety of visual or audito ry stimuli: transparencies, pictures, flashcards, models, etc. (17), and Provide opportunities for students to share experiences and build up on personal or cultural-specific knowledge while problem solving in mathematics and thus buildi ng cross-cultural knowledge (21). The observations also revealed that Ms. Brown rarely asked her students to summarize the key concepts that they have just learned (str ategy 10). Moreover, even t hough she often used an overhead projector to write the bell work on it, she did not utilize the ov erhead projector to draw charts or graphic organizers on it, or show pictures or visual models (strategies 16 and 17). Figure 12 shows that Ms. Brown freque ntly omitted the use of strategy 21. This supports the conclusions drawn from the prev ious graph (see Figur e 11) wherein the ELL students also indicated Ms. BrownÂ’s less frequent use of strategies 16 and 21. Ms. Cortez Ms. Cortez, a Puerto Rican in her late 50s has taught in the USA for 10 years, for five of which she has taught Algebra I classe s. She is certified to teach middle school mathematics, but this year she accepted a position in a high school and is currently working on her high-school mathematics cer tification. She has not yet completed her content area teachersÂ’ requirement of 60 hour s of training toward ESOL endorsement.
168 However, in the interview she indicated that because she has worked with Hispanic students in Puerto Rico, she is aware of their educ ational needs. She finds her Central and South American ELL students to be very motiv ated, disciplined, and responsible. As an example, she showed some of their folders, which were very organized and complete. She commented that even though they have di fficulties learning mathematics in English, they do their best so as to receive a bette r education. They always do their homework assignments, bring their materials to cl ass, and are Â“excellent students.Â” Ms. CortezÂ’ Algebra I class consisted of 17 students, nine of whom were Hispanics, six African Americans, and two multi-racial students. Four of the students were ELLs, three of whom were Spanish-spea king students with different levels of fluency in the English language. The fourth ELL student was from Central America and spoke French. Ms. Cortez allowed the Hisp anic ELLs to work in a cooperative group with students fluent in both Spanish and E nglish and they were he lping each other by sometimes translating directions, or with problem solving. For the ELL student who spoke French, the school had assigned a French language specialist as a teacher assistant (TA) to assist Ms. Cortez with this ELL st udent. There was also another TA who assisted Ms. Cortez with the Hispanic ELLs. The ELLs (and the TAs) were seated in the front right corner of the room and were able to see the lesson on the overhead and participate in classroom discussions. Typical classroom discourse. From the three observed classroom sessions, some patterns typical of Ms. Cortez emerged. For example, unless giving a test, she always started her lessons with a review of the previously learned related concepts (strategy 9) and thus tried to involve he r students (including ELLs) in the mathematical discourse.
169 Next, she collected their homework assignm ents and gave them bell-work which was usually prepared and written on transparencies After collecting the bell-work she usually began teaching the new lesson. And finally, from prepared worksheets she assigned problems for students to work on, individually or in cooperative groups (strategy 19Â— Expose students to different classroom work arrangements ). During that time, she and her TAs assisted the students according to their individual needs. The observations of Ms. Cortez discur sive moves while teaching a new lesson revealed that her teacher-student interact ions are strongly remi niscent of a model recognized in the review literature as IRF (Initiation-Response-Follo w-up). According to this model, Â“[t]he element of structure that is most clearly defined, however, is that of Â‘teaching exchangeÂ’, which typically has thre e phases, involving an Â‘initiatingÂ’ move, a Â‘respondingÂ’ move, and a Â‘follow-upÂ’ mov eÂ” (Ellis, 2000, p. 574). According to this study, under the Â‘initiation moveÂ’ could fall the teacherÂ’s discursive strategies 11 ( involving students in mathematical discussions or problem solving by calling them by name) and 12 ( using different questioning techniques sensitive to the ELLsÂ’ level of English acquisition). Under th e Â“responding moveÂ” fall any an swer the students provide, but because they are not part of the teacher discourse, th ey will not be placed under scrutiny in this study. And lastly, based on this study, under Â‘follow-up moveÂ’ could fall the teacherÂ’s discursive strategy 14 ( provide feedback ) and 4 ( use of repetitions or paraphrasing of studentsÂ’ answers). The following excerpt (lines 1 to 52) will be used to demonstrate and provide evidence of this and so me of the other strategies utilized in Ms. Cortez discourse: Ms. Cortez:  Uhh, who remembers the t opic that we were working with?
170 Student 1 (not and ELL student) :  I remember. Student 2 (an ELL):  Who remember what? Ms. Cortez:  The topic, which one was yesterda yÂ’s topic? Student 1:  CoordinateÂ…( the students speaks the rest unclearly ) Ms. Cortez:  Yes, coordinate-plane. Here the questions in line 1 and 4 fall under the teacherÂ’s Â“initiationÂ” move and demonstrated her use of questions sensitive to ELLs from pre-production and earlyproduction levels of English language acquisi tion, requiring, correspondingly, a oneword response or a list of words (strategie s 12 a and 12b). One student answered (lines 2 and 5) Â— a Â“respondingÂ” move, and Ms. Cort ez immediately did a Â“follow-upÂ” move by providing feedback (strategy 14) and repeating and co mpleting the studentÂ’s answer so that the others could h ear it (strategy 4). Ms. Cortez:  Now, I have a coordinate plane. At this moment the teacher drew the coordinate plane (strategy 16Use of charts or drawings ) on a transparency on the overhead projector (strategy 18Use of technology ). Then, she pointed to the drawing (strategy 15Use of gestures ) and asked the students: Ms. Cortez:  Now, what aboutÂ…wha t about the quadrants?  Who can say something about quadrants?  Remember, you divide itÂ…the gra ph is going to be divided in how many quadrants? Students:  Four. Ms. Cortez:  Okay. Here again lines 10, 11, and 12 provide eviden ce that Ms. Cortez adhered to the IRF Â“teaching exchangeÂ” model. However, for the purpose of this study, it is more interesting for the reader to observe how she gradually decreased the type of her questioning from questions that required an extended response and could poten tially involve ELLs from an intermediate level (strategy 12d), to speech-emergence (strategy 12c) and finally to a
171 question that required a single-word res ponse and thus potentially could involve ELL students from an early production stage of English language acquisition (strategy 12b). However, such a shift in the type of the que stions, as well as the fact that the teacher expressed satisfaction with the one word response, demonstrate that she did not encourage her students to elaborate on their answ ers and thus to begin to experiment with the language which they are just beginning to acquire. Ms. Cortez:  What is going to be this axis right here? Ms. Cortez and Students:  Y Ms. Cortez:  What is going to be this? Students:  X. Ms. Cortez:  X very good.  And we said ye sterday that we have four quadrants.  We are going to start with one, two, three, and four.  This is the fourth quadrant.  The signs over here are going to beÂ…? Here Ms. Cortez raised the intona tion of her voice (strategy 5Â— Change of pitch of voice) and thus indicated that this is a question and the students should finish her sentence (strategy 12bUse of different questioning techniques in this case, a question requiring a one-word response). As expected, a coupl e of students answered aloud: Students:  Positive. Ms. Cortez:  Positive and positive.  And I said yesterday that one way it is easy for you to remember the signsÂ…you go to the first one.  Then go to the opposite.  The oppos ite quadrant is going to ha ve the opposite signs, too.  This is going to be positive and positive, then youÂ’re going to have hereÂ…? Students:  Negative and negative. Ms. Cortez:  Then here, in the second quadrant, you have negativeÂ…? Students:  Negative and positive. Ms. Cortez:  Very, very good.  Okay, then what ar e you going to have in the opposite quadrantÂ…?
172 Here the teacher enthusiastically provided feedback (strategy 14, line 31) that the students are correct and they appr opriately responded to the change of the pitch of her voice (strategy 5 again, line 32) and thus she en couraged them to continue answering her questions (strategy 12b, line 32): Students:  Positive and negative. Ms. Cortez:  Very good.  Something else that you didnÂ’t get yesterday, th at you have a question about yesterdayÂ’s class.  Okay, remember that I said that the first ordered pa ir is going to be the x, the second one is going to be your y  And if you have x the first numbers you have to move to theÂ…? Here the teacher decide d to test (strategy 20Â— Use of an oral form of assessment) a particular (ELL) studentÂ’s knowledge (strategy 22aÂ— Lower level of cognitive demand ) of a previously explaine d concept (strategy 9Â— Review of a related concept) She made eye contact and nodded to that student (strategy 15Â— Use of eye contact and gestures ), thus indicating that he should conti nue her sentence (strategy 12bÂ— Use of a question requiring a list of words ): Ricardo (an ELL) :  To the right, or to the left. Ms. Cortez:  To the righ t, or to the left.  S econd number is going to be the y  The y you move itÂ…? Ricardo (the same ELL) :  Up or down. Ms. Cortez:  Up or down.  Up is going to be positive or negative? Students (a few students say aloud):  Positive. Ms. Cortez:  Very good. [ 47] What about if I m ove to the left side? Students (aloud) :  Negative. Ms. Cortez:  ItÂ’s going to be negative.  You got it. Okay.  TodayÂ’s class we are going to continue working with graphs and the topic for toda y is going to beÂ…scattered...plots.  I am going to pass the paper for todayÂ’s class.
173 At this moment, the teacher was satisfied with the review, and started instructions on the new topic Â— Scattered plots. She passed copies of the same prepared worksheet that Ms. Brown (in case study nu mber 3) was using. Krussel et al. framework. The purpose of Ms. CortezÂ’ discourse was to involve all her students (and ELLs in particular) in whol e-class discussions and individual or groupwork, leading to better unders tanding of the concepts of The Coordinate Plane, Scatter Plots transforming linear equations in Slope-intercept or Point-Slope Forms and then G raphing Lines by applying the concepts of intercepts and slope. For example, Ms. Cortez regularly used repetitions or paraphrasing of her or her studentsÂ’ statements (strategy 4, lines 4, 39, 43, and 49), or often used charts and graphic organizers (strategy 16, lines 24 to 33: which were used as a semantic map for her students to better remember the signs of ordered pairs in diffe rent quadrants), which indicates that Ms. Cortez was attending to the ELLs in her math ematical classroom and that she was trying to involve them in the mathematical disc ourse by Â“visualizing the lessonÂ” (i.e., by providing variety of visual stimuli: transparen cies, charts, and diagrams). Furthermore, she exhibited sensitivity to the level of SLA of her ELLs by trying to use different questioning techniques (strate gy 12), thus encouraging her E LL studentsÂ’ transition from pre-production and early production of mathematics answers in English to speech emergence and intermediate speech However, for example, most of the questions were of the type Â“what isÂ…Â” (lines 13, 15, 21), ge neral questions requiri ng one-word (lines 10, 21) or a list of words as responses (lines 27, 29, 32, 37, and 41), or either/or questions (lines 45, 47, all in st rategy 12b), and did not move the students above the four levels of cognitive demand according to BloomÂ’ s Taxonomy (knowledge, comprehension,
174 application, and analysis; stra tegy 22 a, b, c, d, and e). Ms. Cortez nonetheless also asked questions such as: Â“Who can say something a bout quadrants?Â” (line 9) or Â“What is the next step? To leave alone the b, what do you ha ve to do?Â” which in turn encouraged the student to use longer sentences in English in their answers. Thus, in turn, she encouraged her ELL students to develop a higher leve l of English Language acquisition such as speech emergence or intermediate speech The studentsÂ’ adherence to certain norms of classroom behavior revealed that the setting for classroom discourse was established ea rly in the school year. Moreover, even though there were students enrolled in Intensiv e Math, and others ta king Algebra I, they all demonstrated good responses to the pre-es tablished procedures during the bell-work activity. For example, when the teacher gave the following directions, Â“For the Intensive Math class, if you finish, pass the paper right now. Pass your bell work for Intensive Math. Algebra I, please still work on your bell workÂ…. Three more minutes,Â” the students seemed to be accustomed to the pr ocedures and complied with the teacher. The two TAs present assisted the ELL students by translating the teacherÂ’s directions in French and Spanish and collected the ELLsÂ’ be ll-work. Additionally, fr om the fact that some students occasionally raised their hands and said: Â“Miss, I donÂ’t get this part...Â” and Ms. Cortez or some of the TAs immediately assisted them, it became obvious that the class was imbued with an atmosphere in whic h students felt free to ask questions or seek individual assistan ce if they did not understand something. The form of Ms. CortezÂ’ discour se included both actual teacher talk (verbal) and actions (non verbal) For example, her questions: Â“Who can say something about quadrants?Â” or Â“What is the next step? To leave alone the b, what do you have to do?Â”
175 indicated that she encouraged ELLs (and all other students) to develop their communication in mathematics using English language. However, on very rare occasions she did ask students to reflect on their thinki ng or justify the steps necessary for reaching a solution (strategy 22, e and f). Even t hough she encouraged ELLs to communicate both in their native language and in English, often she missed the opportunities to move the discourse to the higher levels of cognitive demand that synthesis and evaluation require, according to BloomÂ’s Taxonomy. The below exce rpt provides an example of such an occasion in which discourse with an ELL was in fact moved to a higher level of cognitive demand, requiring the student to analyze and ex plain what type of a correlation exists between data: Ms. Cortez:  ( points to paper) EÂ…the amount of ink remaining in a pen.  Do you think itÂ’s a one relation?  Remember, when you are writingÂ…  Okay, when you write a lot, whatÂ’s going to happen with the ink? Miguel:  ItÂ’s going to decrease. Ms. Cortez:  ItÂ’s going to beÂ… Miguel:  Negative. Ms. Cortez:  The more words...the more words th at you have, it means this is going to increase.  What is going to happen with the ink? Miguel:  ItÂ’s going to run out, itÂ’s going to run out. Ms. Cortez:  Huh? Miguel:  There isnÂ’t going to be any more ink in the pen. Ms. Cortez:  Okay, then.  The words are goi ng to increase and th e ink is going toÂ… Miguel:  Decrease. Ms. Cortez:  Decrease, okay.  Then, what type of re lationship are we going to have here?  Positive, negative, orÂ…or itÂ’s going to be no relationship at all. Miguel:  Negative. Ms. Cortez:  Negative, very good. Excellent! As the excerpt above demonstrates, Ms. CortezÂ’ discourse did not exhibit flawless English grammar; however being aware of this, she used repetitions (strategy 4, lines 8,
176 16, and 20) or paraphrasing of her own (lines 3, 4, and 8) or her studentsÂ’ words (lines 6, and 14), and simpler talk and shorter sentences (strategy 1) so as to be better understood by her students. Requesting that her student paraphrase his senten ce (lines 10 and 12), and then also paraphrasing his answers hersel f (in lines 14 and 15), demonstrated that Ms. Cortez was encouraging the ELL student to use specific mathematics vocabulary Â– for example, the words Â“increase/decreaseÂ” or Â“positive/negativeÂ” when talking about the relationship between the number of words and the amount to ink. Ms. Cortez also used different forms of non-verbal discourse (strategy 15Â— Use of gestures, eye contact or demons tration to enhance comprehension ). For example, she often pointed on the overhead to the sections of the prepared transparencies where she wanted her students to focus their attention. She also walked between the studentsÂ’ seats and assisted them or answered questions (if asked) and checked on ELL studentsÂ’ progress with the tasks. Some of the consequences ( intended or unintended immediate or long term) of Ms. CortezÂ’ discourse were classified accordi ng to Krussel et al.Â’ s (2004) framework, as follows below. For example, Ms. Cortez intended to shift the cognitive level of the task performed (for example when determining if there is a correlation between the number of words written and the amount of ink in a pen) by asking the students to explain what they think, but unintentionally she assisted them in doing this and thereby lowered her expectations of their abilities to complete the task on their own. From the other point of view, some of the immediate and long term consequences of Ms. CortezÂ’ discourse can be gleaned from the following instance: in one of the observed classroom sessions, Ms.
177 Cortez performed a folder check and said the following to an ELL student who had been recently assigned to her class: YouÂ’re going to put all graded papers he re, and on the right side the ones that I havenÂ’t graded yet. That way when I open the folder, because youÂ’re new, when I open the folder I will look to the right side. Okay? Thank you. This example illustrates how Ms. CortezÂ’ used alternative forms of assessment (strategy 20) in order to monitor the progr ess of all her students in her class, and especi ally that of the ELL students. She evaluated their work not only on paper-and-pencil tests, but also collected their homework and bell-work fo r grading, and checked their folders and oftentimes examined them orally by as king specific students certain questions. Additionally, by walking around the studentsÂ’ de sks while they performed individual or group work on pre-prepared worksheets, she (a nd her TAs) not only assisted the students in better understanding and completing the task at hand ( immediate consequence of her discourse), but also monitored their progress in her mathematics class in general. This, as she indicated in the interview, provided her with opportunities to modify her instruction and explain a concept again or use more ex amples to model how the concept can be applied in solving a mathematical problem. Therefore, as a long term consequence from her modified discourse to assist her student s better, she increased her ELLsÂ’ chances to become more active participants in mathema tical activities and to improve their fluency in expressing their questions or thoughts in English. Perceptions of classroom discourse. Figure 13 represents the researcherÂ’s evaluation, the teacherÂ’s self-evaluation, and the ELLsÂ’ evaluations of Ms. CortezÂ’ use of each of the strategies found in TTT Form 1 From Figure 13 it is noticeable that Ms.
178 Cortez evaluated herself as always using all of the strategies. On th e form with which she was provided ( TTT Form 2, see Appendix C), next to the printed words 5-Always, she added with her own handwriting: Â“ that [meaning when ] they need. Â” As a result, the pairwise correlations involving the teacher ca nnot be calculated because it results in a division by zero, as there is no deviation from the mean of 5. Nevertheless, the pair-wise correlation between the researcher and ELLs it is .36. Figure 13 also indicates that there are four strategies where ther e is agreement among all of the evaluators as the most frequently used by Ms. Cortez Â– Use of repetitions (strategy 4), Use of clarifications of directions (6), Start a lesson with a review of a related concept (9) and Provide feedback (14) The frequent use of these strategies was demonstrated throughout the excerpts provided above. For example, in the first excerpt, Ms. Cortez asked students to reflect on the lesson she taught previously by asking them review questions (strategy 9, lines 1, 4, 18, 24, 35, and 36). She also frequently repeated her or her studentsÂ’ statements (Strategy 4, lines 17, 23, 39, 43, and 49), and often provided them with feedback, thus indicating to them the validity of the answers they provided (strategy 14, lines 6, 12, 17, 31, 34, and 46). Throughout the observations it was also observed that afte r assigning the students to work individually or in cooperative groups, Ms. Cortez often circulated between their seats and clarified the directions or provide d assistance if the students asked for help (strategy 6). However, the researcher and Ms. CortezÂ’ ELL students both evaluated that she did not incorporate two of the strategies as often as she thought Â— Use of fewer idioms
179 Figure 13. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ eval uations of Ms. CotrezÂ’ frequency of us e of various discursive strategies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
180 and slang words from the mathematics vocabulary (strategy 2) and Provide opportunities for students to share experienc es and build up on personal or cultural specific knowledge while problem solving in mathematics and t hus building on cross cultural knowledge (21). In relation to how ELLs feel in Ms. Co rtezÂ’ classroom, Figure 13 indicates that her ELL students evaluated that two other strategies were not as often incorporated in her teaching style either Â— Use of a slower and simpler speech (1) and Expose students to different classroom work arrangements, su ch as using cooperative groups or partner discussions (19). The difference in ELLsÂ’ opinions of Ms. CortezÂ’ less frequent use of strategy 1 might be due to the fact that most often they (being in very initial stages of English language acquisiti on) actually heard the Fren ch and Spanish/Portuguese translated versions of Ms. CortezÂ’ talk a nd perhaps it was the translators who did not employ slower or simpler speech. For exampl e, during the intervie ws, two of the ELLs indicated that usually they worked in a group and helped each other and/or were aided by the TA who was translating. However, for the mo re fluent ELL (a girl ), it was not hard to understand Ms. Cortez, but it was hard to tran slate Ms. CortezÂ’ speech in Portuguese to her peer (a boy in the stage of English pre-production ) because she did not know the mathematics vocabulary in Portuguese. Add itionally, the ELLs indicated that even though they worked in groups, these groups usua lly consisted not only of them, but also the TAs. Thus, they indicated that they were not provided w ith opportunities for cooperative work or partne r discussions with their English speaking peers. Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 14 indicates the frequency with which Ms. Co rtez implemented each of the discursive strategies found in the TTT Form 1 Figure 14 indicates that the strategies most frequently
181 employed by Ms. Cortez were: Use of repetitions (4), Use of different questioning techniques sensitive to the ELLsÂ’ level of SLA (12), Provide feedback (14), and Use of a slower and simpler speech (1). This information supports two of the conclusions reached by the researcher and the ELLs, as reflected in the previous graph (see Figure 13). In particular, Ms. Cortez often used repetitions or paraphrasing of her statements (strategy 4) or asked students to repeat or restate them, especially when important concepts in mathematics were formulated. Also, Ms. Cort ez frequently provided all her students, and especially the ELLs, with feedback as to whether their answers were correct, in both the mathematics and English language contexts (strategy 14). The excerpts above, and the thorough qualitati ve analysis of the type of questions Ms. Cortez employed (strategy 12) during the ob servations indicated that she frequently switched between questions that initiated one-word responses, gene ral questions that encouraged lists of words, and either/or questions (strategy 12b, lines 1, 4, 10, 13, 15, 21, 27, 29, 32, 37, 41, and 44 in the first excerpt). Sh e also used questions that encouraged ELLsÂ’ speech emergence and intermediate speech development, but not very frequently (strategy 12 c and d; line 8 and line 9 in exce rpt 1; and lines 4, 9, and 17 in excerpt 2). This indicated that Ms. Cortez was awar e of the level of SLA of her ELLs Â– preproduction and early production and provided them with ques tions that led them to the next levels of the subject-specific literacy Â– speech emergence and intermediate speech in mathematics in English. However, she did not challenge them to share their opinions and explore different methods of solving mathem atical problems. She rarely asked her ELL students to justify, criticize, or explain their solutions. However, while in Figure 13 there was di sagreement between the researcherÂ’s
182 Figure 14. Frequency count of Ms. Cort ezÂ’ use of various discursive strategies dur ing the three 20-minute video-recorded sessions. 0102030405060 12. Use different questioning techniques 4. Use of repetitions or paraphrasing 14. Provide feedback 1. Use of a slower and simpler speech 6. Use of clarification of directions 7. Comprehension checks 15. Use of gestures, expressions, eye contact 13. Use wait-time techniques 19. Using cooperative groups 5. Use of changes of tone, pitch, and modality 3. Use of synonyms 11. Math discussions and problem solving 20. Alternative forms of assessment 9. Review of related concepts 18. Use of technology 16. Use of charts, graphic organizers 8. Identify vocabulary, pictures, or models 17. Use of visual or auditory stimuli 21. Use of cultural-specific knowledge 22. Content specific, enriched information 2. Use of fewer idioms and slang words 10. Summary of the key conceptsStrategyFrequency
183 evaluation and ELLsÂ’ evaluation in strategy 1 ( Use of a slower and simple speech ), the actual frequency count of Ms. CortezÂ’ em ployment of this strategy during the observations indicates (see Figure 14) that she indeed applied th is strategy very often. As was conjectured above, this difference in ELLs Â’ opinions might be due to the fact that most often they (being in very initial stag es of English language acquisition) actually heard the French and Spanish/Portuguese tran slated versions of Ms. CortezÂ’ talk and possibly it was the translators who did not employ slower or simpler speech. Figure 14 also reveals the previously discussed omission by Ms. Cortez to Conclude a lesson with a summary of the key concepts (strategy 10), Provide opportunities for students to share experienc es and build up on personal or culturalspecific knowledge while problem solving in mathematics and thus building crosscultural knowledge (21), and to Provide the students with co ntext specific, enriched information, thus exhibiti ng equally high expectations from ELL and non-ELL students (22). The chart also reveals a l ack in Ms. CortezÂ’ talk in her Use of fewer idioms and slang words from the mathematics vocabulary wh ich if used, were not accompanied by a proper explanation or visual representation (strategy 2), and an omission to Use visual or auditory stimuli-pictures flashcards, models, etc. (strategy 17). Mr. Daniels Mr. Daniels, a 60 year old Caucasian, ha s a 12-year teaching experience. He is certified to teach secondary mathematics and has completed the required 60 hours of training toward his ESOL endorsement nine ye ars prior to the date of this study. In his teaching career, he has always taught Algebr a I classes together with Geometry and college preparatory classes. Mr. Daniels is presently teaching Algebra I with the aid of
184 the schoolÂ’s computer lab tutorials, tests, and quizzes developed by the program I Can Learn Lab His Algebra I class consisted of 20 stude nts, 12 of which were African American, five Hispanic, and two White students. Initia lly there were four ELLs (two Hispanics and two African American students) but one of the African Am erican ELLs withdrew from school and one of the Hispanic ELLs was suspended out of school for 10 days. In the interview Mr. Daniels commented on his e xperience of teaching mathematics to ELL students as follows: Â“Having worked with ELL students in the past, the math vocabulary is critical so I emphasize th is especially in their note books. Most [ELLs] have good basic skills, but they have problems when answ ering Â‘wordÂ’ based problemsÂ”. Mr. Daniels reported that his opinion is based on his obser vations that usually when his ELL students call him for help; it is usually when they en counter a word problem. In his opinion, it is because his current ELL students were in more advanced stages of SLA ( speech emergence and intermediate fluency) that he is more successful in helping them. He negotiated with them the meanings of word problems and provided them with contextembedded examples, pictures, or models a nd thus helped them solve the problems. However, he continued: My primary concern is the level I ES OL student who cannot communicate in English at any level. In 12 years of teaching, I have had only one succeed. We need to make speaking basic English the firs t priority before we put them in Math, Science, etc. All we do is set them up to fail. Typical classroom discourse. The excerpt below illustrates some of the natural discourse that took place while Mr. Daniels circulated ar ound his Computer Lab room.
185 During such sessions, the students used tutorial programs that facilitated their individual progress in Algebra I. Usually, Mr. Daniels assisted the stud ents who asked for help, but he also monitored all his students and directed his a ttention more towards the ELLs or those students struggling in mathematics. In the excerpt below, Mr. Daniels assisted an ELL student in understanding the con cept of the slope of a line: Mr. Daniels:  Okay letÂ’s have you do th is one up here on this sheet here ( pointed to the computer sc reen and the studentsÂ’ notebook)  Okay you do the same thing.  Okay, remember your y Â’s are on the tops.  So you put your y Â’s on the bottom (pointed to the yÂ’s ).  ItÂ’s got to be rise over run.  ThatÂ’s the most common mi stake people make is what you did right there.  These y Â’s, that has to be first, okay (pointed again )? Mr. Daniels observed that the student was making an erroneous substitution in the formula and switched the places of rise and run. In his explanation, the teacher used simplified speech and shorter sentences (strategy 1a, lines 2 to 5, and 7). He also used gestures (strategy 15, lines 1, 4, and 7) to better articulate the meani ng of his talk. Mr. Daniels also used simpler synonymous words (strategy 3, lines 3 and 4) such as Â“topÂ” and Â“bottomÂ” instead of Â“numeratorÂ” and Â“denominat or,Â” so as to be better understood by the ELL student. He also used the more informal statement of the slope formula as Â“rise over runÂ” and focused his talk on the procedures of finding the slope, thus fostering ELL studentsÂ’ early production in English (strategy 1b, line 5). (The student tried again by substituting the pointÂ’s coordinates in the formula for slope) Mr. Daniels:  Right.  Okay six, then thereÂ’ s a minus six though, right.  ItÂ’s a negative six minus a negative four.  A nd usually, like I said, youÂ’ve got two signs side by side.  Then you got ne gative two minus four.
186  Okay you only have one of these sign combinations, right? Lester (an ELL): ( nods ) Mr. Daniels:  So you replace that ne gative and negative with a positive.  So youÂ’ve got positive four there. Mr. Daniels:  WhatÂ’s negative six plus four? (the teacher wrote -6+4 while talking) Here is exemplified a situation in which, wh ile teaching the concept of slope, Mr. Daniels found that his ELL student had pr oblems with operations involv ing integers of different signs. Thus, based on his studentsÂ’ needs, he modified his instructions and provided the needed assistance (strategy 6, lines 10, 12, 14 and 15). Lester:  Um Mr. Daniels:  Negative six plus four.  WhatÂ’s the di fference between six and four? Lester:  Six and four?  Four. Mr. Daniels:  You have four dollars you spend six dollars how much are you short? Lester:  Two Mr. Daniels:  So its going to be a negative two.  YouÂ’re two dollars short, right? Lester: ( nods and writes -2 on his sheet of paper) Mr. Daniels:  WhatÂ’s negative two minus four? Lester:  Two As the studentÂ’s answer in line 21 demonstr ated, he really needed assistance with operations with positive and negative intege rs. Rather than directly correcting him (strategy 1bÂ—the teacher needs to model/demonstrate correct responses both in Math and English with students from the early production stages of their SLA, line 22), Mr. Daniels decided to explain the problem by us ing money (a concrete object) instead of numbers (abstract). Thus, he related the pr oblem to a real-life situation that the ELL student more likely encountered in his da ily life (strategy 22 c, line 22 and 25). The student responded correctly to the thus-present ed problem (line 23).Then, when he asked
187 a similar question with abstract numbers ag ain (line 26), and received an erroneous response (line 27), Mr. Daniels made an instru ctional decision to continue presenting the mathematical operations with positive and negative integers via operations with money: Mr. Daniels:  Okay I take two dollars from you, then I take four more dollars from you.  How much have I taken? Lester:  Two. Mr. Daniels:  I took two bucks from you.  I got it in my hand.  I take four more from you, how much do I have in my hand? Lester:  Six. Mr. Daniels:  IÂ’ve taken six do llars from you, right?  The signs are the same.  You add and keep the signs.  Okay a negative divided by a negative is a? Lester:  Um, positive Mr. Daniels:  When you simplify two sixth ( points to 2/6) whatÂ’s that the same as? Lester:  ThreeÂ…uhÂ…one third ( writes 1/3) Mr. Daniels:  One over three.  Okay so is that answer up there? ( points to screen) Mr. Daniels:  Did th at help you, Lester? Lester:  ( nods) Mr. Daniels:  I know it was kind of a math break down, but at least you got something. Throughout this excerpt, Mr. Da niels also occasionally checked the studentÂ’s comprehension (strategy 7, lines 13, 16, 25, 35, and 44), repeated or paraphrased his or his studentÂ’s statements (strategy 4, lin es 3, 5, 7, 18, 19, 31, 32, and 33), and provided him with feedback of whether an operation was perfor med correctly (strategy 14, line 8) or, by asking the student to perform the same operation again (by using an example with money), he indicated to the student that his answer was incorrect. Thus, he used a more subtle form of providing feedback without directly correcting th e studentÂ’s errors.
188 Krussel et al. framework. The analysis of Mr. Daniel sÂ’ Â“discourse movesÂ” using Krussel et al.Â’s (2004) framework reveal that the purpose of Mr. DanielsÂ’ discourse was to assist his ELL students in improving th eir mathematics and language abilities by modeling/demonstrating correct responses, both in mathematics and English language, as was demonstrated in the excerpt above (strategy 1b, lines 5, 10, 14, 36, 37 and 38). He assisted his students, incl uding ELLs, in executing computer adapted activities, thus improving their understanding of the new concep ts in the particular lessons (in the excerpt above, it was the concept of slope and related operations invol ving integers). By analyzing all of the observed sessions, it becam e apparent that Mr. Daniels was aware of the presence of ELLs in his Algebra I class and consciously catered to their specific needs. For example, when formally teaching the topic of slope to the ELL student from the excerpt above, he purposefully used simpler talk and shorter sentences (strategy 1a and b, lines 3, 4, 7, 19, 22, 25, 28, 31 to 33) and paraphrased his sentences and questions (strategy 4, lines 7, 19, 22, 25, 28) to negotiate the meaning of the concept under scrutiny. Frequently, after assisting individual students with 3 to 4 examples, he asked them to complete other sample problems (including some word problems as well) similar to the one whose solution he had just modeled (str ategies 1a to 1c, and 22a to 22c). He oftentimes Â“broke downÂ” the steps of solvi ng a problem (strategy 6) in a manner similar to that exemplified by the excerpt above so that the ELL students or those who were struggling with a certain mathema tical concept to better grasp it. Mr. Daniels established a setting for classroom discourse by instituting certain long-standing expectations and norms of cl assroom behavior. Speci fically, he expected that his students work in their notebooks simultaneously while working on the computer,
189 and he monitored their progress on both as was demonstrated in the excerpt above (strategy 20, line 1). Students seated in ne ighboring computers were allowed to talk, but only on task-related topics. However, some of them often misbehaved and involved themselves in out-of-task conversations a nd moved from their assigned seats. For example, in one of the observations, while Mr Daniels assisted one student, some of the other students talked aloud and thus interr upted their peersÂ’ individual work on the computers. Generally, Mr. Daniels manage d to keep his students focused on the mathematical task at hand, but in order to ma intain such order he often had to interrupt his instructions to deal with a particular student or discipline issue. In the interviews, the students indicated that because they were not exposed to differe nt classroom work arrangement, such as cooperative groups or partner discussions (i.e., lack of utilizing strategy 19), after starting d iligently on their work they s oon experienced boredom and as a result involved themselves in nonmathematics oriented activities. The form of Mr. DanielsÂ’ discourse included both teacher talk (verbal) and actions (non-verbal discourse) For example, after modeling or demonstrating the solution of a mathematical problem (strategy 1b, as demonstrated in the excerpt above), he often encouraged his students (especially ELLs) to try to talk mathematically in English: Â“Ok, so you talk me through the ne xt oneÂ” or Â“Ok, so try this one.Â” This indicated that Mr. DanielsÂ’ discourse took the form of a challenge by encouraging his students (all the while not differentiating be tween non-ELL and ELL students) to move to operations of higher cognitive demand according to BloomÂ’s taxonomy Â– application, analysis (Â“Â…just kind of sketch the points and youÂ’ll seeÂ…Which scatter plot represents a non-linear relationship?Â”), synthesis (Â“predict best price esti mateÂ” or Â“Now stop for a
190 second. What happens if they ask for one that Â’s between 2 and 4? What would we expect the value at 3 to be?Â”; strate gy 22d and 22e). Furthermore, que stions such as the latter revealed Mr. DanielsÂ’ efforts to encourage hi s ELLs to expand their fluency in both the English language and mathematics (strategy 12b to 12d Â— Use of different questioning techniques, sensitive to the ELLsÂ’ level of SLA) and participate in the teacher-student discourse by explicitly voicing the operati ons they perform. However, even though he encouraged his students to dr aw diagrams or write in mathematics, he did not expose them to Â“hands onÂ” activities or work in groups (i.e., he did not use strategy 19Â— Expose students to different clas sroom work arrangement) Furthermore, he did not ask them to justify and perform more critical analyses or further explanations of more complicated steps while problem solving (strategy 22fÂ— Move the discourse to the highest level of cognitive demand according to BloomÂ’s taxonomy ). The consequences of Mr. DanielsÂ’ discourse may be classified according to Krussel et al.Â’s (2004) framework as intended or unintended immediate or long term as follows. For example, as exemplified in the excerpt above relating his explanation of the concept of slope to the ELL student, he intentionally used simplified speech and shorter sentences (strategy 1a) and synonymous words (strategy 3) such as Â“topÂ” and Â“bottomÂ” instead of Â“numeratorÂ” and Â“denominator, a nd Â“yÂ’sÂ” for Â“riseÂ”. He thus demonstrated sensitivity to the level of SLA of his ELL student Â—transitioning between early production and speech emergence However, he unintentionally was Â“taking the floorÂ” and did not provide many opportun ities for his student to artic ulate where his problems in operations of integers arise. For exampl e, when Mr. Daniels understood that the ELL student was more successful in performing operations with positive and negative integers
191 once the mathematical operations were tr ansferred to operations with money, he continued using this technique in subsequent examples. An immediate consequence of this type of discourse was that the stude nt performed the operations more correctly. However, due to the general fact that Mr Daniels was mostly using one type of classroom discourse organization Â– students working individually on the computers and being assisted by Mr. Daniels whenever th e need arose Â– demonstrated that his ELL students were exposed to only teacher-student mathematical interactions. Even though the students were allowed to talk with othe rs in their vicinity, these conversation were rarely mathematics-oriented in nature. As a result, Mr. Daniels did not provide ample opportunities for ELL students to use new math ematics vocabulary in dialogue. Thus, a close examination of Mr. DanielsÂ’ discourse indicated that as long term consequences of his manner of facilitating a single type of classroom discourse, his ELL students were assisted in developing their co nceptual understandi ng of mathematics, but were not given opportunities to be equal partners in cooperative group discussions. Perceptions of classroom discourse. Figure 15 below represents the researcherÂ’s evaluation, the teacherÂ’s self-evaluation, a nd the ELLsÂ’ evaluations of Mr. DanielsÂ’ teacher talk and use of different discursi ve strategies identified in TTT Form 1 The pairwise correlations (Pearson product-moment correlation coefficients ) for Mr. DanielsÂ’ case study are as follows: the correlation betw een the teacher and researcher is .71; between the teacher and ELLs it is .83, and be tween the researcher and ELLs it is .58. There is general agreement that Mr. Dani els most frequently employed strategies 1 ( Use of a slower and simpler speech ), 4 ( Use of repetitions ), 7 ( Use of Comprehension
192 Figure 15. TeacherÂ’s, researcherÂ’s, and ELLsÂ’ evalua tions of Mr. DanielsÂ’ frequency of us e of various discursive strtegies. 0123456 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of cultural-specific knowledge 22. Content specific, enriched information StrategyFrequency Scale (5 as most frequent) Teacher Researcher ELLs
193 Checks ), and 14 ( Provide Feedback ), followed by (slightly less frequently) strategies 6 ( Use of Clarification of directions ), 15 ( Use of gestures, facial expressions, eye contact, or demonstrations to enhance comprehension ), and 18 ( Use of technology ). Mr. DanielsÂ’ frequent use of these strategies was alr eady demonstrated in the excerpt above and discussed in the previous paragraphs. Figure 15 reveals agreement that Mr. Daniels used least frequently the following discursive strategies: Start a lesson with a revi ew of a related concept (9), Conclude the lesson with a summary of the key concepts (10), and Provide the students with opportunities to share experiences and bu ild upon personal or cultural specific knowledge while problem solving in mathematics (21). Interesting disagreement in the evaluations is observed in the following two strategies: Use of fewer idioms and slang words from the mathematics vocabulary (2) and Provide students with alternative forms of assessmentÂ—portfolios, vocabulary banks, oral presentations, and writing or reading in mathematics (20). Mr. Daniels evaluate d himself as using few idioms and that if they were used, were accompanied by a proper explan ation or visual representation. However, the ELL students and the researcher indicate d that even though on occasion Mr. Daniels explained some idioms of the mathematic s vocabulary, this was not done frequently enough. On the other hand, for category 20, the ELLs indicated that Mr. Daniels provided them with alternative forms of assessment su ch as writing or reading in mathematics, whereas the researcher and Mr. Daniels felt that he should ha ve used this strategy more often and included oral presen tations, or portfolios and voc abulary banks as other forms to assess his studentsÂ’ progress in mathema tics (and especially that of his ELL students).
194 In the interviews, the ELL students indicat ed that they had difficulties with word problems in mathematics and that it was usually when presented with such problems that they sought assistance from Mr. Da niels. He usually performed clarifications of the directions (strategy 6) for them and helped by modeling a variety of examples so that they could see the solution pr ocess (strategy 1b). He also asked his ELLs to apply the explained concepts to solve new problems and make predictions as to what would happen in different situations. Thus, he provided them with content specific, enriched information, thus exhibiti ng equally high expectations from ELL and non-ELL students (strategy 22). Figure 16 below also confirms that Mr. Daniels applied this strategy frequently. Summary of the frequency count of the teacherÂ’s discursive strategies. Figure 16 indicates the frequency with which Mr. Daniels implemen ted each of the discursive strategies found in the TTT Form 1 The strategies most frequently employed are: 1 ( Use of a slower and simpler speech) and 14 ( Provide feedback ), as corroborated by the evaluations chart (see Figure 10) It also shows that Mr. Dani elsÂ’ most frequent strategy was 12 ( Use of different questioning techniques ). Additionally, a quali tative analysis of the types of questions employed by Mr. Daniel s reveals that they usually elicited oneword responses, or were general questions th at encouraged a short list of words as a response. This indicates that Mr. Daniel s was aware of the level of his ELLs Â– early production or in transition to speech emergence or intermediate fluency. Furthermore, the qualitative analysis indicat ed that Mr. Daniels used que stions that challenged his ELLs and could potentially lead them to move to higher levels of subject-specific literacy Â– speech emergence and intermediate speech in mathematics in English Â– but such
195 Figure 16. Frequency count of Mr DanielsÂ’ use of various discursive strategies during the three 20-minute video-recorded sessions. 01020304050607080 12. Use different questioning techniques 1. Use of a slower and simpler speech 14. Provide feedback 7. Comprehension checks 4. Use of repetitions or paraphrasing 22. Content specific, enriched information 15. Use of gestures, expressions, eye contact 6. Use of clarification of directions 3. Use of synonyms 5. Use of changes of tone, pitch, and modality 8. Identify vocabulary, pictures, or models 17. Use of visual or auditory stimuli 19. Using cooperative groups 18. Use of technology 10. Summary of the key concepts 11. Math discussions and problem solving 16. Use of charts, graphic organizers 13. Use wait-time techniques 20. Alternative forms of assessment 2. Use of fewer idioms and slang words 9. Review of related concepts 21. Use of cultural-specific knowledge StrategyFrequency
196 questions were not very frequent. Such findi ngs are with agreement with those reported when examining Mr. DanielsÂ’ form of teacher talk according to Krussel et al.Â’s (2004) framework. Figure 16 also confirms the previ ously discussed omissi on by Mr. Daniels to use the following discursive strategies: Use of fewer idioms and slang words from the mathematics vocabulary, or if used a prope r explanation or visual representation is provided (strategy 2), Start a lesson with a review of related concepts (9), Conclude a lesson with a summary of the key concepts (10), and Provide opportunities for students to share experiences and build up on personal or cultural-specific knowledge while problem solving in mathematics and thus building cross-cultural knowledge (21). Summary of Results To summarize the results from the detailed description and analysis of each case study, and to allow comparison between teacher s, Table 3 and Figure 17 were devised (see Table 3 and Figure 17 belo w). Table 3 presents the ge neral level of agreement between the teacher, the researcher, and the ELLs in their evaluations of the strategies used by each teacher. More specifically, Ta ble 3 presents the computed pair-wise correlations (Pearson product-moment correla tion coefficients), that show whether a given teacherÂ’s perceptio ns of his/her own use of strategi es match those of the researcher and the ELLs. As can be discerned from the ta ble, negative correlation coefficients were observed for the novice teachers Â— Ms. Barre ra from Green Bay High School and Ms. Brown from Lincoln High School, who were also recently enrolled in the ESOL certification process. The negative pair-wise coefficients observed between the teacher
197 Table 3 Pearson Product-Moment Correlation Coefficients Teacher Name Teacher (X) Researcher (Y) ELLs (Z) R ( X, Y ) (Tcr.,Res.) R ( X, Z ) (Tcr.,ELLs) R ( Y, Z ) (Res.,ELLs) Green Bay High School Mr. Able Xm=3.77 Sx=0.972 Ym=3.73 Sy=1.08 Zm=3.86 Sz=1.14 Rxy = .62 Rxz = .25 Ryz = .65 Ms. Barrera Xm=3.91 Sx=1.02 Ym=3.73 Sy=0.98 Zm=2.80 Sz=1.13 Rxy = -.07 Rxz = -.23 Ryz = .27 Ms. Chandler Xm=3.23 Sx=1.27 Ym=2.86 Sy=1.36 Zm=2.81 Sz=1.02 Rxy = .77 Rxz = .53 Ryz = .70 Mr. Davison Xm=3.68 Sx=1.21 Ym=3.64 Sy=0.95 Zm=3.33 Sz=1.14 Rxy = .68 Rxz = .17 Ryz = .43 Lincoln High School Ms. Andersen Xm=3.68 Sx=1.09 Ym=3.43 Sy=0.93 Zm=2.64 Sz=1.50 Rxy = .61 Rxz = .12 Ryz = .46 Ms. Brown Xm=4.25 Sx=1.11 Ym=3.97 Sy=1.05 Zm=3.30 Sz=1.22 Rxy = -.14 Rxz = -.26 Ryz = .59 Ms. Cortez Xm=5 Sx=0 Ym=3.77 Sy=1.11 Zm=3.59 Sz=1.06 Rxy = undef. Rxz = undef. Ryz = .36 Mr. Daniels Xm=3 Sx=1.51 Ym=3.11 Sy=1.79 Zm=2.89 Sz=1.54 Rxy = .71 Rxz = .83 Ryz = .58
198 self-evaluation and the researcher evaluation indicate a lack of realistic vision of the classroom approaches. In a similar manner, th e negative correlation coefficients between the teacher self-evaluation and the ELLsÂ’ ev aluations indicate that teachers who had not completed the ESOL certification process lacked or had not yet developed an understanding of their ELL students and thei r educational needs in the mathematics classroom. In reporting the pair-wise co rrelation coefficients an extreme example was Ms. Cortez. In her case, the correlation coefficients between the evaluations of the teacher and the researcher, and the teacher and ELLs, c ould not even be calculated. The formula involves division by the standard deviations from the mean, and because Ms. Cortez had evaluated herself as having used all discur sive strategies from TTT Form 2 with a frequency of 5 (i.e., always/most frequently), the standard deviation from the mean was zero, thus it was not possible to obtain a result. This unreali stic self-evaluation of the used discursive strategies and lack of unders tanding of her ELL students can be attributed to the fact that Ms. Cortez, despite having pr evious teaching experience, had obtained this experience while teaching in a middle school in Puerto Rico. She had a middle school mathematics certification and no ESOL certifi cation. Since this was her first year of teaching in a high school in the USA, she was working on her high school mathematics certification and was not yet enrolled in ESOL certification classes. For the other teachers, the higher positive correlation coeffi cients indicate that teachers with more teaching experience had developed a better sens e of the teaching pract ices they routinely employed and could more realistically eval uate where they needed improvement. For example, in the cases of Ms. Chandler (Green Bay High School) and Mr. Daniels
199 (Lincoln High School), the highest positive co rrelation coefficients were observed across the three evaluators. The more accurate self -evaluation and better understanding of their ELLs could be attributed to the fact that bot h teachers were experi enced and had their with ESOL certification for longer period of time. Figure 17 combines the data from the frequency count of each teacherÂ’s use of different strategies during the observed le ssons (see Figures 2, 4, 6, 8, 10, 12, 14, and 16). On the x-axis, the numbers from 1 to 22 correspond to the categories of teacher talk and other discursive strategies that ar e described in greater detail in TTT Form 1 (See Appendix A). On the y-axis are placed the fr equencies with which each strategy was used by each teacher during the observed classr oom sessions. Above the numbers of each discursive strategy, the different-shaded bar graphs (8 bars corresponding to 8 teachers) represent each teacherÂ’s frequency of us e of each category. For clarity, the legend provided below the graph allows the reader to connect each teach erÂ’s name with the assigned shading. (The eight bars above each category represent each teacher in the same order they were described in the study or in the order as presented in Table 2). Thus, as the detailed description and analysis of each case study has shed light on the specific patterns of strategies typically used by each teacher, Figure 17 allows for comparisons between the teachers in answering the rese arch questions of this study. As Figure 17 indicates, the most frequently used strategi es by all teachers (with minor variations) were: 12 ( Use of different questioni ng techniques, sensitive to the ELLsÂ’ level of SLA ) and 14 ( Provide feedback ). This conclusion is well grounded a nd is based on all the data that was triangulated by using different sources and methods of analysis. However, the additional qualitative analysis revealed di fferences in the types of questi ons the teachers asked their
200 0 20 40 60 801. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes of tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify vocabulary, pictures, or models 9. Review of related concepts 10. Summary of the key concepts 11. Math discussions and problem solving 12. Use different questioning techniques 13. Use wait-time techniques 14. Provide feedback 15. Use of gestures, expressions, eye contact 16. Use of charts, graphic organizers 17. Use of visual or auditory stimuli 18. Use of technology 19. Using cooperative groups 20. Alternative forms of assessment 21. Use of culturalspecific knowledge 22. Content specific, enriched information StrategyFrequency Mr. Able Ms. Barrera Ms. Chandler Mr. Davison Ms. Andersen Ms. Brown Ms. Cortez Mr. Daniels Figure 17. TeachersÂ’ frequencies of used strategies dur ing the three 20-minute video-recorded sessions.
201 ELLs. These differences in questioning techni ques, as well as other differences and similarities between the teachers, will be su mmarized in relation to the studyÂ’s research questions. Q uestion 1 In investigating the extent to which t eachersÂ’ patterns of discourse in the mathematics classroom change as a result of the number of ELL st udent(s) present, the following findings emerged. As indicated in th e beginning of Chapter Four (see Table 2), most of the ELL students in Green Bay High School were assigned to Algebra I classes with computer labs, tutorials, tests and qui zzes (i.e., Ms. BarreraÂ’s and Ms. ChandlerÂ’s classrooms), and were more evenly distributed in Lincoln High School. However, as the analysis of data from different sources ( observations, video-recordings and frequency counts, interviews and the researcher eval uation, teachersÂ’ self-evaluations and ELLsÂ’ evaluations) indicated, to some extent all teac hers changed their patte rns of discourse in the mathematics classroom as a result of simply the presence of ELL student(s), regardless of their number. For example, even in the case of teache rs who did not share their ELLsÂ’ linguistic and cultural backgrounds, and even if there wa s only a single ELL st udent or a couple of ELLs from certain cultural or linguistic groups, there were changes in the classroom setting which subsequently influenced change s in the classroom discourse. As described in the case analyses, for example, Mr. Ab le and Mr. Davison in Green Bay High School and Ms. Brown in Lincoln High School, both of whom did not speak their ELLsÂ’ native languages, tried to seat ELLs from similar linguistic backgrounds in close proximity to each other and thus make sure that they were al so seated next to students that spoke both
202 English and these ELLsÂ’ native language. Thus, these teachers exposed their ELL students to classroom arrangements facilitating peer or group discussions in mathematics in both English and their native language (i .e., utilization of strategy 19). On the other hand, Mr. Daniels in Lincoln High School, w ho also spoke only English, despite not pairing his students in groups (they worked individually on computers), was observed to utilize other strategies from TTT Form 1 : strategy 12 (using different questioning techniques sensitive to the ELLsÂ’ levels of SLA), 1 ( adapting his speech to the level of ELLs present ), and 14 ( providing feedback ). He also often performed comprehension checks (strategy 7) to see if hi s ELL students understood him, and used gestures, facial expressions, eye contact or demonstr ations to enhance comprehension (strategy 15). Strategies 12, 14, and 1 were used often by Ms. Andersen (Lincoln High School) as well, especially when addressing the ELLs who did not speak French (the language she sometimes used to improve the communicati on with two of her ELLs). To be better understood by her ELLs who did not share her linguistic background, she used wait-time techniques after posing a ques tion (strategy 13). Ms. Chandl er (Green Bay High School), who also spoke only English and her ELLs did not work in groups because they worked individually on computers (as in Mr. DanielÂ’s class), also of ten used strategies 14 and 1, but she also often used clarifications of directions and individual assistance when her ELLs were executing specific mathematical tasks on the computers (strategy 6) and was using the technology (strategy 18) to enhance her ELLs Â’ comprehension (just as Mr. Daniels did). The teachers who spoke the language of their ELLs Ms. Barrera (from Green Bay High Schhol) and Ms. Cortez (from Li ncoln High School) with 9 and 4 ELLs
203 respectively, both also used most frequen tly strategies 12, 14, and 7. Additionally, Ms. Cortez also had French-speaking ELLs, beside s the Spanish ELLs with whom she shared similar cultural and linguistic background. She al so seated her ELLs in close proximity to other students or teacher assistants who shar ed their linguistic background and also spoke English fluently. Question 2 In analyzing the data to answer question 2 Â– i.e., to what extent do mathematics teachersÂ’ experiences and teachersÂ’ ESOL e ndorsement relate to their patterns of discourse when teaching mathematics to classes with ELL students present Â– the following findings emerged. As the combined da ta in Table 2 demonstrates, the teachers with more years of teaching experience a nd having an ESOL endorsement for a long period of time had a smaller number of ELLs pr esent in their classe s (Mr. Able and Mr. Davison in Green Bay High School, and Ms. A ndersen and Mr. Daniels in Lincoln High School). Moreover, in both schools, the teachers who had just begun their teaching careers and just completed or were in the process of completing th eir ESOL requirement (Ms. Barrera in Green Bay High School a nd Ms. Brown in Linc oln High School) were assigned to teach classes with the highest num ber of ELLs. In relation to what extent the teachersÂ’ experiences and ESOL endorsement re lated to their patterns of discourse when teaching mathematics to classes with ELL students present, the following patterns emerged: combined data from the frequency count of the strategies used during the 20minute recorded sessions (see Figure 17) revealed that the teachers who just started their teaching careers and lacked practical experien ce of teaching Algebra I to classes with diverse student populations i nvolving a high number of ELLs (Ms. Brown and Ms.
204 Barrera) frequently used almost the same strategies as their more experienced colleagues did. Yet additional qualita tive analysis of the type of modifications to their speech they made (strategy 1a to d), of the type of questions they asked (strategy 12a to d), and the provision of information of higher cognitive demand acco rding to BloomÂ’s Taxonomy (strategy 22a to f) indicated that even t hough all teachers generally needed improvement in using these strategies, the more experi enced teachers (such as Mr. Able and Mr. Davison from Green Bay High School, and Ms. Andersen and Ms. Daniels from Lincoln High School) who had completed their ESOL endorsementÂ’s requirement a long time prior to the observations were applying those st rategies to a fuller extent. That is, at the least, they more often utilized strategies 1 and 12 c, if not d; and 22 c, and d, if not f. Evidence to support this claim was provided wh en analyzing the cases of Mr. Able, Mr. Davison, Ms. Andersen, and Mr. Daniels. For example, Mr. AbleÂ’s questions Â“What would you have to do to get b by itself?Â” or Â“How do we now graph this equatio n?Â” indicated that hi s discourse too often took the form of a challenge. However, he readily provide d assistance in subsequent steps and thus missed opportuni ties to move the discourse to the higher levels of cognitive demand such as synthesis and evaluation as per BloomÂ’s Taxonomy. As another example, Mr. Daniels, after modeling the solution of a mathematical problem (strategy 1b, as demonstrated in the excerpt from his case study), often encouraged his ELL students to try to explain their solutions in English: Â“Ok, so you talk me through the next oneÂ” or Â“Ok, so try this one.Â” This i ndicated that he challe nged his ELL students to move to operations of higher cognitive demand according to BloomÂ’s taxonomy Â–
205 application, analysis (Â“Â…just kind of sketch the points and youÂ’ll seeÂ…Which scatter plot represents a non-lin ear relationship?Â”), and synthesis (Â“predict best price estimateÂ” or Â“Now stop for a second. What happens if they ask for one thatÂ’s between 2 and 4? What would we expect the value at 3 to be ?Â” Â– exemplifying strategies 22d and 22e). Furthermore, with the use of such differen t types of questions (strategies 12b to 12d), Mr. Daniels encouraged his ELLs to expand thei r fluency in both the English language and mathematics, and participate in the teacher-student discourse by explaining the operations they performed. However, he st ill did not ask them to justif y and perform more critical analyses or to provide furt her explanations of more co mplicated steps while problem solving (strategy 22fÂ— Move the discourse to the highe st level of cognitive demand according to BloomÂ’s taxonomy ). Moving the mathematics disc ussions to higher levels of cognitive demand (i.e., analysis, synthesis, and evaluation ) on BloomÂ’s taxonomy creates more opportunities for all students (and ELLs in particular) to become critical mathematics thinkers. However, despite the fact that the teac hers with more teaching experience and ESOL endorsement such as Mr. Able, Mr. Daniels, Ms. Andersen, and Mr. Davison created opportunities for their ELL students to participate in the mathematics discourse, they still did not ask enough questions whic h could provide the ELLs with opportunities to justify and explai n their opinions and, consequen tly, expand on their learning of mathematics and English. They still rarely lead the discussions to a point which could move the ELLs to the highest level of the subject-specific literacy Â– intermediate speech and fluency in mathematics in English.
206 Moreover, it was observed that novice te achers (such as Ms. Brown in Lincoln High School and Ms. Barrera in Green Bay High School) often had problems maintaining discipline in their classrooms, and they confir med this in their interviews (for example, Ms. Barrera in Green Bay High School). As a result, the teachers were switching to discourse that fostered primarily teacher-c entered activities and avoided Â“hands-onÂ” activities, or scaffolding ac tivities involving group discussi ons. Even though case study 6 revealed how Ms. Brown used an activity to teach the concept of scatter plot and data correlation, such instances were scarce a nd generally avoided by the novice teachers because they had problems with their studentsÂ’ behavior and maintaining the focus of the discussions on the mathematical task at hand (a s was observed in the next two sessions in Ms. BrownÂ’s class). Question 3 In reference to how teachersÂ’ own linguist ic and cultural bac kgrounds affect their patterns of discourse when teaching mathema tics in English to classes with ELL students present, the following findings are of particular relevance. Even though in general it is beneficial for the teachers to have a simila r linguistic or cultural background as their ELLs (as in the case of Ms. Barrera in School 1 and Ms. Cortez in School 2), this is not a determining factor for succe ssfully involving th eir ELLs in the classroom discourse (Cahnmann and Remillard, 2002). Research in the field of teaching mathematics to ELLs indicates that more esse ntial factors in involving all students and fostering their active interest and learning would be to incorporate culturally respon sive instruction by utilizing their own backgrounds and cultu re to best suit the specific needs of their students (Cahnmann, & Remillard, 2002; Kersaint, Thompson, & Petkova, 2009, p. 65). For
207 example, Ms. Andersen demonstrated good rapport with her ELLs that was achieved through her own education and teacher devel opment classes. She took advantage of the fact that she speaks French (she has a minor in French) and thus engaged her Haitian students in the classroom disc ourse more directly, for exam ple. In the interviews, she indicated that she Â“is still learning to in corporate more strate giesÂ” in her teaching practices, and that this is an ongoing process for her. The teachers who shared the cultural and linguistic background of the majority of their ELLs Ms. Barrera with 9, and Ms. Cort ez, with 3 Spanish-speaking students (out of 4 ELLs) both also utilized most frequent ly strategies 12, 14, 7, and 4 from amongst the teachers in the sample (see Figure 17). Additionally, Ms. Cortez, who had one French-speaking ELL besides the Spanish ELLs, also seated her ELLs in close proximity to others students (who spoke French) or te acher assistants who sh ared their linguistic background and also spoke English fluently. However, both teachers exhibited the same lack of providing opportunities for ELL stude nts to share experiences and build on personal cultural specific knowledge while pr oblem solving (i.e., lack of implementing strategy 21) as did the rest of the teachers in the sample (see Figure 17). Question 4 In investigating what patterns of di scourse the teachers used when ELL students were present in the mathematics classroom, and what adjustments to teacher talk or modifications of instructions the teachers made, the following findings emerged. For example, Figure 17 shows that the most freque ntly used strategies by all teachers (with minor variations) were: 12 ( use of different questioning tech niques, sensitive to the ELLsÂ’ level of SLA ), 14 ( provide feedback ), 1 ( use of slower and simpler speech ), 4 ( use of
208 repetitions or paraphrasing when important mathematics concepts are formulated ), and 7 ( use of comprehension checks ). This finding is well grounde d and is based on all the analysis of triangulated data. However, additional qualitative analysis revealed that most of the questions the teachers asked their ELLs were of a type that required usually one-wor d or a short list of words in response, or were yes/no or either/o r questions. Further analysis also revealed that the most attempts to move the questio ning techniques to a higher level were made by Mr. Daniels, Ms. Cortez, and Mr. Davison w ith questions such as Â“Why?Â”, Â“What do you recommend?Â”, or questions that elicited thei r ELLs to expand not only their literacy in mathematics but also to develop to speech emergence and intermediate speech in English language Throughout the observations, most teachers, after receiving responses to their questions, usually provided students with feedback (strategy 14) in most cases indicating whether the answer was correct or not. The teachers also adapted their speech to their audience and, being aware that th ere are ELLs in the classroom, used simple commands and shorter sentences, and modeled the co rrect responses both in mathematics and English (strategy 1 a, b, and c). However, the qualitative analysis of their teacher talk also revealed that on more rare occasions when presenting a new concept, the teachers used advanced organizers and at the same time used their talk to lead the students to small group work or Â“hands-onÂ” activities. Further, Figure 17 reveals that the teach ers least frequently used strategies 10 ( Conclude a lesson with a summary of the key concepts ), 2 ( Use of fewer idioms and slang words ), 21 ( Provide opportunities for students to share experiences and build up on personal or culturalspecific knowledge ), and 9 ( Start a lesson with a review of a related
209 concept ). However, while the chart in Figure 17 re veals the frequency count of strategies that were utilized only during the 20 minut es of the observed classroom sessions, the other sources of data collection (e.g., ELL a nd teacher interviews, and the researcherÂ’s observations and field notes throughout the entire classroom sessi on) revealed that strategy 9, and to some degree st rategy 10, were in fact also more frequently utilized by some of the teachers. For example, Mr. Able and Ms. Barr era in Green Bay High School, and Ms. Andersen and Ms. Cortez in Lincoln High Sc hool, traditionally used bell-work in which they included review questions of previ ously learned concepts and thus employed strategy 9. Additionally the classroom observations fo r the duration of the entire sessions, as well as the intervie ws with the teachers and th eir ELLs also confirmed that some of the teachers conclude the lessons w ith a summary of the important concepts the students just learned. For example, Ms. Ba rrera and Mr. Davison in Green Bay High School, and Ms. Andersen and Ms. Cortez in Lincoln High School, were evaluated by their ELLs as using strategy 10 at least a couple of times a week. However, all data collected from different sources (observati ons, video-recordings and frequency counts, interviews and evaluations of the researcher, teachers Â’ self-evaluations, and ELLsÂ’ evaluations) revealed a consistent lack of use of strategy 21 ( Provide opportunities for students to share experiences and build up on personal or cultural-specific knowledge ).
210 CHAPTER V: DISCUSSION AND CONCLUSIONS, LIMITATIONS, AND RECOMMENDATIONS The purpose of this study was to examine the discourse created between a teacher and students in eight mathematics classrooms with ELLs present. The study was an attempt to shed light on current practices esta blished in these classrooms and some of the areas where improvements need to be made to increase the mathematics learning potential of ELL students. Furthermore, this research aimed to provide some information about the impact of student sÂ’ and teachersÂ’ cultural and linguis tic backgrounds upon studentsÂ’ experiences in learning mathematics. The participants of the study were eight teachers and their mathematics classes from two urban U.S. public high schools in the Southeast, with diverse student populations with ELLs from various backgrounds. Discussion and Conclusions In analyzing the data to answer question 1, the results of this study indicated that the teachers changed their pa tterns of discourse due to the mere presence of ELL student(s) in the classr oom, irrelevant of the number of such students present. These observations are consistent with Rhin e (1995a, 1995b, 1999) and DavidenkoÂ’s (2000) findings, who also reported that teachers tend to teach differently when ELLs are present in a group. However, Rhine further reported that teachers often linked the lack of English proficiency to a similar lack of mathema tical knowledge or unde rstanding, and they
211 tended to underestimate the ELLsÂ’ performance. Such teachersÂ’ behavior Rhine related to teachersÂ’ limited understanding of ELLsÂ’ math ematics learning. Davidenko also reported that the teachers often assu med that the ELL students could not handle word problems or discussions in mathematics in English because of their limited proficiency in English. Thus, according to Davidenko, the teachers tended to reinforce computational skills and instrumental learning (learning experiences involving reinforcement of good behavior). In this study, teachers were observed whose e xpectations of their ELLs were both similar and different, in some respects, from thos e reported in Rhine and DavidenkoÂ’s study. Some commented on the limited English abilities of their students and related that to similarly limited mathematical abilities, whil e other teachers clearly stated that while their ELL students might not be very fluent in English yet, they are very motivated students and have good prior knowledge in mathematics. In analyzing the data to answer questi on 2, i.e., to understand to what extent mathematics teachersÂ’ experiences and ESOL endorsement relate to their patterns of discourse when teaching mathematics to cla sses with ELL students present, inconclusive results were observed. More spec ifically, the results of this study did not establish an Â“optimalÂ” learning experience for the ELLs in th e classes of teachers with the most years of teaching experience or having an ESOL e ndorsement for a longer time. Actually, data from the frequency count of the strategies used during the 20-minut e recorded sessions (refer to Figure 17) revealed that the novice teachers frequently used almost the same strategies as their more experienced co lleagues didÂ—more specifically, Figure 17 indicates that the teachers (with slight diffe rences) utilized most often strategies 12 ( use
212 of different questioning techniques ), 14 ( provide feedback ), and 1 ( use of slower and simpler speech ). However, additional qualitative analysis of the type of modifications to their speech (strategy 1a to d), of the type of questions they asked (strategy 12a to d), and the provision of information of higher cognitive demand acco rding to BloomÂ’s Taxonomy (strategy 22a to f) indicated that even t hough all teachers generally needed improvement in using these strategies, the more experi enced teachers who also had completed their ESOL endorsementÂ’s requirement a long time prior to the observations were applying those strategies to a fuller extent. On the ot her hand, despite the fact that they created opportunities for their ELL student s to participate in the math ematics discourse, they still did not ask enough questions which could provi de the ELLs with opportu nities to justify and explain their conclusions and, consequent ly, expand on their learning of mathematics and English. These observations seem to be consistent with the observations in other studies investigating classroom discourse (Bla nton, Berenson, & Norwood, 2001; Brenderfur, & Frukholm, 2000; Nathan, & Knuth, 2003; Renne, 1996). For example, in Brenderfur and FrukholmÂ’s study the two teachers subject to inve stigation were similar in age, attended the same mathematics methods class that prom oted discourse, but wh en assigned to teach in the same school employed different teaching practicesÂ—one encouraged communication while the other used a teach er-centered approach. In RenneÂ’s (1996) study, the teacher initiall y attempted to shift the discussi ons towards one that is more student-centered and to incor porate studentsÂ’ questions a nd initiatives. However, the teacher often converted the st udent initiatives to teacher initiatives and consequently
213 detoured the communications to the traditi onal initiation-reply-evaluation (IRE) sequence wherein the teacher initiates (with a questi on or statement), a student responds, and the teacher evaluates the studentsÂ’ response (verba lly or by a gesture). Further investigations in both studies revealed that the observed differences in teaching patterns could be attributed to the teachersÂ’ initial belief s and disposition toward mathematics and its teaching and learning. On the other hand, Nathan and Knuth (2003) who analyzed one teacherÂ’s patterns of discourse on a more genera l level over a period of tw o school years, reported the following observed change. During the first year the teacher facilita ted teacher-central interactions, but during the s econd year the teacherÂ’s author ity was less evident, and Â“a star patternÂ” emerged. Blanton et al.Â’s ( 2001) study contributes to the notion that Â“a teacherÂ’s developing practice is inherently linked to the social dynamics of the classroomÂ” (p. 228). However, as Renne (1996) also indicated, a lack of details about how to implement discussions, time constraints to complete the course or prepare the students for standardized state tests, the numbe r of students in the cl ass, and the struggle for maintaining discipline were also found to be influential factors in the observed teacherÂ’s behaviors. The multiple factors presented in these studies offer a glimpse as to why a direct correlation between the teachersÂ’ patterns of discourse and their years of teaching experience and years from comple tion of their ESOL endorsement was not found in this study. In analyzing the data to answer questi on 3, i.e. how teachersÂ’ own linguistic and cultural backgrounds affect their patterns of discourse when teaching mathematics in English to classes with ELL stude nts, the results of this study are consistent with those of
214 Cahnmann and RemillardÂ’s (2002) study. In th is study, the researchers indicated that even though it might be beneficial for the teach ers to have a similar cultural or linguistic background to that of their students, this is not a decisive factor in providing equal mathematics experiences to all students. Their study also indicated that all mathematics teachers could use some ideas from resear ch and incorporate culturally relevant instruction in mathematics to diverse student populations. The eight teachers in this study did not utilize strategy 21 (i .e., provide opportunities for st udents to share experiences and build up on personal or cultural-sp ecific knowledge while problem-solving in mathematics and thus build cross-cultural knowledge). Data from the frequency count of the strategies used during the 20-minute re corded sessions (refer to Figure 17) also reveals that some teachers utilized relatively more frequently strategy 22 (i.e., provided their students with content specific, enriched information, thus exhibiting equally high expectations from ELL a nd non-ELL students). In analyzing the data to answer question 4 Â— what patterns of discourse teachers use when ELL students are present in th e mathematics classroom and/or what adjustments to teacher talk or modifications of instruc tions are observed, the present study has reported that beside s above-discussed frequent use of strategies 12 ( use of different questioning techniques ), 14 ( provide feedback ), and 1 ( use of a slower and simpler speech ), the next strategies more often utili zed by the eight teache rs in the sample (with small exceptions) were strategies 4 ( use of repetitions or paraphrasing of teachersÂ’ or studentsÂ’ statements ), 7 ( performing comprehension checks throughout the lesson ), and 6 ( use of clarification of di rections or assistance in executing a mathematical task ) (See Figure 17). The reported findings are consistent with those of Long (1981, 1983),
215 who also found that native speakers (NS) do use more modifications to the input when they interact with nonnative speakers (NNS), as opposed to when they interact with native speakers. Such modifications, according to LongÂ’s studies, include more frequent use of selfand other-repetitions, slow er speech patterns, comprehension and confirmation checks, and explanations. According to Long, the purpose of such modifications is to improve the dialog a nd repair the discourse when troubles in conversations have already occurred. Resear ch in the field of mathematics classroom discourse indicates that teachers improved ELLsÂ’ participation in discussions by using Â“revoicingÂ” (reformulation of st udentsÂ’ statements using formal mathematical terms) and by asking the students to paraphrase their statem ents in order to clarify their meanings (Moschkovich 1999, 2002), or by facilitating a computer-based dynamic instructional environment in which small-group discus sions are encouraged (Brenner 1998; and Moschkovich, 2002). Although six of the teachers fr om the sample frequently utilized the strategy of Â“revoicingÂ” (in th e study referred as strategy 4), and three of the teachers taught Algebra I employing computer-assisted instruction, most of the teachers rarely used small group work. Only two or three of the teachers (Refer to the cases of Mr. Davison, Ms. Cortez, and occasionally Ms. Brown) more often exposed their students to classroom arrangements that facilitated small group wo rk or partner discussions (strategy 19). Research in the field of teachi ng ELLs pointed out the importance of Â“understanding studentsÂ’ cultural perspec tives and backgrounds [because it] might provide insights about behavior s and reactions to instructional approa chesÂ” (Kersaint, Thompson, & Petkova, 2008, p. 64). Furthermore, re search indicated th at Â“when students
216 experience the mathematics in a classroom as not relating to them or their culture, they might feel invisible and unconnected with the contentÂ” (Davidson & Kramer, 1997, p. 139). Moreover, knowing what a student knows about a t opic also helps teachers deal with misconceptions. Frequently students ha ve incorrect background knowledge that can become a powerful impediment to learning. Eliciting studentsÂ’ prior knowledge about a topic helps bring to light misunderstandings, simplistic knowledge, or flawed interpretations. Once brought to light, we can help students repair misconceptions with accurate information. (Santa, Havens, & Valdes, 2004, p. 7) Thus, in light of previous research, th is study furnishes key insights into what improvements in the current teaching practi ces could be implemented in order to encourage ELL students to become active le arners and participants in mathematics classroom discourse by illuminating, for exam ple, that in practice many teachers do not p rovide enough opportunities for students to share experiences and build up on personal or cultural-specific knowledge (i.e. lack of utilizing strategy 21). Research (Goodell, & Parker, 2001) also poi nted out that in order for ELLs to construct their own knowledge in both Englis h and mathematics Â“the teacher must be the facilitator, helping students to construct their own knowle dge by establishing learning situations in which this is possible, for example, through the use of hands-on manipulatives, whole-class disc ussion, group discussion, or pres entation of project workÂ” (p. 419). Research (Campbell & Rowan, 1997) also indicated that in order for ELL students to move to a more advanced level of English language fluency ( speech
217 emergence or intermediate speech ) and cognitive development, they need to be asked more often higher order questions and thus become more equa l partners in the classroom discourse. Santa, Haves, and Valdes (2004) visualized ve ry graphically the following situation: Most of us will remember how it feels to be a student in a classroom dominated by teacher talk and interrogation. The teacher asks the questions. One-by-one, students reel off answers until someone h its the correct one. The teacher remains the sole evaluator and controller of comprehension. Gazden (1988) calls this model of discourse IRE: th e teacher initiates (I) talk by aski ng a question; a student responds (R); and th e teacher evaluates (E) the response. (p. 55) This study reveals that even though the teachers often asked their ELL students questions and thus involved them in classr oom discussions, they did not utilize the full range of questioning techniques available. Most of the quest ions that they asked were Â“yes/noÂ”, Â“either/orÂ” questions or Â“required one word or list of word responsesÂ” (strategy 12a and b). Teachers were found to not p rovide enough opportuniti es for students to enhance both their linguistic and mathematics development by being asked to categorize, predict, explain, justify, or criticize a pproaches to solving mathematical problems (i.e., lack of use of strategies 12c and d, and 22e and f) and did not expose the students to different classroom arr angements such as using group discussions, and hands on activities (i.e., not very often u tilizing strategy 19). Research in classroom discourse (San ta, Havens, & Valdes, 2004; Tomlinson, 2001) underscores the necessity for creating a mathematical classroom environment in which student talk (including ELL studentsÂ’ talk), rather than teacher talk becomes
218 central. Such student-centered discussions enha nce comprehension, facilitate higher-level thinking and problem solving, and improve communication skills (Santa, Havens, & Valdes, 2004). Furthermore, based on previ ous research and current findings, the study indicates that whereas some ELL students can be challe nged by what seems to be a Â“simpleÂ” question according to a non-ELL teach er, Â“all students need to be accountable for information and thinking at high le velsÂ” (Tomlinson, 2001, p. 104) and be asked various types of questions. Teach ers can vary their questions to ensure that they are more open-ended and require explana tions and justifications of answers. By encouraging the students to build upon one anotherÂ’s answers and varying their quest ions appropriately, teachers can Â“nurture motivation though su ccessÂ” (Tomlinson, 2001, p. 104) and in turn become more successful in accommodating their ELL students in the mathematical classroom discourse. Limitations There are two limitations to this study. First, the small sample of high school teachers/participants in the study elicited a qualitative non-relational analysis of the collected data and prohibited th e use of significance tests such as chi-square. In effect, the generalizations from this study are limited in scope. However, as Wood and Kroger (2000) pointed out, Â“because the focus of discourse analysis is language use rather than language users, the critical issue concerns the size of the sample of discourse (rather th an the number of people) to be analyzedÂ” (p. 80). In this study the Â“discursive movesÂ” of eight teachers were analyzed during three 20minute video-recorded classroom sessions, wh ich actually amasses to analyzing very large samples of discourse during the total of 24 video-recorded sessions. As a result, this
219 study involved the analysis of much larger samples of language use than the sample of language users might otherwise indicate. T hus, as Wood and Kroger state Â“[t]he most likely problem for the analysis is that the samp le is too large rather than too smallÂ” (p. 80). They continue: Â“the question about num ber comes down to having sufficient number of arguments of sufficient quality and having sufficient data for those arguments to be well groundedÂ” (p. 81). Therefore, by providing thick descriptions of each case study and thus giving the reader opportunities to judge for him/her-self, the study satisfies its main aim: to shed light on current practices es tablished in the mathematics classrooms under scrutiny, and it illuminates the areas where improvements need to be made. However, even though this study provides th ick descriptions of the data collection and analysis procedures, its claims are still subject to the facets described by Lincoln and Guba: While generalizations are constrained by facts (especially if the facts are the particulars from which the generalization is induced), there is no single necessary generalization that must emerge to account for them. There are always (logically) multiple possible generalizations to acc ount for any set of particulars, however extensive and inclusive they might be. (p. 114) This leads to noting the second limitation of this res earch. As Lincoln and Guba (1985) pointed out, Â“naturalistic inquiry operates as an open system; no amount of member checking, triangulation, persistent observation, auditing, or whatever can ever compel; it can at best persuade Â” (p. 329). Thus, this studyÂ’s criteria for trustworthiness are also open-ended.
220 However, by abiding to the five major techniques proposed by Lincoln and Guba (1985, p. 301) the study is made more persuasive. More specifically, this study is developed, carried out, and described with consistently taking into account the naturalistic inquiryÂ’s criteria for trustworthiness, as expr essed in GubaÂ’s new terms: credibility (as an alternative to internal validity ), transferability (as an alternative to external validity ) dependability (as an alternative of reliability ) and confirmability (as an alternative of objectivity ). (For more details of how exactly the criteria for trustworthiness were sa tisfied, refer to the end of the methodology section of this study Â– Chapter III) Recommendations for Further Research This study furnished valuable insights into the classroom discourse and teacher talk influences on ELLsÂ’ mathematics experiences. The findings lead to further questions that future research can seek the answers to: 1. What changes (if any) in the patterns of mathematics teacherÂ’s discourse would be observed if the study is carrie d over longer periods of time and with a larger teacher and ELL samples? 2. What effects do the cultural and linguis tic backgrounds of the ELLs present in the mathematical classroom have on the teacherÂ’s choice of adopted discursive strategies (i.e., do teachers adopt strategies that differ in accordance with the ELLsÂ’ backgrounds)? 3. Which teaching strategies are most effective in teaching mathematics to ELLs from specific cultural an d linguistic backgrounds?
221 4. What are the effects of long-term in tervention programs offered by teacher development programs to aid teachers in teaching mathematics to classes where ELL students are present? 5. Would the results be different with a different age group sample (such as elementary and middle school students)? This inquiry and any future research as suggested here coul d contribute to the collected knowledge in the field of teachi ng mathematics to diverse classrooms with many ELLs present Â– as is the current a nd emerging situation in U.S classrooms. Findings from such research and recommendations for improvement can directly assist decision-makers to implement the necessa ry changes through cr iteria changes for teachersÂ’ certification programs and/or impr oving opportunities for teacher education and teacher development programs. Furthermore, the interventions suggested by such research can be elaborated in the daily pract ices of the mathematics teachers and can help them function effectively in diverse mathema tics classroom settings, particularly when ELL students are involved.
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232 Appendix A: Teacher Talk Test (TTT) Form 1 Strategies Sample statements Frequency Total Sample teacher statements I. Â“VocalÂ” Strategies: 1. Use of a slower and simpler speech Â—shorter sentences (caregiver speech) to adapt her/ his speech to the appropriate level of ELL students present (pre-production, early production, speech emergence, and intermediate fluency): a) Pre-production Since this is socalled Â“silence period,Â” the teacher should use simplified speech, i.e. simple commands and shorter sentences when explaining things. b) Early Production At this stage the students are just beginning to experime nt with the language, and thus at this stage it is inappropriate to correct errors in grammar and pronunciation. Teachers need to model/demonstrate correct responses both in mathematics and English. c) Speech Emergence At this stage teachers should begin the presentation of new concepts using advance organizers and at the same time focus the teacher-talk on key concepts and use their talk to lead the students to small group work and hands on activities d) Intermediate Fluency The teacher talk should foster conceptual understanding and expanded literacy through content 2. Use of (fewer) idioms and slang words from the mathematics vocabulary, or if used a proper explanation (or visual representation) is provided Right-angled triangle (Unaware that the word right here refers to a particular type of triangle a student might think that there are left-angled triangles) absolute value GCD (greatest common divisor) If and only if 3. Use of synonyms that can be used in the description of mathematical terms and that will help the student better understand the concept behind them greater (bigger) less (smaller) addition (plus) Subtraction (minus) Congruent (equal) 4. Use of repetitions or paraphrasing of his/her statements or asking students to repeat or restate them, especially when important concepts in mathematics are formulated This figure is a parallelogramÂ… The opposite sides in this figure are parallel Â…Juan, would you repeat, a parallelogram is what? In other wordsÂ… 5. Use of change of tone, pitch, and modality to convey better comprehension Change of pitch When a word or phrase that carries the greatest degree of stress in a sentence is said with increased loudness. Change of modality (speaking mode, grammatical mode, instructional mode) 6. Use of clarification of directions and assistance when specific mathematical task or activity is posed for execution Here is what you need to doÂ… This is another way to do thisÂ… 7. Check for comprehension throughout the lesson Trung, do you understand what the next step is?
233 Appendix A (Continued) 8. Identify subject specific and important lesson vocabulary and provide context embedded examples, pictures, or models. Exponent Radical Etc. 9. Start a lesson with a review of related concepts LetÂ’s see what we have learned aboutÂ…yesterday 10. Conclude a lesson with a summary of the key concepts Who would summarize Â… 11. Involve students in mathematical discussions and problem solving What do you think? What would you suggest? How do you know this is true? Tell me more aboutÂ…? Consider thisÂ… Who would explainÂ…? (or just call a student by name) II. Questioning Strategies: 12. Use different questioning techniques, sensitive to the level of ESOL of the students, or their stages of Second Language Acqusition (as summarized by Linda Ventriglia (1982): a) Pre-production point toÂ…; find theÂ…; is this a/anÂ…; etc. Who wants theÂ…? b) Early Production yes/no questions (Is this a square?) Either/Or questions One-word response (What variable is this?) General questions that encourage lists of words (What signs of operations do we use?) c) Speech Emergence Why? How? Tell me aboutÂ…? DescribeÂ… d) Intermediate Speech What do you recommend? What is your opinion....? What would happen ifÂ…? Compare/contrast How are these Â…similar or different? CreateÂ… 13. Use wait-time techniques after posing a question (measured in sec) Provide at least three seconds of thinking time 14. Provide feedback Well done; Hm-m; I see; I agree III. Enhancement to teacher talkÂ’s strategies: 15. Use of gestures, facial expressions, eye contact, (at the same time showing awareness of their culture-specific appropriateness), or demonstrations to enhance comprehension Gestures Facial expressions Eye contact Special proximity 16. Use charts, graphic organizersÂ— (draw) Venn diagrams, tree diagrams, time lines, semantic maps, outlines, etc. 17. Use of a variety of visual or auditory stimuliÂ—(show) Transparencies, pictures, flashcards, models, etc. [The following strategies might be lesson dependent] 18. Use of technology to enrich a concept presentation calculators, computers, Internet, videos, overhead projectors, Power Point presentations, Mathematics application softwareÂ—Geometers' Sketchpad, spread sheets, etc.
234 Appendix A (Continued) 19. Expose students to different classroom work arrangements, such as using cooperative groups or partner discussions Small group work Dyads (pair work and discussions) Collective discussions (scaffolding) Games 20. Provide students with alternative forms of assessmentÂ—portfolios, vocabulary banks, oral presentations, writing or reading in mathematics, etc. Portfolios Vocabulary Banks Oral Presentations Journal writing Research 21. Provide opportunities for students to share experiences and build up on personal or culturalspecific knowledge while problem solving in mathematics and thus building cross-cultural knowledge Tell me what you know aboutÂ… 22. Provide students with content specific, enriched information, thus exhibiting equally high expectations from LEP and non-LEP students. Moves to the higher level of cognitive demand according to BloomsÂ’ Taxonomy: a) Knowledge b) Comprehension c) Application d) Analysis e) Synthesis f) Evaluation Define, describe, matchÂ… Explain, give example, paraphraseÂ… Modify, prepare, relate, Â… Distinguish, outline, identifyÂ… Categorize, predict, design Â… Justify, criticize, explainÂ…
235 Appendix B: Pre-observation Teacher Questionnaire 1. Name:___________________________________________________ 2. Gender: M_____ F_____ 3. Age:_____ 4. Years of teaching experience:_________ 5. Length of time teaching Algebr a I :________ ____________ _______ 6. Type of teaching certification: Te mporary:____ Permanent: _______ 7. Are you Math certified? a. Yes_____ What level? ___________________________________________________ b. No _____ 8. Have you ever had a Math Method course: a. Yes______ What type? __________________________________________________ b. No ______ 9. ESOL endorsement: a. Yes______ Year of completion_________ What type? _________________________ How was it obtained (coursework, inservice points, additional courses, etc.) _____________________________________________________________________ b. No______ 10. List how many ESOL students you have in each of your mathematics classes and their level of ESOL (if known). To what extent can you connect the studentsÂ’ level of ESOL with the stages in Second Language Acquisition Â—pre-production, early production, speech emergence, and intermediate fluency (See the list of definitions for each category): ID Student Name In which class/period ESOL level (if known) Stage in SLA Comments 11. Is English your native language: a. Yes______ b. No______ c. Native Language______________________________________________________ 12. Do you speak any other languages other than your native language? a. Yes______ If yes, specify which language(s) and grade your ability in each: Language: ___________ Reading: ___fluent ___ limited ___ not fluent Writing: ___fluent ___limited ___not fluent Speaking: ___fluent ___limited ___not fluent Language: ___________ Reading: ___fluent ___limited ___not fluent Writing: ___fluent ___limited ___not fluent Speaking: ___fluent ___limited ___not fluent b. No_______ Comment on any concerns you have in your experi ence with teaching mathematics to ESOL students; any positive or negative experiences; recommendations fo r improvement; etc. (Write on additional paper if needed) ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ Thank you for completing this questionnaire!
236 Appendix C: Post-observation Teacher Questionnaire Teacher Talk Test (TTT) Form 2 No: Strategies: Evaluate the extent to which you use the following strategies when ESOL students are in your classroom: (use a checkmark) How Often This Strategy is Used?Â— Rate Using a Frequency Scale from 1 to 5, with 5 as most frequent 1Â—Never 2Â—Rarely (1 or 2 times a month) 3Â—Sometimes (1 or 2 times a week) 4Â—Usually (3 or 4 times a week) 5Â—Always Yes No Needs Improvement I. Â“VocalÂ” Strategies: 1. Use of a slower and simpler speech 2. Use of fewer idioms and slang words 3. Use of synonyms 4. Use of repetitions or paraphrasing 5. Use of changes in tone, pitch, and modality 6. Use of clarification of directions 7. Comprehension checks 8. Identify subject-specific vocabulary and provide context-embedded examples, pictures, or models 9. Start a lesson with a review of related concepts 10. Conclude a lesson with a summary of the key concepts 11. Involve students in mathematical discussions and problem solving
237 Appendix C (Continued) II. Questioning Strategies: 12. Use different questioning techniques that are sensitive to the level of ESOL of the students, or their stages of Second Language Acquisition a) pre-production Â—point toÂ…; find theÂ…; is this a/anÂ…; etc b) early productionÂ— yes/no questions; either/or questions; one-word or two-word responses; general questions that require a lengthy response; c) speech emergence Â—Why? How? Tell me aboutÂ…? DescribeÂ…; d) intermediate speech Â— What do you recommend? What is your opinion....? What would happen ifÂ…? Compare/contrastÂ…; CreateÂ… 13. Use wait-time techniques after posing a question 14. Provide feedback III. Enhancement to teacher talk strategies: 15. Use of gestures, facial expressions, eye contact, or demonstrations 16. Use of charts, graphic organizersÂ—Venn diagrams, tree diagrams, time lines, semantic maps, outlines, etc. 17. Use of a variety of visual or auditory stimuli: transparencies, pictures, flashcards, models, etc.
238 Appendix C (Continued) 18. Use of technology 19. Expose students to different classroom work arrangements, such as using cooperative groups or partner discussions 20. Provide students with alternative forms of assessment 21. Provide opportunities for students to share experiences and expand on personal or cultural-specific knowledge while solving problems in math 22. Provide students with content specific, enriched information Please comment on why you chose to use the te aching practices that you identified. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ __________________ Thank you for completing this questionnaire!
239 Appendix D: Questionnaire for ELL Students [* This questionnaire could be modified in a ve rsion which is more student-friendly, or in the ELLsÂ’ native languages, if possible (or if ELL speci alists or native speakers of that language are available and could contribute as translator s, the interview could be only oral as the interview session is audio and/or video-recorded).] 1. Name:_________________________________________________________________ 2. Boy_____ Girl_______ 3. How old are you? ______ 4. Where were you born? ___________________________________________________ 5. What is your first language? _________________________________________________ 6. What is your momÂ’s first language? ___________________________________________ 7. What is your fatherÂ’s first language ? ________________________________________ 8. What languages do you speak at home? About how much of the time do you use each language at home? 1. ___________-_________% 2. ___________-_________% 3. ___________-_________% 9. How do you describe your knowledge of English in speaking, reading, and writing? (Do you only speak English? Or can you also read English? Can you write in English? Tell me more, please.) _______________________________________________________________________ 10. Tell me about your previous mathematics classes and grades. _______________________________________________________________________ 11. How well do you like Math? _______________________________________________________________________ 12. How well does your mother like math? _______________________________________________________________________ 13. How well does your father like math? _______________________________________________________________________ 14. Think about the mathematics classes where I came to visit or where the video camera was being used. How much did you participate in class? How often did you ask questions? If you didnÂ’t ask questions, why not? _______________________________________________________________________ 15. Was the lesson where I came to visit or wher e the video camera was being used easy or hard for you? [Easy/Hard] Why? What part(s)? Tell me more, please. _______________________________________________________________________ 16. Now, think about your mathematics class this year. For all the lessons, even the ones I did not observe, please fill out the TTT Form 3 for the things your teacher might have done. Your teacher might have used some, but not all of the things that are listed. Thank you!
240 Appendix E: Post-observati on Student Questionnaire Teacher Talk Test (TTT) Form 3 No: Strategies: Comment on your mathematics teacherÂ’s use of the following strategies (use a checkmark) How Often Is This Strategy Used?Â— Rate using a Frequency Scale from 1 to 5, with 5 as most frequent: 1 Â—Never, 2 Â—Rarely (1 or 2 times a month), 3 Â—Sometimes (1 or 2 times a week), 4 Â—Usually (3 or 4 times a week), 5 Â—Always Yes No I. Â“VocalÂ” Strategies: 1. Use of slow and simple talk; short sentences 2. Use of few slang (jargon) words (words connected in sentences or groups typical for the country, region, or technical terms) 3. Use of similar words 4. Use of repetitions in same or almost the same words 5. Use of changes in her/his voice to louder, higher, faster, etc. 6. Use of explanations what you need to do more than once so you would understand 7. Does the teacher ask/check if you understand? 8. Does the teacher write lesson vocabulary words, give examples, or show pictures? 9. Does the teacher start a lesson with a review of (related) similar ideas? 10. Does the teacher ask students to tell what they learned today? 11. Does the teacher ask students to talk and explain their solutions?
241 Appendix E (Continued) II. Questioning Strategies: 12. Does the teacher use any of these types of questions? How often? a) pre-production Â—point toÂ…; find theÂ…; is this a/anÂ…; etc b) early productionÂ— yes/no questions; either/or questions; one-word or two-word responses; general questions that require a lengthy response; c) speech emergence Â— Why? How? Tell me aboutÂ…? DescribeÂ…; d) intermediate speech Â— What do you recommend? What is your opinion....? What would happen ifÂ…? Compare/contrastÂ…; CreateÂ… 13. Does the teacher give you time to think before you need to answer a question? 14 Provide feedback III. Enhancement to teacher talkÂ’s strategies: 15 Does the teacher use her hands or face, or look at students when talking? 16. Did the teacher draw pictures to explain or group ideas? 17. Does the teacher show pictures, cards, or small models to explain words or how to do math problems?)
242 Appendix E (Continued) 18. Are calculators, projectors, or computers used? Or other technology? 19. Does the teacher let you work in different waysÂ— in groups with other students or by 2? 20. When given a grade, is it only from a test written on paper, or you are asked to do different things? If yes, give some examples, please. 21. Does your teacher ask you to talk and give examples from your country or family when solving math problems? 22 Does your teacher explain most of the difficult parts of the lesson so that you can do most of the homework on your own? Thank you!
ABOUT THE AUTHOR Mariana Petkova earned her Bachelor of Science degree in Mathematics and Computer Science from the University of Plovdiv, Bulgaria. Following this, Mrs. Petkova completed her gradua te level research and qualif ying exams in pursuit of a Candidate of Science Degree in Computer Science at the Bulgarian Academy of Sciences, Institute of Mathematics, Sofia, Bulgaria. Currently, she is an International B accalaureate (IB) Mathematics Higher Level (HL) teacher at King High School in Tampa, Fl orida. Prior to this, Mrs. Petkova was a Math Magnet Teacher at Jefferson High School, where she taught students in AP Calculus, Algebra I Honors. Mr s. Petkova has also worked for the Bulgarian Academy of Sciences, the Huntington Learning Center and Youth Environmental Services. A supporter of a student-centered instructional approach, she strives to provide time for discussions, problem solving, and activities in ev ery class period.