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PAGE 1 1 THE INSTRUCTIONAL PRACTICES AND PERSPECTIVES OF HIGHLY EFFECTIVE TEACHERS OF BLACK STUDENTS: CASE STUDIES FROM MATHEMATICS CLASSROOMS By KARINA KAWALL REYBITZ HENSBERRY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF T HE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 Karina Kawall Reybitz Hensberry PAGE 3 3 To Matthew PAGE 4 4 ACKNOWLEDGMENTS My doctoral studies and this dissertat ion are the result of the encouragement and contributions of my family, friends, and colleagues who supported me throughout this process. I want to express my gratitude to each of them, for without their belief in my abilities and constant cheering from th e sideline, I would not have accomplished this immense task. First and foremost, I wish to thank my advisor and chair, Dr. Stephen Pape. His mentorship, support, and friendship have been invaluable to me, and I do not have the words to express the gratitud e I feel for his guidance throughout this journey. I also want to thank my dissertation co chair, Dr. Dorene Ross, as well for her tireless efforts and encouragement. I am a better writer and scholar thanks to Drs. Pape and Ross. Additionally, I wish to th ank my committee members, Dr. Tim Jacobbe and Dr. Walter Leite, for their expert advisement, time, and friendship. I would like to thank Dr. Donald Pemberton and Dr. Alyson Adams of the Lastinger Center for Learning. Dr. Pemberton and his ability to dream big convinced me to pursue a doctorate degree, and I am thankful for the generosity and encouragement he and Dr. Adams bestowed upon me throughout my studies. I would not have survived graduate school without my graduate student support network. The list of the students to whom I am grateful is far too long to include here, but in particular I wish to acknowledge fellow mathematics education doctoral students Sherri Prosser, Yasemin Sert, and Anu Sharma. I also wish to acknowledge my dear friend and colle ague, Dr. Jonathan Bostic, for his unwavering support, encouragement, and belief in my talents. Jonathan was the first friend I made when I began my studies PAGE 5 5 patience wit h my never ending questions, and his friendship. My family has also been a great source of support throughout this journey. My parents, Andrew and Mrcia Reybitz, have always been my biggest cheerleaders and always encouraged me to pursue my dreams, no mat ter how big. Additionally, my sisters, Paula Reybitz and Tarsila Crawford, have been a constant source of friendship and support, and I love them dearly. Finally, my in laws, Robert and Phyllis Hensberry, have provided me with continuous encouragement for which I am grateful. I also wish to thank my participants, Ms. J and Ms. W, and their students for opening their classrooms to me. They always made me feel welcome and gave up a great deal of their time to make this dissertation possible. I appreciate thei r willingness to participate and their honesty with me. Finally, and most importantly, I wish to thank my dear husband and best friend, Matthew Hensberry. I am absolutely certain that this dissertation would not have been possible without him by my side e very step of the way. Matthew pushed me to achieve, encouraged me when I struggled, and never failed to communicate his belief in my including tending to the house a nd caring for our beautiful daughter, Amia, without complaint so that I could focus all my time and energy on my research. No one has done more for me throughout this journey than Matthew has, and I am eternally grateful that I have him by my side. I love him more than words can ever express. PAGE 6 6 TABLE OF CONTENTS page ACKNO WLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ .......... 10 LIST OF FIGURES ................................ ................................ ................................ ........ 11 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 A Comment About Terminology ................................ ................................ .............. 15 Theoretical Perspective ................................ ................................ .......................... 16 Opportunity and Culture: Explanations of Failure ................................ ................... 17 Inequitable Access to Education ................................ ................................ ...... 18 The Home School Cultural Disconnect ................................ ............................ 20 Summary ................................ ................................ ................................ .......... 23 Overview of the Literature ................................ ................................ ....................... 24 Standards based Mathematics Teaching ................................ ......................... 24 Culturally Responsive Teaching ................................ ................................ ....... 28 Statement of the Problem ................................ ................................ ....................... 29 Research Questions ................................ ................................ ............................... 30 Structure of the Dissertation ................................ ................................ ................... 30 2 LITERATURE REVIEW ................................ ................................ .......................... 32 Standards based Mathematics Teaching ................................ ................................ 32 An Example: Integ rating the Standards Through Problem Solving ................... 35 Discourse ................................ ................................ ................................ ......... 37 Standards based Mathematics Teaching and Equity ................................ ....... 42 Culture, Mathematics, and Teaching ................................ ................................ ...... 53 Culturally Relevant Pedagogy ................................ ................................ .......... 54 Culturally Responsive Tea ching ................................ ................................ ....... 55 ................................ ................................ ...... 57 ................................ .......................... 59 Teacher student relationships ................................ ................................ .... 61 experiences ................................ ....................... 62 Supporting students to develop a critical disposition ................................ .. 65 Classroom environment ................................ ................................ ............. 66 Culturally responsive classroom management ................................ ........... 67 Instruction ................................ ................................ ................................ .. 68 Culturally Responsive Mathematics Teaching ................................ .................. 71 Culturally Responsive, Standards based Mathematics Teaching ........................... 78 PAGE 7 7 Conclusion ................................ ................................ ................................ .............. 86 3 METHODS ................................ ................................ ................................ .............. 90 Overview ................................ ................................ ................................ ................. 90 Methodology ................................ ................................ ................................ ........... 90 Procedure ................................ ................................ ................................ ............... 91 Participants ................................ ................................ ................................ ....... 91 Identification of participants ................................ ................................ ........ 91 Description of participants ................................ ................................ .......... 93 Setting ................................ ................................ ................................ ........ 94 Data Sources ................................ ................................ ................................ .... 95 Data Collection ................................ ................................ ................................ 95 Classroom observations ................................ ................................ ............ 95 Interviews ................................ ................................ ................................ ... 97 Summary of data collection ................................ ................................ ........ 98 Data Analysis ................................ ................................ ................................ ... 98 Establishing Trustworthiness ................................ ................................ ................ 100 Credibility ................................ ................................ ................................ ........ 100 Transferability ................................ ................................ ................................ 101 Dependability and Confirmability ................................ ................................ .... 101 Subjectivity Statemen t ................................ ................................ .......................... 102 Structure of the Cases ................................ ................................ .......................... 104 4 CASE ONE: MS. J ................................ ................................ ................................ 105 An Introduction to Ms. J and her Classroom ................................ ......................... 105 Goals for S tudents ................................ ................................ ................................ 107 Learn and Apply Mathematics ................................ ................................ ........ 107 Become Well mannered ................................ ................................ ................. 108 Care About School ................................ ................................ ......................... 109 Psychological Environment ................................ ................................ ................... 110 Cares Deeply for Students ................................ ................................ ............. 110 Classroom is a Community of Respect ................................ .......................... 114 Adopts an Attitude of High Expectations ................................ ........................ 115 Engages in Explicit and Consistent Classroom Management ........................ 118 Summary of Psychological Features ................................ .............................. 120 Teaching Mathematics ................................ ................................ .......................... 121 ......................... 121 Variations from the Typical Day ................................ ................................ ...... 129 Themes in Mathematics Teaching ................................ ................................ .. 131 Breaks down mathematics into easily followed procedures ..................... 131 Emphasizes correct answers ................................ ................................ ... 135 Addresses common errors ................................ ................................ ....... 137 Students compare answers with partners ................................ ................ 138 Allows students to struggle ................................ ................................ ...... 139 Teaches about key words and provides hints ................................ .......... 140 PAGE 8 8 Plays games to increase engagement ................................ ..................... 142 Makes problems relatable ................................ ................................ ........ 143 Supports struggling students ................................ ................................ .... 143 Summary of a Typical Day ................................ ................................ ............. 146 Influences on Teaching ................................ ................................ ......................... 146 Summary of Case One ................................ ................................ ......................... 148 5 CASE TWO: MS. W ................................ ................................ .............................. 150 An Introduction to Ms. W and her Classroom ................................ ....................... 150 Goals for Students ................................ ................................ ................................ 152 Build Mathematical Knowledge ................................ ................................ ...... 152 Become Problem Solvers ................................ ................................ ............... 153 Fee l Cared For ................................ ................................ ............................... 154 Psychological Environment ................................ ................................ ................... 155 Demonstrates Care for and Builds Relationships with Students ..................... 156 Builds a community of respect and honesty ................................ ............. 156 Identifies and overcomes barriers to meet high expectations .................. 158 Uses humor and sarcasm ................................ ................................ ........ 162 Gets personal ................................ ................................ ........................... 163 Grounds Classroom Management in Relationships ................................ ....... 166 Treats students as individuals ................................ ................................ .. 167 Avoids escalating issues ................................ ................................ .......... 168 Provides hints, humor, and sarcasm ................................ ........................ 170 Summary of Psychological Features ................................ .............................. 171 Teaching Mathematics ................................ ................................ .......................... 171 ................................ .... 172 Variations from the Typical Day ................................ ................................ ...... 181 Themes in Mathematics Teaching ................................ ................................ .. 182 Allows for multiple solution strategies ................................ ...................... 183 Emphasizes reasoning over rules ................................ ............................ 184 Engages students in c ooperative learning teams ................................ ..... 186 Provides multiple opportunities to succeed ................................ .............. 188 Supports students to build self confidence ................................ .............. 189 Prepares st udents for the EOC exam ................................ ...................... 190 Uses multiple assessments ................................ ................................ ...... 191 ................................ ................................ ..... 192 Influences on Teaching ................................ ................................ ......................... 193 Summary of Case Two ................................ ................................ ......................... 195 6 CROSS CASE ANALYSIS ................................ ................................ .................... 197 Introduction ................................ ................................ ................................ ........... 197 Culturally Responsive Teaching ................................ ................................ ........... 197 Goals for Students ................................ ................................ .......................... 198 Relationships ................................ ................................ ................................ .. 2 00 Insistence ................................ ................................ ................................ ....... 203 PAGE 9 9 Pedagog y ................................ ................................ ................................ ....... 207 Summary ................................ ................................ ................................ ........ 210 Standards based Mathematics Instruction ................................ ............................ 211 Sequence of Learning Activities ................................ ................................ ..... 211 Focus of Instruction ................................ ................................ ........................ 213 Discourse and Norms ................................ ................................ ..................... 216 Discussion ................................ ................................ ................................ ...... 218 Summary ................................ ................................ ................................ .............. 219 7 CONCLUSIONS ................................ ................................ ................................ ... 221 Discussion ................................ ................................ ................................ ............ 224 Implications ................................ ................................ ................................ ........... 229 Limitations and Suggestions for Further R esearch ................................ ............... 231 APPENDIX A INFORMED CONSENT FORM ................................ ................................ ............. 236 B FORMAL INTERVIEW PROTOCOL 1 ................................ ................................ .. 237 C FORMAL INTERVIEW PROTOCOL 2 ................................ ................................ .. 238 D FORMAL INTERVIEW PROTOCOL 3 ................................ ................................ .. 239 E INFORMAL INTERVIEW PROTOCOL ................................ ................................ 240 LIST OF REFERENCES ................................ ................................ ............................. 241 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 252 PAGE 10 10 LIST OF TABLES Table page 1 1 General comparison of attenti on to process in CCSSM and PSSM ................... 26 PAGE 11 11 LIST OF FIGURES Figure page 2 1 A model of culturally responsive math ematics teachin g ................................ ..... 73 2 2 A revised model of culturally responsive mathematics teaching ........................ 74 2 3 CRMT and Stand ards based instruction (Part 1) ................................ ................ 79 2 4 CRMT and Stand ards based instruction (Part 2) ................................ ................ 80 4 1 Handout: How to find the percent of change. ................................ ................... 126 PAGE 12 12 Abstract o f D issertation P resented to the Graduate School of The University of Florida in Partial Fulfillment of the Requireme nts for the Degree of Doctor of Philosophy T HE INSTRUCTIONAL PRACTICES AND PERSPECTIVES OF HIGHLY EFFECTIVE TEACHERS OF BLACK STUDENTS: CASE STUDIES FROM MATHEMATICS CLASSROOMS By Karina Kawall Reybitz Hensberry August 2012 Chair: Stephen J. Pape Cocha ir: Dorene D. Ross Major: Curriculum and Instruction The purpose of this dissertation was to understand how teachers who are successful with low achieving students of color living in poverty supported their students in learning mathematics. Standards base d instruction and culturally responsive teaching (CRT) have both been suggested as pedagogical approaches that may support traditionally underperforming students of color living in poverty to succeed in mathematics. Some research indicates, however, that t hese students may struggle with elements of standards based instruction such as the open, contextualized nature of problems and classroom discourses (Lubienski, 2002; Zevenbergen, 2000) This dissertation extends prior research by examin ing the potential o f both standard based instruction and CRT to support students to succeed mathematically. This collective case study examined the perspectives and instructional practices of two mathematics teachers identified as highly effective with students of color and whether their instruction aligned or did not align with standards based instruction and CRT. One seventh grade mathematics teacher and one high school Algebra I teacher were identified through a nomination process as highly effective with traditionally PAGE 13 13 un derperforming students of color. Data collection methods included observations, interviews, and the collection of documents. Data were analyzed using qualitative methods to identify themes both within and across the cases. Each case study describe s the te classroom, and the daily classroom practices and beliefs of the two teachers. The cross case analysis examined similarities and differences between teachers and how their beliefs and practices aligned or didn based instruction and CRT. Both teachers were found to have strong, caring and respectful relationships with and one teacher adopted a warm de also aligned with standards was procedurally based and adhered to a pedagogy of poverty (Haberman, 1991). Implications for these results as well as li mitations of this study and further research are discussed. PAGE 14 14 CHAPTER 1 INTRODUCTION The National Council of Teachers of Mathematics (NCTM, 2000) emphasizes that all students can, and should, learn mathematics Unfortunately, many American students are n ot learning at the level at which they are capable. Schools in America are there is a gap between the achievement of students of color and students living in povert y and the achievement of their white, middle class peers (Rothstein, 2002). This gap is especially evident in mathematics, where students of color and students who live in poverty are persistently placed into low track courses (Kelly, 2009) and perform at particularly low levels on standardized tests (Davis & Martin, 2008; Dewan, 2010). Such issues and their effects are problematic, partially because mathematics, and particularly algebra, is a gate keeping subject (Kelly, 2009; National Research Council [NR C], 1989; Schoenfeld, 2002; Thompson & Lewis, 2005). Students who do not perform well in mathematics are less likely to attend college and gain the skills mathematical or otherwise needed for success in our globalized society (Friedman, 2005; NCTM, 200 0; NRC 1989). In an era during which the economy is unstable (Harrington, 2010) and students of color are gaining in numbers (Banks et al., 2005), quickly becoming the majority in some parts of the nation (Dewan, 2010), the issue of the underachievement o f these students is persistent and unacceptable. This dissertation will explore the potential of both standards based mathematics instruction and culturally responsive teaching (CRT) for ensuring an equitable education for all students. Of particular inter est is whether these two teaching approaches intersect in the classroom. The study PAGE 15 15 described examined the teaching practices of successful secondary mathematics teachers of students of color living in poverty A Comment About Terminology In examining the i ssues underrepresented students face related to mathematics, cultural differences, and social class, I use the phrase students of color to describe non Asian students of color, which includes (but is not necessarily limited to) blacks, Latinos and Hispanic s, and Native Americans. Other terms often found in the literature, including minority and non white, imply a deficit perspective that I seek to avoid Similarly, students living in poverty and students from low income families are more appropriate phrases than poor, disadvantaged, or low class students. Finally, I recognize that labels are imperfect: not all students of color live in poverty not all students living in poverty are also of color and not all students in these categories struggle in school The life experience of each child is unique, and there exist many examples of people who are the exception and become quite successful despite the constraints placed on them because of their heritage or social class (consider, for instance, the cases of Ba rack Obama, Frank McCourt, and J.K. Rowling). Because children of color are more likely to come from low income families than white children (Rothstein, 2002) and because they often face similar challenges, students of color and those living in poverty wil l mainly be referred to jointly in this paper. Again, the intention and promising lines of research in education affecting their achievement in mathematics PAGE 16 16 Theore tical Perspective Vygotskian sociocultural theory emphasize s the learning process (Forman, 2003; Gee, 2008; Lubienski, 2002), something reform pedagogy does not inherently do. Some s tudents face challenges in discuss ion rich, standards based classrooms (Lubienski, 2000a, 2000b, 2002; Zevenbergen, 2000) that res into their instructional practices. According to Goodnow (1990, as cited in Forman, 2003) to learn may indicate an unwillingness to identify with a particular teacher; a mismatch among the motives, beliefs, norms, goals, and values of teacher s, students, and their ( p. 336). Gee (2008) additionally argues that social, emotional, and cultural issues must be attended to when considering how knowledge is acquired. For instanc e, for deep learning to occur, new knowledge must be integrated with old, but what of those learners in the same instructional setting who have different prior knowledge bases such as students with different cultural and economic upbringings ? Are they rec eiving the same opportunity to learn (OTL) example highlighted by Gee is the input/intake problem, which contrasts what one is exposed to with what one actually processes (i .e., learn s ). Not everyone tak es up the situation itself is what is causing the affective filter to rise, students will not intake the intended information. Furthermore, academic language builds o acquired vernacular language; it is the same with culture. Students have a culture they bring to school with them, and the school culture must build upon that. If school culture PAGE 17 17 vernacular culture, learning is enh anced. If it conflicts with or fails the vernacular culture of middle class st udents tends to overlap with school culture (Gee, 2008) Underrepresented populations (e.g., students of color and students who live in poverty) are often still very smart and capable, and they engage in poetic and complex discourse, yet they may struggle academically because their vernacular culture is in conflict with that of the school (Gee, 2008 ; Lubienski, 2000a, 2000b, 2002 ). In standards students in deep mathematical learning without also attending to the diverse knowledge and histories of their students. To date, and despite calls for socioculturally based research that addresses the role of culture within reformed classrooms (Gee, 2008; Lubienski, 2002), there is little literature t hat specifically addresses the interplay between culturally responsive and stand ards based mathematics teaching. This dissertation seeks to fill that gap. In the next section, some of the challenges students of color living in poverty face that may affect their academic success will be explored in further detail. Opportunity and Culture: Explanations of Failure Students of color and students who live in poverty often do not receive the benefits from schooling that their white, middle class peers receive (Ba nks et al., 2005; Brantlinger, 2003; Gutstein, Lipman, Hernandez, & de los Reyes, 1997). These students face disproportionately high suspension, expulsion, and incarceration rates (Books, 2007; Banks et al., 2005; Dewan, 2010; Tutwiler, 2007; Wald & Losen, 2007), causing OTL Additionally, these students drop out of school PAGE 18 18 more often (Banks et al., 2005; Books, 2007; Dewan, 2010; Tutwiler, 2007), with many personnel (Hondo, Gardiner, & Sapien, 2008). Th e lack of academic success faced by students of color and students living in poverty, and particularly their struggles in mathematics, are not the result, as some might suggest, of racial differences, incompe tence, laziness, disability, or uninvolved parents (Corbett, Wilso n, & Williams, 2002). Some authors (Burdell, 2007; Hinchey, 2004; Tatum, 1997) argue that c ontrary to popular opinion, America is not a meritocracy so that ot always result in higher levels of achievement Rather, there are certain political, societal, and belief structures in place in American schools that make high levels of mathematics achievement difficult for students of color and those living in poverty to attain (Brantlinger, 2003; Hinchey, 2004). These include inequitable access to a quality school Inequitable Access to Education Gaps in the mathematics achievement of students from low income families and students of color may result from inequitable OTL S chools in impoverished neighborhoods receive less funding because of the economic base of the neighborhoods in which they are located, and thus the schools are typica lly of lesser quality and lack adequate resources in these neighborhoods (Banks et al., 2005; Brantlinger, 2003; Flores, 2007). Additionally, they tend to be staffed by less experienced or less effective teachers (Banks et al., 2005), meaning students who need the most qualified teachers are the least likely to have them (Flores, 2007). Parents PAGE 19 19 financial means to provide the tutoring, after school care, or even the sup plies needed to compensate for the low teacher quality and lack of school resources (Brantlinger, 2007). At the same time, teachers, who tend to be white and from the middle class (Banks et al., 2005; Burdell, 2007; Hagiwara & Wray, 2009), often hold low e xpectations failures rather than their successes (Thompson & Lewis, 2005). A major contributing factor to the low mathematics achievement of students regarding OTL is track ing systems. Tracking is the practice of grouping students of similar achievement levels together. D ecisions about tracking are often made based on standardized test scores on which students of color perform poorly (Dewan, 2010; Post et al., 2008; Rot hstein, 2002; Tutwiler, 2007) and which may be biased against them (Davis & Martin, 2008) as well as on factors unrelated to academics, such as behavior and race (Berry, 2005; Kelly, 2007). The result is that students of color, a nd particularly black stud ents, are disproportionately tracked into low, non college bound mathematics courses so that upper track courses are available predominantly for white, middle class students (Banks et al., 2005; Boaler, 1997; Kelly, 2009; Williams, 2000). L ow track course s do not provide students with the same opportunities to learn mathematics that they may have in higher tracks. The instruction in low level mathematics courses is typically oriented toward skills and procedures rather than concepts, invention, creativity, and experimentation (NCTM, 1999; Web b & Romberg, 1994; Weiss, 1994). There exist critiques of mathematics instruction that is skill oriented for all students, as this instructional approach does little to support deep conceptual mathematical knowledge (Bo aler, 1997; Davis & Martin, 2008; NCTM, 2000). Such a PAGE 20 20 skill oriented approach may be more problematic, however, for students of color than for white, middle class students. For instance, t in lower track courses may contribute to the oppression of underrepresented groups (Davis & Martin, 2008). In fact, teachers of struggling students have been found to develop conceptual understanding and mathematical thinking (Watson, 2002). Thus, once placed in a low track, students will likely never move out of it (Balfanz & Byrnes, 2006; Kelly, 2009; Schoenfeld, 2002), but will instead fall even further behind (Balfanz & Byrnes, 2006). This becomes particularly problematic at the high school level, where algebra is typically a requirement for graduation (e.g. California Department of Education, 2010; Florida Department of Education, 2007) and is a key component of the recently adopted Common Core St ate Standards for Mathematics (CCSSM) (Common Core State Standards Initiative [CCSSI], 2010). Algebra serves as a gatekeeper to higher level mathematics courses, overall school success (Kelly, 2009), high school graduation, college attendance, technologica l literacy (NRC, 1989; Schoenfeld, 2002) and mathematics and science careers (Thompson & Lewis, 2005). A s a result, the policies and practices that exist in schools and which prevent some students from reaching their potential as learners of mathematics and particularly algebra may have long term negative consequences on their opportunities for success in school as well as in college and their future careers. Compounding this problem of inequitable OTL is the issue of culture, the topic of the next sect ion. The Home School Cultural Disconnect Tracking, discrimination, poorly funded schools and low expectations have all been cited as reasons f or the low achievement in mathematics of students of color and PAGE 21 21 students living in poverty. Even when these issues experiences, however, mathematics achievement is still persistently low for these students This may be because the cultural and racial identities of students of color and those living in poverty are not valued in most sch ools (Brantlinger, 2003; Ladson Billings, 1994, 1995a; Tate, 1995; Tatum, 1997; Tutwiler, 2007; Wald & Losen, 2007). Intelligence, the language of mathematics, and what counts as legitimate mathematical knowledge are socially and culturally constructed (St ernberg, 2007; Zevenbergen, 2000). The social norms, practices, behaviors, ways of speaking, and skills valued in school are consistent with those of white, middle class Americans (Brantlinger, 2003), and students who are unfamiliar with that culture often struggle to succeed (Banks et al., 2005; Brantlinger 2003; Tatum, 1997; Tutwiler, 2007; Wald & Losen, 2007; Zevenbergen, 2000). For instance, a common discourse pattern in the mathematics classroom follows the Initiation Response Evaluation (IRE) pattern (Herbel Eisenmann & Breyfogle, 2005; Zevenbergen, 2000). In this exchange, the teacher poses a question 4, Quantavius uld the student respond incorrectly, the teacher typically gives the answer to the class or initiates only perpetuates a power structure with the teacher holding al l the knowledge in the classroom, but these practices are rarely taught to students explicitly (Gee, 2008). They must instead learn appropriate ways of communicating through observation and participation, which can be difficult unless the discourse pattern s one learns at home are somewhat congruent to those used in school (Gee, 2008; Zevenbergen, 2000). In PAGE 22 22 an observation of mathematics classrooms in the middle grades, Zevenbergen (2000) found that middle class students complied with the IRE discourse patter n. In contrast, broke the IRE pattern by calling out, and otherwise did not conform to the expected norms of discourse, ultimately preventing the teacher from completing the lesson in the way she intended. This suggests that behavior problems such as speaking out of turn may be due in large part to a misunderstanding of the linguistic norms of the classroom, not to intentional misbehavior. The result is that students from low contrast, researchers have found that when teachers use the language patterns similar formance improves (Mohatt 2008) to the school culture holds potential for improving achievement levels. Another example of the influence culture can have on student ach ievement is related to what is taught in mathematics courses. Mathematics is portrayed as something created by white Europeans (e.g., Newton, Gauss, Pythagoras and the ir classes, which may prevent them from feeling connected to the subject or from engagement and achievement. While the issues of cultural incongruity will be explored in fu rther chapters these examples highlight the disconnect between the mainstream culture valued in schools (i.e., that of white, middle class Americans) and that of students of color and who live in poverty PAGE 23 23 Summary Students of color and students who live in poverty tend to struggle more in school than their white, middle class peers (Banks et al., 2005; Brantlinger, 2003; Gutstein et al., 1997). For example, these students often perform poorly on standardized tests (Dewan, 2010; Rothstein, 2002; Tutwiler, 20 07) and demonstrate lower levels of problem solving ability than their peers (Post et al., 2008). Furthermore, students of color students tend to enroll in advanced high school mathematics courses at much lower rates than their white peers (Kelly, 2009; Th ompson & Lewis, 2005). There are many reasons for these disparities in enrollment and achievement, including out of school factors such as poor health (e.g., high levels of stress, malnutrition, greater exposure to violence; Fiscella & Kitzman, 2009); lack of stable housing; and income inequality (Rothstein, 2002) as well as in school factors such as lack of funding in impoverished schools (Banks et al., 2005; Brantlinger, 2003; Flores, 2007), high numbers of inexperienced or ineffective teachers in low tra ck courses and high poverty schools (Banks et al., 2007), biased standardized tests (Davis & Martin, 2008), and the practice of tracking students of color and who live in poverty into low level mathematics courses (Kelly, 2009). This dissertation focuses o n the instruction students of color who live in poverty receive. Prior research indicates instruction for high poverty students of color typically focuses on skills and procedures rather than concepts (NCTM, 1999; Webb & Romberg, 1994; Weiss, 1994), and th is low quality instruction is one of the most cited reasons for the persistent underachievement in mathematics of these students. Teachers in low track classes frequently teach to the test (Davis & Martin, 2008), dumb down content (Watson, 2002), and teach procedurally (Webb & Romberg, 1994; Weiss, 1994). For students, the result of this traditional teaching approach is often PAGE 24 24 a lack of deep mathematical understanding (Boaler, 1997; Davis & Martin, 2008; NCTM, 2000). A second reason often cited for the low a chievement of students of color living in poverty relates to the issue of culture: the cultural heritage and racial identities of students of color and students from low income families typically are not valued or represented in schools (Brantlinger, 2004; Tatum, 1997; Tutwiler, 2007; Wald & Losen, 2007). Instead most schools value skills, behaviors, attitudes, and ways of speaking that are typical of white students (and white teachers) but that conflict with the cultural norms of culturally and economicall y diverse students. The literature describes strategies for addressing these two reasons for underachievement (i.e., traditional mathematics teaching and the home school cultural disconnect), and this dissertation focuses on two systems of instruction that may support students in learning mathematics conceptually and promoting equity: standards based mathematics instruction and CRT. The following sections will briefly investigate the literature on these two areas. Overview of the Literature Standards based Mathematics Teaching As described earlier, traditional mathematics instruction, the most common teaching method in the U.S. (Hiebert et al., 2005; NCTM, 2000), is characterized by direct instruction, rote memorization (McKinney & Frazier, 2008), an over em phasis on procedural knowledge, and, as a whole, a focus on lower level mathematical skills rather than conceptual understanding (Hiebert et al., 2005). This type of instruction is prevalent in schools with large populations of students of color and studen ts living in poverty (McKinney, Chappell, Berry, & Hickman, 2009; McKinney & Frazier, 2008; Weiss, 1994) and is problematic for this population because it is incompatible with the way students learn (NRC, 1989) and does not allow them to engage with the PAGE 25 25 ma thematics in a meaningful way. In an effort to improve mathematics instruction and the achievement of all students, the NCTM (2000) issued Principles and Standards for School Mathematics (PSSM). PSSM describes standards for pre kindergarten through twelfth grade mathematics, necessitates a way of teaching that contrasts with traditional instruction, and forms the foundation of standards based mathematics teaching (Schoenfeld, 2002). These standards address content and the processes through which learning sh ould occur. The focus of this dissertation study is on the five process standards, which are: problem solving, reasoning and proof, communication, connections, and representations. These are not intended to be skills taught alongside mathematics content. R ather, they are simultaneously an outcome of learning mathematics (i.e., a product) and a means through which that learning occurs (i.e., a process; NCTM, 2000) Recently, the CCSSI released Standards for Mathematical Practices (CCSSI, 2010). These standar ds are based on the NCTM (2000) process standards as well as the five interwoven, interdependent strands of mathematical proficiency (i.e., conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and a productive disposition ; Kilpatrick, Swafford, & Findell, 2001 ). Like the process standards, the mathematical practices describe outcomes of mathematics instruction that differ from pure mathematical content. These practices can be found in Table 1 1 along with the process stan dards with which they correspond. Much of the literature on reform focuses on particular process standards rather than standards based instruction as a whole. For instance, studies have concluded that choenfeld, 1987), achievement PAGE 26 2 6 (Boaler, 1998, 2002; Charles & Lester, 1984; Ginsburg Block & Fantuzzo, 1998; Silver, Smith, & Nelson, 1995; Verschaffel et al., 1999), and motivation (Charles & Lester, 1984; Ginsburg Block & Fantuzzo, 1998) can all be improv ed when instruction emphasizes problem solving. Table 1 1. General comparison of attention to process in CCSSM and PSSM. Adapted from Making it happen: A guide to interpreting and implementing Common Core State Standards for Mathematics (p. 12), by NCTM, 2010, Reston, VA: Author. CCSSM Standards for Mathematical Practice Principals and Standards Process Standards 1. Make sense of the problems and persevere in solving them. Problem Solving Communication Representation 2. Reason abstractly and quantita tively. Problem Solving Reasoning and Proof 3. Construct viable arguments and critique the reasoning of others. Reasoning and Proof Communication Representation 4. Model with mathematics. Problem Solving Reasoning and Proof Connections Representa tion 5. Use appropriate tools strategically. Problem Solving Representation 6. Attend to precision. Problem Solving Communication 7. Look for and make use of structure. Problem Solving Reasoning and Proof Connections 8. Look for and express regu larity in repeated reasoning. Problem Solving Connections Rather than focusing on one process standard at a time, as is often done in mathematics research (e.g., Davis & Maher, 1997; Turner, Meyer, Midgley, & Patrick, PAGE 27 27 2003; Verschaffel et al., 1999), thi s dissertation extends prior research by focusing on all the process standards that manifest during instruction and which characterize mathematical reform. A common question regarding standards based teaching is whether it is more equitable and better sup ports students who live in poverty to develop deeper conceptual understanding of mathematics than traditional instruction (Boaler, 2002). Several studies suggest that it can (Boaler, 1997, 1998, 2000, 2002; Brown, Stein, & Forman, 1996; Post et al., 2008; Silver et al., 1995). Other studies, however, suggest that students may struggle with the open, contextualized problems and discourse characteristic of standards based classrooms (Lubienski, 2000a, 2000b, 2002). Lubienski (2002) argues that NCTM (2000) mak es strong assumptions about what kind of instruction is best for culturally and economically diverse students, and that learning through the process standards will come more naturally to some students than others because of their cultural background. The c ommunication patterns, preferences, and ways of expressing knowledge common to students of color or who live in poverty may vary greatly from what the process standards expect of them (Brantlinger, 2003; Heath, 1983, 1989; Lubienski, 2002; Mohatt & Erickso n, 1981). This does not mean students cannot learn to communicate about mathematics, engage in problem solving, and reason mathematically in a way consistent with the process standards, but rather that the appropriate supports must be put into place by the ir teacher to bridge the gap the intersection of standards based pedagogy and instruction that is responsive to arning. PAGE 28 28 Culturally Responsive Teaching In addition to standards based mathematics instruction, CRT has been suggested as a main means of overcoming many of the obstacles students of color living in poverty face (Banks et al., 2005; Bonner, 2011; Gay, 2000, 2002; Gutstein et al., 1997; Ladson Billings, 1994, 1995a, 1995b, 1997; Peterek, 2009; Tate, 1995). In essence, CRT is an approach that empowers, or enables, students to develop intellectually, socially, emotionally, and politically by bridging the gap be tween home and school culture, relevant for students (Gay, 2000; Ladson Billings, 1994). The goal is to not only support students to become academically successful but also to build their cultural identity and their ability to critique the existing social order (Ladson Billings, 1995b). This teaching approach is thought to be responsible for the high achievement of traditionally underperforming students of color (Bonner 2011; Ladson Billings, 1994, 1995a, 1995b; Gay, 2000, 2002; Peterek, 2009). While studies on CRT sometimes include mathematics teachers as participants (e.g., Ladson Billings, 1994), most of the literature does not explicitly discuss how this construct i nteracts with, supports, or otherwise influences mathematics instruction (Gutstein et al., 1997; Peterek, 2009). One researcher (Bonner, 2011; Peterek, 2009) 1 has examined CRT within the context of mathematics classrooms to describe a new construct, cultur ally relevant mathematics teaching (CRMT). She provides a model of CRMT but fails to adequately characterize the nature of the mathematics teaching within mathematics classrooms or how CRMT differs from traditional or standards based mathematics instructio n. It is also unclear 1 To clarify, Bonner (2011) and Peterek (2009) were written by the same author, Emily Peterek Bonner. PAGE 29 29 how CRMT differs from broader definitions of CRT. Conversely, Gutstein and colleagues (1997) provide one model of instruction that explicitly connects CRT with reform, but this research occurred prior to the release of PSSM (NCTM, 200 0) and the study was conducted with high achieving students. This study extends research by explicitly examining both standards based and CRT practices within mathematics classrooms and how teachers draw on these two teaching approaches to influence studen Statement of the Problem It is well known that students of color and students who live in poverty struggle in mathematics (Davis & Martin, 2008; Dewan, 2010; Kelly, 2009; Post et al., 2008; Rothstein, 2002; Thompson & Lewis, 20 07; Tutwiler, 2007). Standards based teaching has been theorized as one way to help them overcome these struggles (Boaler, 1997, 1998, 2000; NCTM, 2000), and there is evidence to support such a theory (e.g., Boaler, 1997, 1998, 2000, 2002; Charles & Lester 1984; Ginsburg Block & Fantuzzo, 1998; Post et al., 2008; Schoenfeld, 1987, 2002; Van Haneghan, Pruet, & Bamberger, 2004). Teaching in a culturally responsive way has additionally been suggested as a framework for supporting students of color to succeed academically (Banks et al., 2005; Bonner, 2011; Gay, 2000; 2002; Gutstein et al., 1997; Ladson Billings, 1994, 1995a, 1995b, 1997; Peterek, 2009; Tate, 1995), but prior studies on cultural responsiveness do not adequately characterize the aspects of CRT th at are unique to mathematics, nor do many studies describe the nature of mathematics instruction in culturally responsive classrooms. There are few studies that explicitly focus on both standards based mathematics teaching and CRT (e.g., Gutstein et al., 1 997), and the result is two disparate bodies of literature. There is some literature to suggest that students need PAGE 30 30 more than just, for example, the problem based instruction and class discussions characteristic of reform to support them to learn mathematic s, particularly when the norms, communication and interaction patterns, and ways of thinking viewed as necessary to mathematical success are in conflict with the culture of the student (Gee, 2008; Lubienski, 2000a, 2000b, 2002; Zevenbergen, 2002). This lea ds one to question something else that results in some teachers being particularly effective with traditionally underperforming students. More research is needed to under stand what mathematics teachers who are successful with this population of students do to support them to learn, both mathematically and pedagogically. The purpose of this study was thus to understand how teachers who are successful with low achieving stud ents of color living in poverty supported their students in learning mathematics. Research Questions There are four research questions that guided this study : How do teachers identified as highly effective with students of color living in poverty help th eir students to engage with mathematical content? o o for engaging their students with mathematical content? practices align or not align with standards based instruction and culturally responsive teaching? Structure of the Dissertation The written presentation of this dissertation will include seven chapters. This chapter (Chapter 1) serves as an introduction a nd includes an overview of the literature to provide context for the study. Chapter 2 provides an in depth review of pertinent PAGE 31 31 literature, and Chapter 3 is a formal methods section. The results of the dissertation study will be presented in three chapters: two will be dedicated to a case study analysis of each of the two participating teachers (Chapters 4 and 5), and Chapter 6 will be a cross case comparison of the perspectives and instructional practices of both teachers and a discussion of the findings as related to the literature on standards based instruction and CRT. A final chapter (Chapter 7) will consist of conclusions, implications, and possible future research directions Appendices, which consist of an informed consent document and interview proto cols, will be included at the end of the document. PAGE 32 32 CHAPTER 2 LITERATURE REVIEW Amidst the issues surrounding the low achievement of students of color and students living in poverty are stories of those who defy the odds (e.g., Thompson & Lewis, 2005) and the teachers and pedagogical strategies that help them along the way (Boaler, 1997, 1998, 2000, 2002; Ladson Billings, 1994, 1995a, 1995b; Peterek, 2009; Tate, 1995; Ross et al., 2009). The following chapter will synthesize the literature from two perspect ives on instructional practices that may help to promote equity and support students to engage with mathematics content. Specifically, research literature on standards based mathematics and CRT will be examined. Standards based Mathematics Teaching Over t wenty years ago, the National Research Council (1989) warned educators Unfortu nately, this is still the case. Traditional mathematics pedagogy, which remains the most common teaching method in America (Hiebert et al., 2005; NCTM, 2000), is characterized by direct instruction; rote memorization; infrequent opportunities for group wor k, conversation, or problem solving (McKinney & Frazier, 2008); mechanical procedures, and an overall focus on lower level mathematics skills rather than concepts (Hiebert et al., 2005; Van de Walle, Karp, & Bay Williams, 2010). This stands in s harp contra st to standards based teaching Additionally, while traditional methods of teaching can result in procedural knowledge (Boaler, 1998; Schoenfeld, 1988), which is one important aspect of mathem atical proficiency (Kilpatrick et al. 2001), students taught tr aditionally struggle to transfer that knowledge to new settings and tend to display low PAGE 33 33 levels of conceptual understanding (Boaler, 1998; Bottge & Hasselbring, 1993; Schoenfeld, 1988). On the other hand, research shows that when instruction is characterize d by elements of reform student achievement and particularly that of students of color and students who live in poverty can improve (Boaler, 1998, 2002; NCTM, 2000 ; Post et al., 2008; Schoenfeld, 2002; Van Haneghan et al., 2004 ; see also Kilpatrick, M artin, & Schifter, 2003 ). This improvement, however, is not guaranteed by standards based teaching (Lubienski, 2000a, 2000b, 2002), an issue that will be addressed in detail in a later section. Research indicates that traditional instruction is particular ly prevalent in schools with high numbers of students of color and who live in poverty (McKinney, Chappell, Berry, & Hickman, 2009; McKinney & Frazier, 2008; Weiss, 1994) and may limit nect between the way students learn and the type of instruction occurring in schools, and to encourage teaching practices that are intended to promote equity and ensure the success of all learners, NCTM (2000) issued PSSM In addition to outlining the cont ent that all pre kindergarten through grade 12 students should learn, this document lists five processes through which learning should occur: problem solving, reasoning and proof, communication, connections, and representation. The process standards are no t skill s to be taught alongside other mathematical skills but instead are both a process and a product of mathematics teaching (NCTM, 2000). Teaching mathematics through these five process standards, which are described in detail below, forms the basis of the standards based (or reform based ) movement in mathematics education (Schoenfeld, 2002) and teaching mathematics for conceptual understanding (NCTM, 2000) PAGE 34 34 Problem solving. Students engage in solving problems for which no solution method is known in ad vance. These problems help students to build new knowledge, and are mathematical as well as come from both other disciplines and the real world. Students invent, apply, and adapt multiple solution strategies. Additionally, they monitor and reflect up on the ir solution processes and understand and critique the strategies used by their peers. Reasoning and p roof. Students develop mathematical argumentation skills in addition to understanding the importance of formal proof in mathematics. They think critically about the mathematics presented to them, ask questions, make conjectures, and generalize patterns. Another key aspect of this standard is e Communication. Students c ommunic at e about mathematics as a way to organize and clarify their thinking. They share their thinking with their teacher and peers, and interpret, analyze, and assess the ideas and strategies of others. Finally, students acqui re and use mathematical language. C onnections. Students connect mathematical ideas to informal mathematical knowledge, other mathematical ideas, their own experiences varying contexts, and disciplines outside of mathematics. Students also build conceptual understanding and link concepts to procedures rather than viewing mathematics as only a set of formulae and rules to be followed Representation. Students develop an understanding of multiple types of representations, how and when to use each one, and the benefits of each. They additionall y are able to represent a concept in multiple ways NCTM (2000) argues that these process standards should be integrated throughout all mathematical content areas. Standards based instruction emphasizes the five processes, as well as social interaction and mathematical knowledge. As a product of standards based instruction, students should be able to problem solve, reason about mathematics and prove conjectures and theorems, communicate mathematically, connect mathe matical ideas, and move fluidly between different mathematical representations. In order to achieve instruction aligned with the NCTM Standards, certain conditions must apply: classroom norms must be set and teacher centered instruction is exchanged for a more student centered approach. Moreover, the processes outlined in PSSM (NCTM, 2000) are not mutually exclusive, as PAGE 35 35 each supports and builds upon the other. To illustrate this notion, consider the case of problem solving. An Example: Integrating the Stan dards Through Problem Solving A problem solving situation is one in which students must struggle to find a solution; there is no known solution strategy at the onset of the process (Hiebert, 2003; NCTM, 2000). As students problem solve, they must therefore go through iterative cycles, during which they invent, adapt, test, revise, and carry out a plan (i.e., strategy) for finding the solution (Lesh & Zawojewski, 2007). One important aspect of problem solving is peer collaboration (Schroeder & Lester, 1989) and students who work with their peers on problems tend to perform better (Charles & Lester, 1984; Dees, 1991; Ginsburg Block & Fantuzzo, 1998). The type of task in which students engage during problem solving is also an important consideration for teache rs. Problems should be challenging for students, but not to the point of frustration (Hiebert, 2003). Additionally, they should integrate multiple mathematical topics, relate to other disciplines as well as the real world, and they should be open, allowing for a variety of entry points and strategies (NCTM, 2000). This latter requirement is perhaps the most crucial and difficult to attain as one of the main goals of problem solving is for students to develop multiple solution strategies (Hiebert, 2003; Hie bert et al., 1997; NCTM, 2000; Polya, 1985; Schoenfeld, 1987). The point, however, is not only to develop these strategies for the purpose of solving problems. Rather, much of the time devoted to problem solving must also be centered on developing, discuss ing, explaining, and justifying those solution strategies (Lesh & Zawojewski, 2007; NCTM, 2000; Polya, 1985; Schoenfeld, 1987; Van de Walle, 2003; Van de Walle & Lovin, 2006). This explicit discourse about strategies is intended to facilitate conceptual de velopment by allowing students the PAGE 36 36 opportunity to explain their thinking, make their methods explicit, learn from others, develop more efficient methods for solving problems (Hiebert, 2003; Schoenfeld, 1987) and understand how to interpret new problem solv ing situations (Lesh & Zawojewski, 2007). A later section will explore the nature of discourse in more detail. Studies indicate that p roblem solving as described here poses many benefits to students, including increases in mathematical understanding (Lambd in, 2003; NCTM, 2000; Schoenfeld, 1987), achievement (Boaler, 1998, 2002; Charles & Lester, 1984; Ginsburg Block & Fantuzzo, 1998; Silver, Smith, & Nelson, 1995; Verschaffel et al., 1999), and motivation (Charles & Lester, 1984; Ginsburg Block & Fantuzzo, 1998). A classroom focused on solving problems provides a context in which the other four process standards can be developed. The notion that problems should integrate multiple mathematical concepts and reflect real world issues is intended to allow stude nts to build on the knowledge they already hold and make connections between mathematical concepts and experiences outside of school. This speaks directly to th e Connections process standard. Additionally, encouraging students to invent and adapt strategie s in a problem solving context may support the development of multiple representations (Bostic & Jacobbe, 2010). Finally, the heavy emphasis placed on peer tegrates the standards of comm unication as well as reasoning and proof. The following problem will be used to illustrate these points: There are many vehicles in the parking lot. Some, like motorcycles, have 2 wheels. Other vehicles, such as cars, have 4 wheels. There are even some ve hicles, such as school busses, that have 6 wheels. If there are 46 wheels in the parking lot, how many of each type of vehicle might you have? PAGE 37 37 Students can use either addition or multiplication to calculate the total number of wheels, and can solve this pr oblem with a wide variety of solution strategies, including drawing a picture, making a chart, trial and error, and acting it out. Even students who have not previously solved this type of problem can draw on their knowledge about and out of school experie nces with cars, busses, and motorcycles, then use that knowledge to help them understand the context and to devise a mathematical solution strategy. By doing so, they are connecting their real world knowledge to the mathematical concepts they are learning as they solve the problem. In addition, if the teacher encourages students to share their varying solution method with their peers, the mathematics underlying their solutions is made explicit, which can help students to make connections between different y et related mathematical concepts such as addition and multiplication. By solving this type of problem and discussing and justifying their solution strategies, students are supported to construct connections between and among mathematical concepts as well a s to communicate about the mathematics and engage in reasoning and proof. Additionally, by comparing the different strategies used, students may be supported to develop multiple representations. The se aspects of problem solving thus demonstrate how the fiv e process standards are intertwined. In the following section, the role of discourse, a key aspect of standards based instruction (Lubienski, 2000a) is considered Next will follow a detailed illustration of the potential of reform based mathematics inst ruction to support students of color and lower socioeconomic status (SES) populations Discourse The term discourse refers not only to the communication about mathematical thoughts and information in which teachers and students engage (McCrone, 2005) but PAGE 38 38 a lso to the ways they agree and disagree about mathematics (NCTM, 1991). Thus, discourse encompasses the Communication and Reasoning and Proof process standards. A prerequisite to discourse includes the rules and norms established in the classroom about how communication should occur and what types of thinking, argumentation, and talking are valued (Cobb, Wood, & Yackel, 1993; NCTM, 1991; Sfard, 2003). Discourse in the mathematics classroom has gained increasing attention over the past two decades (Sfard, 20 03; Walshaw & Anthony, 2008) because of the focus on strategies and processes that are an integral part of problem solving (Lesh & Harel, 2003). In fact, problem solving frequently serves as the context in which to engage teachers and students in rich math ematical discussions (e.g., Cobb et al., 1993; Kazemi & Stipek, 2001; McCrone, 2005). Research indicates that discourse in the mathematics classroom is beneficial to students: it may promote conceptual understanding (Cobb, Wood, Yackel, & McNeal, 1992; Ka zemi, 1998; Kazemi & Stipek, 2001; Kilpatrick et al., 2001; NCTM, 2000) and mathematical thinking (Kazemi & Stipek, 2001), as well as support the development of mathematical argumentation skills congruent with the reasoning and proof process standard (Cobb et al., 1992). Discourse also provides an opportunity for students to coproduce knowledge. It allows teachers to observe their reasoning, which can influence their decisions about which approach to take next in their instruction (Wa lshaw & Anthony, 2008). It is important to clarify the characteri stics of discourse that push students to higher levels of mathematical understanding. In a recent review of literature to describe the pedagogical approaches to classroom discourse that prod uce desirable results for PAGE 39 39 learners, Walshaw and Anthony (2008) identified four actions carried out by teachers as described below: Action 1: Engage students in dialogue about mathematics; g; Throughout the first teacher action, Walshaw and Anthony (2008) explained that teachers focused on engaging the entire cla ss in mathematical dialogue and made discourse participation rules (e.g. when and how to contribute) explicit for students by clarifying, establishing, and enforcing those rules. This action is important because it serves to help students come to value dia logue and the sharing of mathematical ideas in the classroom. This is especially important for low achieving students, who often struggle more with participating in discussions than their peers. Teachers must therefore work to ensure the classroom environm ent is a safe one in order to encourage all students to participate. During the second teacher action, teachers focus their attention on the content of carefully sca utilizes appropriate wait time. language, which is important because specialized discourse is a key part of mathematical competence. The teacher creates a context for enculturation into the mathematics community by modeling the appropriate, precise, and generalizable mathematics vocabulary and language use, as well as questioning students to ensure thei r understanding of vocabulary. Over time, students take up the conventional PAGE 40 40 language use demonstrated by the teacher. During this activity, teachers must make an mathemati In the fourth and final action, the teacher works to support students to develop their mathematical argumentation skills through the use of the specialized vocabulary should have the opportunity and space to, for example, interpret, generalize, justify, and Students must be taugh B efore any of these activities can be carried out, however, specific and particular norms must be put into place in the classroom (Yackel & Cobb, 1996). Teacher s must work with students to establish the social and sociomathematical norms that allow for appropriate discourse to take place (Cobb et al., 1993; Yackel & Cobb, 1996). Social norms are those that apply across subjects and content; they are norms that ar e not unique to mathematics (Yackel & Cobb, 1996). An example would be the practice of listening attentively while a classmate or the teacher is talking. In contrast, sociomathematical norms are specific to mathematical discourse and include, for instance, rules about what counts as a mathematically different solution (Yackel & Cobb, 1996). In a study of how conceptual thinking is promoted in reform classrooms with ethnically diverse urban students, Kazemi and Stipek (2001) identified four such sociomathema tical norms : PAGE 41 41 Explanations are focused on processes rather than just procedures T hus, students describe their thinking and provide mathematical justification for their explanations Using multiple strategies to solve problems and describing those strategie s to peers and teacher is required. Students are supported to understand the relationship between strategies Errors are a natural part of the learning process, and they provide opportunities to extend learning and understanding Collaboration among peers to find solutions is necessary. Additionally, groups must reach consensus about their solution through reasoning and argumentation These norms do not necessarily come naturally to students; they must be made clear ( Walshaw & Anthony, 2008; Yackel & Cobb, 1996) and negotiated in the classroom through conversations explicitly focused on the mathematics itself (talking about mathematics), as well as on how to communicate mathematically ( talking about talking about mathematics ; Cobb et al., 1993). T he teacher therefore plays a vital role in establishing classroom norms and guiding the class discussion (Cobb et al., 1993; Kazemi & Stipek, 2001; Lubienski, 2000a, 2000b; Walshaw & Anthony, 2008; Yackel & Cobb, 1996). While all students must learn the appropriate ways of participating in mathematical discourse in the classroom, this learning may be particularly difficult for students of color and students who live in poverty (Forman, 2003; Gee, 2008; Lubienski, 2000a, 2000b, 2002; Walshaw & Anthony, 2008; Zevenberg en, 2000) because their learned ways of communicating may conflict with what communicating about mathematics requires (Boaler, 2002; Zevenbergen, 2000) The following section explores in depth select research on reformed mathematics teaching and the diffic ulties students of color and those who live in poverty may face in reform oriented classrooms. PAGE 42 42 Standards based Mathematics Teaching and Equity An important question is whether standards based curricula and pedagogy support a more equitable education for st udents (Boaler, 2002). S kill based approaches consistent with traditional mathematics pedagogy may be viewed as more explicit about how students should behave and communicate and thus are believed to reduce the inequities faced by culturally and linguistic ally diverse students. This perspective may be a contributing factor that drives teachers of these students to teach to the test, lower their expectations for struggling students, simplify content, and emphasize procedures and skills rather than concepts ( see Davis & Martin, 2008; NCTM, 1999; Watson, 2002; Webb & Romberg, 1994; Weiss, 1994). Direct instruction and an emphasis on procedures are also popular instructional techniques with students who have math ematics learning disabilities (e.g., Jitendra et a l., 2007; Montague, 2007). Taken In a meta analysis of research on mathematics instruction interventions for students with learning disabilities, Gersten e t al. (2009) examined the effect of explicit by step plan (strategy) for solving the problem; (b) the plan was problem specific and not a generic heuristic for solving the problems; and (c) students were actively Explicitness as defined here is common in mathematics classrooms that adopt a direct instruction approach. Gersten et al. (2009) concluded that explicit instruction is an important instructional strategy for students with mathematics learning disabilities. Such a finding has direct implications for students of color and those living in poverty, as they PAGE 43 43 are disproportionately placed into special education programs (Rothstein, 2002) but Other researchers disagree with the argument for explicit or direct instruction. For instance, i n an inquiry into teaching practices that promote equity Boaler (2002) found that traditional instruction did not support learners who live in poverty and that a standards based approach was more beneficial for this population In an attempt to identify the pedagogical practices of teache rs who narrow the achievement gap, Boaler (2002) sought to reexamine two series of studies that identified successful reform oriented mathematics curricula. The first series (Boaler, 1997, 1998, 2000) she reexamined explored the relationship between studen mathematics achievement and their social class over a three year period (from ages 13 16). Boaler followed two groups of students at different schools from their ninth grade to eleventh grade years. The first school, Amber Hill ( N =200), followed a trad itional, procedural curriculum tracked mathematics students according to their perceived ability levels, emphasized teaching through direct instruction and relied heavily on the textbook. The second school, Phoenix Park ( N =110), adopted an open ended, pr oject based mathematics curriculum (Boaler, 1997), used no textbook, and students were often encouraged to think mathematically, pose and extend problems, use their own mathematics, and invent their own strategies (Boaler, 1998). Phoenix Park classes were heterogeneously grouped rather than tracked, and students, who were considered responsible for their own learning, were able to work though problems at their own pace. Both groups of students had the same (individual, booklet based) mathematics instruction during seventh and eighth grades; the change in instruction occurred in the PAGE 44 44 ninth grade when the study began Both schools also had similar populations of students: the majority were white, working class, and low achieving (Boaler, 1997). Using a mixed m ethods design, approximately 100 lessons were observed and students were surveyed, interviewed, and took a wide range of assessments. Standardized test data were also collected for all participants. Interviews and field notes w ere coded using grounded theo ry and data were triangulat ed (Boaler, 1997). When the two groups of students were compared, those at Amber Hill reported feeling inadequate, complained about a lack of relevant problems, and viewed school mathematics as boring, tedious, and disconnected f rom their daily lives. Furthermore, these students believ ing mathematics to be only about rules, formulae and equations. They were not encouraged to discuss the mathematics in which they engaged and were unable to de termine whether different contextual situations were in fact mathematically similar. They also searched for cues when solving thinking on what they thought was expected of them rather than on the mathematics in assertion of the dangers of traditional mathematics instruction: Amber Hill students often stopped working if a problem seemed challenging, too easy, or if it required mathematical procedures other than those which they had most recently learned (Boaler, 1998). In contrast, at Phoenix Park, where instruction was standards based, students were viewed as independent workers and thinkers. They engaged in projects that lasted two or three weeks at a time, and were able to work at their own pace and in unsupervised settings. This resulted in not all students being on task at the same time, PAGE 45 45 but every student eventually completed their assigned tasks. Due to t he format of the lessons, Phoenix Park students were required to regularly monitor and control their own behaviors and were able to discuss mathematics in a meaningful way. They also performed significantly better than their Amber Hill peers on applicatio n/transfer problems; scored equally well on traditional examinations including the General maintained higher grades. In fact, the project based instruction equalized students achievement in terms of both gender and social class, resulting in smaller gaps in student achievement (Boaler, 1998). The second series of studies Boaler (2002) reexamined were a part of the Qualitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project (see Brown, Stein, & Forman, 1996; Silver et al., 1995). Teachers at six urban middle schools developed a curriculum over the course of five years that incorporated problem solving and the encouragement of discourse, including j ustification, into their mathematics instruction. T he QUASAR project resulted in significant gains in achievement for culturally and linguistically diverse students. While Boaler (2002) did not describe her data analysis methods, her reexamination of these studies highlighted three instructional approaches consistent with reform pedagogy that appeared to be associated with these results. First, in contrast to traditional classrooms where students are often expected to interpret text based problems on their own, the participating teachers helped students understand problems through discussions before asking students to solve them. Second, students were required to explain and justify their solutions. Teachers were very explicit about the form these explanatio ns and PAGE 46 46 justifications should take essentially talking about mathematics and talking about talking about mathematics in the manner described by Cobb et al. (1993). Finally, real world problems were used to provide a context for mathematics. Boaler (2002) c autioned, however, about the nature of real world problems used. Good problems require familiarity with the context described, and backgrounds and provide them with the same level of familiarity as their peers ( see also Zevenbergen, 2000). Additionally, when working on realistic problems, students will often bring unexpected information into their interpretation of the problem. For instance, consider the open ended task QUASAR teachers administered to their stude nts (Silver et al., 1995, p. 41) : Yvonne is trying to decide whether she should buy a weekly bus pass. On Monday, Wednesday, and Friday, she rides the bus to and from work. On Tuesday and Thursday, she rides the bus to work, but gets a ride home with her f riends. Should Yvonne buy a weekly bus pass? Explain your answer. Busy Bus Company Fares One Way $1.00 Weekly Pass $9.00 The teachers were surprised when their students chose the weekly pass as the more economical answer. In the ensuing class discussions, students explained the pass could be used on evenings and weekends as well as shared with other family members (Silver et al., 1995). In being provided with a realistic problem, students gave real world d ] their reasoning in the con 251) T eachers must be ready to accept these as valid answers (Silver et al., 1995) and allow students to explain their reasoning, providing further support for the importance of discourse. Boaler (2002) conclude d tha t these instructional practices carried out by the PAGE 47 47 teacher, and not necessarily the standards based curriculum itself, are what ultimately promoted equity and improved student achievement Lubienski (2000a, 2000b, 2002) also conducted a study that examine d the interplay between reform based teaching and equitable instruction ; this research took place i n a seventh grade classroom where students learned mathematics via problem solving (Schroeder & Lester, 1989). The 18 participants in the class were primarily white (one Mexican American female and one black male) but came from socioeconomically diverse backgrounds. Lubienski, acting as the teacher, used the Connected Mathematics Project (CMP) curriculum, which has a focus on open ended, contextualized problems Daily lessons followed the Launch, Explore, Summarize to explore the problem and solve it, and finally, in whole class discussions, the group explorations were summari zed. During these whole class discussions, students were prompted to share their solutions and discuss the mathematical ideas. Over the course of one school year, se veral students were interviewed up to three times, student surveys were administered to the entire class, and Lubienski audio recorded her teaching and kept a daily journal (Lubienski, 2000a, 2000b). These data were transcribed, and themes across SES and gender were identified (Lubienski, 2000b). One question the researcher sought to explore wa perceptions of the CMP curriculum, specifically, how they experienced the open, contextualized nature of the problems (Lubienski, 2000b). The higher SES students clearly preferred the CMP curriculum to traditional mathematics textboo ks. They found problem solving more interesting and easier than computation, were intrinsically PAGE 48 48 motivated to work on the problems, and persisted when they faced challenging problems. The lower SES students, on the other hand, were overwhelmed and frustrate d by the open ended questions and gave up easily. They struggled with interpreting questions and deciding on a strategy, found it difficult to read the problems (in terms of the vocabulary and sentence structure), and reported feeling confused by the probl ems. They were very concerned with using the correct algorithm and finding the correct answer, unlike the higher SES students, who felt comfortable following their instincts when solving a problem. Furthermore, girls were found to put forth more effort on homework and completed many more problems than boys. This effort was correlated with quiz and test scores for higher SES students, but not for the lower SES students; they completed their assignments but were not able to demonstrate knowledge of the mathem atics on tests (Lubienski, 2000b). Finally, while higher SES students were able to extract and generalize the mathematics embedded in the contextual problems, and mathematical principles, the lower SES students often failed to identify the underlying mathematics, instead taking into account many real world variables to solve problems. had a mathe matical goal of comparing volume and unit price, some lower SES students calculated the volume and then responded that the prices increase in order by size, so one should choose which size to buy based on how much popcorn one intend ed to eat (Lubienski, 20 02). These results suggest that while NCTM (2000) calls for realistic, open ended problems, some students may struggle to decontextualize the problems in a way that would allow them to learn the underlying mathematics (Lubienski, 2000b). PAGE 49 49 Another question L ubienski (2000a) explored related to how students of different socioeconomic statuses respond ed to standards based pedagogy, particularly the discourse, and how their cultures may affect their openness to standards based instruction. For this analysis, Lub ienski examined the experiences of six girls of varying socioeconomic and achievement levels B y intentionally focusing on girls only, the researcher sought t o hold gender constant. T he degree of confidence students held in their mathematical abilities inf luenced willingness to participate in discussions and some students did not find discussions helpful to their understanding. Higher SES girls reported that they liked participating in class discussions and felt confident that their thinking was corre ct and s about the problems. The lower SES girls, however, struggled with whole class discussion s and were afraid of sharing a wrong idea. They found it difficult to distinguish between correct and incorre ct answers during the discussions, even when Lubienski provided them with hints, and they disagreed with them. They reported feeling confused and stated that they preferred a traditional instructional style wit h more explicit teacher direction. Higher SES girls were more likely to discuss methods and ideas and to generalize contexts, whereas lower SES girls typically answered straightforward questions they were sure they would get correct (Lubienski, 2000a). Fin ally, the higher SES students articulated that they understood that by allowing students to reason out the mathematics on their own, the teacher was supporting deeper learning On the other hand, lower SES students ly state who was right and who was wrong during discussions was to avoid embarrassing or hurting the feelings of the student with PAGE 50 50 an incorrect answer (Lubienski, 2002). The author suggest ed that by giving authority to the class, certain students took on th e authoritative role and other students were disempowered She conclude d that the classroom culture of standards based pedagogy was better aligned with the preferred ways of communicatin g, knowing, and learning of the middle class studen ts than the lower S ES students (Lubienski, 2000a). These results bring up some troubling questions regarding standards based mathematics teaching and how it affects students of different cultures. There are some studies that suggest that standards based instruction is benef icial for culturally, ethnically, and socioeconomically diverse populations (e.g., Boaler, 1997, 1998, 2000, agree? What was it about classroom, setting, or instruction that ma de the difference? Furthermore, if Lubienski an educational researcher and presumed expert in mathematics instruction, struggled to implement reform in a way that benefitted her lower SES students, what does this imply for classroom teachers? To unders tand and help explain why students of different socioeconomic groups which can be viewed as distinct cultures had such different experiences with the discourse and open, contextualized nature of the pro blems, Lubienski (2002) adopted a sociocultural le ns. She reviewed literature from anthropology and sociology, and suggest ed that the knowledge, skills, and beliefs students enter the classroom with, which are a product of their cultural heritage, influence their experiences with standards based instructi on. Lubienski (2000a) concluded that the practices of sharing, grappling with ideas, and in other words, the defining characteristics of her classroom conflict ed with the values, beliefs, and social norms o f PAGE 51 51 white students who live in poverty. succeed. Furthermore, Lubienski (2002) argued that NCTM (2000) makes strong assumptions ab out how students will experience reform, suggesting that through problem solving, students will abstract powerful mathematical ideas and processes, and that open discourse will make all students feel their ways of thinking and communicating are valued. NCT M (2000) additionally suggests that listening to others will increase about their abilities when solving challenging problems. As Lubienski (2000a, 2000b, 2002) has dem onstrated, these assumptions do not hold true for all students all of the time (see also Zevenbergen, 2000) I n reality, problem solving and discussion as a means of learning mathematics will come more naturally to some students than others (Lubienski, 200 2). If the goal of mathematics reform is to support students to become empower ed mathematically, then the cultural assumptions of discourse need to be examined in order to better address the difficulties some students will face. As Lubienski (2002) argue d, When we understand ways in which a particular discourse differs from dilemmas about whether the discourse we are promoting is inherently valuable as an end in itself or is simply an a rbitrary, value laden means (perhaps a relatively White, middle class means) to an end. In mathematics education, for example, this could mean that we must consider whether whole class discussion of students' conflicting mathematical conjectures is simply one possible means to understanding mathematics or if it is an important mathematical process in its own right. If it is an important end in itself, efforts should be made both to help students understand the norms and roles assumed by such an approach and to adapt the approach to students' needs and strengths. (p. 120 ) PAGE 52 52 contributed to t he struggles the lower SES students had with the class discussions. Similarly, it is unclear in the description of her teaching whether Lubienski took time during instruction to make explicit the underlying mathematics being discussed as required by standa rds based teaching. Teachers thus play an important role in reform classrooms not only in helping students learn the mathematics intended (Lubienski, 2000a, 2000b, 2002), but also in establishing classroom norms for discourse, making those norms explicit, language and culture and the school language and culture (Gee, 2008). wa s conducted with mostly white children, we know that communication patterns and preferences of racia lly and ethnically diverse students also differ from those of the mainstream school culture (Brantlinger, 2003; Heath, 1983, 1989; Mohatt & Erickson, 1981). For example, black children are expected by the adults in their community to use their knowledge ra ther than explain it (Heath, 1983), which may create tensions for them in discussion rich school environments requiring explanation and justification. Hence, it is possible students of color will struggle in similar ways with standards based instruction un less they receive appropriate supports from their teachers To clarify, th e implication is not t hat some students are in capable of engaging in rich mathematical discussions and problem solving, but instead that they must be supported to learn to do so. Tea chers must help to make explicit the mathematics being taught and discussed as well as the norms and expectations for engaging in mathematical discourse. This would serve to support students who enter PAGE 53 53 the classroom with different cultural understandings of what is expected of them in terms of communication than what their teacher and classroom culture dictates. It therefore becomes important to understand not only how standards based mathematics teaching is implemented and how that affects students of color and students who live in poverty but also to understand differing cultures and how those may support their mathematical learning. We turn, then, to an examination of CRT a practice thought by some to be responsible for the academic success of students of color who live in poverty and that may help to address the cultural dilemma described by Lubienski (2002) Culture, Mathematics, and Teaching The previous section explored the literature on the refo rm movement in mathematics education, and the potential of and problems with implementing this type of teaching with students who fall outside of the American mainstream. In this next section, the argument will be made that CRT can serve as a way to su pport students of color and students who live in poverty who are struggling in mathematics. Many scholars contend that attending to the cultural needs of black students may help to narrow the achievement gap (e.g., Delpit, 2006; Hondo et a l., 2008; Ladson Billings, 1994, 1995a, 1995b; Nelson Barber & Estrin, 1995; Tate, 1995) and that 2001 ; Gay, 2000, 2002; Gutstein et al., 1997; Hinchey, 2004; Ladson Billings, 1994, cultures may provide a way to not only integrate cultures and experiences into classroom prac tices in order to empower or enable, students in a system that typically PAGE 54 54 disadvantages them (Ladson Billings, 1994, 1995b; Peterek, 2009), but may also ies to promote equity for all their students (Boaler 2002). The next sections serve to define this instructional approach, referred to as both culturally relevant pedagogy and culturally responsive teaching by beginning with a historical account of the works of Gloria Ladson Billings and G eneva Gay Culturally Relevant Pedagogy In an era where students of color who live in poverty often struggle to succeed in school, Ladson Billings (1994, 1995b) sought to identify effective teachers of this population of children and describe the beliefs, practices, and instructional methods that success. Eight teachers were identified, and through ethnography, teacher interviews, and member checking, she discovered these teachers practiced a unique pedagogy, which she calls culturally relevant that empowers students intellectually, socially, emotionally, and politically by using cultural referents to impart knowledge, skills, and attitudes. These cultural referents are no t merely vehicles for bridging or explaining the dominant culture; they are aspects of the curriculum in their own right. (Ladson Billings, 1994, pp. 17 18 ) The teachers in the study all held the same three goals: (a) to ensure the academic success of all their students; (b) to and affirmation of their culture; and (c) to analyze, and critique the existing social order. This last element, referred to as critical co nsciousness, includes empowering students to take action to change inequities they observe or experience (Ladson Billings, 1995b). The pedagogy of culturally relevant teachers includes scaffolding students to build on what they know and what they need PAGE 55 55 to l earn, a n unwavering focus on instruction and learning, and working to extend Billings, 1994). Additionally, they hold high expectations for students and do not tolerate failure, pushing students and providing approp riate supports until they succeed (Ladson Billings, 1994, 1995a, 1995b). In other words, being culturally relevant means teachers get to know their students and their cultures deeply, and by using that knowledge to build relationships, they help students i n learning mathematics (or other subjects) and to feel proud of their cultural heritage They also support students to feel empowered to understand and make changes in the world around them. This pedagogy is not about accommodation or assimilation of stude nts of color into the mainstream culture, but rather about creating a synergistic relationship between home and school life (Ladson Billings, 1995b). Unlike many teachers who often (and unknowingly) hold prejudices against students of color (Burdell, 2007) teachers who adopt a culturally relevant approach to teaching view all their students as capable and worthy without qualifiers. Through the relationships they build by showing students they know about and value their personalities, families, and cultures teachers let their students know they care for them (Ladson Billings, 1995), which is especially important consider ing that feeling cared for is a key factor for Building on t his notion of cultural relevance, Gay (2000, 2002) posed a theoretical framework for teaching, which she calls culturally responsive Culturally Responsive Teaching Similar to culturally relevant pedagogy, CRT experience s, frames of reference, and performance styles of ethnically diverse students to make learning encounters more relevant and effective PAGE 56 56 thereby enabling students to learn more easily and deeply. Like culturally relevant pedagogy CRT comes from a critical theory perspective, and thus includes the notion of critical consciousness. Students are encouraged to question the status quo, including the portrayal of people of color in mass media, textbooks, trade books, and popular cultur e, and teachers work to build positive examples of diverse cultures into the curriculum. CRT thus places a great emphasis on multicultural education, including multicultural mathematics, in the curriculum (Gay, 2000, 2002). Responsive teaching, however, so metimes seems to lack the notion of praxis an iterative cycle of reflection and action (Hinchey, 2004) that is a key component of cultural relevance (Gay, 2002; Ladson Billings, 1994, 1995b). While questioning and thinking critically about the world are i mportant aspects of both pedagogies, the literature on CRT is inconsistent in its emphasis on the action aspect of praxis (e.g., Banks et al., 2005; Gay, 2002; Scheurich & Skria, 2003). For the purposes of this paper, I adopt the broader definition, which does include praxis as part of CRT A note about terminology is warranted at this point. The pedagogy described here, which is informed by and congruent with the cultures of students of color and (Ladson Billings, 1997), has been called by many names, including culturally relevant, sensitive, congruent, specific, and informed (Gay, 2000; Leonard, 2008). All these terms describe similar ideas (Gay, 2000), and the ways Ladson Billings (1994) and Gay (2000) define this pedagogical approach support each other. T o align with more current literature (e.g., Banks et al., 2005; Bonner, 2011; Hondo et al., 2008; Peterek, 2009; Tutwiler, 2007), culturally responsive teaching (CRT) will therefore be used to re fer to such an PAGE 57 57 instructional approach as described by either Ladson Billings (1994, 1995a, 1995b, 1997) or Gay (2000, 2002) u nless directly quoting a scholar. In the current research on effective teaching of students of color across content areas and grad e levels (Bonner, 2011; Delpit, 2006; Garza, 2009; Gay, 2000, 2002; Gutstein et al., 1997; Ladson Billings, 1994, 1995a, 1995b, 1997; Long, 2008; Nelson Barber & Estrin, 1995; Peterek, 2009; Sheets, 1995; Tate, 1995; Wlodkowski & Ginsberg, 1995), there are eight themes that emerge to characterize culturally be discussed in the following sections: Teacher student relationships; Supporting students to develop a critical disposition; Classroom environment; Culturally responsive classroom management; and Instruction. Culturally responsive teaching is more a way of being a nd believing than it is about specific actions a teacher takes. The discussion of cultural responsiveness that follows is thus not intended to reduce CRT to a list of practices for teachers of students of color to engage in. The organization of themes into separate sections should not imply that each theme is discrete. In fact, many of these topics overlap, but they are presented in effectively teaching students of color. The foremost goal culturally responsive teachers hold is that they want all their students to achieve academically. Academic success which includes technological, PAGE 58 58 social, and political skills ; literacy ; and numeracy (Lads on Billings, 1995a) is non negotiable (Gay, 2000 ; Ross, Bondy, Gallingane, & Hambacher, 2008 ) Teachers believe it is their primary (Gay, 2000, 2002; Ladson Billings, 1995b; Sheets, 1995) Ins truction and academic challenges are the main focus in the classroom (Gutstein et al. 1997; Ladson Billings, 1997), and l earning and mastery of content, rather than grades, is emphasized (Gay, 2000; Sheets, 1995). Similarly, s tandardized tests are not the focus of instruction, but rather a necessary annoyance, and teachers work to help students perform well on them without focusing on them at length (Ladson Billings, 1995b). These teachers view disciplinary knowledge as necessary for helping students to un derstand sociopolitical issues (Gutstein, 2003; Ladson Billings, 1994). Furthermore, culturally responsive teachers believe a cademic achievement must be a goal for students as well as for themselves (Gay, 2000; Ladson Billings, 1995a). In addition to acade mic success, culturally responsive teachers want their students to become empowered socially, individually, and politically. Students are supported to become empowered academically and personally by teachers who nurture their confidence, courage, and will to act ( Bonner, 2011; Gay, 2000). They are also supported to develop self efficacy for learning tasks (Gay, 2000). Additionally, teachers support students to become leaders both within and outside the cla ssroom (Gutstein et al., 1997) and seek to support s tudents in developing their sense of personal and social agency (Gay, 2000; Gutstein et al., 1997). Another goal culturally responsive teachers hold is that they want all their students to develop cultural competence, a positive sense of ethnic identity a nd self esteem. In a PAGE 59 59 review of literature, Ladson Billings (1995b) conclude d that academic success of students of color tend ed to come at the expense of their ethnic identities, with students W (Fordham & Ogbu, 1986, p. 176), having neit her white friends nor friends who share their same cultural heritage. Culturally responsive teachers seek to avoid a negative self identity and loss of ethnic pride for their students, instead supporting students to develop cultural competence, an acceptan ce and affirmation of their culture (Ladson Billings, 1994, 1995b). R ather than learning according to European American cultural norms, students of color are encouraged to maintain their cultural integrity while simultaneously pursuing academic excellence (Gay, 2000, 2002; Ladson Billings, 1995a, 1995b). Teachers and students learn about, value, and praise cultural heritages (Gay, 2000), and s tudents are taught to challenge racist societal views of competence and worthiness of people of color ( Delpit, 2006), with cultural excellence (Ladson Billings, 1994). While this may look different from one classroom to the next, it may include lessons where students are allowed to write a rap song instead of a poem in English class or to learn about the mathematical algorithms developed by different cultures alongside traditional alg orithms. In doing this, culturally responsive teachers support students to self determined by racial and cultural variables and embedded in a social and historical c xpectations for students Culturally responsive teachers hold high expectations for all their students; they expect all students to succeed, rather than some succeeding and others not (Gay, PAGE 60 60 2000) No excuses for low achievement are tolerated (Gay, 2000, 200 2; La dson Billings, 1994, 1995b; Ware, 2006; Wilson & Corbett, 2001). Culturally responsive teachers believe that holding high expectations for students will support the development of intrinsic motivation to work hard (Wlodkowski & Ginsberg, 1995) and stu dents will in turn meet those high expectations (Gay, 2000; Ladson Billings, 1997). T eachers will thus nag, pester, and bribe their students to work hard (Ladson Bil lings, 1995b). They also treat students as capable (Gutstein et al. 1997; Ladson Billings, 1997) and challenge them intellectually and beyond curricular expectations regardless of their home life (Delpit, 2006; Ladson Billings, 1997; Sheets, 1995) in order to push Billings, 19 97, p. 704). academic strengths are identified, valued, and shared with other students (Delpit, 2006; Gay, 2000; Ladson Billings, 1995b), as well as u sed for instructional purposes In addition to academics, culturally responsive teachers hold hi gh expectations related to the persona (Gay, 2000). These teachers additionally view students from an asset rather than deficit based perspective. A deficit based and assumes that and that families (Gutstein et al., 1997). Rather than adopt this perspective, culturally responsive teachers value s g, be lieving, learning, and thinking (Ladson Billings, 1995; Nelson Barber & Estrin, 1995). They believe that l earning styles, not intellectual abilities, are what affect how students learn (Gay, 2002) and that c ulture affects the ways in which students be have, interact with others, and PAGE 61 61 learn, but this is viewed positive ly rather than negative ly (Gay, 2000; Ladson Billings, 1995). Thus, s foundation for future learning (Gay, 2000; Gutstein et al ., 1997; Ladson Billings, 1994). Finally, these teachers never blame students or families for low achievement by a student but rather provide supports for students to overcome the challenges and barriers to their achievement (Bondy & Ross, 2008; Ross et al., 2008; Ware, 2006). Teacher student relationships Teachers who instruct in a culturally responsive manner demonstrate care for and develop personal relationships with students. These te acher student relationships are equitable and reciprocal (Lads on Billings, 1995b). Teachers value students and work to build trusting relationships ( Bonner, 2011; Garza, 2009; Nelson Barber & Estrin, 1995 ; Peterek, 2009), getting to know their students well on both an academic and personal level (Bonner, 2011; Petere k, 2011), learning about their lives outside of school (Delpit, 2006), and treating students with respect (Garza, 2009; Gutstein et al., 1997 ; Nelson Barber & Estrin, 1995). They may also adopt other mothering behaviors, acting as es (Bonner, 2011; Lad son Billings, 1994; Ware, 2006). Also, these t eachers view students as human (Ladson Billings, 1997), see themselves in their students, and act as role models for their students (Gutstein et al., 1997). Culturally responsive teachers intentionally demonstrate care for their students (Gay, 2000; Garza, 2009; Ladson Billings, 1995b) S tudents from different cultures may perceive care in different ways students perceived providing instructional support as the key aspect of a caring teacher, whereas white students perceived a kind disposition as key. Thus, the way teachers demonstrate care toward students may vary from student to student which Garza PAGE 62 62 (2009) calls culturally responsiv e caring Furthermore, from a psychological perspective, enjoyment, effort, and self e fficacy (Sakiz, Pape, & Hoy, 2012). In addition to caring about students, culturally responsive teachers care about activities. Parents and families are valued, honored, respec ted (Delpit, 2006; Gutstein et al., 1997; Peterek, 2009; Wlodkowski & Ginsberg, 1995), and encouraged to be These tea chers work as allies to students and their families (Gutstein et al., 1997) and part ner with parent s to meet their classroom needs Home visits to get to know families be tter are common (Sheets, 1995), and t eachers are often a churches and community services (e.g. grocery stores and barber shops), as well as Billings, 1995a ; Peterek, 2009 ). Connecting with students in personal ways is important for supporting achievement motivation (Patrick, Turner, Meyer, & Midgley, 2003). eriences knowledge of content is addressed and used in instruction. Culturally responsive teachers value non school based forms of knowing in addition to school based knowledge (Nels on Barber & Estrin, 1995), and they experiences ( from both within and outside of school) as a foundation upon which they build learning experiences (Delpit, 2006; Gay, 2000; Gutstein et al., 1997; Ladson Billings 1997; Wlodkowsk i & Ginsberg, 1995). Instructional scaffolding for example, PAGE 63 63 using f and cultures is used to connect what students know to what they are learning at school (Delpit, 2006; Gay, 2000 ; Ladson Billings, 1997). integrated into instruction. Teachers in these studies were found to have respect for all cultures and believe d them to be great resources for teaching l earning (Gay, 2000, 2002; Gutstein et al., 1997) and content worthy of being integrated into the curriculum (Gay, 2002, p. 110). Cultures vary according to communication styles ; nuances ; discourse features ; use of logic, rhythm, and vocabulary; delivery ; the role of speakers and listeners ; intonation ; body movements ; music ; ways of dressing ; how children and ad ults interact ; and whether they prioritize community and cooperation over individualism (Gay, 2002; Ladson Billings, 1995a). Culturally responsive teachers invest time to learn deeply about the culture of their students and the community (Gay, 2000, 2002; Ladson Billings, 1997; Tate, 1995) and use w hat they learn to bridge the gap between home and school cultures (Ladson Billings, 1995b), meet the needs of diverse students (Gay, 2002), build relationships with students (Ladson Billings, 1997), and otherwis curriculum and instruction more reflective of and responsive to ethnic speaking, dressing, and expressing themselves music ally into their teaching and the curriculum (Gay, 2002; Ladson Billings, 1995a; Long, 2008). The purpose of this effort PAGE 64 64 to integrate culture into the curriculum is to maintain that culture and thwart negative effects of the dominant (i.e., white) culture ( Ladson Billings, 1994, 1995b ) In addition to cultural knowledge being integrated into the curriculum, culturally responsive teachers respect and often encourage students to speak in their native languages and dialects, incorporating them into the classroom dialogue ( Gutstein et al., 1997; Long, 2008; Wlodkowski & Ginsberg, 1995). Bilingualism is valued (Hondo et al., 2008; Gutstein et al., 1997; Sheets, 1995) and s tudents are supported to develop competenc e in both languages (Gutstein et al., 1997; Ladson Billings, 1995a; Long, 2008 ). Dual languages may be spoken in class and students are supported to be able to translate between the two languages (Gutstein et al., 1997; Ladson Billings, 1995a) so that they have the cultural capital that non English speakers lack (Gutstein et al., 19 97). For instance, Black English Vernacular (i.e., Ebonics) is treated as a separate language Billings, 199 5a). Furthermore, p rotocols of participation in discourse vary across cultures, a nd teachers address the different communicati on styles of their students (Gay, 2002). In contrast to many traditional classrooms, in culturally responsive classrooms, the teacher is not the ultimate holder of knowledge. Instead, s tudents and teachers shar e knowledge (Ladson Billings, 1995a ; Wlodkowski & Ginsberg, 1995), and the holder of power in the classroom is fluid between students and teacher (Bonner, 2011). Students sometimes act as the teacher (Ladson Billings, 1995a, 1995b) and are expected to be a he or she is the teacher (Gutstein et al., 199 7 ; Ladson Billings, 1995b). PAGE 65 65 Supporting students to develop a critical disposition Culturally responsive teachers encourage students to become critically conscious and social justice oriented. Teachers support students to engage in cultural critique (i.e., develop critical consciousness), in which cultural norms values, mores, and institutions that perpetuate or maintain social inequit ies are critiqued (Ladson Billings, 1994, 1995a, 1995b) and st udents are Sociopolitical and controversial issues (e.g., poverty, racism, powerlessness, hegemony, historical atrocities) are addressed in class to help students recognize, understand, and critique inequities; reflect on their lived experiences to understand how nd explain causes of human differences, such as why some people are rich while others are poor (Gay, 2002; Haberman, 1991; Ladson Billings, 1994, 1995b; Sheets, 1995; Wlodkowski & Ginsberg, 1995) They are also taught to critique mass media and popular cul ture portrayals of diverse cultural and ethnic groups (Gay, 2002; Ladson Billings, 1995b). Is sues important to students and their families (e.g. deforestation, racism, etc.) are addressed in class (Nelson Barber & Estrin, 1995). Content knowledge, and part icularly mathematics, is used as a tool to help students understand and change sociopolitical issues in their lives (Gutstein, 2003; Tate, 1995). Culturally responsive teachers do not believe engaging students in cultural critique and questioning the statu s quo are sufficient and they strive to help students develop a sense of agency to address issues they see in society. Students are taught that with knowledge come consequences and responsibilities necessitating that they take social action to promote equi ty and justice (Gay, 2002). T herefore, t eachers help student s PAGE 66 66 acquire the necessary tools to become active participants in society and develop into agents of social change and democratic citizenship (Gutstein et al., 1997; Ladson Billings, 1994, 1995a, 199 5b; Tate, 1995). This may involve supporting students in learning about local politics, write letters to government officials regarding an issue students are concerned about, or participate in a neighborhood clean up project. Teachers also help students to understand the difference between challenges to authority (such as that of their parent s) and intellectual challenges ( s uch as institutionalized racism; Gutstein et al., 1997; Ladson Billings, 1995b). Classroom environment Students in culturally responsi ve classrooms take an active role in their own learning. Their teachers believe that s tudents must be supported to become intrinsically motivated to succeed (Gay 2000; Ladson Billings, 1995b). Students are encouraged to t ake ownership of their own learnin held accountable for knowing, thinking, The culturally responsive classroom functions as a community of learners. Teachers indicate that a classroom focused on community and the development of a safe learning environment for students of color is crucial (Gay, 2002). The teacher thus creates an environment where the class functions as a cultural family, with teachers and students working as a team toward academ ic achievement (Delpit, 2006; Gutstein et al., 1997; Ladso n Billings, 1995; Sheets, 1995). Interdependence and group effort rather than independence and individual effort are v alued (Gay, 2000; Sheets, 1995), so s tudents work together as a community rather than as individuals, mentoring each other and working cooperatively and learning collaboratively (Gay, 2000; Ladson Billings, 1995b; Sheets, 2000; Wlodkowski & Ginsberg, 1995). Students are responsible not only PAGE 67 67 for their own success, but also for the succ ess of their classmates (Gay, 2000 2002; Ladson Billings, 1995b), and the c lassroom environment is not compet itive (Ladson Billings, 1995b). Culturally responsive teachers insist that all students demonstrate respect. This culture of respect is both betwe en teachers and students as well as among students, and it contributes to high academic achievement (Ross et al., 2008). Culturally responsive classroom management Discipline in a culturally responsive classroom is integrated within instructional practic based disciplinary style of parents, and discipline and pedagogy are interconnected (Peterek, 2009). Teachers act as warm demand ers when disciplining students (Bonner, 2011; Peterek, 2009; Ross et al., 2008 ). Warm demanding, a concept first introduced by Kleinfeld (1975) and a component of CRT Ross, 2008, p. 54). War remain gentle, caring and respectful toward students and do not employ the use of threats, coercion, or the creation of power struggles (Ladson Billings, 1997; Peterek, 200 warm demander may appear tough, harsh, or even mean (Bondy & Ross, 2008; Delpit, 1995; Ladson Billings, 1997; Peterek, 2009; Ross et al., 2008; Ware, 2006). Students, however, do not interpret this as a lack of caring. In fact, the warm demander has been Billings, 1997, p. 703) and some students claim they want a teacher that takes control, exerts authority, and who pushes, disciplines, and encourages them (Ware, 2006; Wilson & Corbett, 2001) because it makes them feel important (Wilson & Corbett, 2001). In other words, these teachers PAGE 68 68 insist though firmly and respectfully that their students meet the high academic and behavioral expectations they hold f or students (Bondy & Ross, 2008; Ross et al., 2008; Ware, 2006). They do this by making their expectations clear; repeating, reminding students about, and reinforcing those expectations; and responding in a calm and consistent way with consequences to cont inued misbehavior (Ross et al., 2008). T he 142). Thus, caring relationships as well as an in sistence on a respectful learning environment and that students meet the high expectations set for them characterize a warm demander stance (Bondy & Ross, 2008; Dixson, 2003; Ware, 2006; Ross et al., 2008) that is part of CRT. High expectations, relationsh ips, and the learning environment were described previously. Instruction Culturally responsive teachers are focused on teaching for understanding. They work to ensure all students learn the basic skills of the dominant society (Delpit, 2006). Basic skills serve only as a foundation however, expectations for students additionally include critical thinking, problem solving, creativity, and higher level thinking skills (e.g., discussion, analyzing, identifying themes, ex amining concepts, and discourse) ( Habe rman, 1991; Ladson Billings, 1997; Sheets, 1995). Furthermore, s tudents are actively engaged in learning (Haberman, 1991; Sheets, 1995) and conceptual understanding is required (Tate, 1995), so students are supported to make connections between concepts ra ther than learning i solated facts (Haberman, 1991). To help students learn conceptually and for understanding, culturally responsive teachers use multiple instructional strategies to engage students and attend to different PAGE 69 69 needs. These multiple instruction al strategies may include discussion, peer teaching, sharing, learning communities, and problem solving (Delpit, 2006; Wlodkowski & Ginsberg, 1995). Students are often given choice s about what 1991), and i nterdisciplinary units and teacher collaboration may occur (Gay, 2000). Students frequently work cooperatively in small heterogeneous groups (Gay, 2002; Haberman, 1991 ; Ladson Billings, 1995b; Sheets, 1995), t echnology is incorporated int o instruction (Haberman, 1991), and s tudent s are provided opportunities to construct their own knowledge (Gutstein et al., 1997). In addition, s tudents are taught more than just content ; values, skills, and the cultural capital necessary for school success (e.g., test taking strategies, study skill s, note taking, time management) and for participation in the society at large are explicitly taught as well (Gay, 2000). By multiculturalizing their instruction, teachers often adapt their instructional strategies to match to the learning styles of studen ts of color (Gay, 200 0, 2002). The content in culturally responsive classrooms is connected to the real world. Relevant world issues that students care about and that frequently reflect ed in problems (Haberman, 1991), and stu dents are allowed to use their real world knowledge to solve real world problems (Tate, 1995). There is purpose Ginsberg, 1995). Culturally responsive classrooms also emphas ize multicultural content in the curriculum, and culturally diverse curricula are used (Gay, 2000, 2002). Finally, culturally responsive teachers use varied, authentic, and formative assessment. In these classrooms, authentic knowledge is valued (Gay, 200 0). S tandardized tests are considered to be only one way to measure academic PAGE 70 70 achievement and do not dominate the focus of instruction (Ladson Billings, 1995). The many types of assessment used by the teacher includ e formal and informal, formative and summa tive, and authentic (Wlodkowski & Ginsberg, 1995). Students are given opportunities to revise and redo their work (Haberman, 1991) and are often encouraged to choose which content and assessment method they prefer, which may include tests, portfolios, peer feedback, observations, and self assessments (Gay, 2000; Ladson Billings, 1995b ; Wlodkowski & Ginsberg, 1995). They are also allowed multiple options for practic ing and demonstrating competence knowledge, and skills (Gay, 2000; Wlodkowski & Ginsberg, 199 5) tactile, auditory), individual and group, competitive and cooperative, active participatory In fact, culturally responsive t eachers strive to learn about consideration when assessing students (Delpit, 2006; Gay, 2000). Assessment processes are intended to types of knowledge, including values, attitudes, feelings, experiences, and ethics. (Gay, 2002; Nelson Barber & Estrin, 1995 ; Wlodkowski & Ginsberg, 1995). Students also sometimes participate in critiquing b iases in tests and testing formats (Wlodkowski & Ginsberg, 1995). There few co nceptions of CRT in this literature that focus directly on CRT in the context of specific gatekeeping subjects such as mathematics rather than effective practices in general (Peterek, 2009 ; Silver et al., 1995). For instance, while some of the participants in Ladson her research does not focus on mathematics teaching per se (Silver et al., 1995). One PAGE 71 71 researcher has studied effective mathematics teachers of students of color in a quest to fill th is gap in the literature and define culturally respon sive mathematics teaching (CRMT) ( Bonner, 2011; Peterek, 2009) Culturally Responsive Mathematics Teaching In an effort to understand CRT in the context of mathematics, Emily Peterek Bonner (Bonner, 2011 ; Peterek, 2009) chose to investigate how highly effective mathematics teachers in high poverty schools with large numbers of students of color structure the instructional practices and interactions in the classroom and establish learning environments that promote student academic success (Peterek, 2009). Similar to the study conducted by Ladson Billings (1994, 1995a), Peterek used a community nomination process to identify an elementary teacher, Gloria Merriex, who was considered highly effective by both s chool personnel and the community. Ms. Merriex was raised in the same neighborhood as were her students, attended the same church, and shared the same cultural heritage. Over the course of several months, Peterek conducted semi structured interviews with M s. Merriex, as well as observed her class several times per week, collected student artifacts, and held informal conversations with students, parents, and other teachers. Using a grounded theory approach that included constant comparison and member checkin g, Peterek identified a model for culturally responsive mathematics teaching (see Figure 2 1) that describes four cornerstones of CRMT: knowledge, communication, relationships/trust, and constant reflection and revision. These cornerstones are dynamic and fluid; each cannot be considered in isolation of the others: Knowledge. This cornerstone includes knowledge of teaching (i.e., pedagogical knowledge and pedagogical content knowledge) and mathematics content ( i.e., content knowledge). Knowledge of students is also essential, which was gained PAGE 72 72 through communication with students and which allowed the teacher to connect with students to better use her knowledge of content and pedagogy Communication. Perhaps because she shared the same ethnic and cultural heri tage as her students, Ms. Merriex was able to communicate with them using discourse patterns with which they were familiar. This was evidenced in the frequent call and response and rhythmic patterns used during instruction, as well her use of specific vern acular in the conversations she held with students and her warm demander pedagogy. Relationships/trust. Ms. Merriex built strong relationships with school community members, families in the neighborhoods near the school, and her students. The students and community knew she cared for them and felt they could trust her Constant reflection and revision. confused, Ms. Merriex stopped t he lesson and probe them to get at the sources of their confusion or misconceptions In addition to these cornerstones, Peterek (2009) describes a cycle of pedagogy and Ms. Merriex wrote engaging and reflective lesson s, and students were constantly moving from one concept to another, responding in unison, dancing, or engaging in call and response chanting, which provided little time for them to act out. T hus, edagogy and discipline occurred almost simultaneously and we re sometimes indistinguishable and t he four cornerstones constantly influenced this cycle Finally, as in Figure 2 1, students are the focal point of CRMT. They are most affected by the pedagogy and discipline cycle, but should they fall out of that cycle (such as when difficulties at home cause a student to feel dis engaged at school), the other cornerstones work to bring the student back to center (e.g., perhaps th r ough her relationship with the student, Ms. Merriex was able to re engage him/her). This mo del was intended to be a working theory that fit within the larger framework of CRT rather than a static depiction of CRMT or a separate aspect CRT (Peterek, 2009). PAGE 73 73 Figure 2 1. A model of culturally responsive mathematics teaching. From Culturally respon sive teaching in the context of mathematics: A grounded theory approach (p. 120), by E. Peterek, 2009, ProQuest Digital Dissertations database (Publication No. ATT 3385979). This line of research continue d with another study that held the goals of refining the theory of CRMT described in Peterek (2009) as well as investigating the commonalities among culturally responsive mathematics teachers. Bonner (2011) identified two more such teachers, again using a nomination process. The first, Ms. H., worked at a w hite majority middle school but her students were mostly of color (particularly Hispanic, some African American), and, unlike Ms. Merriex, she had a different ethnic background from her students. Her instructional focus in the classroom was on differentiat ion. The next teacher, Ms. A., worked at a largely Hispanic Title 1 PAGE 74 74 school and her students were reflective of the student population. She is a foreign national and focused on technology and innovation in the classroom. Using the same data collection and analysis methods as Peterek (2009), and by combining these data with the data from the previous study of Ms. Merriex, Bonner revised her model of CRMT (see Figure 2 2). Figure 2 2. A revised model of culturally responsive math ematics teaching. Adapted from Unearthing culturally responsive mathematics teaching: Using grounded theory to deconstruct successful practice (p. 11), by E. P. Bonner, 2011, Fifteenth Annual Association of Mathematics Teacher Educators Conference, Irvine, CA. While this model includes the same four cornerstones as those described by Peterek (2009), there are a few major differences. Relationships/Trust was moved to the center of the figure to emphasize that this cornerstone is a foundational aspect of Communication Relationships/Trust Power Knowledge Power Pedagogy and Discipline Reflection And Revision PAGE 75 75 CRMT Additionally, the element of power was added. Bonner (2011) describe d that the locus of power in the classroom was fluid (moving between teacher and student at any moment), and that the empowerment of students was a main goal of these three teachers. Pow er was at the forefront of instruction, and teachers purposefully provide d opportunities to support students in develop ing their sense of empowerment. Furthermore, Bonner (2011) identifie d several themes across her participants related to the ways in whic h they embod ied CRMT : Cultural connectedness; A deep understanding of mathematics (content knowledge); Knowledge of self; Warm demanders (how this disciplinary approach manifests differed across classrooms); Mothering; Frequent movement; Fluidity/flexibili ty; and Bonner emphasized that the way in which these elements of culturally responsive mathematics teaching played in individual classrooms differed and that these themes and the model were not intended to be static. She further suggested that there should respond to them will vary. While these two models of CRMT provide us with an understanding of how teachers in a particular context (i.e., mathematics classrooms) might embody cultural responsiveness, there is nothing in either to suggest that how these teachers PAGE 76 76 approach ed the education of students of color was unique to the mathematics classroom. Building relation ships and trust; communicating; reflecting on and revising instruction; holding a deep knowledge of content, pedagogy, and students; and pedagogy and discipline are all important aspects of CRT that might manifest in any classroom. Indeed, Ms. Merriex desc ribed how the rhythmic teaching so characteristic of her mathematics instruction also played a large role in her teaching of vowels and vowel songs to younger students (Peterek, 2009) T his model, therefore, is perhaps not characteristic of culturally resp onsive mathematics teaching but of culturally responsive teaching in general. Furthermore, the nature of the mathematics instruction in these culturally responsive classrooms remains unclear. Bonner (2011) describe d each class as (p. 4) but fail ed to elaborate on what that mean t Wa s the instruction standards based and focused on conceptual understanding? Or, d id we re engaged in challenging, yet still procedurally driven, mathematics ( such as multi step exercises)? When prompted to clarify what was meant by mathematically rigorous, Bonner was unable to give a clear description, suggesting instead that it differed from class to class. Additionally, when asked whether conceptual understan ding of mathematics was a goal of these teachers, school mathematics as evidence (E. P. Bonner, personal communication, January 28, 2011). Achievement tests, however, a re not a main focus of culturally responsive classrooms ( Ladson Billings, 1995b ), and standardized tests and high grades may not necessarily be indicators of conceptual mathematical knowledge This leaves one PAGE 77 77 unsure of the nature of the mathematics teachin understanding in these culturally responsive classrooms. It may be the case that culturally responsive mathematics teachers instruct in a more traditional manner, focusing on drill and memorization without emphasizing un derstanding or it may be that they use standards based instruction with an emphasis on teaching mathematics for understanding (or perhaps there are teachers who serve as examples of both). Consider also that Ms. Merriex was able to use discourse patterns with which her students were familiar (Peterek, 2009), but it is unclear whether she also supported them to learn the standardized mathematical language and discourse described by ccess. experiences is a necessary part of mathematics education (Banks et al., 2005; Forman, 2003; Gay, 2002; Gutstein, 2003; Gutstein et al., 1997; Ladson Billings, 1994, 1995a, 1995 b, 1997; Peterek, 2009; Tate, 1995; Zevenbergen, 2000), particularly within the context of standards based instruction (Gee, 2008; Lester, 1994; Lubienski, 2002; Mohatt & Erickson, 1981; Zevenbergen, 2000). Studies to further refine the model of CRMT and t hat also describe the nature of the instructional practices particular to mathematics teaching that occur in concert with CRMT are needed Additionally needed are studies that examine whether CRT or CRMT supports students to overcome the obstacles of mathe matics reform described by Lubienski (2000a, 2000b, 2002) T his brings up the question however, of whether it is even possible to integrate CRT with standards based instruction in a mathematics setting. In the following section, the work PAGE 78 78 of Gutstein and c olleagues, which provides a depiction of CRT within reformed mathematics classrooms, will be examined Culturally Responsive, Standards based Mathematics Teaching Ear lier sections of this chapter described two approaches to teaching that both hold potentia l for helping all students to engage with mathematics content and support them to succeed academically. Standards based teaching provides teachers with guidelines related to mathematics instruction, but does not address broader instructional issues address ed by CRT such as teacher student relationships, beliefs about students, developing a critical disposition, and classroom management. On the other hand, the literature on CRT and CRMT lacks a clear description of the mathematics instruction of culturally r esponsive teachers and the mathematical practices that occur within their classrooms. Thus, these two approaches are described separately in the literature and seem disconnected, but they may be complimentary. Some of the practices advocated by literature on CRT are theoretically consistent with standards based teaching, for classroom environment in which students are held accountable for their own learning and are expected t o question, reflect, and share their thinking; and using multiple instructional strategies. Few studies exist, however, that specifically examine both culturally responsive and standards based mathematics teaching. In this section, we explore the work of G utstein and his colleagues, who provide a rare glimpse into this area. In an examination of the nature of culturally responsive mathematics teaching in the context of reform, Gutstein and colleagues (1997) conducted a study in a Southwestern U. S. middle school where over 96% of the students were Mexican PAGE 79 79 American, 99% were considered to be from low income families, and nearly half (46%) qualified for English as a Second Language services. At the time of the study, Gutstein worked closely with the middle sc hool teachers in a professional development program on the reform based Mathematics in Context ( MiC ) curriculum and helped them to foster practices congruent with CRT The researchers ( Gutstein et al. 1997) explore d how teachers buil t l mathematical knowledge through their connections with students and knowledge of their cultures and sought to describe the nature of CRT in a Mexican American context. Figure 2 3. CRMT and Standards based instruction (Part 1). Adapted from Culturally r elevant mathematics teaching in a Mexican American context (p. 719), by E. Gutstein, P. Lipman, P. Hernandez, & R. de los Reyes, 1997, Journal for Research in Mathematics Education, 28 Data collection methods included the use of field notes from classroom observations, open ended interviews, artifacts, group conversations, and teacher journals. These data were transcribed and analyzed using a grounded theory approach. What resulted was a three part model of instruction that connects CRT with the NCTM (1989 1991, 1995) Standards documents (see Figures 2 3 and 2 4 ). The first component of the model contrasts critical mathematical thinking with thinking critically PAGE 80 80 Critical mathematical thinking (e.g., explaining and justifying, theorizing, and developing arg uments) falls under the NCTM (2000) C ommunication and R easoning and P roof standards. Thinking critically is a notion in the literature on critical pedagogy and relates to critical consciousness. Teachers who adopted this stance challenged students to quest ion the standard curriculum, consider multiple perspectives, construct their own knowledge, participate in democratic practices, develop a sense of agency, and, as one This critical stance applied not only to mathematics instruction but also to the broader world, and particularly to situations where students were marginalized or disempowered. The authors identified several teachers who encouraged both forms of thinking in their students. Figure 2 4 CRMT and Standards based instruction (Part 2) Adapted from Culturally relevant mathematics teaching in a Mexican American context (p. 727), by E. Gutstein, P. Lipman, P. Hernandez, & R. de los Reyes, 1997, Journal for Res earch in Mathematics Education, 28 teachers use d PAGE 81 81 ng point for lessons in line with a constructivist view of teaching and learning as well as with standards based instruction (NCTM 1989, 1991, 1995). Other teachers added to this, also using the cultural knowledge and experiences of students to enrich the curriculum. This is a key element of CRT (Gay, 2000, 2002; Ladson Billings, 1994, 1995b). Teachers thus used tionship building for some English, fostering a sense of cultural and linguistic power and pride (i.e., cultural competence) The third component of the model cultural and e xperiential knowledge, contrasts a orientation. Those teachers found to hold a deficit orientation may have become familiar an culture and treat ed it as stagnant and folkloric. Additionally, they viewed families as providing insufficient support ficit orientation was characterized by a failure to challenge students academically, perhaps believing that by simplifying instruction, they were helping students who struggle. On the other hand, teachers who adopted an empowerment orientation taught their students more than just mathematics. Teachers who held high expectations, believed every child could learn, and challenged them academically characterized such an orientation. They also understood the fluid, evolving nature of culture that is shaped by pe PAGE 82 82 adversity, oppressed people can develop strengths. T hey worked in solidarity, acting as allies to students and their families rather than saviors of students Finally, these teachers did not enable students but rather supported them to become empowered (Gutstein et al., 1997). The empowerment orientation aligns with the Equity Principle proposed by NCTM (2000) and is consistent with a culturally responsive approach (Gay, 2000; Ladson Billings, 1994, 1995b). The authors conclude d that while aspects of standards based and culturally responsive teaching we re related, one d id not imply the other (Gutstein et al 1997). For instance, t eachers may help students to develop critical mathematical thinking without encouraging a c ritical disposition for effecting social change Also, these perspectives build upon student knowledge, but the type of knowledge valued by each perspective differs. Thus, the connections between reform and CRT do not manifest unless te achers actualize th o s e connections themselves. teachers engaged in (e.g., were problem solving and multiple representations a common part of instru ction?), nor how the approaches to teaching that they investigated supported students to gain access to mathematical knowledge. While this model suggests that CRMT and reform based teaching can be integrated, the effects on success a nd understanding are unclear. In a later study, Gutstein (2003) taught mathematics for social justice by supplementing a standards based curriculum with problems focused on social justice issues a nd offered some insight into the question of achievement T he researcher explore d the use of a standards based curriculum coupled with projects based on social PAGE 83 83 justice issues He also examined the effects consciousness and specific mathematics related objectives. Teaching f or social justice involves supporting students to develop sociopolitical consciousness or conscientization a sense of agency, and positive social/cultural identities. Teaching for social justice, then, is congruent with CRT but does not necessarily focus on using mathematical objectives for students in this study included reading the world with mathematics (i.e., using mathematics to develop conscientization), developing mat hematical power (a notion similar to the strands of mathematical proficiency), and changing their dispositions toward mathematics. The goals for the project were twofold: to explore (a) the nature of teaching and learning mathematics for social justice, an d (b) the relationship of that process to a Standards based curriculum. The focus of was on and the curriculum used, not his pedagogy Gutstein, acting as both teacher and researcher during this study, taught hon ors mathematics to 28 Latino, immigrant students whose fam ilies were working class during their seventh and eighth grade years. He conducted a teacher inquiry using semi ethnographic research methods (e.g., participant observation, surveys, textual documen t analysis, informal student interviews) and engaged in systematic, critical self inquiry. Textual data were analyzed using open coding, and codes were then combined into themes. Gutstein taught using MiC the same Standards based curriculum with real worl d problems used in Gutstein et al (1997). He was concerned the problems were not relevant to students, and thus supplemented the curriculum with projects PAGE 84 84 focused on social justice issues such as racism, gentrification, and poverty. The projects were probl em based, and it was their context, rather than the mathematics, that made them social justice oriented 2 Students were encouraged to solve problems in ways that made sense to them (i.e., use invented strategies), often wrote explicitly about the mathemati cs they used in completing the projects, and examined and (Gutstein, 2003) Gutstein (2003) found that the students shifted from not having a critical of previous been taught and what they thought they knew, and used mathematics to make sense of these ideas. Students realized the value of mathematics for both understand ing and critiqu ing the world, felt mathematically able to understand inequities, and changed their views of the nature of mathematics. Gutstein also suggest ed that 27 of 28 students showed increases in mathematical power: they were able to generalize, solv e non routine problems, explain and reason, and were more confident in their mathematical abilities. They additionally performed well on conventional measures of mathematics success: all passed the eighth grade standardized test, and 15 went on to magnet h igh schools. Finally, most students developed more positive dispositions toward mathematics. Considering the second goal of the study, Gutstein (2003) found that the standards based curriculum potentially support ed a social justice pedagogy, but there we r e some necessary conditions of the curriculum. MiC use d problems with real life 2 Examples of such problems can be found in Gutstein & Peterson (2006). PAGE 85 85 contexts and encourage d multiple perspectives and invented strategies. Gutstein argued that a safe classroom culture where students could openly discuss social justice issues w as important, and while MiC explicitly address the classroom culture, the curriculum did encourage an inquiring habit of mind that play ed a supportive role in creating an environment where students we re comfortable questioning assumptions and power dynamics. Finally, coherence between curricular aspects wa s necessary, and Gutstein worked to ensure the big ideas of projects and MiC aligned well overall. This study is useful in understanding how to supplement a standards based mathematics curriculum t o incorporate social justice teaching, and the results are encouraging. Students showed deep understanding and continued to thrive after leaving already successful mathemat ics students before taking his class. L iterature suggests social justice issues be addressed in classrooms with high numbers of students of color results may not be generalizable to traditionally low achieving students of color. It is uncle ar whether his high achieving students w ould have struggled with the did or whether they would have experienced such success mathematically Furthermore, Gutstein (2003) only focus ed on the curriculum used and did not describe the pedagogical aspects of his teaching. This is a limitation of the study that does not allow one to understand what went on in his classroom over the two years (e.g., what was the nature of the dis course and classroom norms? Did students work individually or in groups on the projects?). In fact, as indicated previously, Boaler (2002) suggested that the pedagogy a teacher adopts is more important than the particular curriculum PAGE 86 86 used. More studies that examine the pedagogy adopted by teachers t o support struggling students particularly those of color and those who live in poverty to engage with mathematics are needed. Conclusion As described earlier, low income students of color often struggle to su cceed in school mathematics (Davis & Martin, 2008; Dewan, 2010; Post et al., 2008; Rothstein, 2002; Tutwiler, 2007) and their achievement levels are far below those of their white, middle class peers (Banks et al., 2005; Brantlinger, 2003; Gutstein et al., 1997). The reform movement in mathematics, which emphasizes standards based instruction, is the predominant strategy suggested in mathematics literature to support these struggling students (see NCTM, 2000). Some research suggests that this teaching appro ach may help to address issues of equity and narrow the achievement gap (Boaler, 1997, 1998, 2000, 2002; Brown et al., 1996; Post et al., 2008; Silver et al., income students enjoyed th e nontraditional mathematics instruction they received. Other research, however, points to problems with standards based teaching. Low income students sometimes not only seem to prefer traditional mathematics instruction (Lubienski, 2000b), but they also r eport feeling less confident and struggle to participate in classroom discourse in reformed mathematics classrooms (Lubienski, 2000a). There may be differences in how teachers implement standards based instruction that account for these discrepancy in find ings. For instance, the ways in which some students have learned to communicate and behave outside of school conflicts with how their teachers expect them to communicate and behave in school (Gee, 2008; Lubienski, 2002; Zevenbergen, 2000), and perhaps some teachers explicitly support their students in PAGE 87 87 learning the school culture in a way that leads to a more successful mathematical learning experience. This attention to culture is not a characteristic of reform mathematics, but rather of CRT. Culturally res ponsive teaching is a pedagogical approach that takes into account students to become empowered and successful academically, socially, personally, and politically (Gay, 2000; Ladson Billings, 1994). Studies of effective teachers of low income students of color indicate that these teachers adopt a culturally responsive approach (Bonner, 2011; Ladson Billings, 1994, 1995a, 1995b; Gay, 2000, 2002; Peterek, 2009). This appro ach is further characterized by an insistence on academic achievement and appropriate behavior, as well as explicit teaching of the rules, roles, norms, and other types of capital necessary for school success (Gay, 2000; Ladson Billings, 1994; Ross et al., 2008). Most studies on CRT, however, are not specifically focused on effective mathematics teaching (Silver et al., 1995; Peterek, 2009). One series of studies (Bonner, 2011; Peterek, 2009) sought to address this dilemma by examining the culturally respon sive practices of mathematics teachers and describing culturally responsive mathematics teaching. In spite of this, this series of studies does little to specifically characterize the mathematics instruction in culturally responsive classrooms and it is un clear if standards based or traditional teaching was occurring. Further, it would be helpful to understand what specific actions if any that culturally responsive teachers take to help students to overcome their difficulties with reform mathematics ins truction. PAGE 88 88 Gutstein and colleagues (1997) suggested that standards based instruction and CRT can occur simultaneously, but teachers must actualize the connections between these two teaching approaches themselves for this to occur. The teachers in that stud y received specialized training, and the literature does not indicate whether classroom teachers who do not receive such training are able to actualize those connections on their own. Furthermore, it brings up the question of whether actualizing the connec tion between standards based and culturally responsive teaching is necessary for effective teaching of low income students of color, or if one of these pedagogical techniques is sufficient for student achievement. Peterek (2009) and Bonner (2011) describe a model for culturally responsive mathematics teaching and Bonner (2011) suggests that this model also needs to be further developed. One way to do this includes examining the practices of teachers of students of color and determining whether their instruction contains elements of both reform and culturally responsive teaching. Another concern such a pedagogical approach on student achievement. A later study (Gutstein, 2003) suggested that a reformed curriculum coupled with some elements of CRT (specifically, teaching for social justice) may support the mathematics achievement of students of co lor and improve their dispositions, but the researcher not the regular classroom teacher was providing the instruction, and the students were in the honors track. This dissertation study seeks to understand whether classroom teachers who are effective with low income students of color adopt a reform based approach, a culturally responsive approach, or an integration of these approaches. I also seek to fill in some of PAGE 89 89 s pecifically on the mathematical practices in classrooms with a high percentage of students of color. By analyzing their instructional practices and perspectives, the goal is to determine whether their instruction is characterized by elements of reform, cul tural responsiveness, both, or neither. This will help the mathematics education community to better understand the instructional practices teachers adopt that contribute to the success of their traditionally underachieving students of color. PAGE 90 90 CHAPTER 3 ME THODS Overview The purpose of this collective case study was to understand how teachers who are succ essful with low achieving students of color and living in poverty supported their students in learning mathematics. The following questions guided this inve stigation: How do teachers identified as highly effective with students of color living in poverty help their students to engage with mathematical content? o o that are important for engaging their students with mathematical content? based instruction and culturally responsive teaching? This study adopted a naturalistic approach and util ized participant observation methodology within the constructivist paradigm (Hatch, 2002). Two teachers were mathematics curriculum specialist. Data were collected through classroo m observations and interviews with teachers. These data were analyzed using qualitative methods to identify themes. Methodology This dissertation perspectives and practices that use d a partici pant observation methodology undertaken within the assumptions of the constructivist paradigm (see Hatch, 2002). Constructivism assumes that individuals build knowledge through interactions with the world and that their experiences and particular perspecti ves are subjective (Crotty, 1998). T his paradigm necessitates the use of naturalistic qualitative research methods (Lincoln & PAGE 91 91 Guba, 1985). This perspective holds the goal of capturing naturally occurring activities in their natural settings because these m extended periods of time interviewing participants and observing them in their natural settings in an effort to r econstruct the constructions participants use to make sense of Thus, p articipant observation was selected as the methodology for this study which allows data to be collected within social settings and enables the researcher to concentrate on a particular area of interest in this case, instructional practices. Data collec tion strategies for participant observation fieldwork including interviews and direct observation with field notes were used for this study The interviews provide a means for the researcher to engage with participants in the coconstruction of a subjecti ve reality to better understand their perspectives. Direct observation enables the researcher to capture the cultural knowledge of participants but perspective from intervie ws (Hatch, 2002) Procedure Participants Identification of participants Participating teachers were purposefully selected from a pool of middle or high school mathematics teachers identified as highly successful in order to represent critical case samples which, according to Patton (as cited in Hatch, 2002), include individuals pre dominantly low income students of color and (b) identified as highly successful with PAGE 92 92 specialist nominated participants for this study based upon student demographic infor mation and achievement test scores. She described each of the nominees as highly successful with the student population in question. broadly defined. First, highly effective teachers are those that have demonstrated ability to habitually secure higher than average performance from traditionally underperforming students on measures of mathematics achievement. tandardized test and the standardized end of course (EOC) examinations administered by the State were used as measures of mathematics achievement. The goal wa s to identify teachers of students who previously demonstrated low achievement in mathematics on the FCAT or EOC but who, with the help of the teacher, made higher than the average gains in the county on the exam While there are certainly other ways to measure teacher effectiveness solving abilities, motivation, attitudes tow ard mathematics engagement ) successful performance on the FCAT and the EOC is necessary for students to graduate from high school and thus it is crucial that teachers make strides in preparing students for the se standardized tests The teachers nominate d by the secondary mathematics curriculum specialist were invited to participate in the study. From the volunteers, two were selected as participants for the proposed study. They were provided with an informed consent form (Appendix A) that provided partic ipants with a description of the research and required their signatures. PAGE 93 93 Description of participants Several teachers were nominated and two subsequently volunteered for participation in this study, Ms. J and Ms. W (all names of people and places are pseud onyms) Ms. J was a 25 year old Hispanic female in her third year of teaching middle school mathematics She was born in Minnesota but moved frequent ly as a child and has lived in several states within the U.S. as well as other co untries including Algeria, is in hospitality and tourism management with a minor in business from State University a large public institution in the same town in which this study took place. Upon graduation, Ms. J became a substitut e teacher; she was later hired as a long term substitute as a high school English teacher in the ESOL program at one of the schools at which she substituted. The following year she worked as a long term substitute mathematics teacher at Southside Middle Sc hool, where she was subsequently hired as a full time mathematics teacher. During the course of this study, Ms. J held a temporary teaching certificate and was taking courses to become alternatively certified as a middle school mathematics teacher. One hun on the FCAT her first year teaching, and she was nominated for the Teacher of the Year award at her school while this study took place. During the course of this study, Ms. J taught seven th grade mathematics and eighth grade advanced mathematics, and she co taught eighth grade mathematics. Her first period seventh grade mathematics class served as the context for this study. Fourteen of the seventeen students in this class were of color. D ata were not available them were on free or reduced lunch. PAGE 94 94 Ms. W was a 31 year old Caucasian female in her ninth year as a mathematics teacher and described herself as a country girl who grew up in a small town that had little cultural diversity. She attended the same large State University as Ms. J, and pursued a five mathematics with a minor in education a s an undergraduate student and received a Ms. W was the County Teacher of the Year in 2010. Ms. W has taught Algebra IA, Algebra I (regular and honors), and Geometry (regular and honors), and during the course of this study she taught both Algebra I and Algebra I Honors. Her third period block Algebra I/Intensive Math course served as the context for this study. Twenty of the twenty one students in the class were black Data were not available on the socioeconomic statuses of these students. Setting The district in which this study took place was located in an ethnically and socioeconomically diverse county in Florida. The student population was just above 27,000 students and was 49% White, 36% Black, 6% Hispanic, 5% Multiracial, 4% Asian/Pacific Islander, and less than 1% American Indian/Alaskan Native. The graduation rate was approximately 65%. Forty four percent of students qualified for free or reduced lunch. This study focused on two teachers at two schools wi thin the district. One school, Southside Middle School, enrolled approximately 1,020 students. The student population was 51% White, 32% Black, 9% Hispanic, 5% Multiracial, 3% Asian/Pacific Islander, and less than 1% American Indian/Alaskan Native. Forty o ne percent of students qualified for free or reduced lunch. Southside housed the Cambridge PAGE 95 95 programs. The second school, Maya Angelou High School, enrolled approximat ely 1,500 students. The student population was 26% White, 60% Black, 3% Hispanic, 3% Multiracial, 9% Asian/Pacific Islander, and less than 1% American Indian/Alaskan Native. The graduation rate was below the district average at approximately 48%. Forty thr ee percent of students qualified for free or reduced lunch. Maya Angelou High was home to two district magnet programs: the International B accalaureate Program and a culinary arts program. Data Sources Primary data sources for this study included direct ob servation with field notes as well as informal and formal interviews. Classroom artifacts provided by the teachers served as a secondary data source. Data Collection The majority of data collection for the present study took place from October to December 2011, with additional interviews in February 2012. Classroom observations Observation was used as one method of data collection. This study focuses in part on the perspectives of teachers, and Hatch (2002) states that observation allows the researcher the opportunity to understand the context being studied from the perspective of the participant. Observation additionally allows the researcher to gain a deeper understanding of the context and learn about that which was not mentioned during interviews (Hatch 2002). Two observations occurred each day, so that one class period taught by each participating teacher was observed over the course of five and one half weeks. The goal was to reach saturation, or redundancy. Lincoln and Guba (1985) PAGE 96 96 described redundanc y as the practice of simultaneously collecting and analyzing data over an extended period of time until findings become repetitive and then collecting observational data once more for good measure. The total number of classroom observations was 33; Ms. J observed 18 times. of the classroom context as well as of the actions and conversations that occurred during the class. Particular care was taken during instruction and conversations to record exactly rather than summarize or paraphrase what was said. My initial impressions, interpretations and feelings were also recorded, but were bracketed so that they were clearly distinguishable from the raw field notes. The goal when taking field notes is to create accurate descriptions of what was observed, but these descriptions are usually incomplet observation for the day was completed. This process involved rewriting the raw field notes into an organized format (i.e., research protocol) while simultaneously expanding on the details to create as accurate a record as possible of everything that was observed. Because I was working from memory as well as from the raw field notes, the research protocols were filled in as soon as possible after an observation occurred, and always occurred before the ne xt observation in that classroom. Similar to the note taking process during observations, any impressions, interpretations, and feelings that occurred to me during this filling in process were bracketed in the protocol to separate them from the descriptive data (Hatch, 2002). PAGE 97 97 Analysis of the research protocols from observations began as the data were collected. These observations informed later interviews with the participants. For instance, I asked the teacher to explain specific classroom interactions or instructional decisions she made during the observations. Furthermore, artifacts (e.g., teacher handouts) were collected during classroom observations and served as a secondary data source; they were not explicitly analyzed but instead were used to suppor t my understanding of the instructional practices being observed. Interviews Interviews with participating teachers focused on their perspectives about practice and teaching philosophy. There were three formal interviews per teacher using a semistructured interview protocol (Hatch, 2002). These formal interviews lasted approximately forty five minutes to one hour and focused on (a) background and general teaching practices and perspectives; (b) knowledge of mathematics teaching and how the teacher used this knowledge in the classroom; and (c) knowledge of culture and CRT and how the tea cher use d this in the classroom (see Appendices B, C, and D for interview protocols). Some interview questions have been adapted from Adkins (2006) and Peterek (2009). These f ormal interviews were scheduled at a time convenient for the teacher, and were audio recorded and transcribed verbatim. Additionally, there were multiple informal interviews that occurred throughout the course of the study as needed, typically immediately before or after an observation. The informal interviews allowed me to probe the teacher about something I observed during the class period. As an example, I asked a teacher to explain her thinking behind a particular activity, why she responded in a certa explain classroom procedures that were unclear. Thus, the questions for the informal PAGE 98 98 interviews varied depending on what aspects of the observation I felt need to be clarified or further investigated (see Appendix E). Field notes were taken during these informal interviews. After each informal interview, these field notes were reviewed and expanded in the same manner described in the section on classroom observations. Summary of data collection Over the course of thi observations. Additionally, three formal interviews and multiple informal interviews were conducted with each teacher. Ar tifacts, including handouts distributed to students, were also collected throughout the study. Data Analysis As suggested by Hatch (2002) and Lincoln and Guba (1985), I kept a daily research journal throughout the analysis process. This journal provided d ocumentation of the affective experience of conducting the study, as well as served as a running record of what has been done each day during the study. It further provided a place for me to record relationships and patterns emerging from the data that I w anted to further explore during analysis (Hatch, 2002). An abundance of data were collected for this collective case study. To make meaning of these data, both within case and cross cases analyses were conducted. For the within case analysis, I followed a series of steps to identify emerging themes in a single case. These steps combined several analysis methods described by Hatch (2002). First, I began by reading through the data (i.e., interview transcripts, expanded field notes, and artifacts) multiple ti mes to get a sense of the context of the data as a whole. As data collection continued and new data were added to the data set, all the PAGE 99 99 data were reread repeatedly Next, I identified frames of analysis. These conceptual frames of analysis are segments of text that contain ed one idea, piece of information, or occurrence (Hatch, 2002; Tesch, 1990) and which broke up the large data set into smaller, analyzable units. For this study, the frames of analysis of observation data typically began when the teacher p osed a question for students and ended when the problem was solved. Thus, each observed lesson included several frames of analysis. Next, I read through bracketed comments and impressions I made during the course of data collection and which were recorded in the research journal. These notes to yourself about the thoughts you have about the data and your understanding of he frames of analysis several times and recorded new impressions in additional memos. The next step in data analysis involved rereading the memos and examining them for emerging themes. As suggested by Hatch (2002), some memos were combined while others w ere kept in their original form. For instance, there were several memos about observations in which I noted Ms. J was guiding students through the procedures for solving a mathematics problem. There were additional memos about interview data where I noted able to solve problems on their own. These memos were combined into a theme that describes the ways in which she broke down mathematics into easily followed procedures. After identifyin g themes from the memos, I reread the data and identified instances in the data that supported my interpretations. This step also involved a search for PAGE 100 100 counterevidence in the data, and my interpretations were modified as necessary. Finally, I read the data once again in search of excerpts to support my findings. In this final step, I conducted a final check to ensure the data were adequately represented in the findings. A detailed summary of my interpretations was written from the memos and included the exc erpts I had identified. For the cross case analysis, I drew on a CRT and standards based instruction framework to compare the observed teaching practices of the participating teachers. Specifically, I examined the cases for similarities and differences in the ways each based instruction. I again recorded memos and examined those memos for themes, conducted a search for counterevidence, and wrote a summary of my interpretat ions. Establishing Trustworthiness In any study, it is important to ensure the study is rigorous. Lincoln and Guba (1985) suggested that researchers conducting naturalistic studies attend to credibility, transferability, dependability and confirmability to establish the trustworthiness. The following sections describe how each of these was attended to in the present study. Credibility Credibility refers to the degree to which findings accurately represent the realities of participants. There are five tech niques one can use to establish credibility: engaging in activities that increase the probability that credible findings will be produced (e.g., prolonged engagement, persistent observation, and triangulation); peer debriefing, negative case analysis, refe rential adequacy, and member checks (Lincoln & Guba, 1985). For the present study, data collection occurred over an extended period of time (five and one half weeks) and observations occurred on a nearly daily basis. PAGE 101 101 Additionally, multiple sources were use d to triangulate the data, such as formal and informal interviews, observations, and classroom artifacts. Furthermore, the interviews were audiotaped. Interpretations of the data were shared with participants at interviews, and they were provided with the opportunity to confirm these interpretations or provide further clarifying information. Finally, data analysis included the search for counterexamples. Transferability Unlike establishing external validity in quantitative studies, establishing the transfe rability of results of naturalistic studies to other people and settings is not the responsibility of the researcher. Instead, the researcher must provide a thick description of the setting, participants, context, and time of the study so that the reader i s able to determine whether results are transferable (Lincoln & Guba, 1985). The thick description of participants, setting, and data required for establishing transferability is included in the present chapter and expanded upon in the case studies (Chapte rs 4 and 5). Dependability and Confirmability Establishing whether results of a study are dependable and whether they could be activities engaged in during the desig n of the study as well as collecting and analyzing the data that an outsider could later review (Lincoln & Guba, 1985). This includes keeping records of raw data (e.g., recorded data, field notes), data reduction and synthesis products (e.g., description of findings and their connections to relevant literature), process notes, and materials related to intentions and dispositions (e.g., research proposal, research journal; Halpern, as cited by Lincoln & Guba, 1985). To PAGE 102 102 ensure the dependability and confirmab ility of this dissertation study, a detailed explanation of each step of the research process has been provided. Furthermore, the method chosen for data analysis required the production and retention of many of the records suggested by Halpern. Subjectivi ty Statement Before presenting the results of this study, my personal perceptions, incoming experiences, and expectations related to the education of youth in America must be bracketed. I am a 28 year old, married mother who has much in common with both th e participating teachers in this study as well as the students they teach. First, I am a Ph.D. candidate majoring in Curriculum and Instruction with a concentration in Mathematics Education. As such, I certainly have my own opinion about what form mathema tics instruction should take that is informed by the research I read about and conduct. In particular, I believe mathematics teaching should engage students in problem solving and mathematical discourse, and should provide students with an opportunity to l earn about mathematics conceptually before they learn about procedures. Furthermore, my studies are funded by an organization that provides professional development to teachers in low performing schools with high numbers of low income students of color, an d my doctoral experience involves educating and supporting mathematics teachers to improve their instruction. I am also a former high way the participating teachers are, I did teach mathematics to struggling, low income students of color just as they do. My parents are educators who have worked both in public schools and after school programs for youth living in poverty. I spent much of my childhood in these settings and listening to my parents discuss the difficulties teachers PAGE 103 103 face in a poorly funded, conservative, standards driven school system. These takes to be a teacher, and I under stand on some level many of the struggles (and successes) these teachers face on a daily basis. Second, I am a U. S. citizen born abroad to an American father and Brazilian mother. I also claim citizenship of Brazil. I am bilingual and consider myself bicu ltural. I have seen my mother, who is dark skinned and bears a heavy accent, struggle at times with language issues and prejudice. I have thus struggled in the past as many students of color do with maintaining the Brazilian in me while attending schoo l in a the fact that both my parents are teachers, however, I am fortunate to have grown up that I believe is part of schooling in America. This has required me to often set aside my parents moved to the U. S. with their four young children, they struggled financially for several years. These past experiences with poverty and the struggle I still have to maintain my cultural identity, I am in a sense (or at least was in my younger years) similar in background and feel a connection to some of the participati students. My identity and these experiences certainly colored the way I perceived the instruction I observed and the conversations I had with the teachers in this study, as one can never completely disconnect oneself from the participants and data or be truly objective. As a researcher, however, I strove to put aside my assumptions and PAGE 104 104 preconceptions as much as possible in an effort to be receptive to the practices and perspectives I was trying to understand. Structure of the Cases Each case (C hapters 4 and 5) begins with a description of the teacher, her students, the classroom setting, and why she believes she was selected for the study. Next, the goals the teacher held for her students are described, followed by an examination of the psycholo gical environment in the classroom that helped the teacher reach those goals. A description of how the teacher approached mathematics teaching each case concludes with a discussion of the influences on instruction identified by the teacher. Excerpts from interviews and field notes will be included as evidence to support my interpretations of data. These excerpts were coded using a system to identify the participant the data source (e.g., interview or observation protocol) and the date. For occurred on October 31, 2011. Similarly, [W_Int3_120111] refers to the third interview with Ms. PAGE 105 105 CHAPTER 4 CASE ONE: MS. J An Introduction to Ms. J and her Classroom Ms. J is an alternatively certified mathematics teacher. She is an Hispanic woman who was 25 year s old and in her third year of teaching middle school mathematics at the time of this study. Her background differed from those of her black students who lived in poverty She was born in Minnesota but moved frequent ly as a child and has lived in several states within the U.S. as well as other countries including Algeria, Indonesia, and Brazil. As a child she dreamed of becoming a tea cher and often played school with her sisters but was convinced by her parents to study something other than education when she began d egree is in Hospitality and Tourism Management with a minor in Business. Following an in ternship during her final year of studies, however, Ms. J realized she wanted to pursue teaching after all. She applied for the Masters in Education program at her university and took up substitute teaching while she waited for the next semester to begin. Ms. J was then hired as a long term substitute as a high school English teacher in the ESOL program at one of the schools where she worked and decided not to pursue her Master s degree. Ms. J admit ted and indicated that this made her initially apprehensive about participating in the study. She additionally had no formal training in advanced mathematics and when questioned about the differences between mat hematicians and mathematics teachers, she stated I struggled a lot in math .... I had to go home and practice, PAGE 106 106 so I feel like I sometimes can relate to some of these studen ts who get frustrated with [J_Int2_013012] Ms. J became a mathematics teacher almost by chance. She worked as a long term substitute mathematics teacher at Southside Middle before being hired full time. She enjoyed the other subjects she taught as a substitute and in particular English for ESOL students and stated, rtunity. So perhaps if I [J_Int2_013012] Even so, Ms. J Aware of her lack of formal education in both content and pedagogy, Ms. J stated that she sought out every possible opportunity to attend workshops. She indicated that m ost of the teaching strategies she used were learned in workshops particularly in a Kagan 3 worksh op she attended over the summer or came naturally to her, a gift that s [J_Int1_110711]. Perhaps it was her lack of formal training in education and mathematics that made Ms. J very modest about her teaching abilities. Though her colleagues knew her as a successful teacher who cared deeply for her students, Ms. J indicated surprise at being identified as highly effective and being nominated for participation in this study, stating, suming maybe ... my principal FCAT scores are unbelievable, like 100% [made AYP] Ms. J was 3 See Kagan and Kagan (2009). PAGE 107 107 participant, even though she repeatedly expressed disappointment in the fact that many of her students were making S he explained, however, that whenever the standardized tests come ay Even though her students demonstrated growth on the FCAT, Ms. J claimed to have never actually looked at sample FCAT test questions, though she did utilize resources designed for the benchmark assessment administered by the district. When asked to describe what she thought about when it came to teaching, Ms. J something where the y walked in not knowing how to do it and hopefully by the time they was to show or explain to students how to solve problems. The following section describes the goals Ms. J held that influenced the way she taught and interacted with students. Goals for Students Ms. J held three overarching goals for her students. First, she indicated students needed to learn mathematics and be able to apply it in the real world. Second, she stated that she wanted students to become well mannered. Third, Ms. J indicated it was important for her students to care about school. Learn and Apply Mathematics master the PAGE 108 108 indicated that being able to use mathematics outside of the school setting and apply the H lls that they can apply, not just in math cl ass but outside of you know the classroom.... I try to make a bunch of applicational [ sic ] prob lems ... and so trying to bring it back to real world and not just you know figures in their textbook. So yeah, my go al for them is to acquire the math skills that they need in life. [J_Int3_020612] This statement is supported by the classroom observations during which most of problems. Sh e explained that the reason she assigned word problems was because they better prepare students for their futures than learning to solve bare numerical problems would: [Word problems are] what they need to know T hi tested.... T [J_Int1_110711] Finally, Ms. J indicated that in addition to learning mathematics, it was important to her that students enjoyed it and said, ... to just like math Become Well mannered Ms. J was committed to the growth and success of her students in life, not just in mathematics. To her, this meant equipping students wi th social skills, and she sought to I feel like I need to teach these skills to students. For example, du ring games or when students were assigned new seats (and new partners), she asked students to practice shaking hands and introducing themselves to someone. She spent a great deal of time at the beginning PAGE 109 109 of the year teaching social skills because she belie [J_Int1_110711]. Ms. J found it challenging, however, when teaching students of varied cultural backgrounds: she gestures to her side. Then, gesturing toward her face: Ms. J stated that she believed these ideas were so important that she wanted to teach a class on morality, manners, and social and life skills. She even asked for permission from the principal to offer such a course as an elective, but in the meantime did her best to incorporate these skills into her mathematics class. Care About School Ms. J believed many of her students did not care about school, so one of her goals realize what a privilege it was that they are able to attend school at all: [I want them to] enjoy coming to school and realize it really is a privilege to be here. In America they com e to school for free, you know? A lot of kids [J_Int3_020612] Ms. J indicated that a lack of care was more of a problem with her students of color than their white counterparts, stating that perhap this lack of care was a result of the difficult lives many students of color who live in poverty faced and that it was her responsibility to get them to care about school: concern with them is getting them to care [about sch ool.] A lot of times they PAGE 110 110 have other things outside of school that intervene with them being able to come to school because of some of the situations at home. So you know you work w ith the students individually I do what I can. [J_Int3_020612] In summary, Ms. J sought to support students to become learners of mathematics who mastered content standards and were able to apply the mathematics knowledge they learned in real world settin gs outside the classroom. She suggested that it was important for students to learn manners, morals, and social skills. Finally, Ms. J stated that her students needed to learn to care about and appreciate school. These goals impacted the way Ms. J approach ed the teaching of mathematics and the psychological environment in her classroom. This psychological environment is the topic of the next section. Psychological Environment l themes that characterized the psychological environment in her classroom. The data related to each will be discussed in the following sections: Cares deeply for students; Classroom is a community of respect; Adopts an attitude of high expectations; and E ngages in explicit and consistent classroom management. Cares Deeply for Students Ms. J cared deeply for her students. When asked about her students of color, she [J_Int3_020612 ]. When her first year FCAT results became available and she found out anybody lo PAGE 111 111 about her students, and it was her students that Ms. J liked most about teaching. She asserted, know, like Jesus bring each one back to school, please. Let nothing happen to them. that makes me the happiest [J_Int3_020612] Ms. J believed that letting her students know she cared was crucial to her success as a teacher: time with who was here in firs great time with you [on the field trip] Ms J. I did my homew [J_Int3_020612] Ms. J stated she wished she had more time in her day to talk with students believed, however, that demonstrating care was a necessary part of her teaching, she comm unicated her care in many ways. She greeted students excitedly as they entered her class every morning, frequently asked them about their lives outside of school, and expressed concern for students when they were out sick. Ms. J noticed when students got t heir hair cut, made sure to congratulate them when they won an award, and shared personal stories about herself with her students. extended beyond the school walls. She regularly vis ited the homes of students who ere doing, still encourage him.... One time the guidance counselor told me that she went to go visit him and as soon as s he went to visit him he pulled out a card that I had written It was just very PAGE 112 112 encouraging and he was so excited about that card a remembered. [J_Int1_110711] Ms. J also took a personal interest in some students who were having a parti cularly hard time: M y first year teaching [one student] was just really a troubled child, getting into a lot of trouble, poor home life situation, and I just kind of took her under my wing.... t In fact, Ms J stated that many of her students did not have positive role models in their lives and part of her job was to serve as t hat role model for them. She suggested this was especially important since she believed a lack of parental support was a contributing factor to the lack of success in mathematics of students of color: I guess what contributes [to the struggles of students of color] is you know, the lack of parent support sometimes. I remember one time I called a parent ... because I care arents used to be more involved and used to ... [do] something about their schoolwork, [do] something a step dad or foster parent, or cousin, you kn ow, grandma. So I guess the lack of parental support in the homes probably contributes [J_Int1_110711] majority of did have parental support: [Parents] do supply them with materials and extra help and they have a place where they sit down and do their homework and they have to get it done before [being allowed to participate in extra curricular activities]. [J_Int3_020612] believed this was more the exception than the rule. By serving as a positive role model PAGE 113 113 for students of color, Ms. J provided support to students that she believed they did not receive at home and thus demonstrated care. Once, when Ms. J was telling the class that she would be out the next day for a wedding, many of the students started to complain. They made comments that the Ms. J explained, his student decided to write their essay about you. worthwhile. [J_Int1_110711] mutual feelings for her and her willingness to go above and beyond what most teachers do in order to support her students: This one [student] that I teach, she struggles with English and with the FCAT and s he also struggles with health suffering physically, emotionally, but for whatever reason she loves being in this class I counselor said you to a I was like made me feel happy but at the same time okay now I have to work extra hard with this student on one. [J_Int3_020 612] Ms. J loved her students, and they cared for her in return. She used a variety of methods to communicate her feelings toward her students, including taking an interest in their personal lives, acting as a role model, and making home visits. PAGE 114 114 Classroom is a Community of Respect Ms. J insisted on a classroom environment in which students were respectful. She modeled use of manners on a regular basis and complimented students who worked well together. Additionally, she did not tolerate negative comments or interruptions. The next example illustrates that while students sometimes misbehaved, they knew what Ms. J expected of them: Chantilly: [ To another student :] Shut up! [ Ms. J calls her name 1] When students were exceptionally talkative, Ms. J did not treat it as misbehavior but rather as a sign of disrespect: Ms. J: fun, but you have to be respectful to each other.... [ At the end of the period Ms. J dismisse d students individually. The loudest students were left at their desks.] Guys, please, tomorrow, be respectful. Do your homework. [ The students were dismissed.] [PJ_112811] This statement also exemplifies how Ms. J drew on her relationship with students, and particularly her expressions of care for them, to encourage them to behave more respectfully. Ms. J explained that she spent a great deal of time teaching students about respect and morality in order to support stud ents to become well mannered: The first week of school ... we just focus on social skills and team building and manners, respect, responsibility, classroom roles and ... I model it, we practice it, you know respecting one another. ... I try to incorporate things like that ... engaging on the material [J_Int2_013012] ect also meant mutual care. At the end of one class period, I noted: PAGE 115 115 Then Ms. J tells the class that today is ince [PJ_110911] This example illustrates how Ms. J took time to build a sense of community among the students. Ms. J insisted that students treat each other and herself with respect i n her classroom. She modeled use of manners for students and took time away from teaching mathematics to educate students about etiquette and social skills. Adopts an Attitude of High Expectations Although Ms. J stated that her students fa ced many challeng es to school success, attitude with her students, held high expectations for them, and suggested that students held the responsibility for their learning. She warned her students at the beginning of the [J_Int3_020612]. Ms. J also stated that she challenged her students, expecting them to solve the hardest math problems: I really like to challenge my students. out of the textbook [ sic ] [J_Int2_013012] Ms. J provided multiple opportunities for students to meet her expectations. For instance, she allowed students to ma ke corrections on graded quizzes for partial credit: I f they get [ all the corrections] right they get up to an 80%. So it helps a lot going back and doing the problems that they di PAGE 116 116 [J_Int3_020612] Additionally, Ms. J allowed students to submit late work during the first semester and she provided frequent reminders. She stopped this practice as the year progressed, however, expecting students to be responsible and complete their work on time: S k ay when I ask ... S o, hopefully get them a little bit mo re responsible and accountable for their stuff. [J_Int3_020612] Ms. J used a variety of strategies to motivate students to work toward the high expectations she held for them, drawing on both intrinsic and extrinsic factors. She frequently communicated to them a belief in their abilities and regularly told the students that she held high expectations for them: Ms. J: [ Ms. J was about to give out grade reports. with some of your grades. All of you are capable of getting an A. If y better. [PJ_120111] This excerpt also illustrates another common strategy Ms. J used in an attempt to motivate students: using her relationship with them to stress their ability by communicating he r disappointment. For example, when Ms. J was helping students with [PJ_120711]. She also frequently praised students, pointed out positive behaviors, or PAGE 117 117 As additional reinforcement Ms. J gave students rewards, sometimes returning assignments to high scoring students with a piece of candy attached to it. She also held the highest homework completion rate for the grading period was rewarded with a pizza party. Even though Ms. J accepted late work, allowed students to make test corrections, and used multiple strategies to motivate students, they did not always take advantage of these opportunities and many students did not complete assignments. For Ms. J, holding high expectations and accepting no excuses for not completing work meant that her students often received poor grades when they were unsuccessful at meeting her expe ctations. She indicated that her high failure rate was, in part, due to her refusal to accept anything but the best from them: My kids are learning in my classroom and ... perhaps I am a little harder grading because they tend to do better on the FCAT and standardized tests as opposed to other classrooms.... M aybe I should be a little more lenient I think [I] have high expectations for my [J_Int3_020612] She was unhappy about the high fa ilure rate, but remained firm in her conviction that students needed to work for their grades: expressed this to students by consta ntly reminding them in class to turn in their homework and study hard, telling them, PAGE 118 118 Friday. If I take the [PJ_112911] re Ms. J sometimes felt as if she pushed her students too hard by insisting that they learn, but she did not believe she had another option, nor did she seek out barriers to school as a place for students to focus themselves and get away from the negative distractions in their lives: I really push them .... I really do. And again, not to diminish the situations that they may be going through at home, but if nobody at home has high .... Y learn to get out of that hole by ... focusing your energy on your sch oolwork. [I] ve to do anything. into your school work as opposed to drugs or whatever a lot of these kids get involved in. ... I set the bar high. [ J_I nt 1_110711 ] Overall, Ms. J sought to support her students in learning mathematics by telling them she had high expectations for them, reminding them to turn in missing assignments, and not accepting a difficult home situation as an excuse for not turning in work. She motivated students to work toward her expectations by usi ng her relationship with them to stress their ability and communicate her disappointment. She was also encouraging and supportive and rewarded students with candy and pizza. Engages in Explicit and Consistent Classroom Management Classroom management was b uilt on top of the climate of care and respect in Ms. but as a sign of disrespect. Further, when students misbehaved, Ms. J communicated her disappointment to them clearl y: PAGE 119 119 Ms. J: activity we have to do today and no one was listening.... It really bothers me you were talking during announcements. [PJ_111611] Ms. J was very explicit and consistent about he r expectations for behavior. For instance, she frequently pointed out positive student behaviors and explicitly stated her expectations of them: Ms. J: By the way, you guys did an incredible job yesterday staying quiet during the morning announcements. I expect the same thing today. [PJ_110111] Ms. J also used a variety of strategies for managing that behavior. For instance, [J_Int3_020612]. She also dealt with behavioral problems in private, perhaps to preserve her relationship with students. She knelt down next to a student to remind him to turn in his homework or assigned a problem to the class and spoke with a student in the hallway about her behavior. Ms. J was also selective in which matters she chose to address and which to ignore. Some issues, such as tardiness, she chose not to take up even though school policy dictated that excessive tardies warranted a referral. She preferred for students to come to class late and be present to learn than for them to not come to class at all: T here are other more important reasons why they should be going into [in school detention] all day [The Dean] them remind them, please be on time. [J_Int3_020612] Similarly, Ms. J did not always scold a talkative student for h is or her inattentiveness. paying attention. If the behavior continued, however, she did address it by moving the eir behavior (as described PAGE 120 120 above), playing a game that allowed students to get out of their seats, or bribing them. behavior. On the ticket, it said, wide practice, and tickets could later be redeemed for rewards. Ms. J asserted that classroom management was less of an issue in classes where the majority of students were white: [With white, middle class students,] I still do the hands up, I still tell them G though. Most times kids are in their seats doing what L [students of color] [J_Int3_020612] She did not see these differences in the behavior between the two groups of students as necessarily negative. She described some of her students of color as very loud and r [J_Int3_020612]. Overall, Ms. J situated classroom management in the community of care and respec t. She used a variety of strategies to address classroom management issues. She did not blame students of color for misbehavior. Instead, she was explicit in her Summar y of Psychological Features Ms. J felt a deep sense of care for her students. She communicated this care frequently and believed that the relationships she built with her students were PAGE 121 121 paramount to her success as a teacher. Her classroom was a community of respect, where good manners were encouraged and social skills were taught explicitly. She felt that she challenged her students, held high expectations for them, and adopted an explicit and consistent approach to classroom management. Teaching Mathematic s In this section, the ways in which Ms. J addressed her goal of supporting students tics teaching. Ms. J stated that her assistant principal encouraged her to set up her lessons consulting with a fellow teacher about what this meant, she realized this was al ready how she structured her lessons. After learning about this method of sequencing a so that it was explicitly clear to observers in which phase of the lesson she was engaging [J_Int2_013012]. The typical day example will highlight these portions of the lesson. This section will conclude with a discussion of the themes that emerged in Ms. cs Class First period at Southside Middle School always began with televised announcements. Ms. J had a student helper whose job it was to turn on the television every day when the tardy bell rang. A warm up assignment, or what Ms. J called previous lessons. Most often, as in the following excerpt, the problem was practice of the procedures for solving the type of problem taught the previous day. Other times it PAGE 122 122 was a review of warm up during announcements, after which Ms. J always explained the solution to the class: [ was on the Smart Board: ] Your dinner at a restaurant costs $13.65 after you use a coupon for a 25% discount. You leave a tip of $3.00. a) How much was your dinner before the discount? b) What percent tip did you leave if your tip is based on the price before the discount? Round answer to the nearest tenth. [ Ms. J wrote on the boar d: Ms. J circulate d sto pping homework for completion The announcements conclude d and Ms. J greet ed the class .] Ms. J: Thank you for being qu iet, I really do appreciate i t. eyes be? [ Ms. J point ed at the board .] Who can read it for me? [ A student read the introduction to the problem and question a ] What are you solving for? S 1 : The orig inal price. Ms. J: [ Ms. J wrote on the board : ] If you had a 25% discount, how much do you pay? [ Several students call ed out 75%. Then Ms. J ask ed how much was paid, and students respond ed with $13.65. Ms. J wrote on the board ] Over x. I want to solve for that original amount. [ Ms. J ask ed the students to solve the equation on their own and then check their answers with their shoulder partner. As th ey work ed Ms. J walk ed solutions. She gave one student a high five for having the answer correct Ms. J went to the board .] Cross multiply guys. S 2 : You get equals [ Ms. J wrote this equation on the board She finished the problem by writing ] [PJ_111511] PAGE 123 123 In the excerpt above, Ms. J approached solving the problem in a procedural way by setting up the proportion rather than writing a mathematical sentence. Ad ditionally, she students toward the procedural strategy she expected them to use to solve the problem. Finally, she encouraged students to check the correctness of their a nswer with a partner. Next, Ms. J continued to go over the solution to the Special: Ms. J: hat percent tip did you leave if your tip is based on the price before the discount? Round answer to the nearest tenth What are you looking for? Ss: The percen t. Ms. J: Okay, listen up. If this is my whole amount [ Ms. J point ed to $18.20 to use the percent equation. [ J proceed ed by writing the percent equation on the board, .] Wha t is the whole? Ss: Eighteen twenty. Ms. J: And what is the part? Ss: Three. Ms. J: [ Ms. J wrote on the board : S he ask ed what the next step was and students call ed out, telling her to divide the 18.20 and then multiply by 100 ] Go ahead and do that on your papers. [ She pause d as students work ed .] Students call ed out several different answers, including 5.5% and 6.25% ] What?? So we divided three by eighteen point two, then multipl y by 100 [ Ms. J wrote the steps as she talk ed ] and I asked you to round to the nearest what? Ss: Tenth. Ms. J: So we get 16.5%. [PJ_111511] Ms. J began the problem by identifying what to solve for and thus what formula they needed. Here, Ms. J asked stu PAGE 124 124 with which they were to begin every solution. She frequently emphasized choosing the correct formula in this manner. Then Ms. J proceeded by asking questions about the values for each parameter in the for mula and setting up the solution path she had taught students to take in order to solve the problem. Following the warm up exercise, Ms. J went over the homework by calling out the answers and then asking students if they had questions. On quiz days or wh en questions arose, Ms. J sometimes asked a student to write the correct solution on the board. Other times, she went over the problem herself in the same manner she did on the warm up: she began by asking students what formula to use, wrote down that form ula, and then asked step by step questions to students while guiding them through her intended solution strategy. Several times during the course of the observations, Ms. J played a game with students at this point in the lesson. These games were non mathe matical and required students to get out of their chairs and move around, similar to a game of tag. Ms. J used these games to engage students as well as to teach them the social skills she believed were important for them to learn. Next, Ms. J shifted the focus to new material. She began by asking the students, [J_Int1_110711]. Th ey answered her question by reading the agenda that was posted or individual pr actice. Rules, procedures, formulae and correct answers were explicitly PAGE 125 125 taught and reinforced. Consider, for example, the following excerpt. The focus of this lesson was on finding the percent of change. Ms. J began with vocabulary by writing the definitio n of percent of change on the board. She then illustrated this idea by asking for a real world example: Ms. J: S 1 : A bag of apples. Ms. J: How much does that bag of apples cost? S 2 : $1.25. Ms. J: Then you go S 3 : $ 2.35. Ms. J: Can we find out how much it changed? [ Students told her to subtract the two amounts, resulting in $1.11 .] Is that the percent it changed? No! [PJ_110311] In this episode, notice also that Ms. J engaged students by asking them to suggest by displaying a PowerPoint slide on the Smart Board that listed the steps for finding the percent of change. Students were pr ovided with an identical handout to glue into their notebooks (see Figure 4 1) so they did not have to copy the procedures themselves. second blank. She proceeded by pu tting a word problem on the board and asked: Did the amount increase or decrease? What is the new amount? What is the old amount? What is the value when we subtract those amounts? As she asked these questions, students called out answers and she wrote th em on the PAGE 126 126 procedure outlined in Figure 4 1, writing on the board as students copied what she wrote in their notebooks. In other words, Ms. J asked step by step question s that guided students through the procedure she presented them with. This was similar to the way she reviewed the solution method in the warm up exercise as described above. Steps: FIND THE PERCENT OF CHANG E 1. Determi ne if change is an increase or decrease 2. Find the amount of change: New amount Original amount (subtract new minus original) 3. Divide amount of change ( your answer in step 2 ) by the original amount. 4. Convert to percent: Multiply by _____ OR M ove decimal 2 places to the _____ Figure 4 1 Handout: How to find the percent of change. [PJ_110311] Following the initial introduction of a formula and procedures, Ms. J engaged in the several mathematically s imilar problems on the board and explicitly guided students through the procedures again by asking step by step questions: [ The following problem was posted on the Smart Board: ] In 2000 the price to for [ sic ] a full day pass to Disney world was $50 for a n adult. In 2011 the price to for [ sic ] a full day pass to Disney world is $65 for an adult. ???What is the percent of change??? [ Ms. J ask ed students what to write for each step: ] Ms. J: Did the amount increase or decrease? PAGE 127 127 S: Increase. Ms. J: What i s the new amount? Ss: 65. Ms. J: What is the old amount? Ss: 50. Ms. J: What is 65 minus 50? [ Ms. J continue d to ask questions guiding students through the steps for finding percent of change. As she as ked the questions, she wrote s on the board: ] 1. Increase 2. 3. 4. increase Ms. J: You always need to write if it is an increase or decrease. [PJ_110311] After this guided practice the students were given problems to solve on their own answers, provided assistance to struggling students one on one, and offered encouragement to those whose answers were correct. For example, Okay a by the original amount! [ Inaudible. ] Why are you dividing by the 20? Is that the original amount? [ Inaudible. ] Thank you, finally one student got it right. Jarvis! You got i t! [ PJ_110311]. This example also illustrates two other common practices. First, Ms. J emphasized correct answers over the process that students used to find the correct answer. Second, Ms. J frequently pointed out common student errors. PAGE 128 128 Once students pr acticed a solution on their own, Ms. J reviewed the answer by asking step by step questions to guide students through the procedures and wrote [ The following problem was displayed on the Smart Board :] ???What is the percen t of Change??? In 1980 there were 460 students at Southside Middle School. In 2010 there were 990 students at Southside Middle School [ There are 2 pictures of students at Southside ]. Ms. J: [ The students solved this problem at their desks individually. Ms J walk ed around and check ed Ms. J wrote the percent of change equation on the board. ] Ss: 990. Ms. J: Minus? Ss: 460. Ms. J: Divided by 460 times 100! You have to do ALL TH ESE STEPS!! [ Ms. J wrote She ask ed for the difference, wrote it down, and then she calculate d the answer. Ms. J wrote the final answer on the board: Ms. J almost always began a solution by either stating the formula or procedure needed or by asking students to state the formula or procedure. At times she then put the solution on the board as in the excerpt above. Sometimes she asked a student to write his or her solution on the board rather than writing it he rself, but she explicitly guided the student through the procedure she intended in a similar manner as described above. She did not ask students to express their mathematical thinking or provide reasons for why they answered the way that they did. PAGE 129 129 After th e introduction of new material in which Ms. J showed the class how to solve a given type of problem, she engaged students in drill and practice by assigning problems very similar in structure to the problems practiced on the board with teacher guidance. Sh e indicated that allowing them an opportunity to practice was important to [J_Int1_110711]. Following this practice, Ms. J asked students to call out the answers, sometimes reviewed the solution on t he board, or walked around checking answers before assigning homework. When time permitted, students were instructed to begin their homework in class before being dismissed. Variations from the Typical Day There were several variations in the way the mathe matics instruction proceeded in a quiz, during review of previously taught material, and on days during which Ms. J supported students to generalize a rule. On a day whe format. Prior to the quiz Ms. J went over every homework problem in detail rather than just reading out the answers, assigned students review problems to solve in pairs, and then went over these problems. She called out each question, students responded with the answer, and Ms. J wrote the answer on the board before assigning the quiz. Changes in instruction were mostly superficial, however, as many of the elements of the instruction descri bed above were still present. The problems on the review that was assigned for homework and on the in class review just before the quiz were similar in structure to problems Ms. J used as examples during instruction and on homework the days leading up to t he quiz. Thus, the review was additional practice of the formulae PAGE 130 130 and procedures emphasized during instruction. Additionally, students were allowed to use their notes on the quiz. Ms. J also provided feedback as to the correctness of the quiz and gave them the opportunity to correct their mistakes answers. Another variation in instruction occurred when reviewing previously taught material. Whether this r eview occurred before the introduction of new material or was When reviewing, Ms. J did not begin by showing students how to solve a given type of problem. Instead, after posing a question, either she or the students stated the formula or procedure that could be used to solve it, and then she assigned one or several problems for students to practice individually. Finally, Ms. J went over the answers at the board, aski ng students step by step questions to lead students through the procedure they were to follow. Thus, she substituted values into the given formula and new material, with the exception that students solved more problems individually before she went over the solution. The final variation in instruction occurred over the course of the tenth and eleventh observations. Instruction followed the same pattern on both days. Ms lessons was to support students to generalize a rule about the relationship between the dimensions of similar geometric figure and their perimeters and areas. Thus, she did not begin with the rule or procedure. Instead, she first guide d students with her questions through a few carefully chosen examples that focused on calculating perimeter and area PAGE 131 131 and then writing ratios comparing those to the sides of the figure. Next, she asked students to solve several similar problems and asked th em to notice a pattern among the solutions to their answers. Students were thus supported to compare the ratios of the perimeters (or areas) of similar figures to the ratios of the sides and generalize a rule about the relationship between those ratios. In essence, these lessons had the same components as lessons on other days namely, students were shown how to solve a particular type of problem and then solved several similar problems on their own with the main difference in instruction being that Ms. J did not begin with the rule to be instruction will be discussed. Themes in Mathematics Teaching The typical day example highlighted several themes in how Ms. J taught m athematics. These themes are the topic of this section. Breaks down mathematics into easily followed procedures through procedures, and practicing those procedures. There were m ultiple posters around the classroom displaying the formulae being taught, and when solving word problems, students were encouraged to always begin with a formula. In one example, Ms. J commented on the solution one student wrote on the board, exclaiming, [PJ_110311]. Similarly, Ms. J prompted students to begin with a formula in the following scenario: Ms. J: Ss: Percent of change. PAGE 132 132 Ms. J: [ ] So immediately what does that tell you to do? Ss: Write the equation! [ Ms. J wrote the percent change equation on the board. ] [PJ_110711] In the typical day example, Ms. the formula with which they were to begin every solution. This formula was usually chosen based on a key word in the problem. She explained during an interview that writing the formula was the first st ep in solving a problem: [I tell the student,] read the directions read the problem. ... What does it ask ? rite out the formula. Okay shape is it ? So ... f ind the formula of a cylinder, write it down. [J_Int3_020612] After writing down the appropriate formula, Ms. J asked questions to guide students through the procedures for substituting values in for the parameters and then for solving the expression or equation. By guiding students through the procedures, Ms. J was setting up the solution path she wanted students to take in order to solve the problem. In fact, Ms. J insisted that students solve problems using the procedures and formulae she taught them, and other strategies were considered incorrect. In the following vignette, students were asked to find a simple interest rate: [ Ms. J pose d a problem on the Smart Board: ] Another example: Find the annual simple interest rate. I = $18 P = $200, t = 18 months Ms. J: [ She ask ed students to set up the problem on their own, then review ed ed them out. ] What did you put for I ? Ss: 18. PAGE 133 133 Ms. J: What did you put for P ? Ss: 200 Ms. J: What did you put for r ? S: r over 100. Ms. J: What did you put for time? S: 18 over 12. [ Ms. J has written the following equation on the board : She told students to finish solving it on their own and then walk ed aroun d checking answers After a few moments, she addressed the class ] Ms. J: All right guys, what did we get? Ss: 6 percent. Ms. J: 6 percent. The fastest way I found to do this guys is multiply 200 by 1.5. What do we get? Ss: 300. [ Ms. J wrote ] Ms. J: This is where I do the inverse operation. A lot of you guys are doing 300 divided by 100 T [ She told students to first divide both sides of the equation by 300, then multiply both side s of the equation by 100. ] [PJ_111511] procedure by asking step by step questions about what values to substitute into the formula. Furthermore, Ms. J told students that their strate gy of simplifying the equation by dividing 300 by 100 was incorrect. This strategy was not mathematically incorrect, PAGE 134 134 but it did not follow the inverse operations procedure Ms. J had shown the class several times earlier in the lesson and thus was not an ac ceptable strategy. Ms. J indicated that her role as a teacher was to break down the mathematics into [J_Int3_020612] She suggested that by guiding students through the procedures for unfamil iar problems until they were able to solve that type of problem individually, she was supporting them to learn. She stated, nt1_110711]. Breaking down the mathematics manifested as Ms. J showing how to solve a type of problem, practicing a few of these problems with students, and then allowing them an opportunity to practice more of the same type of problem on their own: If a k you know well ok ay show them how to do it ... watch what I do ... and then they until they can do it by themsel ves. [J_Int2_013012] Additionally, Ms. J regularly told students that a problem was tricky or hard and she demonstrated how to solve the problem for them. In fact, she made certain to never showed students how to solve a problem when she believed they were struggling. She said to students, Fi nally, as Ms. J explicitly taught formulae and procedures, she did not emphasize understa nding of the mathematical concept. For example, PAGE 135 135 Ms. J: Why do we multiply by 100? S: [PJ_110711] In this example, the student clearly knew that the concept of percent and the number 100 were somehow related, but Ms. J did not push him beyond this simple answer to clarify his underlying reasoning. In another example, when asked about the difference [PJ_113011]. Based on this one response and emphasized the procedure for calculating area or perimeter of a rectangular shape rather than emphasizing the fun damental mathematical differences between these two mathematical concepts. By encouraging students to begin every problem with a formula and by breaking down mathematics into procedures, Ms. J believed she was helping her students learn. She required stud ents to follow the strategies she outlined for solving problems and did not explore the conceptual meaning underlying the mathematics. Emphasizes correct answers suggested to st udents that if they knew the formula, they would be able to solve the problems she assigned correctly: Ms. J: Ms. J. circulate d stopping at students who had their hands up. to do is plug into the formula.... be looking for what formula? [ PJ_110711_54 ] PAGE 136 1 36 Ms. J often asked the class for a show of hands of who got the correct answer, [PJ_113011; PJ_120111]. Furthermore, Ms. J taught procedures in a way that led to correct answers. She asked students questions to guide them through the procedures and students called out responses to these questions. If someone called out a response was usually to correct the student or ignore his or her answer, then state the correct response or call on another student. This frequently followed the Initiate Respond Evaluate, or IRE, format. In the following example Ms. J asked a student area. The student, Chantilly, began by explaining how she found the original perimeter: Chantilly: I did three times two Ms. J: [ She drew the figure on the board ] How do we find perimeter? Chantilly: [ Realizing her mistake : The other students called out that the perimeter of the original figure should be 14, so the new perimeter should be 28. Ms. J asked fo r the area .] Three over four? [ Ms. J ignore d this incorrect response and call ed on another student who respond ed that they should do three times four. Chantilly loo ked confused. She sat, thinking, for a few moments. ] Oh! I was finding the ratio! [PJ_120611 ] Chantilly is a bright, engaged student and she was able to figure out on her own where her mistakes were. In the days leading up to this observation, however, the emphasis of the lessons was on calculating ratios of the sides of figures and it is possibl e that other students in the class made the same mistake. Ms. J did not take up the incorrect call out the correct answer. In this excerpt, Ms. J pushes Chantilly to expl ain how to find perimeter, illustrating her tendency to question students about their responses when PAGE 137 137 they were incorrect. In contrast, when an answer was correct, Ms. J usually accepted the answer without questioning it, possibly suggesting to students tha t if they were questioned about a response they gave, they must be incorrect. Other times, Ms. J clearly became frustrated and reprimanded students when they were incorrect: Ms. J: [ Ms. J was She said t o one stude nt: To the class: ] If you [PJ_113011] Here, Ms. J stated that incorrect answers were because students had not paid attention. As will be discussed fur ther in the section on struggle, once Ms. J explained how to solve a problem, she expected students to remember the procedures or figure them out without her help. Addresses common errors Ms. J pointed out common procedural errors on a daily basis. At tim es, she noticed a particular common error as she circulated and then took time to address the entire class about it, such as in the typical day example. Other times, she anticipated a common error and pointed it out to students before they had the chance t o make the error. The following excerpt highlights both these practices: Ms. J: This is what I saw a lot of students do. [ Ms. J. wrote write this down, this is wrong. Jaquanda, tell me why this is wrong. A lot of students fo have to be careful. Is the 6 the longer side or the shorter side? Ss: Longer side. Ms. J: Ss: Four. Ms. J: So the 6 and the 4 are corresponding sides. [ Ms. J se t up two proportions on the board, told students that both will work. Then PAGE 138 138 she gave another example: corresponding to the 4 and the two corresponding to the 3? Ss: Yes. Ms. J: So why is this wrong? [ Response is inaudible. Students did not give ] You have to put both values from [Figure] A on one fraction and both values from [Figure] B on the other fraction. You have to take your time setting up the proportion [PJ_112811] Ms. J was adept at anticipating common procedural errors and frequently presented them as non examples to the class during instruction. In doing this she noted the error and taught or re taught the correct procedure with a focus on identifyin g the correct numbers to use in the formula or the correct order of steps to solve the problem. Students compare answers with partners answers with a partner. Occasionally Ms. J encouraged them to check their answers answer. Ms. J also sometimes asked one student to explain how to solve a problem to their teammates when they were struggling, be Ms. J suggested that by working cooperatively, students took on the responsibility of teaching their peers, resulting in deeper understanding: W teach something they learn it better.... I could teach the whole period but if a peer is sitting down next to another peer sometimes that interaction is a lot more helpful to understand than having a tea cher in front of the classroom.... They have to reall [J_Int2_013012] PAGE 139 139 Observations, however, did not entirely support this statement,. Students did not usually take the time to share their thinking with each othe r. Sometimes students explained their solution method, but most often they compared answers and waited for Ms. J to show them how to solve the problem if there was disagreement. Thus, cooperative work typically took the form of comparing answers rather tha n discussion of solution strategies. Allows students to struggle Ms. J believed that once she had taught procedures it was important for students to work independently to solve problems during practice: I really do try to break it down for them and just r epeat myself and reinforce the important things and have them repeat it to me and have them pick up the pencil and struggle to figure it out on their own. [J_Int3_020612] She always showed students procedures for solving a problem, but subsequent questions on the topic were often left to students to figure out on their own: A H know, you tell me, e to figure it out .... thing because they need struggle [because] t I think [J_Int2_013012] Ms. J believed that by allowing students to struggle, she was challenging them and teaching them to persist through difficult problems: down to them. ... I want them to struggle I want them to figure it out, I want to challenge them. A lot of them seek that approval real fast and sure every now and then you give it to them or whatever, but other times no, work at it, ng but I really push them. [J_Int1_110711] problem long enough to figure it out without teacher help. Examples of her comments PAGE 140 140 more than that. You have to figure it make these statements when students were particularly talkative and inattentive: Ms. J: [ Ms. J was stating a rule at the board. She told the class to make sure to write it down in their notes. Jaquanda paying attention. disappointed. [ Jaquanda indicate d that she understand the lesson. ] I want you to stare at your paper until you know how to do nd give you a lesson all by yourself. [PJ_120111] As illustrated in these examples, Ms. J often asked students to stare at their paper or the board. This suggests that by encouraging students to focus and be attentive, Ms. J thought it would allow them to understand the mathematics and overcome their struggles. Teaches about key words and provides hints Ms. J used the key word strategy and provided hints as a way of simplifying problems for her struggling students. She told students to look for key words, from which they would know which formula or strategy to employ: Ms. J: Why did you subtract? S: Ms. J: No, because the key word says deleted. Delete means subtract.... add or subtract your answer? S: Subtract. [PJ_110711] Consider also the following example, from a day during which Ms. J taught about simple interest. She assigned a problem in which students were asked to calculate the interest PAGE 141 141 iven amount of money into a savings account. Ms. J had into an account meant that person deposited money: Ms. J: put put mean? [ J underlined the word write down? Ss: I equals p r t. Ms. J: So you see how I use the key words? I see simple interest, I see years, formula. [PJ_111511] Ms. J also hinted to students about how to solve a problem: Ms. J: [ Ms. J call ed on Jaquanda, who give an answer. Ms. J pushe d her for an answer.] If you double the sid e length, the guide students toward the answer. Other times her hints provided the answer: Ms. J: How do I find perimeter? S: Add up the sides. Ms. J: 4. [ Ms. J wrote on the board. ] How do we find area? I just gave you guys a hint. Ss: Length times width. [PJ_120111] break down the she explained that by hinting and teaching the key wo rd strategy, she from a belief that her students did not have the prerequisite skills necessary for success in middle school mathematics: PAGE 142 142 dation s to then be able to also things getting them to learn that. I want to say that students [who] two. Two plu I [J_Int1_110711] simplify the mathematics as much as possible for them. She did this through hinting and teaching the key word strategy. Plays games to increase engagement Ms. J frequently engaged students in nonmathematical games. She indicated that since she was out sick for several days during the study, the class fell behind on the pacing cal es when you [the researcher] were here of their seats and sometimes run around the classroom. They also usually required students to sha ke hands or make eye contact with other players, thus practicing the etiquette Ms. J was so insistent on them learning. When asked about the games, Ms. J indicated they were one strategy she learned at a Kagan workshop that she used for dealing with inatt entiveness T hose kids who [J_Int3_020612]. She suggested that her mathematics lessons sometimes like we need to get this content out you need to know these steps, you need to know how to solve this problem, so unfortunately [J_Int3_0 20612]. The games were thus her way of addressing the lack of engagement with the mathematics instruction. PAGE 143 143 Makes problems relatable Ms. J often asked students to suggest values for examples she presented at the board as a way of engaging them. She also rew rote word problems to make them more In another instance, Ms. J displayed a name with her own, and then told the studen ts to continue working on the problem knowledge about the contextual elements in a problem. For example, When I talked Students seemed to enjoy seeing references to their names, school, and the local increas not on building on mathematical background knowledge. Instead, she focused on engagement. Sup ports struggling students Ms. J cited several strategies for supporting struggling students. She indicated that one of her main strategies was working with them one on one and was observed many times doing just that: PAGE 144 144 All kids learn better one on one. In my opinion, like when I sit down with one student like after school or whatever they pick it up, you know, because ... probably [understanding] it. [J_Int1_110711] O ther strategies Ms. J cited as important for supporting struggling students include d hands on and structured cooperative learning activities (although these were not observed during the course of the study), Rally Coach (another Kagan strategy in which stu dent A solves a problem with only one correct answer and student B either students had adequate materials such as calculators and pencils, playing non mathematical games, a nd working with a partner (the latter two having been described above). Additionally, Ms. J often called on students she knew were struggling: ell why are they going to explain the problem if none of them knew how to do it? going to help them as a class, like how us how [J_Int3_020612] involved Ms. J explicitly guiding the student through the procedures required to solve the proble m, as is illustrated in the following scenario: [ Ms. J ask ed a student Kendra, to solve the problem on the whiteboard. Kendra s aid she know how to do it.] Ms. J: Do you know how to set up a proportion? Kendra: No. Ms. J: Yes, you do... What are the corresponding sides? The six corresponds to what? Kendra: PAGE 145 145 Ms. J: proportion is a fraction. Something over something equals something over something. [PJ_120611] In this exampl e, Ms. J explicitly instructed Kendra on what to write. By doing this, Ms. J led Kendra (and the other students) to the correct answer. Another strategy Ms. J used often was moving a struggling student to a single desk that she had placed at the front of the room, directly in front of the board: This is the hot seat I make my students like want to sit here because they .... they want to sit up front [J_Int1_110711] During observations it was noted that it was always the same two students being moved quanda and Jarvus. At times Ms. J moved their seat to help them understand, but she also put them in the hot seat and called on them frequently when they were off task. Additionally, Ms. J cited the Smart Board and individual student white boards as import ant tools for supporting students who were struggling in mathematics. In particular, when she assigned practice problems, she frequently asked students to write their answers on the white boards and hold them up so she could assess the entire class at once stay after school to attend the tutoring sessions she held several times a week, stating, the bus to school and could not attend activities after school. She encouraged these students to come to her classroom at lunch for extra help. PAGE 146 146 Summary of a Typical Day previously learned co ntent followed by a review of homework. Ms. J introduced a new topic and showed students the procedures for how to solve problems related to that topic, then provided time for them to practice the procedures they learned. Indeed, Ms. J agreed with this cha racterization of her instruction when she described a typical day in her class: warm formulae and procedures and an emphasis on practice, correct answers, and teacher demonstration. Ms. J strove to make problems contextually relevant, address common errors, and break down problems for struggling students. Influences on Teaching The ways in which Ms. J taught were driven by several factors. First, there was the issue of the pacing calendar assigned by the district. Ms. J felt pressure to stay on track with the pacing calendar, which did not, in her opinion, allow students enough time to practice the skills she was teaching and had a negative effect on what they learned: I look at the pacing calendar e sure that they actually get this because at the point of rushing thr ough l, you know, facts or material. [J_Int2_013012] Ms. J also struggled with time management because she wanted to spend as much time as necessary with each student but found it logistic ally impossible. She PAGE 147 147 r goals for her future, Finally, Ms. J asserted that teaching mathematics to students of color living in poverty was not very different than teaching white, middle class students. She indicated that the two populations of students acted differently, but her instructional approach did not change. She taught procedures in the same way with bo th populations but, when compared to students of color, did not have to conduct as much review or break down the mathematics as explicitly while teaching white, middle class students. For instance, Ms. J stated that black students viewed mathematics differ ently or did not have the oreover, she explained that black students struggled and her white, middle class students learned quicker: have to drag desks to the front. Like, they pick it up which is kind of nice, they learn it faster so we don't spend so much time on the actual lesson and we can do more activities and other things. ... My method of teaching is still the same, it j ust seems to go by much faster. [J_Int3_020612] Thus, Ms. J maintained that her overall approach to teaching mathematics remained unchanged regardless of which population with which she is working. She added a same lesson some things might not work with certain students, so differentiated PAGE 148 148 ou have to find ways to differentiate the instruction and teach them t he same thing, like what are other ways for these kids to see it During the course of this study, however, Ms. J did not differentiate instruction. In fact, students were always assigned the same problems and expected to solve them th e same way. Furthermore, when a student didn't understand something, she frequently restated the procedure they were to use rather than explaining it differently. Summary of Case One Ms. J held several goals for her students. She indicated she wanted stud ents to learn mathematics and be able to solve real world problems, become well mannered, and care about school. Ms. J cared deeply for her students and believed that her relationships with them were crucial to her success as a teacher. The students clearl y felt cared for and returned the feelings toward Ms. J. She also built a community in her classroom that demanded respect and insisted that students be well mannered. She also believed that she held high expectations for her students. Ms. J pushed her stu dents, did not accept excuses for poor achievement, and challenged them with word problems. She drew on her relationships to motivate students in addition to offering them external rewards. Finally, Ms. J built classroom management on top of the climate of care and respect in the classroom and took time to explicitly inform students about her expectations for behavior. She was selective about which issues to address, but was consistent in how she addressed those issues. haracterized by an emphasis on formulae and explicitly guiding students through procedures for solving problems in an effort to break down mathematics into something manageable for struggling learners. She PAGE 149 149 engaged students regularly in drill and practice r elated to these formulae and procedures. Furthermore, Ms. J emphasized correct answers and encouraged students to check with partners to ensure the correctness of their solutions. She engaged students by playing nonmathematical games with them and making p roblems contextually relevant. Finally, Ms. J addressed common errors, provided hints, emphasized the key word strategy, and used a variety of methods to support struggling students. PAGE 150 150 CHAPTER 5 CASE TWO: MS. W An Introduction to Ms. W and her Classroo m Ms. W was a traditionally certified mathematics teacher and her background differed from those of her black students who lived in poverty. She described herself as a 31 year old Caucasian country girl who grew up in a small town with little cultural dive rsity. She came to Maya Angelou High School as an intern and was hired directly following this initial experience. Ms. W shared that she always wanted to be a teacher: When I was five I knew I wanted to be a teacher. ... I liked the idea of you know having this place where kids can come in and feel safe or be something more than they came in with. [W_Int2_111711] Ms. W regularly hosted visitors and interns, and data for the present study we re collected when an intern was placed in her classroom. While the intern taught two during these instances There were also two student aides, Matthew and Asia, present they typically try to put a student in here that has gone through the major program successfully, who ... re you are T they worked on a computer program, described below. order for students to obtain credit for the course, the State required they pass an end of course (EOC) exam. Ms. W explained how the EOC worked to her students: nd an entire year working on that class. What if you have an F but do pass? The EOC gives you PAGE 151 151 permission to keep your grade. Your grade is spank but you still have it in your record that you took the exam. [PW_110311] Ms. W taught on a block schedule that followed the A day/B day pattern, and class periods lasted 80 minutes. Although students had algebra every other day, the mathematics department decided students should receive mathematics instruction daily, so Ms. W taught these students both Algebra I an d Intensive Math, which counted as an elective credit. They received separate grades for the courses however, she treated each period as Algebra I. The instruction and content did not differ when she taught Intensive Math. ually in the computer lab on a self paced computer program, Carnegie Learning, for the first 40 minutes of the period. The program provided individually tailored instruction and focused primarily on word just ... great practice for test demonstrated mastery of the first semest er content in Carnegie, they were exempt from their semester exam. The second 40 minutes of the class period involved introduction of new material, practice, tests, and quizzes. Sometimes during this second half of the period, the intern taught a lesson. M ore details about the teaching that occurred in these two components will be described in a later section. Ms. W spent a couple of hours planning each week. She mostly used the textbook only as a resource for test items, or additional support: The way tha resource than a driving force. [W_ Int2_111711] PAGE 152 152 On most days Ms. W taught with the aid of a Smart Board using resources adapted from the New Jersey Center of Teaching and Learning. She learned about the resources at a conference and described them as lessons developed by multiple teachers a s part of a lesson study [W_Int2_111711]. Finally, on the topic of resources, son plans] from the year before. m always kind of starting fresh and so I use a lot of the same strategies but ... I kind of tweak it a s The curriculum specialist for secondary mathematics and several educators at State College nominated Ms. W for this study When asked why she believed she was nominated, she cited her open door policy to visitors and the constant culture of learning in her classroom as likely factors. Ms. W also explained that she was known for her efforts to build strong relationships with her students. Ms. W was well known in the community for being a successful teacher who had strong relations hips with her students. The following section describes the goals Ms. W held that influenced the way she taught and interacted with students. Goals for Students Ms. W stated during interviews that she held three overarching goals for her students. First, s he indicated she wanted students to build their mathematical knowledge. Second, she said she wanted students to become problem solvers. Third, Ms. W indicated the importance of enabling each student to feel comfortable and cared for. Build Mathematical Kno wledge Ms. W indicated in interviews that the most immediate goal she held for her students was for them to learn the mathematics being taught and earn credit for the PAGE 153 153 course, which required successful completion of the EOC exam. More than gaining credit, h improvement in their mathematical knowledge: If I feel like my kids are engaged and my main goal [W_Int2_111711] improvement or progress students made in a year varied by student: they walk out somewh from. [W_Int2_111711] This flexibility in her defini tion of improvement, however, did not mean that students were not expected to achieve their potential. In fact, Ms. W held high expectations for all her students, which will be described in further detail in a later section. Become Problem Solvers In addit ion to wanting students to build mathematical knowledge, Ms. W held a goal for students to become problem solvers. To her, that required her to provide a context in which students had the opportunity to learn practical mathematics that could be applied in real life S he suggested that her students should and [look] at price points on items look at ounces on the package or whatever you long te rm, this ability to use mathematics in daily life and to reason through problems was something students should learn in addition to the day to day topics covered in an d reason and order that you understand PAGE 154 154 she indica ted her students needed to be able to do: that looks familiar and if I needed to go back and look it up I cou ld find how the answers, but who was an expert problem solver [W_Int2_111711]. She strove to one she was committed to nonetheless. Feel Cared For that students should feel that the classroom was a safe environment. She clarified that nd to be view the classroom as a place where they felt free to be themselves. Ms. W cited three reasons why feeling cared for and comfortable was an important goal for stud ents. First, she maintained that when students felt cared for, they were more willing to do the work assigned to them, I think kids will do anything for you if they PAGE 155 155 felt car ed for, they made an effort in her class that they may not have made for another teacher. always receive the support outside of school that was necessary for their success and sta ted, I supportive in a Finally, Ms. W argued that feeling supported and cared for woul d enable her students to defy stereotypes about poor achievement of black students and to become active and contributing citizens: e stereotypes or [W_Int2_111711] ay in which she taught. The first two goals, build mathematical knowledge and become problems solvers, influenced the way Ms. W approached the teaching of mathematics. The final goal, to feel cared for and supported, impacted the kind of psychological envi ronment she created in her classroom. This psychological environment is the topic of the next section. Psychological Environment psychological environment in her classroom. The data rel ated to each will be discussed in sections that communicate that demonstrating care and building relationships with PAGE 156 156 students was a priority and classroom management was grounded in her relationships and the community of care. Demonstrat es Care for and Bui lds Relationships with Students supportive relationships with her students. These caring relationships were the essence iety of ways, including building a community of respect and honesty, identifying and addressing barriers to meet high expectations, using humor and sarcasm, and interacting with students on a personal level. Builds a c ommunity of respect and honesty One wa y Ms. W demonstrated care and built relationships with students was by treating students with the utmost respect. She stated that sh e tried to treat them as adults communicated re spect and a belief in their abilities. In one instance, Ms. W was talking impressed and amazed. ... [PW_120711]. Thi s example high lights how Ms. W expressed care. A dditionally, it building respect. Also, Ms. W praised students when they were respectful and polite. Once, a student told Ms W that his mother and she commended this student for his manners in this example, reinforcing respect. She also spoke positively about his family, further strengthening the relationship with her student. PAGE 157 157 In addition to encouraging manners as a way of reinforcing respect, Ms. W maintained that it was important to be honest and straightforward with her students. She explained that being honest with stu dents was an essential aspect of getting them to respect her and to maintaining positive relationships. She stated, face just as quick as you lay it out there. If I am fake in any wa y they will call me out on that and then that translates into a lack of respect, which then Being honest and respec tful, according to Ms. W, meant not holding back, even when what she had to say was not something her students wanted to hear. She told students, [W_int1_110311] Thus, sometimes the way Ms. W spoke to her students may have given the impression that she was harsh or rude, but her students did not seem to take it that way. Perhaps this approach worked for Ms. W because she balanced her honesty with commendation for their successes. She was explicit wit h students about her reasons for being honest: Hey, And today, You guys did awesome today. They need to know both sides of that. [W_Int1_110311] R elationships with her s tudents were important to Ms. W, and her honesty with them was not only about building respect but also connecting with students and supporting them to feel cared for. Also, Ms. W did not demand respect and expect students to give it to her for no reason. Instead, she treated them wit h respect first, which in turn helped to build a respectful relationship between students and teacher. PAGE 158 158 Identif ies and overcom es barriers to meet high expectations Another way Ms. W demonstrated care for and built relationships with students was by holding high expectations and insisting that students meet those expectations She drew on her relationships with students to communicate her high expectations and scaffolded them to meet those expectations: I try to have like a safe place, to create a place whe mess up or whatever, but not a place where they feel like they can get away with half way work either. ... Like still having a standard and an expectation and saying for this. ... T mor e! ... L Ms. W did not accept excuses for failure S he maintained all her students were capable and explicitly taught students strategies for reaching her expectations. She also provided reminders when necessary: done it? ... sufficient. But yeah, I know I will have to r inclination. [W_Int3_120111] She further indicated that being explicit about expectations and the reasoning behind those expectations helped students to better accomplish the tasks set forth for them. This suggests that it was important to Ms. W to teach students what behaviors were expected of them rather than penalizing them if they did not know what was expected. Ms. W also supported students to suc ceed by identifying and addressing barriers to their success. One such barrier was instability at home: Kids come with so much baggage. ... and in and out of the hospital. H er older brother was not the best role model; her home s ituation was not great. She moved three times during the school year. She PAGE 159 159 [ sic ] so many little things that keep coming at these kids. [W_Int1_110311] Another barrier was that students were not expected to achieve at high levels in their past, though she did not adopt a blaming attitude toward previous teachers or parents: I think a lot of times kids get into this position because somebody has caved somewhere along the way. ... something for a kid than to make them do it for themselves. [W_Int1_110311] She also noted that a pattern of unsuccessful experiences in school mathematics stated that at t imes, white, middle class students were more willing to work and they persisted in spite of the teacher, but students of color living in poverty did not persist: major difference here is with a lot of kids coming from maybe not so successful background in math are just used to kind of shutting down at the first sign of you know, being not successful, and I hope that this kind of gets them over that and they keep trying. [W_Int3_12011 1] Ms. W adopted several strategies to support students to overcome these barriers : I try to find what works with that kid and if praise works with that kid then praise I You told me you we If that works with them, do that, and if joking around with them works then do that and whatever it is that really connects with that kid. [W_Int2_111711] This excerpt also illustrates that when students reached the expect ations Ms. W held, she was not satisfied. She pushed them to reach even higher expectations. Additionally, Ms. W encouraged students and reinforced effort by congratulating them for what might appear to be minor achievements, such as turning in an assignme nt on time or being present at school. For instance, she told one student, Ms. W supported her students to feel successful. PAGE 160 160 Ms. W challenged studen ts to hold each other accountable as another way of supporting students to meet her high expectations. In the following exchange, a visiting student was talking to Tyrion as he tried to work. Ms. W said to the visitor: Ms. W: You friends with Tyrion? [ The student nodded outside of school? [ The student indicate d ] Make sure he does his homework. [PW_111611] In fact, some of the students began to take on this challenge of holding each other accountable and encouraged their peers witho ut prompting from Ms. W: S: Makai: Fifteen was already due. Sixteen is due Friday. I want you to be doing better. [PW_120711] Thus, the community of care and respect was not just shared between the teacher and her students, but among st udents as well. Finally, in order to convince students to make an effort so that they were able to overcome barriers to success and reach the expectations she set for them, Ms. W supported her students to attribute their achievement to the amount of effort they put in: They need to see that direct and immediate positive consequence to O h look, my grade went up a point. Oh look, I knew how to do the problems on the quiz today because I did my [homework And even homework and your grade just dropped. Look at that direct relation D o you Ms. W tried to support students to attribute their grades to effort by telling them: hat are some translation becomes apparent. [W_Int3_120111] PAGE 161 161 expectations, however, related to more than mathematics. She expected her students to learn vocabulary and learn to speak Standard English and she scaffolded them to do so. In the excerpt below, Ms. W suggested that students solve a problem she assigned i Ms. W: The students were unsure. Ms. W describe d a trip to Orlando. ] You can get up, get gas, and drive down I 75 and take the Turnpike to Orlando. Or, you can wake up, you forgot something there, then stop to get past. [ Ms. W explain ed that efficient means quicker, easier. ] It efficient. [PW_1 10111] denominator, and Ms. W responded, i Ms. W: [ She wink ed Students made several comments, teasing Ms. W. we say in the country about to divide [PW_103111] By referring to her own background, Ms. W kept the situation light and did not imply that the student was wrong for not speaking Standard English. She personalized the correction to help students underst and that there are different vernaculars that need to be learned. Ms. W addressed this topic often in class and also supported students in learning standard pronunciation of words. casual, nonthreatening manner, Ms W preserved her caring relationship with students while also supporting them to learn the standard language and vocabulary that she viewed as important. PAGE 162 162 Ms. W adopted an attitude of high expectations and no excuses for failure to succeed with her studen ts. She included difficult home lives, low expectations, a history of unsuccessful mathematics experiences, and a lack of effort. She then regularly encouraged students, reinforced their efforts, supporte d them to feel successful, and held them accountable so that they could overcome those barriers. She also supported them to learn to speak Standard English in addition to the vernacular they spoke. These practices communicated to students that she cared ab out their futures, thus strengthening her relationship with them. Uses h umor and sarcasm advantage with students in order to build relationships with them. Laughter was common in Ms Ms. W: Are and equivalent ratios? ... The Land Before Time [PW _111411] Ms. W also frequently teased students and used sarcasm with them: Dominique: [ To Ms. W: ] Did you just hear [Matthew]? He told me to shush up or [ Ms. W stare d at them for a moment, then rock ed her arms as if rocki ng a baby. She walk ed over to them and they all start ed laughing. ] [PW_110311] In another instance, she teased one student for being the only person off task, stating, Octavius, sometime today, please. Look at [PW_111811]. PAGE 163 163 Sarcasm can come across as hurtful if all involved do not recognize that it comes from a place of care, but because of the relationship she held with students, this was teasi ng and jokes with smiles and jokes of their own. In this way, the sarcasm and humor helped to reinforce the connection Ms. W had with her students. If there were times when was aware of th ese situations and remed ied them quickly. In the following scenario, Ms. W had just handed out a quiz: Reagan: This whole thing the quiz? Ms. W: No, just the first problem. ... [ After class, Ms. W approache d Reagan. ] I apologize for being sarcastic. I hop mean just the first problem. [PW_112811] This excerpt again demonstrates that relationships were she thought she might have damaged a relationship, she worked to repair it. Ms. W joked, teased, and was sarcastic with students as a way of letting them know she cared about them. Ge ts personal Ms. W made an effort to interact with students on a personal level in order to strengthen their relationship. She explained that by taking an interest in their lives and getting to know students on a personal level, she was letting them know she cared: If I know that the kids has [ sic ] a game one day and then I ask them the you did notice me She also questioned students who had been absent or suspended about why: The school will send out discipline r PAGE 164 164 [W_Int2_111711] Ms. W also often talked to students about who was dating whom, their pets, and other personal topics. She enjoyed getting to know her students, indicating: where are they going from here, what sports are they involved in, who T hey have all these choices and to be a part of that for a little while in their lives with me is pretty cool. [J_Int3_120111] ared about her students and made an effort to convey that to each of them, even after they were no longer in her class, by remembering them and continuing to build her relationship with them: As they get older and I see them in the hallway I try to remembe r them and ... I just try to make it a priority to keep in touch with the kids. [W_Int3_120111] Finally, Ms. W interacted with her students personally by being physically affectionate toward them. She frequently put time! [PW_112811]. By asking stude nts about their personal lives, attending their extra curricular events, and being physically affectionate toward them, Ms. W built caring, respectful relationships with her students. These efforts to build relationships with her students in order to creat e a classroom community in which students felt comfortable and cared for were successful. Students entered her classroom happily, joked with her, and clearly PAGE 165 165 liked Ms. W. They also felt comfortable sharing about their personal lives with her. She recalled: It makes me happy when I know the kids and I feel like they want me to be pierced. ... They want to share thin gs with you. [W_Int3_120111] with her students was so strong that at times she served as a mother figure to them. She described the first time she realized students viewed her in that role: It was maybe my third or fourth year here and a kid was raisin g their [ sic ] ... If it gets to O that connect ion. [W_Int1_110311] Ms. W stated that she valued the relationships she built with students because those were what made teaching meaningful. Ms. W worked hard to create a classroom community in which students felt comfortable and cared for. To do this, s he was respectful and honest with students, who supported them to overcome those barriers so they could reach the high expectations she held for them. Ms. W also used humo r and sarcasm to strengthen her relationships with students, got to know them on a personal level, and was physically affectionate toward them. These efforts were worthwhile, as students clearly perceived the care Ms. W demonstrated. PAGE 166 166 Grounds Classroom Ma nagement in Relationships When it came to classroom management, Ms. W sought to create a productive, respectful environment in which everyone could learn. For example, she insisted that students paid attention I f they asked a question she already addresse d, she did not [W_Int3_120111] and advised them to turn to their teammates for information. In this way, Ms. W taught students to not rely solely on the teacher for information and made [a] [W_Int2_111711]. Ms. W indicated that she spent a great deal of time at the beg inning of the year teaching students her expectations of them and then built on more expectations as time progressed. In particular, she focused on routines and procedures that students needed to learn in order for the class to run smoothly and for student s to reach the expectations set in the class. One way she did this was by reminding them at the end of a class period what they would be expected to do the next day at the beginning of class. Attention to these routines and procedures early in the year mea nt that students learned quickly what to do each day as they entered the classroom. By the time observations began in October, Ms. W did not tell students what was expected of them on a daily basis. For example, it was noted that students usually turned in their homework and gathered materials as they entered the class without prompting. Ms. W was not lenient and expected students to follow rules. In fact, she made certain to enforce the rules she found important and held students accountable for their b ehavior. For instance, Ms. W was very strict when it came to being respectful and PAGE 167 167 20111]. Ms. W rationalized this when she stated, The relationship of care and respect that M s. W established was paramount to how she addressed classroom management issues. Specifically, by drawing on her relationship with students, she was able to convince them to do what she asked: I think that because ... I have that sort of rapport or whatev er that the kids willing to do things that I ask them to do. ... like [a chore] ... [W_In t3_ 120111] Unsurprisingly, Ms. W did not want to damage her relationship with her students when she chose to address misbehavior. Instead, she tried to communicate to them that she was not upset with them, but merely unhappy with the behaviors in which the y were jus [W_Int3_120111]. The following sections describe the ways in which Ms. W a ddressed classroom management issues. Treats students as individuals Ms. W knew her students well and had a close relationship with them. S he realized students were individuals who may not have always acted the way one hoped they would, but it did not mean they were acting inappropriately, either: His head down is not PAGE 168 168 defiance H with that because otherwise it becomes a battle with him. [W_Int2_111711] Her relationship with students also allowed Ms. W to better understand the reasons behind why s tudents behaved the way they did. A s their teacher, she tried to be aware of those reasons and considered them when making management decisions: This kid is having a bad day, probably not because of anything I did or It through class is doing that because they [sleep in a car at night.] Okay, let me not ride their case about that. [W_Int3_120111] Thus, by drawing on her relationship with students and the knowledge she held about them, Ms. W was able to consider each case separately and treat students as individuals when addressing classroom management issues. Avoids escalating issues Ms. W avoided overreaction that would escalate minor issues Again drawing on her knowledge of students, she sometimes ignored misbehavior and instead responded in a casual way, acting as if the behavior was not of major concern to her. Ms. W ntion. ... Everybody has off days, you know. ... I Sometimes, when a student acted inappropriately, Ms. W let him or her know but did not reprimand the student. An example occurred when Strom retu rned to class after a very long visit to the bathroom: Ms. W: [ Ms. W gave Strom a disapproving look and loo ked at her wrist, indicating toward her watch. ] Let me just be straight with you. When you disappear that long during B lunch [ Strom smile d ] and whe n you smile that big, that really makes it look like you went to B lunch. PAGE 169 169 On another day, Ms. W walked over to a student, pulled his hood off his head (a dress code violation), took the MP3 player he was listening to, wound up the cord, and put it next to him. She did not say a word to the student during this encounter [PW_111711], but her actions let the student know that he had broken rules. In this scenario, Ms. W ons. Ms. W did not get upset when students broke rules. She remained calm when a student misbehaved, especially if the student was not. Ms. W explained that by not ... Sometimes, being calm and rational was all Ms. W needed to do to diffuse major classroom di sruptions. For instance, she told two students who were threatening to fight: fighting in my class. You want to fight you take it outside. ... here. ... If I write you allowed to have six unexcused absences. ... Is that person that important to As a re sometimes shifted from disruptive to well behaved. As an example, Ms. W described an interaction with a new student who had received multiple referrals from other teachers. H is first day in her class, he looked at his assignment and started yelling profanities. Ms. W responded: [that kid, he] started out just like oh my goodness and then by the end of the school year [he was] putting forth effort, [was] trying to stay out of trouble, PAGE 170 170 She further explained that when students did misbehave, she did not take it personally: A lot of teachers can be in that point where t ... I try not to get to that p oint class and you. So I try to keep it real light. [W_Int3_120111] Ms. W asserted that she did not have many disruptions and this was supported by obser vation data. There was only one instance during the classroom observations that could have become a major disruption, but Ms. W quickly diffused it by sending one of the students involved into the hallway to take a break and later suggesting to the student s involved they needed to learn to work out their differences [PW_120211]. She remained calm, insisted they treat each other with respect, and let them know that their behavior was unacceptable. Provide s hints, humor, and sarcasm Similar to the way she u sed humor and sarcasm to build a relationship with her students, Ms. W joked with students and provided hints to them as a way to address classroom management issues. For example, students were off task and asked Ms. W about the difference between diet sod to get back on task. This statement also illustrates how Ms. W drew on the sarcastic aspect of her personality to address disr uptions. Ms. W claimed that her use of humor was her most effective classroom management tool: The most effective thing is to laugh stop. [W_Int3_120111] PAGE 171 171 Summary of Psychological Features believed relationships were key to her success as a teacher be cause if students felt cared for, they were more willing to behave and complete the ir work. She modeled respect was honest toward her students, and expected them to do the same. Ms. W also demonstrated care by setting high expectations. She identified bar riers to her expectations and scaffolded students to overcome those barriers. She also used her humorous, sarcastic personality to her advantage in building strong relationships and demonstrated that she cared for the students on a personal level. view of classroom management involved a productive, respectful environment in which everyone could learn, and management was grounded in her high expectations and the climate of care and respect that was established. She scaffolded students to succeed by explicitly teaching students appropriate behaviors so they could become self sufficient. She also used a variety of strategies to address classroom management issues including drawing on her relationships with students avoiding overreaction that would lea d to escalation and using humor and sarcasm. Teaching Mathematics In this section, the ways in which Ms. W addressed her first two goals, building mathematical knowledge and becoming problem solvers, will be described. First, a mathematics teaching. Even on a typical day, however, there was great variation in Ms. to create lesson plans that followed the Gradual Release Model (GRM) which she described in an interview: PAGE 172 172 partner,] now do this by yourself. ... At the end of class we do like a daily quiz or whatever but th own. [W_Int2_111711] Ms. W viewed this model for lesson plans as flexible and therefore included variation in her instruction. M and [W_Int2_111711]. Her rationale for this was that she found it boring to teach in the same way every day and explained, ... I know By regularly varying instruction, Ms. W kept students engaged. Even with this variation in instruction, however, there were themes that emerged and some are highlighted in the example. In reality, each element may not have occurred every day. This section concludes with a discussion of the themes in mathematics instruction. A Typical Day in Ms. Ms. W greeted students cheerfully when they entered the classroom. They turned in homework and a student helper returned graded work. Ms. W made announcements to the class about what to expect fo r that day or the upcoming week and when assignments were due. She also reminded them to turn in any late work: Ms. W: I was trying to run [grade] reports and some of you are still missing homework. ... You need to check online for assignments that [are missing] and you need to get thos e things turned in. I want to make sure you get all the grades that you have earned [and] worked for. [PW_111411] After Ms. W made announcements, students worked on the Carnegie program on their computer or completed missing assignments. During this time, Asia graded homework and Matthew assisted students with their work. Students assisted each other PAGE 173 173 as well. Ms. W typically circulated, answering questions, or sat at her computer to take attendance, input homework grades, or request missing assignments from students: Ms. W: If there is anybody who was absent Thursday and you did not already ask me about making up your test, you need to do that right now. ... Tabrishia, you working on that one homework you owe me? ... Ruby, you only have two assignments that you owe me. ... Tyrion, do you have Internet at home? [ He indicate d that he did ] Do the homework now if you promise to do Carnegie at home. [PW_111411] hounded and pestered s tudents to turn in late or missing assignments. During the time that students worked on the Carnegie computer program, former students often came into the classroom to visit Ms. W. She chatted with these visitors and encouraged them to help her current stu dents. Ms. W also used this time to check in with students about their grades and to remediate them individually or in teams. She explained that the purpose of these conferences was to build their skills for the EOC: [Meeting with me] is not in any way, sh want you to think 10 th grade. ... ... I love you too much to want to see you again next year. [PW_110811] following example, she spoke to a small group of students: Ms. W: You all missed it for the same reason, because you put minus you eat hotdogs and they disappe But each package contains eight hotdogs. Each said I'm going to give each person in this class $8? How would I find out how much money I needed? [ S 1 began to say something S 2 cu t her off. ] Yeah, but listen to her S contribute. S 1: Multiply. PAGE 174 174 Ms. W: Right, so I count how many people in the class, multiply it by eight dollars. [PW_111011] Other times Ms. W remediated students on their homework. In the following excerpt, Ms. W met with a student to discuss her incomplete assignment: Ms. W: What happened with number two, you got nervous? [ The student indicate d yes. ] ... Okay, I understand. Th e taxi driver sets the meter at $2.00 so you gotta pay him what? S: $2.00 Ms. W: Then, it adds $1.50 for every mile. Then you give him a $2.00 tip S: Oh, I got it. Bye. [ The student took the paper back to her desk. ] Ms. W: n I see that ambition. [PW_110911] There are several things worth noting. First, Ms. W talked the student through the word described but did not tell her the answer or a procedure to solve the problem. Second, Ms. W allowed the student to redo the problem, which provided another opportunity for her to succeed. Finally, Ms. W was encouraging S he did not put down the student or reprimand her for not completing the assignmen t. In another example, Ms. W met with several students and stressed that they need not worry about making mistakes: Ms. W: Octavius, can I see you please? You know that I cannot grade this ... g Confused is one thing, not done is another. Just write something so it. ... ... It says to write an inequality. So, what i s an inequality? S: A comparison. PAGE 175 175 Ms. W: is, can you take this information in this problem and make some sort of comparison? I just want to see that you can try it. We can move forwar d from there. [PW_110911] Ms. W stressed to the student in this scenario that it was important that he try so that she had something to discuss with him If students did not do the work, she could not talk to them about what they knew or help them with wha t they did not know. By encouraging students and telling them it was okay if they made a mistake, she was supporting students to build their self efficacy for mathematics. After the first 40 minutes, the class moved classrooms for the second half of the p eriod. Desks were arranged in groups of four. One student from each team gathered the materials listed on the board (e.g., calculators, clickers). On some days, Ms. W began by going over homework, perhaps beginning with student requests or recording answer s on the board. Ms. W might choose which problems to go over, which were typically problems on which many students had made mistakes. In other instances, Ms. W asked students to compare their solutions with their classmates and discuss the problems they di sagreed on. Sometimes she would answer unresolved questions, but other times students were left responsible for sorting out their difficulties as a team. The way students were grouped when going over homework changed often. They might work with their team or an informal group assigned by Ms. W. For example, students After reviewing homework, Ms. W began the lesson, usually by first informally i [PW_110311] and that they were preparing for something they would need to know in PAGE 176 176 t Furthermore, Ms. W treated students as experts who already knew the content she was teaching: Ms. W: Here we go! In a math class, you know how your teacher will t each day. You already know this stuff from middle school, so today went to middle school. And if you were paying attentio n. [PW_111411] In this excerpt, Ms. W encouraged students as a way to build their self confidence. She did this often as a way of making the introduction of new content casual and nonthreatening. knowledge about a topic. On a typical day, this included defining vocabulary. In one instance, Ms. W asked groups to determine what the difference was between a ratio and a proportion: Ms. W: he get written down in our notes. Strom. Strom: Ms. W: Good She prompted students for more details. ] S 1: Out of. [ Students started calling out words and Ms. W wrote them on board: ] fraction 9/12, 9:12, 9 to 12, 9 out of 12 Ms. W: What are you doing with those numbers [in a proportion]? S 2: Are we comparing them? Ms. W: comparison, making some sort of relationship. [ She wrote and ask ed PAGE 177 177 that before. ] [PW_111411] In this ex cerpt, Ms. W questioned students about what they knew about a topic. She took their input, validated it, and, through questioning, led them toward definitions for the vocabulary word, filling in the gaps in their suggestions when necessary. Next, Ms. W co ntinued to activate their knowledge by posing a review problem for students to solve. Ms. W asked students whether and were equivalent ratios: [ The students suggested dividing 2 by 3 to tur n i nto a decimal on the calculator. Ms. W reviewed how to divide a fraction on the calculator, because, she stated, some students on the test did 3 divided by 2. ] Ms. W: Can I set up a proportion that is true? Ss: Yes. S: Put an equal. [ Ms. W wrote ] [PW_111411] Next, Ms. W often posed questions, and students shared ideas of how to solve them. In the following excerpt, she wrote a problem on the board, for students to solve. Problems of this form (i.e ., proportions with monomials in the numerators or denominators) were prerequisite material for the course: [ Students suggested turning the fraction into a decimal, multiplying through by 10 (i.e., doing the inverse operation), or cross multiplication. Ms. W wrote each of these suggestions on the board. Next, they solved the problem using each of these strategies S tudents call ed out each step and Ms. W wrote it down. ] Ms. W: Can anybody think of another way to solve it? [ S 1 suggested finding a common den ominator. ] What numbers do 8 and 10 both go into? S 2: 80. PAGE 178 178 [ Ms. W solved the problem using this fourth strategy: they multiplied through by 80 and then students told her what other steps to take to solve the problem. For each strategy suggested, they rea ch ed a point in the solution where the equation was .] Ms. W: All four solution methods were on the board. Ms. W reminded them of their trip to O rlando, says there are many ways to get there and she has shown them at least four ways to solve this problem. They can choose the way that works for them. She assign ed two more problems to complete in teams. ] [PW_111411] It was not uncommon for Ms. W to b egin lessons with problems that reviewed material with which students were already familiar as illustrated in the above excerpt This excerpt also highlights regularly discuss ing multiple solution strategies for a problem, compared thos e strategies, and encouraged students to choose the method they preferred. Sometimes she also made an effort to point out to students when one strategy was more appropriate than another. In the following excerpt, students were working in teams: Ms. W: Mak for your [exit] quiz. I see people doing solo stuff. ... [ To one group: ] Go back and look at the examples W hich one is your favorite? [ The student pointed ] Do it that way then. ... [ After a ti me, Ms. W wrote the problem on the board and asked how to solve it. ] Makai: Multiply by 8 so the 8 and 8 cancel. Ms. W: [T he class considered this sugge stion momentarily. Ms. W reminded them that with the reciprocal strategy sugge sted by Makai, they have to be careful. ] The 8 is being divided by x not the x divided by 8. This problem is tricky ... because you have to we want a method that always works. These four strat egies are good, but you might want one that always works. ... [ She recall ed ed ] but which She told them the means extremes method will always work. ] [PW_111411] PAGE 179 179 As illustrated above, rather than beginning the lesson by telling students to use the means extremes method to solve proportions, Ms. W let students suggest many strategies and slowly led them toward a strategy that resulted in less student errors, which she suggested made the strategy more effective. Usually, as in the example above, Ms. W allowed this strategy to come from the students, but if no one suggested it she presented it as an alternative strategy. She did not insist students used the most effectiv e strategy but discussed the merits of each strategy with the class before Next, Ms. W posed a problem that appeared different, but she supported students to transfer their strategy and apply it to the new p roblem. In this example, after discussing the means extremes method as the most effective, Ms. W asked the students to solve a proportion that contained a binomial in one of the denominators. The students agreed to try to solve it using the means extremes method and noted that they needed to write parentheses around the binomial and apply the distribution property [PW_111411]. Thus, Ms. W built on a problem type with which students were familiar and supported them to apply the strategy they knew to a new pr oblem type. Similarly, Ms. W asked students to solve problems with simple numbers and then supported them to apply the same strategy in problems with variables. After supporting students to transfer what they knew to new problems, Ms. W assigned problems f or students to practice. There were instances when students practiced individually, but most often they worked on problems in their teams. She also told them Ms. W explained that sh e allowed students to choose which problems to solve as a way PAGE 180 180 approach on tests and There were occasions during the course of the study when students did not practice problems on their own or in a team. Instead, Ms. W put one multiple choice questi on on the board and students solved it individually and submitted an answer via their clicker. Ms. W then displayed the results on the Smart Board screen, and the class discussed the solution. At times, Ms. W also focused on the most common incorrect answer and asked how someone may have come t o that conclusion, therefore supporting students in learning from incorrect answers. Typically, when discussing a solution, students suggested what strategy to use, Ms. W revoiced the strategy, and students called out each step as Ms. W wrote what they sa id. In a few instances, Ms. W suggested the strategy and students called out the steps, or she walked students through the entire solution. When this occurred, however, Ms. W explained the solution and focused on reasoning rather than simply stating the pr ocedures. This emphasis on reasoning was common. For example, when students were struggling to understand when the intern, Ms. G, was solving a problem at the board by applying the rule that any variable raised to the power of zero is one, Ms. W stepped in to help them reason through the problem instead of expecting them to apply the rule they did not understand: [ Ms. G is solving She cited the rule that so Students appear ed confused. ] Ms. W: Ss: Three. PAGE 181 181 Ms. W: [ She point ed to .] S: Oh, zero. Ms. W: So th Some days, at this point in the lesson, Ms. W posed word problems to students similar to the types of problems they might encounter on the EOC ex am. She solved these problems at the board, thinking aloud and making her solution strategies transparent. Class concluded with Ms. W assigning homework and an exit quiz (a one question, one point assignment) that covered content taught that day. Before ad ... [PW_110811]. Again, Ms. W was conveying to students that it did not matter to h er if they did the problem incorrectly. She wanted them to try so that she could evaluate their knowledge. The next day during Lab, Ms. W met with students who did not perform well on the quiz. Variations from the Typical Day There were several variation s in the way the mathematics instruction proceeded in These opportunities for divergence occurred wh en students were to take a quiz and when the class engaged in an end of semester review. Twice, instruction varied from the typical format wh en students took a quiz during the second half of the block period. On the first quiz day, Ms. W reviewed homework before administering an open note quiz. Students could select which problems to solve as long as they completed a set number of problems in t otal. The second quiz occurred finishing notes, the PAGE 182 182 open note exit quiz, and a worksheet. Next, the class participated in a brief review game during which students solved a problem with a p artner, Ms. W announced the answer, and they switched partners before being assigned another problem. Finally, the quiz was assigned. Another variation occurred when Ms. W provided review for the end of semester exam by playing a review game. The students solved a problem individually and then came together as a team to decide upon one answer to submit. Ms. W then reviewed the solutions and compared multiple solution strategies at the board. The students did not take an exit quiz that day. In the next secti instruction will be discussed. Themes in Mathematics Teaching The description of typical pedagogy instructional practices. These themes are the topic of this section. Begins with w hat students know knowledge. She asked students to solve a familiar problem, discussed their strategies, and scaffolded students to apply those strategies to an unfamiliar problem. She explained how she approached activating prior mathematical knowledge: ... try to W hat did you do last time? How is that related here? What can you predict will ha [W_Int1_110311] PAGE 183 183 [making] it a lot easier to access that prior knowle dge and ... draw on stuff that they In addition to supporting students to transfer strategies, Ms. W drew correlations daily between money and the problems students were working on. In one example, Strom was struggl ing to write a representation for one lap if four laps were equivalent to one mile: Ms. W: Think about money. ... Four make a dollar? Strom: Quarters. Ms. W: Yeah. A quarter is what? Strom: Zero point two five? Ms. W: You got it. [PW_120211] S he explained that by suggesting that student think about a problem in terms of [like money] and then ... [taking] that Allow s for multiple solution strategies Ms. W regularly discussed multiple solution strategies for a problem. She told her ... Some will take you longer than others, but moved students toward using the most effective strategy: I do try to go back and tell them ... ways lead to some issues sometimes which is why ... if you want a way that ... not to m She also encouraged them to choose the method they knew how to apply without error PAGE 184 184 [PW_110811]. Ms. W explained that encouraging students to choose a n effectiv e strategy that they were comfortable with rather than requiring the most efficient method ensured their success: do the problem in 30 seconds but they can do it in a minute a nd get it right, ... or allowing students to choose for themselves which method to use side. ... Do things that make sense to you. ... What is the point of making them do it this way when th students that it was important for them to understand the strategy they chose and be able to apply it without error [W_Int2_111711]. Finally, Ms. W explained that giving students options about which solution method to use was a step toward supporting them to be able to handle unfamiliar problems: ... they can reach into the Thus, by allowing students to choose from a variety of solution strategies, Ms. W was working toward her goal of supporting students t o become problem solvers. Emphasize s reasoning over rules Reasoning was emphasized over rules, formulae and procedures. Ms. W explained it was the role of mathematics teachers to help students understand mathematics : PAGE 185 185 I think mathematicians are excellent a t math and I think that math teachers are excellent at se this An example of emphasizing reasoning over rules occurred in the discussion regarding multiple solution methods, when Ms. W made it clear that students needed to understand the strategy th ey chose. She also did not teach the rules for exponent operations but rather supported students to understand how to reason through exponent operations, explaining, do on [exponents] but I think tha t my goal is at least that that will produce more students needed to understand and derive rules rather than memorize them. This was evident during an observation when th e class was solving Students [ The following was displayed on the board: ] Ms. W: She indicate d toward the on the left hand side of the equation. [PW_11 0111] In this scenario, Ms. W did not emphasize the inverse operations rule that is often used to solve two and then asked students to think about what five plus five was; the resul t would not be PAGE 186 186 zero, as they needed it to be. Ms. W elaborated on this situation, stating she repeatedly asked students to think about the meaning behind procedures: Otherwise work. [W_Int1_110311] Engage s students in cooperative learning teams Ms. W encouraged students to work in heterogeneous team s every day. When she began teaching, she did not utilize the practice of teaming: My first year I pretty much only did direct instruction type things. ... [Over a team, just b ecause I think they gain so much from the discussions. [W_Int3_120111] Now, students completed the majority of their assignments (even some tests) as teams. Ms. W explained that she used a Kagan strategy to assign students to their group so there was one high performing student, two mid range students, and one low performing student per team. The factor she used to group them changed periodically. First, students were grouped based on pretest scores, but later in the year this shifted to focus on grades, h taking abilities. Ms. W tried to b alance teams in terms of gender and, when she changed groups, she preferred not to partner students with someone they had already worked with. She explained, I try to seat them or work with shoulder partners that the high is never like having to drag Ms. W also did not give students a choice about who they were partnered with and taught students about working together by stressing that they needed to approach their teamm ates with a positive attitude. PAGE 187 187 There were several reasons Ms. W believed teaming was an effective teaching strategy. First, she suggested teaming rather than lecturing was more consistent with how she believed students of color learned: e of a conversation type of learning. [Teaming] goes along with backgrounds, you know, it goes along with how are they learning when ... How can I meet the strengths of you know the culture? [W_Int3_120111] This contrasted with her belief a bout how white, middle class students learned, which was more consistent with direct instruction: [That] population is very used to like a structured non verbal kind of school and coming from my experience, the way that families operated and the ways that you know just community operated was less based on conversation and more based on just taking in information and different from the [black] kids here. [W_Int3_120111] This was not to imply that white, middle class students could not learn we ll from teaming but rather that their culture had taught them to learn from a teacher directed approach. Second, Ms. W stated that teaming worked because it was congruent with how the real world functions and provides practice for the real world: That [is ] very counter intuitive, to me, to have kids in rows raising their hands and doing that back and forth thing. ... works. ... I want to mirror the real world as much as I can. [W_Int1_110311] [W_Int1_110311] and helped keep students accountable. At times she assigned a team test with too many problems for one student to complete individually so all the team members had to contribute. If a student was not prepare d, that affected the entire team: ... how t time to show somebody else how to do it. [W_Int2_111711] PAGE 188 188 Finally, Ms. W engaged students in cooperative learning teams because it allowed them to demonstrate their knowledge to peers when s he was unavailable: comfortable talking with those people and Provide s multiple opportunities to succeed Ms. W provided students with multiple opportunities to meet the high expectations she set for them. She constantly reminded students to turn in missing assig nments and accepted late work. She expressed that she was especially persistent before grades were due because it was important for students to complete assignments even if they homework, students were more willing to complete those assignments if she awarded them partial credit for their late work: ... but then there were so many arguments still have to do it et 100, they get a 20%, but they did the work. [W_Int3_120111] Ms. W further explained that it would be inconsistent for her to be flexible in some of her decisions if she did not also allow students with multiple opportunities for them to demonstrate thei r success by accepting late work: give them several strategies to solving a problem. ... d have learned it Tuesday W S o okay, it took them two days longer to get it. L they got it. [W_Int3_120111] PAGE 189 189 Support s students to build self confidence Ms. W did s everal things to support students to build their mathematical self confidence. First, she often began a lesson by telling students they were capable and introducing a topic in a casual manner. She expanded on this in an interview: I try to let them know li Y essional. Now we know how to do [this.] o they kind of get to that level where like level of confidence or whatever in themselves. [W_Int1_110311] For example, when Ms. W taught students how to find the degree of a polynomial, she encoura served to make new material less intimidating and helped students feel more confident in their mathematics abilities Ms. W also focused on what students did well in a problem during reme diation sessions before correcting them as a way of building their self confidence for helping them understand what they did wrong. Furthermore, Ms. W let students know that it was okay for them to make mistakes. During one observation, after the solution to W tried to support students to build confidence by asking them to try in non threatening situations: and PAGE 190 190 the answer just kind of very low under your breath just say it and see if you [W_Int2_111711] confidence over time, parti cul arly focusing on assessments, and stated, and scaffold it really so they get Ms. W state d that supporting students to build their sense of self confidence was particularly important when working with students who traditionally did not perform well in mathematics. During her daily consults, she told students m trying to build confidence in them I students that they were capable, thus supporting their self confidence to succeed in mathematics. Prepare s students for the EOC exam The way Ms. W taught was not determined solely by the EOC exam, but the exam did play a role in influencing her instructional pract ices. For instance, Ms. W taught test taking strategies. As an example she asked students to consider the reasonableness of story problems Y ou can th ink if the answer is [PW_111411] At other times, Ms. W focused on eliminating incorrect answers when possible: [ The question require d students to solve and graph Each of the four multiple choice options ha d a graph with an endpoint at and various combinations of open and closed PAGE 191 191 circles and arrows shaded in different directions. Ms. W said that the answer must have something to do with their only numerical option. ] Ms. W: with the fraction (though we could turn it into decimal) because we can eliminate answers. [ They eliminate d the graphs with closed circles, then they eliminate d (i.e., shaded to the left). This left one option that they select ed as the answer. ] ... We just eliminated the wrong answers. [PW_110811] Another way Ms. W sought to prepare students to take the EOC was by giving students regular practice with w ord problems in order to support them to become comfortable and confident in their ability to pass the exam: question. I just want them to b e aware more some ways that. That kind of lets them into the test th [W_Int3_120111] Teaching tes t taking strategies and providing EOC practice occurred with more frequency as the semester progressed. Ms. W explained that in the past she had students who did not try on the EOC exam and by addressing it frequently, she hoped to change that. Use s multip le assessments Ms. W used several types of assessments to reinforce student learning. First, she gave a daily exit quiz and explained that the purpose was to determine whether or not students understood the material: worth such a tiny, little amount of point that I hope taking chances there and then ... we remediate the next day. [W_Int2_111711] PAGE 192 192 [W_Int2_111711]. She also used the clicker systems for instant feedback and administered traditional assessments such as tests and quizzes. The exit quizzes, homework, and clickers served as formative assessments, as Ms. W used them to mak e decisions instantly about how to proceed with instruction and remediation. Ms. W occasionally allowed students to use notes on tests Finally, Ms. W stated not like teacher student [all the time Like today in c hat shows me more assessment than anything else. If they can catch a mistake or that. th e interactions between students during discussions were her greatest form of formative assessment a self paced computer program while Ms. W reminde d students repeatedly to turn in missing work and met with students for remediation. Half way through the period, the students and teacher moved to a standard classroom. Ms. W began by informally about a topic by asking strategies to the problem, then posed another problem and supported students to transfer their strategy to that new problem type. Later, Ms. W as signed practice problems, and students completed these problems either in their teams or individually. PAGE 193 193 Individual solutions were often submitted via a clicker system, which allowed the class to discuss the solution and incorrect answers. Finally, Ms. W ass igned students wor d problems as EOC exam practice and completed the lesson with a one question exit quiz. Throughout the lesson, Ms. W often gave students choices about which problems knowledge and their self efficacy for mathematics, as well as understanding the reasoning behind rules and procedures. She also frequently utilized the practice of teaming and encouraged class discussions. Influences on Teaching The ways in which Ms. W tau ght were driven by several factors. First was the issue of time, which Ms. W struggled to use efficiently: and teach, you know? And given unlimited time, I feel like I could accomp and trying to get the things done that need to happen. [W_Int2_111711] She did, however, have an extended class period with her students each day, as well as student aides who helped her with time consuming tasks such as grading, which afforded her the opportunity to meet one on one with students. In particular, the student aides checked homework and returned it immediately so students could discuss the assignment th at same day. Ms. W expl ained her struggles about whether it was appropriate to use student aides to assist with grading: but for me to be able to remediate [and do consult with] the kids who need it that time has to c ome from somewhere and so what I try to do is you know provide that discussion time in the class and then let them ask me questions, so even if I can talk to each other about that. Th best use of teaching time I could take all those papers home and grade PAGE 194 194 them but then I think with homework kids need that [immediate] feedback. [W_Int2_111711] Ms. W also explained she tried to balance the time she put into teaching with taking personal time for herself. Asking aides to help her with some of her more time consuming tasks allowed her to better balance that time and focus her energy on being the best teacher she could be: I have a lot of external t hings that pull me away from teaching that I enjoy ... ... But the flip side of that is ... ... and that does not make for an effective teacher either because then you come in the next day dead and ... when I can have to give that up for being fresh, being energized, having the ideas that I need to be successful at teaching. [W_Int3_120111] Additionally, Ms. W struggled with how much time to spend on class discussions. She felt compelled to curtail conversations for th e sake of time even though she believed students learned a great deal from the conversations: I have to be very aware of ... how far we can go down a road of thought, because I want them to be able to flesh it out themselves and be able to generate a conce that thought but then at some point in time I feel like I have to call this and H Related to the issue of time management was the pacing calendar distributed to teachers by the district. Ms. W did not feel confined by the pacing calendar I n fact, she stated in an interview that she purposefully veered off the pacing calendar to present topics in a sequence that would better facilitate understandin g of the meaning behind rules. PAGE 195 195 Finally, as described earlier, the requirement that students pass the EOC exam influenced the way in which Ms. W taught. Specifically, she referenced the EOC much more frequently than she did in the past when students were e xpected to take the state standardized achievement test: When we were doing FCAT ... seemed like such a separate thing. Now I feel like [the EOC is] more Also, the EOC exam caused Ms. W to alter some of her instruction during the course of this study. She indicated during the stu dy that the school had achieved a 47% passing rate on the Algebra I EOC exam. This was concerning to the mathematics department, so all the Algebra I teachers changed their teaching strategies in an effort to improve that score for the current school year. Specifically, that was when Ms. W began administering a daily exit quiz and using that quiz as the basis of the small group remediation. Summary of Case Two Ms. W held several goals for her students: to build mathematical knowledge, to become problem solv ers, and to feel cared for. These goals directly influenced how Ms. W approached teaching as it related to the psychological environment and mathematics instruction. Ms. W strove to create an atmosphere in the classroom where students felt comfortable, sup ported, and cared for. She held high expectations for students and scaffolded them to reach those expectations. She also frequently used humor and sarcasm when communicating with students and believed honesty and respect were important virtues for her to c onvey to them. Furthermore, Ms. W remained calm when PAGE 196 196 disruptions occurred and conveyed to students that it was their behavior, not the students themselves that she disapproved of. In this way, classroom management was grounded in the caring relationships M s. W had with students. understanding rules, teamwork, multiple solution strategies, and learning test taking strategies for the EOC exam. She sought to support students to build confidenc e in their mathematical abilities, as well as to attribute success in mathematics to the amount of effort one put in. Ms. W met regularly with students to remediate them and gave them multiple opportunities to submit work. PAGE 197 197 CHAPTER 6 CROSS CASE ANALYSIS Introduction The findings presented in Chapter 4 (Ms. J) and Chapter 5 (Ms. W) provide a detailed picture of the way two teachers identified as highly effective supported their students of color to learn mathematics. The focus was to examine the perspecti ves and practices of these teachers pertaining to how they engaged their students with mathematics content. This chapter integrates the findings from the case studies. The cross case analysis was conducted using the lens of the two pedagogical approaches d escribed in the literature review (Chapter 2), culturally responsive teaching and standards based mathematics instruction. The findings will be described in terms of these two pedagogical approaches. Culturally Responsive Teaching The description of CRT in cluded in Chapter 2 (Literature Review) contained eight themes: Teacher student relationships; Supporting students to develop a critical disposition; Cl assroom environment; Culturally responsive classroom management; and Instruction. One theme, supporting students to develop a critical disposition, will not be addressed in this chapter because neither teacher demonstrated evidence of this behavior. The o ther themes have been clustered into four sections to reduce overlap and facilitate cross case analysis. The four clusters are: a) goals for students; b) relationships PAGE 198 198 (encompasses teacher student relationships and classroom environment); c) insistence (en and instruction). These four clusters form a framework for CRT. Using this framework, the observ ed teaching practices of Ms. J and Ms. W were compared. Goals for Students In her study of effective teachers, Ladson Billings (1995b) outlined three goals of culturally responsive teachers: (a) to ensure the academic success of all their students; (b) to their culture; and (c) to sense of critical consciousness, or ability to understand, analyze, and critique the existing social order. These teachers also seek to suppo rt students to become confident, courageous, and have a sense of personal and social agency (Bonner, 2011; Gay, 2000; Gutstein et al., 1997) There were similarities and differences in the goals Ms. J and Ms. W held for their students. Both indicated it was important for students to learn the mathematics being taught and to perform well on their end of year standardized tests (FCAT or EOC). Additionally, both believed students should to be able to apply the mathematics they learned in out of school contexts, and Ms W further indicated her students needed to become problem solvers who could reason, think logically, and solve unfamiliar problems. Additionally, Ms. J held a goal that her students learn to like mathematics. These goals for students align with the first goal of culturally responsive teachers: to ensure the academic success of all their students (Ladson Billings, 1995b). Ms. W also held a goal that her students felt cared for at school. This goal was a driving force behind many of her actions as a teacher, and was enacted in terms of her PAGE 199 199 priority of building relationships with her students. Ms. J cared for her students and, like Ms. W, viewed caring relationships as necessary to her success as a teacher. By supporting students to feel cared for, bot h Ms. J and Ms. W demonstrated an mand students for speaking Black competence, which aligns with Ladson responsive teachers. Ms. J held two other goals for stude nts. First, she stated it was important for her students to care about school, believing that if they cared more they would be more successful. Second, Ms. J sought to enable students to learn social skills. Some teachers believe social skills are necessar y for the future success of students (Gay, 2000), but Ms. J did not express it this way. She simply suggested that everyone should learn manners and morals, and she took it upon herself to teach these to her students. Ms. W also modeled the use of manners for her students, but the reasons behind this were different. For Ms. W, teaching students manners was not a goal in the same way that it was for Ms. J, but rather was one way she demonstrated respect and supported her students to feel cared for. These act ions are not necessarily contradictory to the goals of culturally responsive teachers outlined by Ladson Billings (1995b) but they are not necessarily characteristic of CRT. Furthermore, Ms. W supported her students in developing their sense of self confi dence in mathematics in service of the first goal of mathematics achievement. In PAGE 200 200 this way, she was supporting students to feel empowered to achieve, which is consistent with culturally responsive teaching (Bonner, 2011; Gay, 2000). She did this by casually introducing topics so as to not intimidate students, telling them they were capable, and focusing on what they did well during remediation. Relationships Culturally responsive teachers, and warm demanders specifically, have caring and personal relationsh ips with their students (Bondy & Ross, 2008; Dixson, 2003; Ladson Billings, 1995b; Ware, 2006) and consciously demonstrate care for students (Gay, 2000; Garza, 2009; Ladson Billings, 1995b). Ms. J frequently expressed to her students how much she cared abo ut them, Ms. W made building relationships a priority in her classroom, and students in both classrooms returned the caring feelings toward their teachers. students visited her dai ly. Both teachers built their relationships by getting to know students on a personal level, learning about their interests outside the classroom, and asking questions to show stud ents they remembered something about them; this is consistent with how culturally responsive teachers build relationships (Bondy & Ross, 2008; Delpit, 2006; Garza, 2009; Gutstein et al., 1997; Ladson Billings, 1995a; Peterek, 2009; Nelson Barber & Estrin, 1995; Ware, 2006). Culturally responsive teachers, and particularly warm demanders, may be described by outsiders as tough, harsh, strict, or mean, but students do not interpret this as a lack of caring ( Bondy & Ross, 200 8; Delpit, 1995; Ladson Billings, 1997; Peterek, 2009; Ross et al., 2008; Ware, 2006). PAGE 201 201 insistence that students met their behavioral and academic expectations. Furt hermore, students in both classrooms such as fighting over who got to hug Ms. W first or suggested they liked their teacher s and perceived the care their teachers displayed. At times, the caring relationship between teacher and student may mirror that of parent and child, and there are several studies (e.g., Bonner, 2011; Ladson Billings, 1994; Ware, 2006) that provide examples of culturally responsive teachers who act as mothering stance toward her students, evident particularly in her physical affection toward them. She did not seek to replace viewed her in a mothering role, she knew her co nnection with them was strong Conversely Ms. J strove to be a role model for her students because she believed students had insufficient support from home. S he conducted home visits like many teachers who adopt a CRP approach (Ladson Billings, 1994; Shee ts, 1995), but s he also believed her students of color living in poverty did not have someone at home who held high expectations for them or who cared about their education. These beliefs are are characteristic of holding a deficit perspective of students, which is inconsistent with a culturally responsive approach. The environment in a culturally responsive classroom also promotes relationships. The classroom serves as a safe learning environment (Gay, 2002), interdependence and group effort are valued over independence and individual effort (Gay, 2000; Sheets, 1995), and students are encouraged to accept responsibility for the success of their PAGE 202 202 classmates (Gay, 2000, 2002 ; Ladson these characteristics. In particular, she worked hard to create an environment where students felt cared for and safe. She also required students to work in groups and complete joint assignments, thus h success. At times, students also encouraged each other without prompting from Ms. W, and the community of care and respect in her classroom was shared among the students. Ms. J and Ms. W both stated that relati onships with their students were an essential aspect of their teaching and they would not have been as successful as they were had it not been for those relationships. For example, similar to the teachers in grew to kn ow students well and thus were able names and interests into the problems she assigned as a means of engaging them with the problem, and Ms. W made a point to connect with s tudents through praise, joking, teasing or sarcasm in order to encou rage them to put forth effort. Ms. W and Ms. J also drew on their relationships with students to facilitate classroom management. For instance, Ms. J communicat ed her disappointment whe n students did not behave appropriately and drew on those relationships to stress humor and sarcasm and to better understand the reasons behind the ways in which stu dents behaved Ross and colleagues (2008) stated that warm demander s ground classroom management in their relationships with students, and Dixson (2003) PAGE 203 203 ine approach is effective because students trust the teacher and feel cared for. Insistence Culturally responsive teachers insist upon the achievement and appropriate behavior of their students. Specifically, w arm demanders hold a pervasive belief in their abilities (Ross et al., 2008), belie ving all stude nts can learn regardless of pre vious behavior or grades (Gay, 2000; Ware, 2006), and they challenge their students ( Delpit, 2006; Ladson Billings, 1997; Sheets, 1995; Ware, 2006). These teachers communicate d high expectations for both achieve ment and behavior and they stress ed the importance of perseverance and persistence (Bondy & Ross, 2008). Thus their pedagogy was consistent with that of warm demanders, who are explicit about their expectations, use varied strategies during instruction abo ut expectations to keep students engaged, provide examples and nonexamples of appropriate behaviors, and expectations (Ross et al., 2008). Warm demanding and high expect ations enable teachers to insist, firmly, respectfully, and in a calm, warm tone that students achieve (Bondy & Ross, 2008; Irvine, 2003; Ross et al., 2008). Teachers who adopt a warm demander stance also assume the responsibility for making sure students learn (Ware, 2006) and never give up on their students (Ross et al., 2008; Wilson & Corbett, 2001). This was not did not accept late work after the first semester and told students it was their responsibility to learn In doing this, Ms. J was holding students to high standards, but she failed to scaffold the students to success For instance, after providing instruction PAGE 204 204 on a topic, she di d not repeat herself or support stud ents in learning how to work through a difficulty. Instead, she conveyed the content and then told students who were ensure students master the material ( Ross et al., 2008; Wilson & Corbett, 2001 ). By taking these actions, Ms. J seemed to communicate to students that once she fulfilled what she believed to be her duties as a teacher, she was no longer responsible and it was up to her students to succeed without her help Ms. J at times also blamed students for their lack of success, stating that students failed because they didn't care about school. She additionally and difficult home situation s Culturally responsive teachers do not blame students or their families for their lack of success but instead work to overcome any identified barriers and create the environment necessary to support students in meet ing the high academic and behavioral ex pectations they hold (Bondy & Ross, 2008; Ross et al., 2008; Ware, 2006). Despite this blame, Ms. J maintained that she challenged students academically by assigning difficult word problems. Her instructional approach, however, suggested she did not believ e that students could solve problems without her first showing them how, which calls into question whether she in fact held high expectations for her students. Ms. J also stated that students. To her, no excuses meant that she did not tolerate excuses for not completing assignments and students deserved a zero if the assignment was not turned in. Ms. J suggested that by failing students, she was holding them to high expectations, but she did not take time to unde rstand the reasons why students were not understanding the PAGE 205 205 content or completing assignment s This is inconsistent with the approach adopted by reasons for students to not le arn ( Gay, 2000, 2002; Ladson Billings, 1994, 1995b; Ware, 2006; Wilson & Corbett, 2001), and they dedicate themselves to removing any barriers to learning in order to ensur e By blaming students and their families and not sup porting students to overcome the barriers to their success, Ms. J did not, from a CRT perspective, have high expectations for her students. Ms. W, on the other hand, adopted a stance of high expectations for academics and no excuses that was consistent wi th CRT Each time students reached an expectation she held for them she raised that expectation higher. She addressed lack of success those barriers uggles in mathematics as societal issues, suggesting that society at large expected these students to fail and that their previous teachers held low expectations for them. She also stated that traditional mathematics instruction, during which students are typically lectured to and which does not allow them opportunities to work with their peers, conflicted with how students of color learn. To overcome some of these barriers Ms. W met with struggling students regularly fo r remediation rather than leaving th em to figure out how to succeed on their own and did not seat students in rows or teach through a direct instruction model, instead allowing students to work in teams a practice also consistent with standards based instruction Ms. W also created a safe and comforting space in her classroom as a way of addressing the instability some students faced at home. Furthermore, she pushed PAGE 206 206 students to reach their full potential to combat the low expectations she believed their previous teachers might have held for them. She did not tolerate excuses constantly reminded students to turn in missing work and provided multiple opportunities for students to demonstrate their knowledge and become successful. Culturally responsive teachers will nag, pester, and bribe the ir students to work hard (Ladson Billings, 1995b). This does more than just support students to meet high expectations. By fussing at students and taking an authorita tive stance, these teachers are communicating that they believe in their students and expe ct more from them (Dixson, 2003), thereby strengthening the relationships that are so characteristic of warm demander pedagogy and CRT in general that students would behave appropriately in her class Ms. J noted differences in the behavior between students of color and their white, middle class peers, and she did not blame students of color when they did not engage in appropriate behaviors. Instead, similar to warm demander s (Ross et al., 2008), Ms. J explicitly taught students how to reach her expectations for behavior by providing reminders, pointing out positive behaviors, and modeling for students what she wanted them to do. Ms. W engaged in these actions as well. She also remained calm and rationa l during disruptions and did not interrupt instruction to address discipline. These actions are consistent with the ways in which warm demanders encourage students to meet their behavioral expectations (Bondy & Ross, 2008; Ross et al., 2008; Ware, 2006). I n addition to high expectations, warm demanders are characteri zed by a n insistence on respect (Ross et al., 2008). For the warm dem ander, respect is PAGE 207 207 nonnegotiable Students are expected to demonstrate respect for the teacher and their classmates, and the t eacher shows respect for students (Bondy & Ross, 2008; Ross et al., 2008; Ware, 2006). An insistence on respect was a defining characteristic of the Ms. W modeled respec t for their students. Furthermore, Ms. W was deliberately honest with her students as a way of building respect, and Ms. J took time to explicitly instruct her students on how to be respectful. Both teachers also modeled and encouraged the use of manners a s a means of demonstrating respect. Pedagogy based mathematics teaching will be described in detail in a later section. There are, however, some characteristics of pedagogy common to all culturally respon sive teachers regardless of content. The ways this section. Culturally responsive teachers emphasize conceptual understanding (Tate, 1995) and strive to support stude nts in learning to think critically and creatively, problem solve, analyze, make connections between concepts, and engage in discourse (Haberman, 1991; Ladson Billings, 1997; Sheets, 1995). Teachers use multiple instructional strategies including discussio n, peer teaching, problem solving, and grouping (Delpit, 2006; Gay, 2002; Haberman, 1991; Ladson Billings, 1995b; Sheets, 1995; Wlodkowski & Ginsberg, 1995) and they incorporate technology into instruction (Haberman, 1991). These teachers also use students which to build learning experiences (Delpit, 2006; Gay, 2000; Gutstein et al., 1997; PAGE 208 208 Ladson Billings, 1997; Wlodkowski & Ginsberg, 1995) and provide students the opportunity to construct their own knowledge (Gutstein et al., 1997). Many of the pedagogical practices of culturally responsive teachers are consistent procedures, engaged students in cooperative learning activities and classroom discourse, and used the clicker system on a regular basis for formative assessments. what they knew, thereby supporting students to construct their own knowle dge. Ms. W additionally recognized the different learning needs of her black students and implemented practices such as grouping in order to meet those learning needs. This is another practice consistent with CRT (Delpit, 2006; Gay, 2000, 2002). Ms. J, on the other hand, did not believe students of color had different learning needs than white, middle class students. She utilized the same instructional practices with both groups, though she acknowledged that the white students responded more quickly and mo re willingly to her instruction. Culturally responsive teachers teach more than just content to students; they provide opportunities for students to acquire the cultural capital (e.g., ways of interacting, test taking strategies, note taking skills, stand ard English) necessary for school success (Gay, 2000; Ladson Billings, 1995a). Ms. J and Ms. W both modeled the use of manners for students, and Ms. J made an explicit effort to teach students the manners and other life skills she believed were important. Furthermore, Ms. W addressed test taking strategies and note taking skills on a daily basis, and taught PAGE 209 209 students to speak Standard English without demeaning their use of Black English Vernacular. Providing students with multiple opportunities to practice, demonstrate knowledge, and revise and resubmit work, as well as the use of varied and formative assessments are other practices common to culturally responsive teachers (Gay, 2000; Haberman, 1991; Wlodkowski & Ginsberg, 1995). Ms. W and Ms. J both provide d students the opportunity to submit late work (though Ms. J ended this practice after the first semester), and Ms. J encouraged students to revise incorrect answers on assessments so that they could recover points lost. They also both used a variety of as sessment techniques, including traditional tests, quizzes, and homework. In addition, Ms. J employed the use of personal white boards, and Ms. W frequently used the clicker system, daily exit quizzes, and class discussions as sources of data on student und erstanding. Another characteristic of culturally responsive classrooms is that the teacher does not act as the ultimate holder of knowledge and power; students and teachers share these roles and students act as arbitrators of knowledge (Bonner, 2011; Guts tein et al, 1997; Ladson Billings, 1995a; Wlodkowski & Ginsberg, 1995). Ms. J believed it was her role to show and tell students how to solve problems and never assigned a problem for which she had not provided procedures. Ms. J thus was the authority of k nowledge in the classroom. In contrast, Ms. W often asked students to provide suggestions for which strategies to use, and she built on these suggestions rather than telling students how to solve problems. PAGE 210 210 Finally, culturally responsive teachers utilize s learn as resources for teaching and content (Gay, 2000, 2002; Ladson Billings, 1995b, 1997; Tate, 1995). Similarly, these teachers emp hasize multicultural content (Gay, 2000, 2002) and address relevant world issues that students care about and that reflect actions. Ms. J did alter word problems to include topic s of interest to students (e.g., their names, the school mascot, locations they were familiar with). These changes, however, were superficial and did not address critical issues of interest to students and their families. Summary es and perspectives aligned entirely with CRT. Ms. J made the learning of mathematics her main priority, had caring relationships with students, insisted on respect in the classroom, and held high expectations for behavior. She also provided students with multiple opportunities to demonstrate success. These practices are aligned with CRT, but this pedagogical stance as a whole, however, was for lack of achievement, di d not support students to overcome the barriers to success that students of color living in poverty face, often did not always provide students support in working through difficulties, and acted as the authority on knowledge in the classroom. ruction as a whole was not entirely characteristic of CRT. She did support students to acquire the cultural capital necessary for their future success, PAGE 211 211 instruction to mee t the needs of her black students. Like Ms. J, however, she did not critical disposition. Even so, Ms. W was firm yet caring, built strong relationships with students, ins isted that they meet her expectations, and worked to remove barriers to same. She also provided students multiple opportunities to achieve success and did not give up when st aligned with those of a warm demander even though the entirety of her instruction was not characteristic of CRT. In the next section, the perspectives and practices of each teacher as they re late to standards based mathematics instruction will be described. Standards based Mathematics Instruction The literature review described many aspects of standards based instruction. These include attention to the process standards of problem solving, re asoning and proof, communication, connections, and representation (NCTM, 2000) and the Standards for Mathematical Practices (CCSSI, 2010) that align with the process standards (NCTM, 2010) and strands of mathematical proficiency (Kilpatrick et al., 2001). Discourse was also highlighted as an important aspect of reform. Using this description of standards based in mathematics instruction, the observed teaching practices of Ms. J and Ms. W were compared. The results will be described in sections that communic ate a) the sequence of learning activities, b) the focus of instruction, and c) discourse and norms. Sequence of Learning Activities The participating teachers adopted a dissimilar sequence of learning activities. Ms. taught PAGE 212 212 through direct instruction. Lessons stated was required by her administrators. She began by stating a formula and explicitly outlining a step by step procedur e for solving a particular type of problem, usually a word problem. Students were required to memorize the formula, and Ms. J taught them to look for key words that would help them determine which formula to apply. She then showed students how to apply the procedure in one or more example word problems, guided students through several examples, and assigned additional problems for students to solve individually. than memorizing ru les and procedures. She stated that her lessons followed the GRM format, but Ms. W implemented this format flexibly and varied many of her instructional knowledge about a top ic and guiding them through solutions in a way that supported students to understand how to reason through unknown problems and apply familiar strategies in new situations. Ms. W provided guidance to students as they solved problems rather than explicitly showing or telling them how to proceed through the solution. The GRM that Ms. W stated she followed to sequence learning activities is administrators. The GRM, widely used for reading and literacy instruction, is based on Pearson and Gallagher 3) Gradual Release of Responsibility Model and stipulates situation in which the students assu PAGE 213 213 p. 211) through a gradual fading of support. The FDOE ( n.d. ) described the GRM as having four steps: (a) explicit instruction, (b) modeled instruction, (c) guided practice, and (d) independent practice. step by step how to use the strategies and concepts being taught. Guided practice is hase during which students work on problems under the hase. Based on this description and the fact that the county in which both teachers taught adopted the GRM lesson sequence, Ms. J was thus used the GRM as a structure for modeling procedures in a manner consistent with traditional mathematics instruction (Hiebert et al., 2005; McKinney & Frazier, 2008). Ms. did not explicitly show and tell students what procedures to follow but rather modeled her thinking, encouraged students to suggest strategies, and guided them through problem solutions by using their input and building upon it. These practices are consistent with standards base d instruction as described by NCTM (2000). Focus of Instruction There were few similarities in the way the participating teacher s approached the problem and it almost a lways required starting with a formula. If students did not follow the procedure outlined by Ms. J then their solution was considered incorrect. Ms. W, on PAGE 214 214 them to sugge st multiple strategies and choose one of these strategies when solving problems. Her only requirement regarding multiple solution strategies was that students understood the strategy they employed. Most often, Ms. W wanted students to find the solution str ategy that they understood best and to know when to apply it, even if that strategy took longer than applying a rule or algorithm. Inventing, applying, and adapting multiple solution strategies is an important part of problem solving and standards based ma thematics (NCTM, 2000). The type of mathematics teaching in which Ms. J engaged is what Gersten and colleagues (2009) refer to as explicit instruction. Specifically, she routinely demonstrated problem specific step by step procedures for students to solv e problems and did not provide students with a heuristic for solving multiple types of problems. Additionally, she encouraged students to use the procedure she demonstrated rather than an invented strategy or one they learned elsewhere. Explicit instructio n is common in classrooms that adopt a direct instruction approach. Furthermore, explicit and direct instruction are characteristic of traditional mathematics teaching, which is also typified by other instructional strategies employed by Ms. J, including i nfrequent opportunities for group work, conversation, or problem solving (McKinney & Frazier, 2008); emphasis on procedures ; and an overall focus on lower level mathematics skills rather than concepts (Hiebert et al., 2005 ). The traditional mathematics i nstruction adopted by Ms. J adheres to what in schools with high numbers of students of color and who live in poverty (McKinney et al. 2009; McKinney & Frazier, 2008; Weiss, 1994) Teachers in these settings often lower PAGE 215 215 their expectations for struggling students, simplify content, and emphasize procedures and skills rather than concepts (see Davis & Martin, 2008; NCTM, 1999; Watson, 2002; Webb & Romberg, 1994; Weiss, 1994) M the mathematics for her students as a way to support them to learn, but Watson (2002) procedural. This approach to mathematics teaching also unde to solve real world problems, which, by their very nature, are problems to which the solution method is not immediately evident. Scholars argue that traditional instruction, or a pedagogy of poverty, does little to support the mathematical learning of students of color living in poverty. While traditional methods of teaching mathematics can result in procedural fluency (Boaler, 1998; Schoenfeld, 1988), this is only one aspect of mathematical pro ficiency. Conceptu al understanding, strategic competence, adaptive reasoning, and a productive disposition are also strands of mathematical proficiency that mathematics educators should strive for (Kilpa trick et al., 2001). Traditional instruction, which by definition does not take into account all five strands of mathematical proficiency, does not support diverse learners (Boaler, 2002) because they are not pushed to develop their mathematical thinking (Watson, 2002) and thus struggle to transfer knowledge to new settings a nd display low levels of conceptual understanding ( Boaler, 1998; Bottge & Hasselbring, 1993; Schoenfeld, 1988 ). require all students to follow that one procedure. Instead, Ms. W began lessons by PAGE 216 216 with standards based instruction (NCTM, 2000). She did not believe memorizing rules or procedures was necessary for (or beneficial to) her students, and she spent much of her instructional time supporting students to understand mathematical concepts so that they could derive formulae. Ms. W stated repeatedly that sense making was important to her and she sought to support students to reason and think math ematically. Being able to apply a strategy to unfamiliar problems and provide a rationale for their choices was an indication to Ms. W that her students had learned the content taught. Her instruction thus emphasized understanding rather than rules and pro cedure s, which is consistent with standards based instruction (Kazemi & Stipek, 2001; Kilpatrick et al., 2001; NCTM, 2000). Discourse and Norms classrooms varied greatly. Ms. J rarely asked students why they chose to solve a problem in the way that they did but when she did ask, she accepted surface level responses that were usually one or two words rather than responses that were focused on mathematical meaning. In fact, exchanges bet ween teachers and students often followed the IRE questioning pattern (Herbel Eisenmann & Breyf ogle, 2005; Zevenbergen, 2000) that is common in traditional mathematics classrooms. The norms or understanding. processes rather than just procedures, multiple strategies for solving problems were discussed, and students were supported to understand the relationships between those strate gies. These norms were identified by Kazemi and Stipek (2001) as sociomathematical norms that promote conceptual understanding with ethnically diverse urban students in reform PAGE 217 217 classrooms. Ms. W did not require multiple strategies, but she did allow for and value them. Students often suggested several ways to solve a problem, and Ms. W wrote these on the board. She then engaged the class in a discussion of the multiple solution strategies including when one strategy would be more appropriate than another. Sh e also encouraged students to choose the strategy with which they were most comfortable as long as they could explain how and when to apply it. This explicit discourse about strategies is an key aspect of standards based instruction (Lubienski, 2000a; Scho enfeld, 1987 ) and it allows students the opportunity to explain their thinking, make their methods explicit, learn from others, develop more efficient methods for solving problems (Hiebert, 2003; Schoenfeld, 1987) and understand how to interpret new proble m solving situations (Lesh & Zawojewski, 2007 ). Also, developing, discussing, explaining, and justifying solution strategies are important part s of solving problems (Lesh & Zawojewski, 2007; NCTM, 2000; Polya, 1985; Schoenfeld, 1987 ), and encouraging stude nts to invent and adapt strategies may support the development of multiple representations (Bostic & Jacobbe, 2010). Kazemi and Stipek (2001) also found that errors were a natural part of the learning process and provided opportunities to extend learning a nd understanding in these classrooms. Unlike Ms. J, who pointed out common errors to ensure students came up took up incorrect answers and discussed them with students. She questioned students about why someone may have chosen that answer, thereby providing the opportunity for students to learn from their mistakes. A final norm identified by Kazemi and Stipek was that collaboration among peers was necessary, and groups were required to reach PAGE 218 218 consensus about their solution through reasoning and argumentation. As described in the cases, students in both classes were grouped heterogeneously according to a y engage in collaboration or mathematical discourses within their teams. Instead, students mainly class, students engaged in cooperative learning activities in their groups on a daily basis. They were given multiple opportunities to work and communicate with their peers, and groups often had to come to consensus about problem solutions before submitting their assignment. Studies have shown that students who work with their pe ers on problems tend to perform better (Charles & Lester, 1984; Dees, 1991; Ginsburg Block & Fantuzzo, 1998). Discussion different ways in which each teacher viewed mathematic s and mathematics teaching, which were evident not only in the way they spoke during interviews but the ways in which they taught. For example, Ms. J displayed what Boaler (1998) referred to as a rule following behavior, believing mathematics to be only ab out rules, formulae, and equations that lead to correct answers, and she stated that mathematics teaching is about explaining to students how to solve unfamiliar problems. Thus, Ms. J believed her role was to show them how to solve problems when they did n ot know how. This view of the teacher as the holder of knowledge, which emphasizes teacher demonstration of procedures and correct answers, is characteristic of a traditional view of teaching. Ms. based mathematics teaching but adhered to a pedagogy of poverty. In contrast, Ms. W viewed mathematics as a PAGE 219 219 creative process and suggested that mathematics teaching was about sense making and helping students to understand the content. Her instruction thus highlighted the importance of understanding the meaning behind rules and procedures, allowed for multiple solution strategies, and stressed communication with peers. Norms regarding m that guided the way students explained their thinking, made their methods explicit, and allowed students to develop more efficient methods for solving problems. Furthermore, errors were a natural part of the learning process, instruction of new material built on the knowledge students held, and students worked cooperatively in heterogeneous groups was aligned with standards based mathematics teaching With its emphasis on teacher demonstra tion and modeling of procedures, the GRM is not well aligned with reform mathematics teaching and is more aligned with direct instruction and a pedagogy of poverty. One might argue that Ms. J adopted a pedagogy of poverty because of the requirement that she structure lessons around the GRM, but Ms. W was also required to utilize the GRM and she did not adopt such a pedagogy. Instead, Ms. W found the GRM flexible and adapted it suit her needs. For example, she required students to conduct the practice requ ired by the GRM in groups, and she modeled thinking rather than modeled the correct application of procedures. The GRM does not then explain the differences in instruction between the two teachers. Summary Ms. J and Ms. W both had strong relationships wit h their students and elements of employed to support students of color to engage with mathematics differed greatly. Ms. PAGE 220 220 J, who had a high course failure rate, adhered to a pedagogy of poverty inconsistent with reform mathematics teaching, and students were taught little more than procedural of teaching, and given her high failure rate c oupled with the literature that describes the negative effects of traditional mathematics instruction (Davis & Martin, 2008; Watson, 2002; Webb & Romberg, 1994; Weiss, 1994), it is unclear whether Ms. J was consistently effective, if her first year scores occurred by chance, or if the pedagogy of poverty aligned well with the assessment. Even so, Ms. J was identified by the secondary mathematics curriculum specialist as a highly effective teacher and was well liked and respected by students, other teachers, and her administrators. Conversely, Ms. W can be characterized as a warm demander because of her caring relationship with students and insistence on respect and achievement. The warm demander stance can support the academic engagement and achievement of students of color living in poverty (Bondy & Ross, 2008; Ware, 2006) Additionally, Ms. W taught in a way that was more closely aligned with standards based instruction. Data on her test scores were also not available, but again, she was identified for the study as a highly effective teacher. Ms. W also worked closely with mathematics teacher educators at State University and regularly hosted college interns, suggesting she was respected in the educational community for her instructional practices. PAGE 221 22 1 CHAPT ER 7 CONCLUSIONS The purpose of this dissertation was to understand how teachers who are successful with low achieving students of color living in poverty supported their students in learning mathematics. Students of color and students who live in poverty have struggled with academics and tend to perform poorly in mathematics, especially when compared to their white, middle class peers (Dewan, 2010; Post et al., 2008; Rothstein, 2002; Tutwiler, 2007). Prior research indicates mathematics instruction for hig h poverty students of color typically focuses on skills and procedures rather than concepts (Davis & Martin, 2008; McKinney et al., 2009; McKinney & Frazier, 2008; NCTM, 1999; Webb & Romberg, 1994; Watson, 2002; Weiss, 1994). Standards based instruction ha s been suggested as one solution to the struggles these students face in mathematics. This type of instruction is characterized by a focus on processes such as problem solving, reasoning and proof, communication, connections, and representations (NCTM, 200 0). Additionally, standards based instruction requires students to engage in the mathematical practices outlined by the CCSSI (2010), which include practices such as learning to reason abstractly, model with mathematics, make sense of problems, and perseve re in solving problems. To achieve the goals of the NCTM (2000) and CCSSI, teachers need to consider a way of teaching that differs from the traditional direct instruction model (Kepner, 2011), and research indicates that standards based instruction may be beneficial to students (Boaler, 1998, 2002; Kilpatrick et al., 2003; Lesh & Zawojewski, 2007; Post et al., 2008; Schoenfeld, 2002; Silver et al., 1995; Van Haneghan et al., 2004 ). Other researchers (Gee, 2008; Lubienski 2000a, 2000b, 2002; Zevenbergen, 20 00) point out, however, that students of color or who live in poverty PAGE 222 222 may struggle with standards based instruction if teachers do not provide the appropriate Culturally responsive teac hing is a pedagogy adopted by effective teachers of cultures. CRT may support students of color to achieve academically (Banks et al., 2005; Bonner, 2011; Gay, 200 0, 2002; Gutstein et al., 1997; Ladson Billings, 1994, 1995a, 1995b, 1997; Peterek, 2009; Tate, 1995). Gutstein and colleagues (1997) examined the role of standards based instruction and CRT in middle school classrooms, but more research focusing on both t hese approaches to mathematics teaching is needed. This study extends prior research by explicitly examining both standards based and CRT practices within mathematics classrooms and how (or whether) teachers draw on these two pedagogies to influence studen This study sought to understand how highly effective teachers of students of color living in poverty helped their students to engage with mathematical content. Specifically, I examined the perspectives and practices of these te achers. There were two participants, Ms. J and Ms. W. Ms. J is alternatively certified and taught seventh grade mathematics education, taught high school Algebra I. Both different from those of their students. Data were collected through classroom observations and interviews and were analyzed using qualitative methods. Both Ms. J and Ms. W had caring relationships with their students. They believ ed these relationships were essential for their success as teachers. Furthermore, Ms. W acted as a warm demander ( Bondy & Ross, 2008; Dixson, 2003; K leinfeld, 1975 ; Ross PAGE 223 223 et al., 2008; Wilson & Corbett, 2001) because she insisted that students act respectfu lly and meet her high expectations for academics and behavior. She additionally supported them to overcome barriers to their success in a demanding yet caring manner. Ms. J also insisted upon respect and held high expectations for behavior but, unlike Ms. W, sometimes blamed students and their families for lack of achievement and did not While both teachers utilized the GRM required by the county in which they taught to structure their lessons, their m athematics instruction differed greatly. Ms. J believed mathematics into easily followed procedures. She required students to follow the procedures she outlined and emphasized corr ect answers and key words. She played nonmathematical games to engage students and strove to make problems interesting and explicit demonstration followed by drill and practice characterized her instruction as a pedagogy of poverty (Haberman, 1991). Despite this, Ms. J was well liked and respected as a quality teacher by students, colleagues, and her principal. In contra supported them to build on what they knew and to apply known strategies in new contexts. Students were given multiple opportunities to succeed and were supported to build their sense of se lf confidence in mathematics. Ms. W emphasized understanding and reasoning over formulae and procedures, established norms for communication about mathematics, modeled her thinking as she solved problems with students, PAGE 224 224 required students to work regularly i n heterogeneous groups, and encouraged the use based mathematics teaching. Discussion This study sought to extend prior research by examining the ways in which CRT and standa rds based instruction manifest in the classrooms of mathematics teachers identified as highly effective with traditionally underachieving students of color. The two char acterized as adhering to a pedagogy of poverty (Haberman, 1991). She did not acknowledge the societal barriers to school success that students of color face and sometimes blamed students or their families for their failures. Conversely, Ms. W adopted the i nsistent and caring stance of a warm demander. She understood that her black students had a different culture and way of learning than white, middle class students, and she adapted her instruction to match their learning needs. She taught in a way that val ued understanding over efficiency, engaged students in cooperative learning, and encouraged them to express their unique ways of thinking by allowing them to choose one of several solution strategies when solving problems. The main similarity between the p articipating teachers was the relationships they had with students. They both made a point to let students know they were cared for, and the evidence suggests that students perceived and returned these feelings. Why is it that even though the pedagogy of p overty that Ms. J adhered to is directly contradictory to standards based instruction, she was a viewed as a successful, respected, and well liked teacher? Haberman (1991) provided an explanation: PAGE 225 225 The pedagogy of poverty is sufficiently powerful to undermi ne the implementation of any reform effort because it determines the way pupils spend their time, the nature of the behaviors they practice, and the bases of their self concepts as learners. Essentially, it is a pedagogy in which hout becoming involved or thoughtful. (p. 292) Explicit, traditional instruction is what students are familiar with and it takes the guesswork out of what to do in order to successfully solve a problem. Furthermore, mathematics reform, and particularly cla ssroom discourse, may be difficult for students whose cultural backgrounds are not aligned with the cultures valued in school (Gee, 2008; Lubienski, 2002; Zevenbergen, 2000). Standards based instruction requires students to take an active role in their own learning in a way that the pedagogy of appropriate supports to ensure their success with reform based mathematics may encounter the struggles that Lubienski (2000a, 2000b, 200 2) and others described, where students were not only resistant to classroom discourse but explicitly stated they would prefer more directed instruction. When we consider the question of the nature of standards based instruction and CRT in the classrooms of highly successful teachers of students of color, Ms. W can provide us with some insight. First, she is a warm demander. This pedagogical stance is one aspect of CRT but it does not define it. CRT is characterized by teachers who ensure the academic succ ess of all their students and competence (Ladson Billings, 1995b). Ms W achieved both of these goals The data suggest she supported students to become successful academically by creating a safe environment in her classroom in whi ch students felt comfortable trying and motivated to put forth effort. She also reminded them constantly to turn in assignments, and by supporting students in learning to reason mathematically she may have improved their PAGE 226 226 chances of success when encounterin g unfamiliar problems (e.g., on the EOC). used instructional practices such as teaming that were grounded in their cultures rather than clashed with them, thereby potent ially supporting students to build their sense of cultural competence. Ms. W did not, however, work to understand, analyze, and criti que the existing social order. This, coupled with praxis, the iterative cycle of reflection and action (Hinchey, 2004) is a key component of CRT (Gay, 2002; Ladson Billings, 1994, 1995b). Thus, Ms. W taught in such a way that was aligned with but not precisely defined by CRT. based instructio n but she did not enact every element of reform. For instance, her students did not engage in true problem solving. Problem solving requires students to work (individually or with peers) through problems for which no solution method is known in advance (NC TM, 2000). Instead, the class often solved problems as a whole, and Ms. W let ideas come from her students, took up their suggestions, and guided them toward the application of a strategy for unfamiliar problems. She did, however, establish norms for class room discourse and encourage students to use and discuss multiple solution strategies, which is one important aspect of standards based instruction (Lubienski, 2000a; Schoenfeld, 1987 ). Ms. W also required students to work in heterogeneous groups, took up knowledge of mathematics; all of these practices are consistent with standards based instruction. PAGE 227 227 While not a perfect example of CRT or standards practi ces embodied elements of each. Gutstein et al. (1997) stated that the two pedagogical approaches are complementary but that teachers need to explicitly actualize the connections. Ms. W did this by adapting her instruction to align with the way she believed students of color learn best, specifically, by engaging students in strong, caring relationships she had with her students, and she drew on these relationships in an eff ort to support students in learning mathematics. The participating between these teachers and align with research that suggests relationships and trust are a foundational aspect of CRMT (Bonner, 2011). Models of effective mathematics teaching of students of color need to highlight the importance of these relationships. self efficacy, enj oyment, and effort (Sakiz et al., 2012), and teachers who fail to connect in personal ways may be unable to support achievement motivation (Patrick et al., 2003). Research indicates that care, a component of CRT, is a crucial element of student success ( Ga rza, 2009; Tutwiler, 2007; Wald & Losen, 2007 ). In contrast, the literature on standards based instruction does not address the importance of care or the psychological environment (Patrick et al., 2003) in the classroom. Without strong relationships, it ma y be that the mathematics instruction standards based or not is less effective. Perhaps the way in which Ms. W taught is what is realistic and attainable for practicing teachers. Researchers and mathematics educators paint a picture of what an PAGE 228 228 ideal te acher is, but it may not be practical for teachers to embody everything educational researchers, teacher educators, NCTM, and politicians say they should. Teachers have innumerable constraints placed upon them limited time, standardized tests, paperwork, even requirements for the type of lesson plan format to use and these constraints certainly make focusing on teaching students more difficult. As Ms. W teach t2_111711]. Teachers strive to do the best they can with the knowledge that they have. Ms. J, for example, was aware of her lack of formal education in both content and pedagogy, and she sought out every possible opportunity to attend training or collabor ate with more veteran teachers so that she could improve the way in which she taught. She also asked me on a weekly basis during observations what I was learning about teaching from the other participating teacher (I promised her I would share my results o nce the study was complete). What we might glean from these cases is that even teachers identified as highly successful teach in very different ways. In addition, both teachers expressed a willingness and eagerness to improve. Are the ways in which Ms. W e Ladson Billings (1994) noted that cultural responsiveness is not pedagogy specific, but this is not the case for the NCTM standards (NCTM, 2000) and CCSSM (CCSSI, 2010). The State of Florida supports a model of instruction called gradual release (see FLDOE, n.d. ) that, as described by the state, is aligned with a direct instruction approach and is contradictory to the problem based, discussion rich, student driven nature of PAGE 229 229 mathematics reform. Some teachers, such as Ms. W, may implement this model flexibly, but overall this type of mandate undermines the pedagogy that mathematics educators ar e striving for. It is important for mathematics teacher educators to consider the challenges teachers face and work to identify ways to integrate elements of reform into their instruction in a realistic manner. Implications This study provides a thick desc It may offer concrete examples for teacher educators as they prepare future teachers or conduct professional development with inservice teachers. Specifically, the case study of Ms. W characterizes one w ay that standards based teaching can occur in a setting with high numbers of students of color. The case may be a model for conducting mathematics instruction that aligns with mathematics standards (e.g., CCSSI, 2010; a perfect example of standards based instruction but it provides a realistic illustration of the way one teacher effectively supported traditionally underperforming students to become successful in mathematics. The case additionally provides an example of a warm demander. This may afford teachers insight into how one might structure the psychological environment of a classroom to promote strong relationships and a feeling of belongingness as a way of improving mathematics achievement. This study may also he lp teacher educators to support preservice and inservice teachers to develop an assets based belief system as described by Gutstein and colleagues (1997). The findings indicate a problematic issue related to teaching students of color effectively. Ms. J ta ught in a way that was not entirely aligned with standards based instruction nor CRT. Her instruction, characterized by elements of a pedagogy of PAGE 230 230 poverty, undermined the kind of teaching that the literature suggests is necessary for the success of students of color who live in poverty. Ms. J, however, was nominated as an effective teacher, her superiors and colleagues believed her to be a successful and caring teacher, and her students performed well on standardized tests. Furthermore, her instruction was a ligned with GRM that the FLDOE ( n.d. traditional approach to instruction that is inconsistent with standards based reform and which does not support students in learning much more than procedural knowledge ( Boaler, 1998; Schoenfeld, 1988). This suggests that what society values regarding teacher practice and educational outcomes is not aligned with what mathematics educators seek for the children of our society (e.g., problem solving and communication skills, conceptual knowledge). Students who do not perform well in mathematics are less likely to attend college and gain the skills necessary for success in our globalized society (Friedman, 2005 ; NCTM, 2000; NRC, 1989), and they need more than just the procedural knowledge they gain from a traditional model of instruction. Teaching that does not prepare students to engage in critical mathematical thinking (Gutstein et al., 1997) certainly is not preparing them for future success outside of the school setting. Mathematics educators thus need to attend not only to the practices of teachers but also to the beliefs in our society about what constitutes mathematics, effective teaching of mathematics, and mathematical pr oficiency in order to support teachers to make the changes the education research community is calling for. This study also highlights the issue of teacher education, certification, and the d held degrees in both PAGE 231 231 mathematics and education. Ms. J, on the other hand, was alternatively certified and she had no advanced mathematics training. Moreover, all her formal pedagogical training occurred in professional development sessions provided by he r district or state. Ms. W taught in a manner aligned with standards were contradictory to mathematics reform. It is possible that other teachers are following a path similar to that of Ms. J so it is important for state s to examine the pedagogy that is being perpetuated through the professional development sessions they offer and the policies they implement to determine their consistency with standards that are being described as the end goal. For instance, the CCSS requ ire students to construct arguments and critique the reasoning of others (CCSSI, 2010) but the GRM adopted by the State of Florida does not provide students with an opportunity to engage in this practice. If the policies and professional development provid ed by states perpetuate a way of teaching that does not afford students an opportunity to engage in the mathematical practices necessary for their success in mathematics, then teachers who have not learned alternative methods of instructing may struggle to implement reform in a manner that would scaffold their students to succeed. Limitations and Suggestions for Further Research There are several limitations to this study that need to be addressed. First, the nomination process employed necessarily limited the findings of the study. This study relied on nominations provided by only one person, the secondary mathematics of verifying whether the teacher was indeed perceive studies (e.g., Ladson Billings, 1994; Peterek, 2009) utilized a process that allowed for school personnel as well as community members to nominate participants. Related to PAGE 232 232 passing rates on standardized tests or low course failure rates, but other ways of measuring teacher success exist, including student motivation, conceptual understanding, self efficacy, attitudes toward mathematics, and problem solving ability. The participant pool may have differed had community members been part of the willingness to participate. Seven teachers were nominated but only two met all the requirements for this study and agreed to participate. There are certainly other depictions of effective mathematics teaching of traditionally underperforming students that were not captured in this study. Another limitation is that I was unable to acquire FCAT and EOC scores for the teaching, but this study occurred during her third year and it is unclear whether her success occurred by chance or was consistent year after year. A further limitation is Learning computer program. This is a confounding factor, as effectiveness as a teacher may be attributed to Carnegie rather than to the practices in which she engaged. Given the literature that suggests feeling cared for and standards based instruction may contribute to student success (Boale r, 1998, 2002; Garza, 2009; Kilpatrick et al., 2003; Post et al., 2008; Schoenfeld, 2002; Silver et al., 1995; Tutwiler, 2007; Van Haneghan et al., 2004 ; Wald & Losen, 2007 ), this may not be the case but this alternative hypothesis needs to be acknowledged PAGE 233 233 In addition to correcting these limitations in future studies, there are additional areas that warrant further research. To begin, the design of this study did not utilize interviews with students as a form of data collection. There are many studies on effective teaching ( Boaler, 1997, 1998, 2000, 2002; Ladson Billings, 1994, 1995a, 1995b; Peterek, 2009; Tate, 1995; Ross et al., 2009 ) but few that turn to students as a source of data. Future research that includes this method may be of benefit for adding the voices of traditionally underperforming students of color to the conversation on effective teaching practices. One goal of this dissertation study was to examine how standards based teaching and CRT intersect in a mathematics classroom. These two ped agogies are theoretically compatible but literature on them is rather disparate, with few studies examining the way both reform and CRT are manifested by teachers (e.g., Gutstein et al., 1997). Other researchers suggest that when adopting a standards based approach for teaching mathematics to students of color or who live in poverty teachers need to be cognizant of the struggles students may face with this approach due to the differences between school culture and vernacular culture (Gee, 2008; Lubienski, 2 000a, 2000b, 2002; Zevenbergen, 2002). Thus, the cross case analysis aimed at not only comparing the two cases, but also understanding whether reform or CRT characterized the instruction of each teacher. Ms. W adopted a warm demander stance and, although h er instruction did not incorporate all elements of standards based teaching, it was fairly well aligned with a reform oriented teaching approach. She thus provides us with some detail about the way reform and CRT may be enacted jointly. More studies in set tings to which these PAGE 234 234 results are not transferable are needed to help us to further understand culturally responsive mathematics teaching. Additionally, a study of this kind would benefit from a discourse analysis of classroom practices. Lubienski (2000a) revealed that the lower SES girls in her study struggled with whole class discussions. They found it difficult to distinguish between correct and incorrect answers during the discussions react when a classmate disagreed with the ir a nswer or mathematical statement. Lubienski (2002) and Gee (2008) suggested that the difficulties students of color face in school may be sroom may provide insight into the ways in which that teacher was able to bridge that cultural gap to successfully engage students of color in the mathematical discourse that is an integral part of standards based instruction. A final area of potential s tudies includes examining conceptions of teacher effectiveness and what it means for a student to be successful in mathematics. Ms. J was identified as an effective teacher: her students showed growth on the FCAT; she was well respected and liked by her st udents, colleagues, and superiors; and she was were failing her course, however, and her instruction was skill based and procedural. She held a goal for her students to lea rn to solve real world problems, but she did not provide them with an opportunity to solve such problems, nor did she engage them in the mathematical thinking, discourse, and problem solving that mathematics educators have been calling for for many years ( CCSSI, 2010; NCTM, 2000; NRC, 1989). PAGE 235 235 Furthermore, students may want teachers to provide explicit, procedural instruction (Lubienski, 2000a). This contradiction suggests that school administrators, parents, and even students have a different perspective of what and how students should learn than mathematics educators. In particular, it would appear from these results (and the current climate surrounding standardized testing and teacher evaluations) that society at large values students who can recite formula e and apply rules and teachers who explicitly show students how to solve problems. Mathematics educators, on the other hand, value students who are able to reason through unfamiliar problems, communicate their thinking, and derive and understand rather t han memorize formulae. They suggest that teachers must engage students in processes that allow students an opportunity to learn the type of knowledge they value. Studies that further examine these discrepancies in effectiveness and mathematical success w ould help researchers and mathematics educators to better understand the challenges they are facing in regards to changing the way mathematics is taught to better enable all students to learn. PAGE 236 236 APPENDIX A INFORMED CONSENT FOR M Dear Educator, I am a docto ral candidate in the School of Teaching and Learning at the University of Florida conducting research for a dissertation on the teaching practices of successful teachers of low income students of color. I am conducting this research under the supervision o f Dr. Stephen Pape. The purpose of this study is to understand how teachers who are successful with low achieving students of color who live in poverty support their students to learn mathematics. I am asking you to participate in this study because you ha ve been identified as a highly successful teacher. With your permission, I would like to observe one of your classes for a total of six weeks. Together, we can select a class. During data collection, I will observe the class every time it occurs as well a s conduct several informal interviews and three formal interviews with you about classroom practices and student interactions. I will take field notes during these observations and interviews. Informal interviews may be audio recorded but will not be trans cribed. Formal interviews will last no more than one hour, will be audio recorded, and will be scheduled at your convenience. You will not have to answer any question you do not wish to answer. Only I will have access to the audiotapes. The formal intervie ws will be personally transcribed by me, removing any identifiers during transcription and replacing your name and any other names mentioned with pseudonyms. The tapes will be kept locked in a cabinet in my office. Your identity will be kept confidential t o the extent provided by law and will not be revealed in the final manuscript. There are no anticipated risks, compensation or other direct benefits to you as a participant in this study. Your participation is voluntary and you may withdraw your consent a t any time without penalty. If you have any questions about this research protocol, please contact me at (352) 359 8417 or my faculty supervisor, Dr. Stephen J. Pape, at (352) 273 4230. Questions or concerns about your rights as a research participant may be directed to the IRB02 office, University of Florida, Box 112250, Gainesville, FL 32611; (352) 392 0433. If you agree to participate in this study, please sign and return this copy of the letter to me. A second copy is provided for your records. By sig ning this letter, you give me permission to report the data I collect in interviews with you and observations in your classes. This report will be submitted to my faculty supervisor as part of my dissertation requirements. Also, by signing, you give me per mission to use these data in academic presentations and publications. Thank you, Karina K. R. Hensberry ______________________________________________________________________ hing practices. I voluntarily agree to participate in the interview and I have received a copy of this description. ___ ______________________________ _________________ Signature of participant Date PAGE 237 237 APPENDIX B FORMAL INTERVIEW PROTOCOL 1 1. Tell me abou t your academic background. Where did you attend college? What was your major? Did you attend graduate school? If so, where? What was your major? 2. How long have you been teaching? How long have you been at this school? 3. Describe a time you felt proud to be a teacher. 4. What are your goals for your students? What do you do to help them reach these goals? 5. Can you think of a student you taught who was a real success story academically? Tell me about this student. Where was he/she at the beginning of the year? At the end of the year? (Probe for definition of academic success.) What factors do you Can you give me another example? 6. Think of a time when a student was struggling to understand the content being taught. How d id you figure out that he/she was struggling? What do you think contributed to his/her struggle? What did you do to help the student gain stronger understanding? Can you give me another example? 7. Think of one student of color who stands out to you as a suc cess story. What do you think was the key to your success with this student? Can you give me another example? Are the keys to success different with this student? Are there any guiding principles that you think define successful teaching for students of co lor? 8. What types of teaching methods have given you the most success with students of color and who live in poverty? Where did you learn to do that? As a group, what do you feel it is that your students need from you as their teacher? 9. I noticed that you [ pr ovide specific example of classroom interaction ] Can you talk to me more about that? What was going through your mind during that interaction? ( back on it, would you have do ne the same thing if you had a chance to do it over? If so, why? ( Repeat question s as necessary. ) PAGE 238 238 APPENDIX C FORMAL INTERVIEW PROTOCOL 2 1. When did you decide to become a teacher? What made you choose mathematics? What do you think are the differences be tween being a mathematician and being a mathematics teacher? 2. What grades have you taught? What other mathematical content areas have you taught? How do those experiences compare to what you now teach? 3. Describe a typical day in the math class I am observing 4. today or yesterday ]. What did you think about when planning the lesson? Is planning a time consuming process for you? Why or why not? What resources do you use when planning your lessons ? 5. Think about the unit you are currently teaching. What are your goals for math teaching? What are your goals for students as math learners? How do you evaluate these goals? Are there some goals you feel successful in reaching fairly consistently? Which and why? Are there some that are hard to reach? Which and why? 6. student. What is she/he able to do? What characteristics/traits help him/her be successful? In what ways do you support your studen ts to become like this ideal mathematics students? (Probe for specific examples.) 7. Describe a student who has struggled in your class. How did you handle it? Do you have other strategies you use to support struggling students? How did you learn to do that? 8. Think of a time when you would say your students were highly engaged in a math lesson. Tell me about that lesson. What did you do to support their engagement? (Probe for specific examples.) How did you learn to do that? Can you describe another time when students were highly engaged perhaps for different reasons? (Go through all the questions again.) 9. During the last interview, you mentioned [ ]. Can you tell me more about that? (Repeat question as necessary.) 10. During the last i nterview, you mentioned [ ]. Can you help me understand what you meant when you said [ here ]? Would you give me an example of that? (Repeat question s as necessary.) 11. During the last interview, you mentioned [ ]. Would you give me an example (of that/of what you meant by) [ ]. (Repeat question as necessary.) PAGE 239 239 APPENDIX D FORMAL INTERVIEW PROTOCOL 3 1. Describe your approach to teaching. Has it changed since you be ga n teaching ? If so, how? 2. What characteristics, if any, do diverse students bring to the classroom? 3. You have been identified as a highly successful teacher of low income students of color. What do you think you do that makes you so successful? What else? (Probe for specific examples.) What else? Do you think this would differ if you were teaching predominantly white, middle class students? If yes, how might it differ? 4. Do you think the experiences in math class of white students in middle class communities /classrooms differ from those of low income students of color? If so, how? 5. What strategies do you use for classroom management? What routines and procedures do you have set up? (Probe for specifics.) How did you learn to do that? (Repeat question as necess ary.) How do you deal with lateness to class ? How do you deal with failure to do homework? How do you deal with inattentiveness? How do you handle major classroom disruptions? Would these strategies differ if you were teaching predominantly white middle cl ass students? If so, how? 6. I noticed that you [ provide specific example of classroom interaction ] Why did you decide to do that? (Repeat question as necessary.) 7. Was your instruction in the class I observed over the last few weeks typical of your instructio n in other classes this year? In previous years? If so, how so? If not, how was it different? 8. During the last interview, you mentioned [ ]. Can you tell me more about that? (Repeat question as necessary.) 9. During the last intervie w, you mentioned [ ]. Can you help me understand what you meant when you said [ here ]? Would you give me an example of that? (Repeat question s as necessary.) 10. During the last interview, you mentioned [ use te ]. Would you give me an example (of that/of what you meant by) [ ]. (Repeat question as necessary.) 11. Based on the feedback I have shared from observations, is there anything you Was there anything I saw that surprised you? PAGE 240 240 APPENDIX E INFORMAL INTERVIEW PROTOCOL class observations. They are open ended and semi structured to allow for flexibility in terms of the actual teacher behaviors that will be examined. The forms of the questions are reflected below. 1. I noticed that you [ state classroom behavior ] Can you tell me more about what was happening there? What was going through your mind when you d id that? (Probe for would you respond differently if you had the opportunity to do it again? If so, why? (Repeat questions as necessary.) 2. I noticed that [ ] as ked a question about [ question ]. Can you tell me about that? What was going through your mind when [ ] asked that? What were you thinking about when you responded? back on it, would you respond differently if [ ] asked that question now? If so, why? (Repeat questions as necessary.) 3. 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Sociomathema tical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27 458 477. Zaslavsky, C. (1996). The multicultural math classroom: Bringing in the world. Portsmouth, NH: Heinemann. success as a function of linguistic, social, and cultural background. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning: International perspectives on mathematics education (pp. 201 224). Westport, CT: ABEX. PAGE 252 252 BIOGRAPHICAL SKETCH Karina K. R. Hensberry graduated from the University of Florida in August 2006 with a Bachelor of Science in mathematics. After graduation, she taught mathematics at Gainesville High School for two years. Kar ina also acted as an instructor for the Engineering GatorTRAX program at the University of Florida. While teaching at Gainesville High, she earned a Master of Education in curriculum and instruction from the University of Florida in August 2007. One year later, she enrolled at the University of Florida to begin working on a Doctor of Philosophy in curriculum and instruction with an emphasis in mathematics education. Karina graduated in August 2012 and joined the School of Education at the University of Col orado Boulder as a Research Associate for the PhET interactive simulations project. She continues to research equity in mathematics education and mathematics simulation design and use. 