What Mathematics Education Might Learn from the Work of Well-Respected African American Mathematics Teachers in Urban Schools

by Daniel Chazan, Andrew Brantlinger, Lawrence M. Clark & Ann R. Edwards - 2013

Background/Context: This opening article, like the other articles in this special issue, is situated in scholarship that attempts to understand the racialized nature of mathematics education in the United States and to examine the racial identities of students and teachers in the context of school mathematics. It is designed to respond to the current (mathematics) education policy context that largely ignores teachers’ experiential and cultural knowledge while stressing the importance of teachers’ content knowledge and academic achievement.

Purpose/Objective/Research Question/Focus of Study: This article presents theoretical perspectives and research questions concerning the knowledge and other resources that African American teachers bring to teaching mathematics, perspectives and questions that are taken up in the five subsequent articles in this special issue.

Setting: The cases developed in this special issue were developed from observations of the introductory algebra classes of, and interviews with, two well-respected African American teachers in one neighborhood high school in a large urban school district that serves a predominantly African American student population.

Research Design: This opening article frames two case study papers and two analysis papers that report on findings from a large-scale qualitative study of the racialized identity and instructional approaches of two of the six African American mathematics teachers studied in the Mid-Atlantic Center for Mathematics Teaching and Learning Algebra 1 Case Studies Project.

Conclusions/Recommendations: Together with the other articles in this special issue, this work contributes to the development of more sophisticated attempts to integrate understandings of race into the work of the mathematics education community. It challenges taken-for-granted notions of the knowledge base and resources needed to be an effective mathematics teacher of African American students in underresourced large urban schools.

In The Social Turn in Mathematics Education Research Stephen Lerman (2000) suggested that since the mid-1980s, research in mathematics education has moved away from theoretical frameworks almost solely grounded in mathematics and psychology and toward ones for interpreting the social origins of knowledge and consciousness . . . that see meaning, thinking, and reasoning as products of social activity (p. 23). He called for research in mathematics education to develop accounts that bring together agency, individual trajectories, and the cultural, historical, and social origins of the way people think, behave, reason, and understand the world (p. 36).

In the United States, race plays an important role in how people understand the world and in how lives are structured and lived (Cashin, 2004; Conley, 1999); understandings of race in the United States are therefore the focus of both journalists and authors of fiction (e.g., Ellison, 1952; Lelyveld and correspondents, 2002; Morrison, 1970). For mathematics educators based in the United States, a major, and seldom addressed, challenge in responding to Lermans call is, as Danny Bernard Martin (2009) so eloquently articulated, that it requires that we attend to, and grapple with, race:

Very few researchersincluding those working within the so-called mathematics learning as social perspectivehave attempted to unpack their use of race in the context of mathematics learning and participation . . . most studies of differential outcomes in mathematics education begin and end their examination of race with static categories and group labels that are used for the sole purpose of disaggregating data . . . students are reported as belonging to certain races, and the performances of these races are then compared. (p. 313)

Drawing on critical race theory, Martin (2006) would like us to consider mathematics classrooms as racialized spaces, as spaces in which the meanings of race in our society both shape interactions and are constituted by these interactions.

With this special issue, we seek to respond to Lermans call, and Martins challenge, by providing case descriptions of two African American¹ teachers who teach in contexts in which most of their students are Black, offering a framework for understanding how teachers can influence student mathematics identity formation and address issues of race directly, and presenting a historically grounded conceptualization of the role of the African American mathematics teacher that connects the work of the case study teachers to the work of their predecessors. The articles in this special issue highlight the experiences that African American teachers bring with them to the work of teaching and use as resources for creating the instruction they offer to students; we focus in particular on how these teachers convey to students a sense of purpose for engaging with school mathematics. We build on the work of Project 2061 that centers on how textbooks convey a sense of purpose to students (see DeBoer et al., 2004) and on the description in Lampert (2001) of the task of teaching students to be people who study in school.

This work intends to contribute to the development of more sophisticated attempts to integrate understandings of race into the work of the mathematics education community. At the same time, given the social realities of the contemporary United Statesfor example, racially segregated housing patterns and implications for intergenerational patterns of the accumulation of wealth (Shapiro, 2005), as well as concentrations of poor communities in center cities and inner-ring suburbs (Hanlon, 2009)we are conscious that even though race is a strong factor in contemporary U.S. schooling, it is often indirectly referenced in policy discussions by the term urban; in such discussions, the descriptor urban for schools is not simply a descriptor of the population density of the surrounding community, but, among other things, describes schools with many students of color, schools for which many contemporary policies are designed. Therefore, even though we have reservations about the term urban (and often feel it necessary to qualify this term), we engage it and see our work as contributing to the growing body of literature in mathematics education that focuses on urban teachers, teaching, and reform (e.g., Boaler, 2006; Boaler & Staples, 2008; Cwikla, 2007; Gutiérrez, 1999, 2000; Gutstein, 2003, 2006; Silver, Stein, & Nelson, 1995; Webb & Romberg, 1994, as well as earlier scholars, e.g., Forbes, 1970; Mendelsohn, 1970).

With the opening article of this issue, we create the context for the remaining articles. This article is organized in four sections. It begins with theoretical and conceptual frames that structure our utilization of cases to explore the use of knowledge and other resources in mathematics teaching. Like Cook and Brown (1999), we are interested in how people bring tacit and explicit, as well as individual and group, resources to bear when they act in patterned social interactions. Like Hill, Rowan, and Ball (2005), we are focused on teachers and are interested in accounting for differences in teachers actions (and ultimately effectiveness) by describing individual teachers as bringing resources to the tasks of teaching. In particular, we focus our efforts on understanding the resources that teachers bring to bear on the task of conveying to students a sense of purpose for engaging with school mathematics. We consider a range of resources, explicit knowledge of mathematics for teaching, as well as tacit resources that result from racial identity and experiences growing up as an African American in the United States.

In the second section of the article, with these frames in place, we articulate overarching exploratory research questions of the Mid-Atlantic Center for Mathematics Teaching and Learning (MACMTL) Algebra 1 Case Studies Project, from which these articles stem, as well as a rationale for the value of pursuing these questions in the school contexts sampled from in this study; these research questions structure the presentation of the teaching of Floyd Lee and Madison Morgan in the two articles that follow. As we lay out our conceptual frames and research questions, we highlight ways in which this project breaks with common ways of thinking about teachers: This project seeks to understand the work of well-respected, but not extraordinary (in contrast to Matthews, 1988), practicing teachers in racially and economically segregated urban settings, and this study seeks to cast a teacher knowledge net more broadly than teachers content knowledge or pedagogical content knowledge. As a part of highlighting these aspects of the study, and stimulated by another piece of Danny Martins (2007), we consider how this work might inform policies about the question of who should teach mathematics in nonselective urban schools that are segregated along race and class lines. We contrast our perspectives on teachers and teaching with perspectives prominent in current discussions of school reform.

The third section of this article focuses on the criteria that were used to select the contexts in which this research was carried out and provides further detail about these contexts. The fourth and final section closes by outlining the structure of this special issue, highlighting relationships between the remaining five pieces, and explicating the central role of cases describing the work of Floyd Lee and Madison Morgan.

PART 1: THEORETICAL AND CONCEPTUAL FRAMINGS

At the heart of this special issue are cases of the teaching of two well-respected African American mathematics teachers who teach in an urban school where student performance on an end-of-course assessment has high stakes both for the students taking the test (for a description of the status of graduation-linked exams, see Center on Education Policy [CEP], 2009) and for their school. Students must do well on these tests to graduate from high school; the school is graded on the percentages of students with proficient scores and can be penalized if those percentages are not high enough.

Though they are both well respected in their district and school community (when we observed, they were teaching in the same building), the two teachers teach quite differently. Like many of the teachers in our sample, Floyd Lee follows the districts pacing guide closely. A unique aspect of his teaching is that he conveys a sense of purpose for engaging with school mathematics by taking substantial class time to exhort his students to make use of their educational opportunitiesto study in school and prepare for the high-stakes test they will take at the end of Algebra 1. His way of speaking to students about these issues evidences his familiarity with both urban youth culture and modes of sermonizing from the Black church, and is expressed in language grounded in Black English Vernacular. In the speeches he gives to his students on making use of their educational opportunities, he is assertive, yet positions himself as caring and as an advocate and supporter for students in ways that Brown (2003) articulated as central to culturally relevant classroom management.

By contrast, in her efforts to convey a sense of purpose for engaging with mathematics, Madison Morgan focuses her energy on creating a coherent curriculum for her students, even though to do so, she departs from the order and the pacing of the districts curriculum guide. As described in her case study, in an effort to give students a reason to solve for unknown variables (like x), Madison inverts relationships between equations in one variable and equations in two variables, as outlined in the district guide and in most typical textbooks. Her instruction focuses on a smaller number of tasks than suggested in the guide and is notable for the connections it makes among data analysis and algebra, real-life contexts, and student-generated solutions. In the cases, in addition to representing what these teachers do, we also share some of the reasons that they express for teaching in these ways, as well as how what it means to them to be African American, or Black, in the United States at this time is connected to their work as teachers.

In the rest of Part 1 of this article, we outline the theoretical and conceptual orientations that structure our construction and use of the cases of Floyd Lee and Madison Morgan and provide our rationales for the importance of focusing research on African American mathematics teachers in urban schools.

AFRICAN AMERICAN TEACHERS AS INDIVIDUALS WITH SHARED EXPERIENCE AND CULTURE

Given the prominence of the National Council of Teachers of Mathematics (NCTM) reform vision (e.g., NCTM, 2000), over the last two decades, many mathematics education researchers have sought to explain why some teachers carry out reform visions of mathematics education with fidelity and others do not, and to understand what might need to improve in order to better equip teachers to teach in particular ways (McNaught, Tarr, & Sears, 2010). Other researchers have investigated what makes some teachers more effective than others (Campbell et al., 2011; Hill et al., 2005). Thus, the focus of much existing mathematics education research on teachers is on how teachers differ one from another. Typical explanations for what differentiates one teacher from another make recourse to beliefs (Leder, Pehkonen, & Törner, 2003) or teacher knowledge (Lloyd & Wilson, 1998). Until recently, social categories (like race or gender) have not been as widely used for these purposes, even though there are widely cited examples from outside mathematics education that do so (e.g., Ballenger, 1992; Delpit, 1988).²

Particularly when it comes to insight into teachers adoption of new practices, in contrast to research focused on teacher knowledge and beliefs, some researchers have argued for the potential explanatory value of teacher identity (Ponte & Chapman, 2008), though there are many differing conceptualizations of teacher identity in the research literature. One such perspective frames teacher identity as the constellation of interconnected beliefs and knowledge about subject matter, teaching and learning as well as personal self-efficacy and orientation toward work and change (Collopy, 2003, p. 289) that stem from teachers personal experiences and, specifically, the stories and narratives they construct of their experiences both as learners and in the profession (e.g., Connelly & Clandinin, 1990; Drake, 2006; Hammerness, 2003). Another perspective on identity, more recently emerging in the literature on teacher identity (and as yet less well represented in mathematics education), draws largely from sociocultural and poststructuralist theories of identity and frames identity as a nexus of multi-membership (Wenger, 1998) involving a constant social negotiation in specific situational contexts (Britzman, 1993; Gee, 2000; Holland, Lachicotte, Skinner, & Cain, 1998). Under this view, who I am involves the negotiation among multiple arenas of our lived experiences. How one understands oneself as being a teacher stands in relation to other ways of being (e.g., a parent, a student, an African American). Although these notions of identity differ, both offer ways of understanding the relations among experience, teachers knowledge and beliefs, and their enactments of practice.

In this special issue, we conceptualize teachers as social actors who bring individual knowledge, personal experiences, and group memberships to teaching and use these as resources in their instruction (following Cook & Brown, 1999). We are interested in the ways that specific teachers teach differently than other teachers. In particular, our cases focus on key aspects of Floyd Lees and Madison Morgans instruction that differentiate their instruction one from the other, as well as from the other four well-respected urban mathematics teachers we have studied. But at the same time, we also conceptualize both Floyds and Madisons work, as well as the work of the other four, as mathematics teaching in the context of urban schools; thus, it shares some dimensions with the work done by other mathematics teachers in urban schools, as well as with the work of mathematics teachers in other sorts of schools. Similarly, we are interested in ways in which some of the experiences that Floyd and Madison use in crafting their instruction are ones that they share with other African Americans in the United States, such as experiencing institutional racism or attending African American churches. Although we are conscious of the heterogeneity of the African American experience and do not expect that all African Americans will share these same experiences, we seek out some aspects of Floyds and Madisons experience that are common to the experiences of many African Americans, although not a necessarily part of the individual experience of every African American. This attention to typical experiences of being African American in U.S. society leads us in the fifth piece in this issue (Clark, Jones, & Davis, 2013, this issue) to view Floyds and Madisons teaching in the context of the historical role of African American mathematics teachers in the education of Black students.

LEARNING FROM WELL-RESPECTED MATHEMATICS TEACHERS IN URBAN SCHOOLS

There are a number of challenges to developing accounts of teaching of the kind we just described and to using them to understand and improve schooling in the United States. Many of these challenges are conceptual and linguistic ones that involve constructing an appropriate object of study, for example, unpacking what we mean by urban schools and how the label urban is related to race in the United States. We illustrate these challenges, and the potential benefits of focused research, by examining discussions of who should teach in nonselective urban schools and by exploring how the contexts of urban schools might influence mathematics teaching. Finally, in an effort to learn from current practice in urban schools, we articulate our strategy of focusing on the work of well-respected teachers.

LIMITATIONS OF URBAN AS AN EDUCATIONAL CATEGORY

Many educators might be quite comfortable with the notion that the teaching of mathematics in urban schools has problems or challenges that are not present in mathematics teaching in other settingsindeed, that urban mathematics teaching might be defined by such challenges. As we will demonstrate next, a strong existing rhetoric claims that urban schools are categorically different from other schools (e.g., Haberman, 1991). However, for us, much of this rhetoric seems problematic. It does not accord with day-to-day experiences in urban schools or, for that matter, suburban or rural schools. For example, many of the contextual issues associated with rhetoric about urban schools are equally true for lower track students in what people might call suburban schools (Chazan, 2000; Page, 1991). And such a definition of urban in terms of challenges paradoxically makes schools no longer urban schools if they successfully meet these challenges.

Although the term urban may be a more neutral-seeming alternative to such clearly disparaging terms as ghetto, at-risk, and disadvantaged widely used in the past (Ladson-Billings, 1999), in terms of its connotations, urban is a problematic descriptor of communities and their schools. Unless it is unpacked, the word urban conjures up many of the same associations as the earlier terms it replaced, ones that are potentially obstacles to research endeavors. For example, Popkewitz (1998) argued that, rather than signifying properties of a geographical place, as more narrowly intended, in the educational literature, urban usually gives reference to certain unspoken qualities of the child and community who belong in that space (p. 10). Hence, without direct reference, urban invokes particular associations related to race, social class, segregation, family difference, poverty and economic status, and, supposedly, disproportionate rates of crime, violence, and drug use (Box, 1983; Theobald, 2005). In encompassing these aspects of social life, and in that sense responding to Lermans call, the word urban tends to assign deficiencies to urban places and urban people, a counterproductive effect, rather than to conditions such as racism, exploitation, segregation, lack of adequate resources, and corporate divestment and underemployment in cities that influence urban schools and students (Noguera, 2003; Rothstein, 2004). As a result, urban student failure and school failure surface as the sole factors in need of remediation and redress.

Usage of the term urban also implicitly, and explicitly, compares nondominant to dominant groups. These comparisons almost inevitably lead to generalized assumptions about the abnormality and inferiority of non-White and poor urban people. Hence, Popkewitz (1998) asserts that the urbanas used in education, rather than in architecture or urban planningis a place that lies outside of reason and the standards of the normal (p. 10). This conceptualization of being outside the normal quickly slips into deficit thinking about the dysfunction of urban communities and the flawed nature of students in urban schools.

A final problem with the term urban is that it masks considerable diversity between urban centers and heterogeneity within them. For example, Oakland, Milwaukee, and New York are cities with distinctive histories, economies, geographies, politics, population densities, ethnic mixes, and traditions. Although it is often taken as obvious that selective and nonselective schools in urban areas are different, often overlooked is the considerable diversity among nonselective city schools, over which the term urban school glosses. As is the case with some nonurban schools, some are small, and some are large (Council of Great City Schools, 2009). Some are well run, others are not. Some are racially diverse, some are racially segregated (Kozol, 2005). Some prioritize discipline over learning, whereas others do not (Devine, 1996).

Despite the ways in which this term, if not unpacked, can lead researchers astray, we consider the discourse of which the term urban is a part an important conversation to which educational research must contribute (though we will qualify the term to telegraph our lack of comfort). In particular, we offer our research as a resource to those considering desirable teacher profiles for the staffing of nonselective urban schools (what Prince, 2003, and others referred to as hard to staff schools).

WHO SHOULD TEACH IN NONSELECTIVE URBAN SCHOOLS?

The federal No Child Left Behind Act (NCLB, 2001/2002)with its mandates, incentives, and sanctions designed to raise student achievement, close achievement gaps, and ensure that teachers were highly qualifiedwas in full effect at the time of our study. As was the case in other large urban districts, there was a range of efforts to accomplish these goals. Like many urban districts, the district created pacing guides to ensure that teachers covered the content that would appear on end-of-the-year exams and to ensure that students who moved from school to school would experience fewer problems with the transition. There also was particular concern for raising achievement in the highest poverty urban schools that tended to be staffed by the least experienced, least effective, and often uncertified teachers.

The creation of alternative certification and alternative routes is another policy initiative adopted by many states to address the mandates of NCLB. Between the early 1980s and recent years, urban public school districts have relied on thousands of emergency certified teachers to staff high needs schools in such critical shortage areas as mathematics and science; these teachers are often portrayed as lacking strong subject matter backgrounds but frequently sharing the cultural, linguistic, and racial backgrounds of their students (Goodnough, 1995, 2004; Urban Teacher Collaborative, 2000). In the current high-stakes accountability era, one in which highly qualified teachers figure in sanction and reward regimes, districts are discouraged from hiring emergency certified teachers, and their numbers have dwindled (Boyd, Lankford, Loeb, Rockoff, & Wyckoff, 2008). To address persistent teacher shortages, states and urban school districts have turned instead to alternative routes to teacher certification (Liu, Rosenstein, Swan, & Khalil, 2008). In contrast to emergency certification, alternative certification allows new teachers to meet the NCLB (2001/2002) definition of highly qualified as long as they are working toward standard certification. In this context, the prominent recruitment organizations for alternate routes to teacher certification, namely Teach for America (TFA) and The New Teacher Project (TNTP), the latter of which recruits candidates for more than 20 Teaching Fellows programs (TNTP, 2011), have attracted considerable attention and resources for recruiting the best and the brightest to teach mathematics and other critical shortage areas in high-needs schools (Darling-Hammond, 1994; Levy, 2000; Paige, 2002). Proponents of alternative certification suggest that TFA corps members, Teaching Fellows, and participants in similar academically selective alternative route programs are a superior breed. In their view, in contrast to the typical urban teacher, this new breed brings to teaching a strong combination of the following: (1) subject matter knowledge, (2) subject-relevant professional experience, (3) verbal ability, and (4) a high-achievement work ethic (Kopp, 2001; Levy, 2000; McKinsey & Company, 2010; Paige, 2002; Walsh & Jacobs, 2007).

We find this rhetoric about the characteristics of the new breed of urban teacher to be problematic. To begin with, claims about the strong subject matter backgrounds of the new breed are inflated. As a case in point, TNTP and TFA have difficulty recruiting prospective teachers with strong postsecondary backgrounds in mathematics. Both TFA and TNTP have applied for waivers in a number of states (e.g., New York, Maryland) to allow teacher recruits who do not meet the required number of university courses in mathematics and science to be counted as certified (Boyd, Grossman, Lankford, Loeb, & Wyckoff, 2009; Goodnough, 2001). For example, less than 25% of mathematics teachers in the New York City Teaching Fellows program have majored in mathematics, or a cognate area (e.g., engineering, physics), in college (Donoghue, Brantlinger, Meagher, & Cooley, 2008).

Although it is true that the new breed of urban teacher has prestigious academic credentials, we question the assumption that high-achieving novice teachers from selective universities will disseminate a culture of achievement (Farr, 2010, p. 103; TNTP, 2005, p. 211) among students who do not share their economically and racially privileged backgrounds. The transfer of achievement theory is tied up with a denial of privilege, myths about meritocratic schools and society, and deficit narratives that suggest that urban students and families do not care about school (Brantlinger, Cooley, & Brantlinger, 2010).

We also find the claims about prior experience problematic, given that teaching is the first professional job for many new breed teachers (Donoghue et al., 2008; Veltri, 2010). Research suggests that the ostensibly content-relevant professional experience that a few of the remainder do possess is of limited to no instructional value (Scribner & Akiba, 2010). Recent research also has cast doubt on earlier studies that appeared to demonstrate that teachers with greater verbal ability produce larger than typical gains in student achievement (Aloe & Becker, 2009; Andrew, Cobb, & Giampietro, 2005). Beyond these empirical issues, such a focus on verbal ability as captured by standardized tests overlooks that there are many forms of communicative competence (Heath, 1983; Labov, 1972; Lee, 2004; Johnson, Nyamekye, Chazan, & Rosenthal, 2013, this issue) and that standardized tests used to measure verbal ability (e.g., SAT, ACT) may well be biased in favor of the cultural mainstream (Au, 2009) and therefore miss communicative competences that might be especially useful in teaching in race- and class-segregated urban schools.

Finally, where they have been implemented, these recruitment practices and certification policies designed to attract the new breed to teaching have had racialized and classed effects on urban and minority education. Because the new breed is majority White and middle class, such practices and policies have whitened and privileged the urban teaching force (Boyd, Grossman, Lankford, Loeb, & Wyckoff, 2006; Donoghue et al., 2008; Veltri, 2010). Since their inception, TNTP and TFA combined have recruited and trained more than 60,000 alternative-route teachers to teach in hard-to-staff urban schools (Kramer, 2010). In New York City, approximately 5,000 Teaching Fellows and TFA corps members annually were being certified to teach in the period from 2002 to 2008. During the same period, the number of new African American teachers working in New York City fell from 27% to 13%, whereas the number of new White teachers rose (Green, 2009). This whitening of the urban teacher workforce has effects beyond mere demographics; for example, a sizable number of preservice mathematics candidates in the New York City Teaching Fellows program openly articulate deficit views about the students and communities they work with (Brantlinger et al., 2010). Fast track preservice preparation in TFA and Teaching Fellows programs may contribute to this because issues of diversity are afterthoughts in much preservice preparation (Brantlinger & Smith, in press; Hammond, 1994; Popkewitz, 1998; Veltri, 2010).

By contrast, as part of its Woodrow Wilson-Rockefeller Brothers Fund Fellowships for Aspiring Teachers of Color, the Woodrow Wilson Foundation (n.d.) argued,

Current trends indicate that, by 2020, the percentage of teachers of color will fall to an all-time low of 5 percent of the total teacher force, while the percentage of students of color in the system will likely exceed 50 percent. . . . Research also shows, overwhelmingly, that students of color perform betteracademically, personally, and socially-when taught by teachers from their own ethnic groups.

We expect that the research reported in this special issue will encourage others to examine questions of race and teaching and that such research will provide a basis for examining such policy differences and designing policies to help urban schools improve students opportunities to learn.

Urban mathematics teaching as a resource for mathematics education.

Some educators might be quite comfortable with the idea that nonselective urban schools are different from other schools. However, skeptics may want to know specifically how we see mathematics teaching in such schools as different from the teaching of other subjects in such schools, or from mathematics teaching in other schools. For example, do we believe that the tasks carried out by mathematics teachers in nonselective urban schools differ from those carried out by their colleagues in suburban schools or selective urban schools? Or is it that these tasks of teaching mathematics are present in all settings but shaped by aspects of the school context in important ways? We would anticipate that it is more the latter, but, over time, with the data that we have collected, we seek to make such questions ones for empirical exploration rather than treat such questions solely as conceptual ones.

In subsequent sections of this introductory article, we define what we mean by nonselective urban schools and articulate how our selection procedures and the resulting data were designed to locate schools that could be described as nonselective urban schools. At the same time, we challenge discourses in the literature about teaching in urban schools and provide resources for scholars with particular interests. In particular, we do not define an urban school as simply any school situated in a geographical area that would be described as urban on the basis of population density; we seek a definition, and selection procedures, that will capture the ways in which the word urban is used in educational discourse, and we challenge empirically these usages.

Moreover, our intent is not only to study teaching in nonselective urban schools for lessons about teaching in such schools but also to seek insights into mathematics teaching more broadly. We believe that the teaching and learning of mathematics in nonselective urban schools has much to teach mathematics educators about mathematics teaching and learning more broadlythat mathematics educators can learn from well-respected mathematics teachers in nonselective urban schools. What remains is to argue what it is that the field as a whole can learn from studies of teaching in such schools. We do so by focusing on teacher knowledge.

Implicitly in this article, we have been operationalizing urban mathematics teaching as the ways in which mathematics teachers in nonselective urban schools structure learning experiences for their students in response to unique conditions of these schools and the communities they serve. Typical discourse about the conditions and contexts of urban schools portrays them as dysfunctional (Noguera, 2003). From this perspective, urban mathematics teachers pedagogy is portrayed as being shaped by the following sorts of conditions:

1.

In light of patterns of low mathematics achievement in urban schools, urban mathematics teachers must prepare and facilitate mathematics lessons for large groups of students who do not have the requisite mathematical skills, experiences, and preparation and who may have less than healthy perceptions of their mathematical abilities and capacities.

2.

In that the pressure to increase student mathematics achievement is acute in these environments, urban mathematics teachers are tasked with managing multiple curricular reforms and interventions simultaneously.

3.

In that urban schools are cast as places that are marked by high student truancy and absenteeism, urban mathematics teachers must create learning environments where a consistent number of students attend irregularly.

4.

In that urban schools are scrutinized for providing opportunities for students to learn and that teachers in urban schools are often characterized as underprepared, with weak content knowledge for making curricular decisions, urban mathematics teachers are held to exacting standards for covering curriculum and must learn to cope with strong accountability measures like common curriculum guides and regular districtwide assessments.

5.

In that urban schools are cast as places where parental involvement and adult guidance are lacking, urban mathematics teachers must rely less on the assistance that students might receive outside of the school walls and even counteract harmful beliefs and dispositions learned in the home.

6.

In that urban schools are cast as places where students are less engaged in academic pursuits, urban mathematics teachers must be particularly skilled at creating and facilitating mathematics tasks and lessons that capture students attention and interest. This could mean contextualizing tasks in familiar settings and through the use of cultural referents; it might also mean mastering discourse patterns and managing micro-interactions with urban students that produce desired student engagement.

7.

In that urban schools are characterized as places where students disrespect authority, are prone to disruptive behaviorincluding violenceand lack important social skills (e.g., conflict management), urban teachers are tasked with asserting strict authority and creating a no excuses classroom before mathematics learning can take place.

But, as we suggested earlier, these are issues of teaching in a variety of contexts, especially in lower track classrooms. Turning these seven points on their heads and removing the student and community as the problem connotations, urban mathematics teaching can be considered as involving the mastering of the following skill set. Well-respected teachers of mathematics in nonselective urban schools may very well possess the following capacities and knowledge base:

1.

The capacity to establish and facilitate mathematics learning environments that serve a large group of students who possess a wide range of mathematical experiences and levels of performance; highly skilled at differentiated instruction and making mathematics accessible to a wide range of students; possess a depth of knowledge related to students mathematical identity development and formation.

2.

The capacity to identify the mathematical content, processes, and state standards associated with tasks in various curriculum materials and documents and to select and assemble these tasks into a coherent program that students can manage.

3.

The capacity to create classroom management and learning systems that do not penalize students for absenteeism, yet encourage students to improve their attendance.

4.

The capacity to meet both external accountability measures and their own assessments of the needs of their students.

5.

The capacity to build, facilitate, and coordinate before school, during school, and after school mathematical support services for students; the capacity to support the capacity of parents, guardians, and community members to assist students mathematical learning.

6.

The capacity to recognize and utilize engaging yet appropriate discourse techniques, strategies, and tactics when facilitating mathematical tasks and to convey to students a sense of purpose for engaging with school mathematics.

7.

The capacity to build relationships of respect and care for students who have experienced little institutional respect, create classroom routines and structures that teach institutionally expected behaviors, and balance sometimes contradictory instructional and disciplinary goals.

Thus, we posit that explorations of the knowledge of well-respected teachers of mathematics in nonselective urban schools may have much to teach mathematics educators. With the two cases in this special issue, we explore this possibility by focusing on instructional strategies that involve conveying a sense of purpose to students for engaging with school mathematics.

A STRATEGY FOR EMPIRICAL STUDY OF URBAN MATHEMATICS TEACHING

With this special issue, we pursue a strategy for empirical study of some the realities of teaching and learning in nonselective neighborhood urban schools, particularly some aspects related to race. In the context of describing the procedures that led us to select the teachers whose cases we present, we define what we mean by calling a school an urban school. Our definition does not turn a blind eye to the challenges in our society faced by many people who live in urban areas, but neither does it define urban settings as, of necessity, sites of dysfunction.

Similarly, we aim to learn about the teaching and learning of mathematics from teachers of mathematics in such schools rather than assume either that results and findings from other settings will automatically transfer to urban schools, or that they automatically will not. And though we are sensitive to the strength of contextin this case, the societal dynamics related to race that are encoded in the word urbanwe also hope, through our research in urban schools, to gain broad insights that will serve us in a wide range of contexts (e.g., conceptualization of how teachers can aid students in identity construction; see Clark, Badertscher, & Napp, 2013, this issue).

In particular, with this study, we seek to learn from well-respected African American teachers in nonselective urban schools about what they do when conveying a sense of purpose to students for engaging with school mathematics and improving their achievement, and perhaps in this way identify overlooked and undertheorized elements of teachers work. We explicitly seek to avoid selecting troubled teachers or teachers sometimes described as heroic or exceptional (Matthews, 1983). Instead, we sought to select teachers who were well respected by administratorssolid, good citizens whose practice will shed light on what is possible in neighborhood urban schools by virtue of the fact that they are able to carry out such possibilities. The nominated teachers ranged in years of experience, including two teachers in their first two years of teaching and veterans with more than 15 years of experience.

In studying the work of these teachers, we do not seek tips or strategies that might be imported wholesale into other classrooms. Instead, in contrast to those who would portray teachers in urban schools as an important component of the underperformance of these schools, we seek an understanding of the nature of the work done by the sorts of teachers whom educators in urban school districts value. Similarly, by presenting two cases, we do not intend to compare the two teachers or simply to present their teaching in a nonevaluative manner. Instead, we aim to describe characteristic strengths of these two teachers that we also find present, to greater and lesser degree, in the other case study teacher participants. By studying well-respected teachers, our aim, writ large, is to help urban districts build on the strengths of the teachers they already have, strengths that may otherwise be overlooked in efforts to improve urban schools.

AN INSTRUCTIONAL FOCUS: CONVEYING A SENSE OF PURPOSE

In her seminal book on mathematics teaching, Magdalene Lampert (2001) identified a number of tasks of teaching that she argued face every teacher wherever he or she teaches, though school context may shape how a task of teaching reveals itself in a particular class (as argued and illustrated in Chazan, 2000, with respect to high school mathematics teaching). Sensitive to the ways in which human knowing is fundamentally a social act and how participation and identity are key constructs in understanding learning, Lampert suggested that an inescapable task of teaching is teaching students to be people who study in school (pp. 265328). She suggested that although some students show up at school as intentional learnerspeople who are already interested in doing whatever they need to do to learn academic subjectsthey are the exception rather than the rule (p. 265). In her view, an important challenge for many students is to develop a sense of oneself as a learner that makes it socially acceptable to engage in academic work (p. 265), and

the goal of school teaching is not to turn all students into people who see themselves as professional academics, but to enable all of them to include a disposition toward productive study of academic subjects among the personality traits they exhibit while they are in the classroom. . . . It is the teachers job to help them [students] change their sense of themselves so that studying is not a self-contradictory activity. (p. 265)

One initial, key step in this process is how the teacher communicates the learning goals in a way that is motivating to students, how the teacher provides students with a clear sense of purpose for what they are learning (DeBoer et al., 2004, p. 3).

Lamperts (2001) book goes on to illustrate challenges that she came up against as she took on this task of teaching in a fifth-grade classroom where some of her students did not come to class as intentional learners. But such issues are expressed differently in the context of different teaching settings. In the introduction to his description of the dynamics of lower track secondary classrooms and the academic trajectories that his students were traversing, Chazan (2000) illustrated how this task of teaching is context dependent. When he arrived at Holt High School, in a rural/suburban community with a strong working-class component, he wrote,

I was used to teaching students [in a lower track classroom in a wealthy suburban school] who took initiative in academic matters . . . I was planning to use the textbook for guidance about the topics to be covered, but to prepare my own materials and to require students to collect and organize these papers in notebooks which would be their central record of class work. I was also contemplating asking students to purchase a calculator for use in the class. But, suburban and lower-track do not mean the same thing everywhere; they are not well-defined terms with widely shared meanings&. Students in a lower-track mathematics classroom may not have particular issues with the learning of mathematics; their issues may concern the larger enterprise of schooling. I had stumbled into a different sort of teaching problem. (p. 27)

This larger problem involved skepticism on the part of students about the academic goals of the school. By contrast, early in his teaching career, Chazan taught students in a private K9 school. Although he did not suggest that all his students at that time knew how to study mathematics effectively, they came to school as intentional learners, and students, teachers, parents, and the school shared images of the futures for which students were preparing and a sense of the importance of schooling in attaining those futures.

On the basis of Chazans earlier work, we anticipated that the task of teaching identified by Lampert might take on a different cast in urban schools with a large majority of African American students. Throughout U.S. history, African Americans have experienced assaults on their intellectual capacity in general, and mathematics ability in particular, and have consistently been portrayed as intellectually inferior to humans of European descent (Gould, 1996; Miller, 1914, esp. pp. 146147). Although decades of social and structural efforts have certainly reduced collective perceptions of African American intellectual inferiority, themes from this narrative remain, including in contemporary discourse related to achievement gaps, disproportionate graduation rates, and disproportionate college attendance (Martin, 2009). Thus, we hypothesized that the task of teaching students to be people to study mathematics in school might take on a particular cast in a setting with a majority of African American students; helping students change their sense of themselves so that studying is not a self-contradictory activity might have different dimensions than it had for either Lampert or Chazan in their teaching. Moreover, we hypothesized that around this task of teaching, we might find teacher knowledge that would be particularly consequential for student achievement in nonselective urban schools and useful for those thinking about teacher preparation and teacher certification for teaching in such schools.

With our interest in teacher knowledge, but with a desire to see how that knowledge plays out in practice, we decided to focus a good portion of our observations and interviews on this task of teaching and on the related teacher action of conveying a sense of purpose. As a result, our observations of teachers were designed to be dispersed throughout the year and thus to be able to capture evolution in how teachers convey a sense of purpose to students throughout the year, across different mathematical content.

PART 2: RESEARCH QUESTIONS AND RATIONALES FOR THIS FOCUS

The MACMTLs focus on teacher knowledge plays a key role in shaping our project. Other projects in the center include: examinations of the nature of the learning of mathematics content by undergraduate secondary preservice teachers in an innovative sequence of courses designed to promote conceptual understanding (e.g., Heid, 2008); examination of the teaching skills and dispositions of undergraduate preservice elementary teachers who take a sequence of three specially designed mathematics-for-elementary-teachers content courses as these undergraduates take the courses, do a teaching internship, and then become teachers (e.g., Hiebert & Morris, 2009); and a large-scale quantitative examination of links between upper elementary school teachers content knowledge and pedagogical content knowledge and their students achievement (Campbell et al., 2011). Within the set of research projects sponsored by the MACMTL, the Algebra 1 Case Studies Project contributes to the Centers attempts to understand how teachers use knowledge, or perhaps other resources, in instruction. Rather than examining teacher knowledge outside of the classroom through interviews or surveys, our project tackles the challenges of examining teacher knowledge in action (see Boyd et al., 2009; and Peressini, Borko, Romagnano, Knuth, & Willis, 2004, for similar attempts), as teachers enact lessons. Thus, our project initially articulated two research questions:

"

In order to convey a sense of purpose to students for engaging in the algebra and data analysis to be taught in introductory algebra classes, what mathematical and pedagogical knowledge do well-respected teachers in nonselective urban schools draw on and continue to develop?

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How do such teachers draw on and continue to develop their knowledge?

Although we had initially proposed these questions, our early experiences in our research setting began to suggest a shift in our questions. All the teachers nominated in the first year of our study were African American; in the second year, one European American teacher was nominated but declined to participate. As a result, we had a sample of six well-respected African American teachers. Second, the work of teaching in the context of a high-stakes course with a district-mandated curriculum guide and districtwide quarterly exams seemed to leave less room for teachers to do the kinds of task selection and design so often conceptualized in mathematics education as central to the work of teaching (NCTM, 1991).

As a result of observing these six well-respected teachers, we began to wonder about some of the assumptions common in mathematics education that are implicit in our initial questions. For instance, our questions do not incorporate the kinds of social or tacit knowledge often held by organizations rather than individuals (Cook & Brown, 1999). The use of research on teacher knowledge to inform policies about matters like preparation and certification of mathematics teachers is built on commonsense notions about links between explicit teacher knowledge and student achievement and on a particular view of the role of knowledge in the work of teaching. Although that research has shown that teacher knowledge can explain portions of the variance in student achievement (Hill et al., 2005), such studies look at a narrow range of the resources that individual teachers bring to instruction, all but ignoring, for example, cultural ways of knowing and being in the classroom. We began to consider the possibility that other resources might play as large a role, or even a larger role, in explaining variance in student achievement. In particular, supporting Martins (2007) questions about who should teach Black students, there is some research suggesting that the performance of Black students is stronger when they have African American teachers (Dee, 2004; Ferguson, 1998). This research raises the possibility that there are aspects of racial identity, culture, and experience that African American teachers may use as resources in improving student achievement.

Thus, in the cases in this issue, we seek to cast our net more broadly than teachers content knowledge or pedagogical content knowledge. Influenced by arguments that connect learning and identity (e.g., Wortham, 2006), we constructed our data collection to be open to the possibility that other kinds of resources might also prove to be relevant to teachers decision making about classroom action. We were also open to the idea that these resources might ultimately turn out to be more consequential for student learning. For example, one might want to examine teachers experiences and community memberships side by side with more traditional dimensions of the knowledge that teachers draw on to teach mathematics. Such experiences and memberships might constitute teachers funds of knowledge, in González, Moll, and Amantis (2005) sense, that might offer useful resources for what is often described as the task of motivating students.³ Perhaps these experiences and memberships, as they relate to issues of student identity, might also be important elements in a set of factors that interact with each other to produce student achievement outcomes.

These shifts in our thinking caused us to orient increasingly to the importance of tackling the difficult challenges of conversation about race and racialization in mathematics education. As members of our research group completed graduate school or changed institution, new members joined the project, and the project staff became more diverse. The researchers who joined the project and who identified as either African American or Black scholars helped the project deepen its relationship with participants; like all of us, they brought their own racialized experiences in our society, as well as their insights into urban schools, to the task of analyzing the interviews and observations. As outlined by Clark, Johnson, and Chazan (2009), this work is challenging. It required, and continues to require of us, that we, as a diverse research team:

"

Guard against constructing Blackness as a static social category and, as a result, masking the heterogeneity of what it means to become and be Black in the U.S.;

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Attend to our positionality in society as mathematics educators, and often teacher educators or educational reformers, with respect to the teaching of the teachers we study; and

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Continually re-evaluate our judgments, as people who live inside different racial categories in this society, about the influence of race in the interactions and teaching we observe. (pp. 4344)

As a result of our ongoing conversations, we articulated two additional and more specific questions that form the basis for the two exploratory cases that follow this article:

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In the context of an introductory algebra course with a statewide end-of-course exam that is part of a system of exams used to determine high school graduation, what instructional strategies do well-respected African American mathematics teachers teaching in nonselective urban schools use to convey to their students a sense of purpose for engaging with mathematics?

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What experiences as African Americans in our society seem to influence these teachers in selecting and crafting these instructional strategies?

PART 3: SELECTING SCHOOLS, SELECTING TEACHERS WITHIN SCHOOLS, AND DATA COLLECTION

To investigate our two research questions, our selection of the schools and then of teachers within these schools was purposeful (in the sense of Patton, 2002, see especially Exhibit 5.6 on p. 243). We operationalized what we wanted to mean by urban schools and then chose schools that fit these criteria. We then relied on administrator nomination, as well as self-nomination, to identify teachers who would illustrate what is typical of well-respected teachers in these schools and who taught students in high-stakes courses. (Next, we will show that the two teachers whose cases we present are well-respected teachers in nonselective urban schools who teach a course whose outcomes have high stakes for both the students and their schools.) Resources, chiefly time, have limited our data collection to the work of six teachers, two of whom are described in this issue. In the next four sections, we describe our selection criteria and the data collected from each teacher.

SELECTING URBAN SCHOOLS WITHIN WHICH TO SELECT TEACHERS

To select the well-respected urban teachers who are the focus of our work and whose cases we will present, we needed to identify the nonselective urban schools in which such teachers teach. Thus, one of our first challenges in doing the study was to articulate what we meant by urban schools and how we were going to choose districts and schools from which we would sample teachers. As outlined earlier, the connotations of the words urban school are often negative (see Martin, 2009, for further criticism of such thinking about urban schools). Locating our work in the worst of urban schools would simply reinforce negative connotations; locating our work in exceptional urban schools would likely lead to a different shortcomings, mainly that the exceptional learning environments then identified and studied would be interesting, yet hardly representative of schools of the sort intended by the descriptor urban. We sought instead to find places that would be seen as representative of many urban schools. We reasoned that, based on much of the rhetoric about urban schools, the realities of these schools could potentially challenge some the negative preconceptions about urban schools found in the literature. Could we use the word urban simply as a descriptive term, then find schools meeting these descriptions and see what we would find in such schools? If so, what kind of descriptors might be used?

The U.S. census defines urban areas as areas that have a population density of 10,000 people per square mile (which includes much of the Eastern seaboard). Would it make sense to suggest that any school located in an area of such population density be considered an urban school? We decided that such a definition would not do. Many selective schools can be found in urban settings; cases from such schools would not lead to a reexamination of discourse about the urban school, or phenomenon related to teacher knowledge typically overlooked by researchers. But, at the same time, we also did not want to choose as our selection criteria measures that might be thought of as outcome measures, like student achievement, student mobility, or teacher turnover rates. Such selection criteria by definition could not help us create a sample with a diversity of outcomes in it and a chance to challenge preconceived notions of urban schools. We sought descriptions that were not outcome measures associated with nonselective urban schools, but that might correlate with those outcomes.

To capture our sense of the sorts of schools within which we wanted to sample from teachers, we were interested in issues of relative size and student selectivity, as well as relative wealth and relative racial and ethnic demographics, at the level of district, school, and classroom. In terms of size and selectivity, based on literature about the bureaucratic challenges in urban schools (Weiner, 1999), we sought typically sized classrooms in large nonselective schools in large public school districts. In these contexts, we then sought teachers to study. The district in which we chose to situate our study is located in an area that meets the census definition of an urban setting. It is large by national standards, it is one of the 50 largest school districts in the United States,⁴ and it does not have many selective schools or charter schools. The school district has more than 20 high schools, which offer a variety of specialized academic and vocational programs. Because we felt it was valuable to have some teachers in our sample whose building context would be the same, with the resources we had, we limited our data collection to 3 teachers at each of two schools. Ultimately, our project was carried out in the context of regular track Algebra 1 classes in two large nonselective schools: Erasmus High School and McBierney High School. Both of these schools are among the larger high schools in the district; each schools mathematics department has more than 20 teachers. The classes we observed enrolled students from these schools in a nonselective way.

In terms of wealth and racial and ethnic demographics, we looked at wealth per pupil, free and reduced meals (FARMs) rates, and the percentage of minority students in the district and school. This district has lower than average wealth per pupil for its state, higher than average FARMs rates, and higher than average percentages of minority students. Almost three quarters of the students in the district are Black. The next largest racial or ethnic group in the district consists of Latino students. More than 30% of the students in the district are eligible for free and reduced meals. Both of the high schools we selected enroll a higher than average percentage of students eligible for free and reduced meals and a higher than average percentage of minority students. The two schools have approximately the same racial/ethnic composition as the school district as a whole (Black and Latino student populations combined), but the racial/ethnic breakdown in the two schools has slightly more Hispanic students and slightly fewer Black students than the district percentages would indicate.

Given our definition of urban and our selection criteria, it is not automatic that student achievement in these schools would be low, student mobility high, and teacher turnover large, and that there would be over-enrollment of students, though both schools struggle with all these issues. At the same time, like many, if not most, high schools in large urban school districts, and in tension with some depictions of urban schools, Erasmus and McBierney offer a wide range of academic courses, including Advanced Placement and honors mathematics courses. Erasmus and McBierney also offer a wide range of athletics programs, including soccer, golf, baseball, indoor track, basketball, track and field, tennis, softball, volleyball, wrestling, swimming, cross country, and football.

Thus, we are confident that these schools, as contexts for the work of the teachers we will profile, can be called urban schools, though they may also challenge some existing descriptions of nonselective urban schools. Although the schools have some of the characteristics often ascribed to schools in the literature, they also present us with much more complex and compelling stories, ones that challenge us to develop more useful and nuanced ways of understanding urban schools as contexts for the work of teachers.

SELECTING WELL-RESPECTED TEACHERS WHO TEACH A HIGH-STAKES CLASS

Rather than expect that improvements to urban schools will come about from the outside, we seek to ground our understandings of nonselective urban schools in the day-to-day realities of such schools and to understand existing strengths in such schools. We sought well-respected teachers in these contexts from whom both we in the project and other educators might learn. We expect that as a result, for example, in addition to speaking to researchers in mathematics education who are interested in teacher knowledge, the results of our work may be of use to policy makers, practitioners, and teacher educators who must grapple with what it means to be highly qualified to teach mathematics in urban settings.

Given our focus on collecting useful information about urban mathematics teaching, one might imagine that we simply sought to study the teachers whose student achievement was highest. Practically, this was not feasibledistrict mechanisms for linking teacher and student data were not yet fine-tunedbut even had they been, given the impossibility of anonymity, concern about the uses to which the data would be put would have made it unlikely that we would receive access to such data. In selecting teachers, our notion was to select teachers from whom we could learn and who, in terms of their teaching, would be in the center of the distribution of teachers one might find in nonselective urban schools, not at one of the extremes. We did not want to choose teachers whose experiences, knowledge, and commitments would make the teaching they did far out of the reach of most teachers and thus of little use for those making policy for urban schools, nor did we want to choose teachers whose practice was so poor that it is not worthy of emulation. We sought teachers who are well-respected in their settings, teachers likely to be a part of the ongoing work in urban schools. In the end, we relied on principal and math department chair nomination and, in the first year, one self-nomination. We asked the principals and department chairs to nominate successful algebra teachers for participation in our study, though we were not privy to the considerations that influenced their nominations (e.g., did they consider the race of the teachers, their capacity to manage their classrooms, their involvement in the life of the school outside of their classrooms?). Our rationale for selecting the two cases we highlight in this special issue is the ways in which the practice of these two teachers differs. We hope with this choice to portray a diversity of strategies for conveying to students a sense of purpose for engaging with mathematics. Our purpose is not to have readers choose one set of strategies over another, but instead to articulate the resources that African American teachers bring to the task of creating such instruction. In the context of each of the cases in this issue, we outline the ways in which the teachers are well respected in their school and in their district, including that in the year following our observations, they were both promoted out of the classroom to coach other teachers.

Because we designed our research to be useful to those interested in mathematics teaching in urban settings, given the recent dramatic changes to the nature of high school graduation requirements (CEP, 2009), as well as the accountability pressures associated with adequate yearly progress (AYP), we decided to situate our study in a context that had high stakes for both students and schools. During the years we observed, in the context of the state within which this school district is located, Algebra 1 was a high-stakes course. Beginning with the students who were high school freshmen when we began this study, the state implemented a policy that required the passing of high-stakes exams as a part of the states graduation requirement. An end-of-course Algebra 1 exit exam (that includes both algebra and data analysis content)as opposed to a comprehensive or minimum competency exam (CEP, 2009, pp. 1517)was one component of this policy. Student performance on this exam also had high stakes for schools. During the 2 years that we observed, the scores of first-time test takers were used to establish AYP in high school mathematics. This context is one in which there is great pressure on teachers to convey to students a sense of purpose for studying mathematics and where the high stakes associated with the end-of-course exams are a part of the context that may be used for this purpose.

Other contextual aspects of the urban schools in which we collected the data that are related to the high stakes of Algebra 1 influenced features of our data corpus. For example, as in many urban districts, to support teachers in teaching high-stakes courses, the district created curriculum guides to support teachers as they taught the course. Teachers were asked to pay careful attention to the guide. The district also provided 9-week benchmark tests in the style of the end-of-course assessment. The district scored these assessments and provided results to schools and teachers so they might assess their students progress against that of other students in the district and the expectations of the exam. The existence of this guide and widespread adherence to the guides pacing shaped our observations, allowing us to observe teachers teaching the same lessons. Given the centrality of the use of curriculum guides in accountability strategies and their greater prevalence in urban schools, the ways in which well-respected teachers interact with these guides is one lesson we might learn from teachers in urban settings.

In sum, our selection process resulted in the participation of 6 well-respected teachers of Algebra I who teach in large nonselective public schools in a large urban district with comparatively low scores on per-pupil wealth and high enrollments of minority students. An interesting note is that although we did not set out to recruit a sample of African American teachers, all the teachers in the sample are African American. Our corpus concentrates on the work of the teachers and their rationales for their actions.⁵ In this issue, we concentrate on the work of 2 of these teachers from one high school.

THE COLLECTED DATA

Consistent with frameworks that focus on individual teachers decision-making power (e.g., Lipsky, 1983), we sought to collect enough data about each teacher we studied to support the creation of cases around each teacher and the making of claims about that teachers instruction patterns and focus, as well as his or her rationales for the instruction he or she provides to students. In line with work like Rowan, Camburn, and Correnti (2004), which suggests that variance in curriculum implementation by lesson is high and that it would take 2030 observations of a teacher to identify patterns in a particular teachers curriculum implementation, the team that designed the study called for roughly 25 observations of each teacher during a particular year (about one visit every two work weeks on average), and 910 interviews, in addition to debriefing around each observation.

These classroom observations and interviews were done by three groups of researchers. Each teacher was assigned a sense of purpose researcher who served as contact point and who visited at least twice a month throughout the year. In addition, there were algebra and data analysis teams that, using the district-mandated curriculum guide, carried out focal observations of particular lessons across all teachers. With the statistics lessons, we chose lessons that would allow researchers interested in, for example, how instruction does or does not incorporate a statistical perspective (Konold, Pollatsek, Well, & Gagnon, 1997) to examine this issue across the instruction of each teacher. To help understand the instruction in these lessons, two interviews focused on the teaching of data analysis. Similarly, stimulated by work on how the algebra curriculum asks students and teacher to interact with variables (Marcus & Chazan, 2010), the algebra observations focused on instruction on solving equations in one variable and then later on the solving of equations in two variables. Two algebra interviews focused on understanding the teaching in these lessons.

In 20052006, we followed 3 teachers from the more than 20-person mathematics department of Erasmus High School, and in 20062007, we studied 3 teachers from the more than 20-person mathematics department of McBierney High School. Follow-up interviews of all the teachers that focused explicitly on race were done in spring 2008 and fall 2009; these interviews asked teachers about their understandings related to race in their teaching and in their own personal experience of schooling. As Table 1 indicates, across the 6 teachers, we have about 155 seventy-five-minute classroom observations and 54 one-hour interviews. The cells in the table indicate the numbers of observations (and interviews in parentheses) carried out by each research team for each teacher.

Observational data consist of an audio or video recording of a lesson, the transcript of that lesson, a timeline that chunks the lesson into pieces and that links the observation to the curriculum guide, and observers notes, as well as artifacts from the class (teacher handouts, and so on). Interview data consist of a video recording of the interview, a transcript of the interview, and a timeline that divides the transcript into parts and connects it to the relevant interview protocol. In addition, the data corpus includes memos written by the researchers who visited the classrooms about their observations and patterns they saw in instruction, as well as policy artifacts and documents that illustrate larger currents in national, state, and district efforts in mathematics education.

Table 1. Numbers of Observations and Interviews in MACMTL Algebra 1 Case Studies Data Corpus by Research Focus and Teacher

Explicit research focus Pseudonym (School)	Sense of Purpose	Algebra	Data Analysis	Race	Total
Susan Miles  (Erasmus)	10 (4)	8 (2)	4 (2)	0(1)	(9)
April Lincoln  (Erasmus)	13 (4)	8 (2)	5 (2)	0(1)	(9)
Dorothy Hicks  (Erasmus)	15 (4)	8 (2)	8 (2)	0(1)	(9)
Madison Morgan  (McBierney)	9 (4)	9 (2)	7 (2)	0(1)	25 (9)
Floyd Lee  (McBierney)	13 (4)	9 (2)	6 (2)	0(1)	28 (9)
Terri Martin  (McBierney)	10 (4)	9 (2)	4 (2)	0(1)	23 (9)
TOTAL	70 (24)	51 (12)	34 (12)	0(6)	155 (54)

Note: Interviews in parentheses. Bolded rows indicate the teachers whose cases are presented in this volume. MACMTL = Mid-Atlantic Center for Mathematics Teaching and Learning.

There are particular challenges in developing accounts of urban mathematics teaching in the United States that respond to Lermans (2000) call. We believe that one of the reasons mathematics educators struggle to create these kinds of accounts is the difficulty of having the requisite data available. As the foci of the observations and interviews suggest, from the beginning, a central aim of the project was to collect data that could be analyzed to support a variety of research interests and frameworks and to organize the data and prepare them in ways that would allow researchers who did not collect the data to analyze them. We are now exploring the creation of an archive of materials to support historical examinations of the mathematics teaching of African American teachers. These data from the MACMTL Algebra 1 Case Studies Project could form one cornerstone of that archive.

PART 4: THIS SPECIAL ISSUE: CASES AND CROSS-CASE ANALYSES

To begin fulfilling the promises outlined earlier, for the purposes of the analyses presented in other articles in this monograph, 2 of the 6 teachers, Floyd Lee and Madison Morgan, were purposefully selected because of considerable differences in their life histories, years in teaching, pedagogical approaches, and perspectives on mathematics teaching and learning. The next article presents a case of Floyd Lees teaching (Johnson et al., 2013, this issue), followed by an article that presents Madison Morgans teaching (Birky, Chazan, & Farlow Morris, 2013, this issue). Both of these teachers are well-respected teachers of mathematics in the same high school within the same district, yet how they tackle the task of teaching students to be people who study in school is quite different.

The cases we present of the teaching of Floyd Lee and Madison Morgan select data from both classroom observations and interview data. In the case of Floyd Lee, a young African American male who has just begun teaching Algebra 1, we focus on a kind of event that was prominent in his classroom but not in the other classes we visited. We call these events speeches to capture the way in which he exhorts his students at these times to make use of their educational opportunities and improve their life chances. The speeches are characterized by speaking patterns that are grounded in Black English Vernacular and give evidence of knowledge of both urban youth culture and the Black church.⁶ We consider the capacity to use these culturally familiar communication modes as one of the resources that Floyd, as an African American teacher, brings to teaching.

In the case of Madison Morgan, an African American female who is a veteran teacher, we focus on her curricular decision making when she departed from the curriculum guide in ways the other teachers did not. Although her teaching and her talk about her teaching invoked many aspects of the NCTM reform vision for mathematics classrooms, in discussing her rationales for teaching in this way, she did not refer to that vision. Instead, she made regular references to her experiences as a learner, one key aspect of which was feeling that she, as an African American, was underprepared for collegiate study. These references on her part raise the question of whether her racialized experiences as a learner are at the heart of her reasons for teaching in the way that she does, and more generally, whether such racialized experiences may be a resource for efforts to challenge what Haberman (1991) called the pedagogy of poverty and strengthen mathematics teaching in urban settings.

Juxtaposing the work of these two teachers is potentially of great value; together, Floyd and Madison remind us not to conceptualize racial categories as static with predefined meanings that would lead all African American teachers to teach in the same ways. At the same time, such juxtaposition can lead to comparison in ways that would not fit our purposes. By juxtaposing their work, we do not seek to value the work of one teacher over the work of the other, but rather to explore key dimensions of the task of conveying a sense of purpose to students for engaging with school mathematics; the knowledge, commitments, and experiences that underlie the actions of teachers in response to this task; and different ways in which individual African American teachers support their students in important tasks of identity construction as members of ethnic and racial groups and as doers of mathematics. By highlighting aspects of what each of the teachers does, even where his or her goals may seem contradictory, we aim to stimulate discussion about the range of instructional goals that are important in urban schools, as well as the range of strategies that help achieve those goals.

Following the two cases, we present two analytic pieces that build on the presented cases and respond to Lermans call for accounts of mathematics classroom activity that bridge the individual and the social, as well as Martins challenge to incorporate understandings of race as a social category in such accounts. Clark, Badertscher, and Napp (2013, this issue) conceptualizes the doing of mathematics in classrooms as a social interaction involving individual agency and grounds itself in Martins (2000) framework for describing students mathematical identities. Where Martins interview study of mathematics learners offers mathematics educators a framework for understandings students mathematics identity development that is responsive to their experience of race in society, this article uses the data from the two cases and its focus on how teachers convey a sense of purpose for studying mathematics to develop a framework for understanding socialization practices that teachers can use to influence student identity development as conceptualized by Martin. Like Martins work, this framework also pays explicit attention to race by exploring how teachers can use their own experiences and membership in social groups as resources for influencing student identity formation.

As suggested earlier, such analytic work may well be of relevance to understanding mathematics teaching broadly. For example, development of this line of reasoning (Clark, 2012) suggests further elaboration of the subcategory of mathematical knowledge for teaching labeled by Hill, Ball, and Schilling (2008) as knowledge of content and students (KCS); Clark (2012) proposed a category of knowledge of students mathematics identity development and formation. Such theoretical developments can then be tested (as done with other kinds of mathematical knowledge for teaching by Hill, Rowan, and Ball, 2005) by putting them to the challenge of, for example, creating items to assess newly proposed elements of teachers knowledge (Hill, Schilling, & Ball, 2004).

This work may also be useful to mathematics educators who want to better understand issues of context in teaching in nonselective urban schools, issues often acknowledged but then left undertheorized. African Americans in general, and Black students in particular, engage in a unique, continuous process of negotiating multiple traditions, cultural frames, and identities, some of which arguably have been perceived and/or constructed as incompatible, in an effort to function in U.S. society (Bell, 1993; Boykin, 1986; Du Bois, 1903; Williams, 1997)identities such as being African American and becoming doers of mathematics (Martin, 2006, p. 147). It is reasonable to believe, therefore, that teachers, particularly African American mathematics teachers, play a role in assisting their Black students to negotiate and reconcile real or perceived conflicts or dilemmas in their identity formation (Wenger, 1998). If students are in the process of structuring identities in practice (Nasir, 2007), in what ways might African American mathematics teachers influence this process? In what ways might African American teachers influence their Black students perceptions of themselves as doers of mathematics? Might African American mathematics teachers serve as bearers of counter-messages that challenge the historical and contemporary narrative of African American students as being less able than students in other racially defined groups in the United States?

Whereas Clark, Badertscher, and Napp (2013, this issue) focus on how teachers influence students mathematics identity formation and development as a key task of teaching, and explore teachers racialized experiences and group memberships as resources for this task, Clark, Jones, and Davis (2013, this issue) focus historically on race in the work of teachers and bring what is often discussed as context or background into the foreground. Clark, Jones, and Davis begin by juxtaposing Floyd Lees teaching with a historically based but fictional account of an African American mathematics teacher teaching in segregated schools prior to Brown v. Board of Education. In response to Lerman, with this juxtaposition, Clark, Jones, and Davis propose conceptualizing the work of Madison Morgan and Floyd Lee not only as the teaching of individuals with particular instructional practices and rationales for these practices, but also as examples of the historical work done by African American mathematics teachers of Black students. The article situates Floyds and Madisons work in the experience of previous African American mathematics teachers and argues, given the nature of our cultural notions about race, for conceptualizations of the work of the teacher that include the race of a teacher not as a static demographic characteristic but as a socially negotiated, fluid construct. We view such conceptualizations of race and teaching as important both in the context of current policy debates about the characteristics of teachers who can improve schooling in urban settings and in the context of greater understanding of the historical roots of contemporary trends in schooling.

This issue concludes with the perspective of outsiders to the research team that collected this data and produced these cases. Against the backdrop of their current examination of district practices with regard to mathematics education, Paul Cobb and Kara Jackson (2013) endorse that the cases presented in this issue speak to mathematics education in a variety of settings by enriching the images we have for how teachers can convey to students a sense of purpose for engaging with school mathematics. They, however, raise questions about the implications of the differences in the learning goals pursued by these two teachers for the reform of mathematics education in urban schools.

Notes

1. In this article, we will refer to the teachers as African Americans, rather than Black. With this usage, we seek to emphasize that the teachers in this study are not immigrants to the United States; they have grown up in the United States and have been socialized to race in this country and cultural context. We will refer to students as Black, because regardless of country of origin, they are being socialized into racialized experiences in the United States on the basis of the color of their skin.

2. Such a focus on the individuality of teachers is not the only possible focus to take when studying the work of teaching; one might study what it is that, across their differences, teachers share. Such work has most typically been done by sociologists (see Metz, 1993, for an articulation of teachers dependence on their students, and Cohen, 2011, for an analysis of the predicaments of teaching). Recent empirical work has focused on the mathematical work of teaching and has surfaced, for example, the tacit norms that geometry teachers share about what mathematical work must be done in a classroom to install a theorem as a piece of collective knowledge in a high school geometry class (Herbst, Nachlieli, & Chazan, 2011).

3. Hill et al. (2008) explicitly suggested that such a focus is parallel to the one that they explored with the construct knowledge of content and students (KCS).

4. To maintain our promises of confidentiality for the teachers in the project, in what follows, when we describe the schools and the district, we provide ranges of demographic data rather than absolute numbers. Similarly, we also do not provide detailed information about the evolution of district policies in mathematics education.

5. We did not collect data directly from students in the classrooms of the teachers we studied.

6. Our usage of the term is modeled on that of Lincoln and Mamiya (1990) in their book The Black Church in the African American Experience.

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