Journal of Urban Mathematics Education December 2015, Vol. 8, No. 2, pp. 87–118 ©JUME. http://education.gsu.edu/JUME MELVA R. GRANT is an assistant professor in the STEM Education & Professional Studies Department at Old Dominion University, 257-B Education Building, Norfolk, VA 23529; email: mgrant@odu.edu. Her research interests include exploring student mathematics identity development and developing teacher leaders and coaches capable of supporting mathematics teachers working with underserved and underrepresented math- ematics learners. HELEN CROMPTON is an assistant professor in the Department of Teaching & Learning at Old Domin- ion University, 145 Education Building, Norfolk, VA 23529; email: Crompton@odu.edu. Her research interests include K–12 teacher preparation and technology integration for mathematics learning and teaching. DEANA J. FORD is a doctoral student in the Department of Teaching & Learning at Old Dominion Uni- versity, 145 Education Building, Norfolk, VA 23529; email: dford005@odu.edu. Her research interests include secondary students’ mathematics learning and the intersection with English language literacy. Black Male Students and The Algebra Project: Mathematics Identity as Participation Melva R. Grant Old Dominion University Helen Crompton Old Dominion University Deana J. Ford Old Dominion University In this article, the authors examine the mathematics identity development of six Black male students over the course of a 4-year The Algebra Project Cohort Model (APCM) initiative. Mathematics identity here is defined as participation through interactions and positioning of self and others. Data collection included nearly 450 minutes of video recordings of small-group, mathematics problem solving in which student actions, coded as acts of participation, were tallied. These tallied actions were conceptualized descriptively in terms of mathematics identity using the lenses of agency, accountability, and work practices. The analyses suggest that the APCM students’ confidence in self and peers increased over the 4 years, they consistently chose to engage in mathematics, and their reliance on knowledgeable others less- ened. Opportunities for future research and implications for policy makers and oth- er stakeholders are discussed. KEYWORDS: Black male students, mathematics identity, mathematics teaching and learning, The Algebra Project any urban high school mathematics classrooms have disproportionate num- bers of students who are often described in policy reports and media as “at risk” (Durbin, 2012). This imbalance is especially true for Black male students who are often labeled as learning deficient, targeted for disciplinary action, and positioned for future incarceration (Booker & Mitchell, 2011; Gregory, Skiba, & Noguera, 2010). Many Black male mathematics learners have been historically, and continue to be, underserved by schools and society at large, especially those attending urban schools and qualifying for reduced-price meals (Anyon, 2006; Haberman, 1991/2010). Nevertheless, research has shown that when Black male students become aware of and have opportunities to learn mathematics in cultural- ly receptive climates they take on productive mathematics identities (Berry, Ellis, M http://education.gsu.edu/JUME mailto:mgrant@odu.edu mailto:Crompton@odu.edu mailto:dford005@odu.edu Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 88 & Hughes, 2014). Conversely, Black male students positioned in restrictive school climates with limited learning opportunities often experience negative out- comes (Gibson, Wilson, Haight, Kayama, & Marshall, 2014). Furthermore, when Black male students take on productive mathematics identities they are better equipped to self-advocate for more positive learning opportunities and improved outcomes for themselves, their communities, and society at large (Hope, Skoog, & Jagers, 2015). To understand how Black male students might take on productive mathe- matics identities, we explored the mathematics identity development of six Black male students who chose to participate in The Algebra Project Cohort Model (APCM) initiative during their 4 years of high school. The overarching research question and accompanying sub-questions that guided the exploration were: How did the mathematics identity of six Black male students participating in The Algebra Project Cohort Model initiative develop over their 4 years of high school? i. What types of agency were students observed exercising and how did their agency evolve? ii. How did students’ observed work practices (i.e., small-group prob- lem solving) influence their mathematics identity development? iii. To whom were students observed being accountable to and how did their accountability evolve? Review of Literature There has been substantial scholarship over the past 20 years that explores identity from many perspectives. Cultural and social psychologists, anthropolo- gists, sociologists, and social scientists in general have reframed how we think about identity. In the mathematics education literature, this reframing has been driven by concepts derived out of a variety of theories such as critical theory, crit- ical race theory, feminist theory, sociocultural theory, poststructural theory, and so forth (see, e.g., Berry, 2008, Gutstein, 2007; McGee & Martin, 2011b; Stinson, 2013). Nonetheless, for the study reported here, we take a narrower view of iden- tity. We define mathematics identity simply as participation. Specifically, we ex- plore how six Black male students’ mathematics identities developed over 4 years of high school using nearly 450 minutes of video recordings of small-group, mathematics problem solving. To contextualize our study, we discuss two con- nected areas of research: (a) “reform” in mathematics education, and (b) Black male students and mathematics identity. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 89 Reform in Mathematics Education To position Black children in reform efforts, we drew on Berry, Pinter, and McClain’s (2013) critical review of K–12 mathematics education reform efforts from the mid 1950s to the early 2000s. Their review of reform mathematics focused on what was taught, how it was taught, who taught it, and, most importantly, who got access to it. They concluded that the needs of Black children in mathematics education reform efforts have not been attended to over the decades. Segregation has been re-enacted through testing and tracking in many schools, and the brilliance of Black children has been largely ignored by the majority of mathematics educa- tors and researchers. Recently, Martin (2015) argued that mathematics education reform for several decades has yielded few benefits for the collective Black1 as he critiqued the Na- tional Council of Teachers of Mathematics’ (NCTM) latest policy document Prin- ciples to Actions: Ensuring Mathematics Success for All (NCTM, 2014) at the 2015 NCTM Research Conference held in Boston, Massachusetts. His critique included categorizing the long-standing rhetoric about equity and “mathematics for all” as political, questioning the audience for whom the document was written, and calling for mathematics educators to consider revolutionary reform designed for the collec- tive Black.2 For the most part, extant reform efforts have neither targeted nor yielded sub- stantive improvements for the collective Black, in general, and Black male students, in particular. In this article, we discuss aspects of mathematics education reform in spite of this oversight because that is what exists (for now) and these efforts are per- tinent for situating our project. Over the last several decades, national organizations such as the NCTM (e.g., 1991, 2000, 2014) and the National Research Council (2001) have called for signif- icant cultural changes in mathematics classrooms. The NCTM, for example, called for classrooms that are co-created by teachers and students, “where students of var- ied backgrounds and abilities work with expert teachers, learning important math- ematical ideas with understanding, in environments that are equitable, challenging, supportive, and technologically equipped for the twenty-first century” (NCTM 2000, p. 4). The latest national call for change is embedded in the Common Core State Standards for Mathematical Practice (CCSS, 2010). Specific recommenda- tions for mathematics education reform efforts have also emerged from mathemati- cians and mathematics educators. Mathematicians have suggested that Black stu- dents, in particular, need opportunities to engage in doing mathematics in ways that 1 This term was used by Martin (2015), defined as African American, Latin@, Indigenous, and poor; he attributes the term and definition to Eduardo Bonilla-Silva. 2 It appeared that the predominantly White audience received his remarks with loud silence. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 90 are both social and cultural (Fullilove & Treisman, 1990; Maton, Hrabowski, & Greif, 1998). Similarly, mathematics educators have advocated for pedagogical ap- proaches that are student-centered and collaborative versus the traditional didactic approaches that have persisted for decades (Franke, Kazemi, & Battey, 2007; Hiebert et al., 1997; Lampert, 1990/2004). The consensus among many: students participating with peers in ways that fosters sense making while using their cultural experiences and ways of knowing within and beyond school settings supports mathematics learning (e.g., Moses & Cobb, 2001; Walker, 2006). Critical mathe- matics educators have also stipulated that these student-centered, collaborative ap- proaches are more effective for Black students when they are carried out by teach- ers who are culturally aware versus those who believe that learning and teaching are race neutral (Martin, 2012; Matthews, Jones, & Parker, 2013; McGee & Martin, 2011a; Stinson, Jett, & Williams, 2013). In addition to student-centered, collaborative pedagogical approaches, math- ematics educators have advocated for using high-level, cognitively demanding tasks (e.g., Stein, Smith, Henningsen, & Silver, 2000). These educators claim that the cognitive level of the task affords different types of teaching and learning opportu- nities. High-level tasks that require students to engage mathematically, to seek con- nections to other mathematical ideas, and to prove their approaches, require teach- ers to facilitate learning differently than low-demand tasks that only require stu- dents to recall memorized facts that teachers, in turn, validate. The types of peda- gogies needed for facilitating high-level tasks are typically more student-centered, such as examining student work and listening to their explanations to inform in- structional decisions, and requiring students to use mathematical processes and practices in learning (CCSS, 2010; Doerr, 2006; Henningsen & Stein, 1997; NCTM, 1991, 2000). However, classrooms where students engage collaboratively in cognitively demanding tasks are not available to all students, in particular Black students (Ladson-Billings, 2006). In fact, too many Black students attend poor performing schools. According to Balfanz and Legters (2004), in 2002 almost half (46%) of Black students attend- ed high schools with weak promoting power3 where graduation was not the norm; most of these schools were in urban areas with high poverty. Few reform efforts have been meaningfully enacted in schools, in general, and urban high-poverty schools, in particular, for many reasons that are beyond the scope of this article (for a complete discussion see Marrus, 2015). Mathematics education in urban, high- poverty schools typically manifests as perpetual remediation, discipline, and other authoritative actions (Bracey, 2013; Ladson-Billings, 2006; Love & Kruger, 2005; Patterson, 2014). Bracey (2013) captured the essence of mathematics education re- 3 Promoting power is an indicator of high school dropout rates, calculated as a percentage compari- son of seniors to freshmen 4 years earlier; 60% fewer seniors than freshmen represent weak promot- ing power (Balfanz & Legters, 2004). Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 91 form within the current climate of accountability: “The net result is a lack of oppor- tunity to engage Black children beyond prescriptive remediation to pass annual yearly performance … mandates” (p. 173). In summary, effective mathematics ed- ucation reform efforts for Black students must provide opportunities for and access to: (a) pedagogies that are student centered and collaborative; (b) teachers who are culturally aware and well prepared; and (c) high-level mathematics courses with cognitively demanding tasks. Black Male Students and Mathematics Identity Supporting positive and productive mathematics identity development for Black male students requires they have access to teachers who: (a) explicitly and publicly hold high expectations for them to learn rigorous mathematics; (b) create receptive, engaging, and supportive learning environments; and (c) are culturally aware and responsive while exercising decentralized teaching authority (Ladson- Billings, 1994, 1997; Stinson, Jett, & Williams, 2013). From this perspective, we review literature about Black male students’ mathematics identity development. Martin (2009, 2013) argued that discussions about Black students’ mathemat- ics identities cannot be independent of discussions about race and racism in the United States. The historical rhetoric in the United States around mathematics teaching and learning often positions Black students implicitly and explicitly as mathematically deficient compared to White students who are positioned as the norm. This positioning, unfortunately, is often supported by mathematics education research and educational polices (Martin, 2013). Therefore, for Martin (2009), mathematics identity refers to the dispositions and deeply held beliefs that individuals develop about their ability to participate and perform effectively in mathematical contexts and to use math- ematics to change the conditions of their lives. A mathematics identity encompasses a person’s self-understanding and how others see him or her in the context of doing mathematics. (pp. 136–137) Central elements of mathematics identity that emerge from this definition include perceptions by others and beliefs about self in relation to mathematics learning and doing. Perceptions by others influence the ways we think about ourselves and the ac- tions we take. One perception about Black male students held by others is the stere- otypical image of the non-academic, street “thug.” This stereotypical image not on- ly influences but also can threaten Black (male) students’ mathematics identity de- velopment (Steele, 1997, 2006; Steele, Spencer, & Aronson, 2002). Steele (2006) referred to this phenomenon as stereotype threat and defined it as “the threat of be- ing viewed through the lens of a negative stereotype, or the fear of doing something Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 92 that would inadvertently confirm that stereotype” (p. 253). Researchers have re- ported ways that Black male students have navigated the peril of stereotype threat in mathematics learning contexts (e.g., Berry, 2008; McGee, 2013a; Stinson, 2008). For instance, McGee (2013a) analyzed interviews with 11 successful Black male high school juniors and seniors. McGee described the students as having defiant reactions to the stereotype; some ignored the threat and persevered and others worked harder to attain academic achievement. In either case, students developed productive mathematics identities using internal coping mechanisms when faced with negative perceptions by others. Watson (2012) uncovered another type of perception that is more covert by nature. She studied mathematics teachers whom she described as norming suburban when asked to describe their students. The act of norming suburban uses middle- class, White cultural perceptions as the standard by which all other groups “are compared, judged, and subordinated” (p. 987). Neither innocent nor objective com- parisons emerge when norming suburban because it requires one “to posit, either implicitly or explicitly, that teaching in suburban schools is better, and base this be- lief on the perceived inferiority of urban students,” all the while not using “race language” (p. 988). Watson outlined a three step suburban norming process: (a) as- sume groups are monolithic with respect to behaviors, values, and beliefs; (b) de- cide if these cultures are negative or positive; and then (c) establish hierarchies among groups. Norming suburban appears to be a form of stereotype threat that does not attend directly to characteristics such as race and class. Students, particu- larly those in the lowest hierarchical group, however, are likely to notice teachers who adopt norming suburban practices and discourses (Berry, 2008). Stinson (2006) reviewed historical and theoretical perspectives surrounding Black male students schooling experiences and presented three discourse clusters often used by others when discussing Black male students: the discourse of defi- ciency, the discourse of rejection, and the discourse of achievement. The discourse of deficiency is the perception that Black children are products of genetics, families, communities, and sociocultural spaces that are historically lesser than and not suffi- cient. This discourse leads to perceptions by others that Black male students, in par- ticular, are incapable, lacking, and otherwise deficient with respect to mathematics learning and achievement. School officials and policy makers who adopted defi- ciency perceptions for Black students often select intervention options that are typi- cally segregating and anti-intellectual, such as labeling, tracking, isolating remedia- tion, and authoritative pedagogies. The discourse of rejection is the perception that Black male students reject either a productive intellectual identity or the collective Black identity; the intervention here is often nurturing support programs, such as African-centric rites of passage programs. The discourse of achievement is the per- ception that Black students are able to achieve intellectually and mathematically. Leonard and Martin (2013) took up the discourse of achievement to compile their Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 93 edited volume The Brilliance of Black Children in Mathematics: Beyond the Num- bers and Toward New Discourse, which approaches mathematics learning and identity development of Black children from the perception of brilliance, omitting the discourses of deficiency and rejection altogether. Beliefs about self as articulated by successful Black male students starkly con- trast the mathematical identity descriptions presented about them received via edu- cational research, policy reports, and media outlets (Ladson-Billings, 2006; Martin, 2012); and there are too few of these stories told within extant literature (Martin, 2013; McGee, 2013a). Berry (2008) reports one such student’s self-account who described mathematics as “an easy subject for him to learn because he likes it and he loves the challenge of problem solving” (p. 464). This mathematically talented and engaged Black male student’s account was shared during middle school; he had been identified as academically gifted in the fourth grade. The account reported by Berry described the student’s relationship with his father that included mathemati- cal challenges with games and puzzles done at home. In sixth grade, however, he encountered a teacher with whom he did not connect. This teacher appeared set on removing him from her class and presumably the gifted program. The student with parental advocacy persevered and passed the teacher’s class earning a B. Accounts of Black students’ mathematics learning experiences that include social and cultural influences using students’ “voices” (e.g., Berry, 2008; Jett, 2010; McGee, 2013b; Stinson, 2013) or strongly influenced by students’ voices (e.g., Grant, 2014; McGee & Martin, 2011a, 2011b) are adding new positive perceptions and characterizations for how Black students see themselves in relation to doing and learning mathemat- ics. Conceptual Framework: Mathematics Identity as Participation Mainstream education scholars have explored the notion of identity develop- ment to better understand how people think about themselves or how others per- ceive them in relation to learning (e.g., Cobb & Hodge, 2002; Gee, 2000; Gilpin, 2006; Greeno, 1997). In these cases, mathematics identity is conceptualized as mathematics participation: the ways that students interact with others and position themselves and others in relation to mathematics engagement. These mainstream conceptualizations, however, most often do not consider the socio-cultural and -political contexts of learners and of learning. With this limitation in mind, Varelas, Martin, and Kane (2013) used a socio-cultural and –political critical lens to develop the content learning and identity construction framework for researching learning in mathematics and science classrooms. This framework considers content learning and identity construction as requisite. They described identities as “lenses through which we position ourselves and our actions and through which others position us” (p. 324). Positioning influences learning opportunities in which students may en- Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 94 gage, and changes in positioning result in different learning opportunities. Opportu- nities are essential for an exploration of identity as participation. Students’ self-perceptions are central to the actions (or inactions) they pursue within social systems, such as mathematics classrooms (Gresalfi, 2009; Nasir & Hand, 2006; Nasir, McLaughlin, & Jones, 2009; Varelas et al., 2013). Within the lens of mathematics identity, self-perceptions, the perceptions of others, and the situated contexts converge and influence identity related processes (Esmonde, 2009; Esmonde, Brodie, Dookie, & Takeuchi, 2009; Nzuki, 2010). Esmonde and colleagues defined three identity related processes, referred to as work practices for cooperative groups: collaborative, individual, and helping. In this study, we explore student mathematics identity development within the context of small-group, math- ematics problem solving, and through observation we sought to interpret their mathematics identity development in terms of mathematics agency, accountability, and work4 practices (with a focus on collaborative and individual practices only). Mathematics Identity and Participation This study characterizes participation as observable mathematics engagement and uses participation as the overarching construct for students’ mathematics identi- ty. This two-tiered construction of students’ mathematics identity has foundations in educational psychology and mathematics education literature: (a) exercised agency (Bandura, 2006; Gresalfi, Taylor, Hand, & Greeno, 2009; Gutstein, 2007; Hand, 2010) and (b) student accountability (Ares, 2006; Cobb, Gresalfi, & Hodge, 2009; Cobb & Hodge, 2002; Yackel & Cobb, 1996). These constructs, agency and accountability, manifest as observable student actions (i.e., agency) or inactions in mathematics learning contexts, and students chose participation or non-participation based on afforded opportunities that are influenced by feelings of accountability. Mathematics identity and agency. Our perspective of agency is grounded in Bandura’s (2005) agentic perspective of social cognitive theory: “To be an agent is to influence intentionally one’s functioning and life circumstance. In this view, people are self-organizing, proactive, self-regulating, and self-reflective” (p. 9). In other words, people make intentional choices in their self-interests, which, from our perspective, are manifestations of identity as (observable) participation, or non- participation, which is also agency exercised. Gresalfi and colleagues (2009) ex- plain the possession and exercise of agency: It is important here to dispel the notion that people “have” or “lack” agency. In virtual- ly any situation, even the most constrained, people are able to exercise agency; at the basic level, by complying or resisting. The ways that agency can be exercised, and the 4 The work in this study is mathematics problem solving within a small group of three to four stu- dents. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 95 consequences for doing so, are what change in a particular context. Said differently, an individual can always exercise agency, it is the nature of that exercise that differs from context to context. (p. 53; emphasis in original) Here, Gresalfi and colleagues are suggesting that close attention be paid to issues of power and authority within the mathematics classroom when considering distribu- tion of agency. Many critical researchers have acknowledged that mathematics classrooms and mathematical tasks are not neutral or without power dynamics, eq- uitable access, and opportunity for engaging (e.g., Esmonde & Langer-Osuna, 2013; McGee & Martin, 2011a; Tyner-Mullings, 2012; Varelas et al., 2013). While power dynamics and equity are not prominent in this study, we recognize that these dynamics influence agency and accountability and the importance of being mindful of such within both research and practice. Otherwise, there is no commitment to social justice and the status quo continues. Mathematics identity and accountability. Accountability is a prominent ele- ment of construction of competency as participation (Cobb et al., 2009). Cobb and colleagues articulated competency in terms of curricula with respect to agency dis- tribution (i.e., accountable for what) and in terms of the culture for discourse in terms of accountability (i.e., accountable to whom). Similar to the other participa- tion components described thus far, this component is observable and interpreta- tions can be made to categorize what was observed. The second portion of this par- ticipation component, accountable to whom, includes five levels: (a) teacher only or class only; (b) teacher and peer; (c) small group only; (d) teacher and small group; and (e) teacher, small group, and class. For this study, as students were situated in small groups for problem solving and the proctor followed a non-helping protocol (discussed later), our focus for whom students were accountable included: (a) ex- pert: directs discourse to knowledgeable other, in this case, proctor or a peer posi- tioned by the student as expert or more knowledgeable; (b) peers: expressed con- cern for peer in relation to mathematics at hand; or (c) self: positioning self as ex- pert/knowledgeable or expressed disinterest in peer or others’ perspectives. In summary, mathematics identity as participation was framed using agency and accountability. Observable incidents of participation were used as the overarch- ing construct that situated actions of agency and accountability related to mathemat- ics problem solving. We connected our study to recommended reforms for improv- ing mathematics teaching and learning and to extant understandings about Black male students and their mathematics identity development. Methods Interpretive qualitative analyses were employed for the purpose of under- standing student mathematics identity development as related to mathematics par- ticipation (Schwandt, 1994). Descriptive statistics were also used to aid in pattern Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 96 discovery and recognition. The social and cultural contexts selected for explora- tion were the small-group, problem-solving assessments observed throughout the 4 years of the APCM initiative. Video recordings of these assessments were the primary data source. The Algebra Project The underlying genesis of The Algebra Project was influenced by concerns of mathematical equity and access (Moses & Cobb, 2001). Therefore, the primary goal for the APCM initiative was to transform urban and rural students’ percep- tions of themselves from adopting mathematical identities given to them by others (e.g., at risk students) to mathematics learners and leaders who possess mathemat- ics literacy. In other words, “young people finding their voice instead of being spoken for is a crucial part of the process” (Moses & Cobb, 2001, p. 19). The APCM initiative was designed for accelerating mathematics under- standing for mathematics students who are likely to be underserved by schools and society at large. It was comprised of three parts: a cohort structure, curricu- lum and pedagogy, and community outreach. Worth noting explicitly, The Alge- bra Project consistently seeks to work with students from the lower quartile,5 but “interventions” neither advocate for nor include remedial approaches, and stu- dents are not positioned as deficient. Instead, The Algebra Project curriculum be- gins with students sharing an experience from which mathematical understand- ings are developed and abstracted, an experiential learning approach (Kolb, 1984). The APCM initiative is built on 15 years of experience in middle and high school pilot programs that included instructional materials development funded by the National Science Foundation (Moses, Dubinsky, Henderson, & West, 2008). A robust discussion of The Algebra Project curriculum6 would likely be interesting, but is beyond the scope of this article. The APCM initiative endeavors to create opportunities for students to actively engage in mathematics that devel- ops mathematical identities while building mathematical literacy. Participants and Context The first author, in Years 1 and 2 of the project, visited participants’ class- room several days per month to work with the APCM teacher and the local univer- sity mathematician, the principal investigator for the local project. In Years 3 and 4, 5 How one measures and determines hierarchies that order students and relegates some to the lower quartile is of no consequence because The Algebra Project seeks to work with all students perceived as underserved or otherwise labeled through deficiency discourses. 6 The Algebra Project curriculum is available for inspection and comment through a Public Curricu- lum Portal accessible at http://www.algebra.org/curriculum/. http://www.algebra.org/curriculum/ Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 97 she visited the classroom once or twice per year, and attended two of the four sum- mer institutes.7 During site visits, she supported the teacher and sometimes took an active role during instruction with the students and she collected data to support the research project. Through these activities, she got to know the students and they got to know her. In Year 1, she conceptualized the need for and developed the small- group assessment protocol (discussed later). The study reported here was part of a larger The Algebra Project research pro- ject that spanned five urban and rural sites across the United States. The goals for the larger APCM research project included: (a) students graduating from high school in 4 years; (b) students, upon graduation, enrolling in credit-earning mathe- matics courses for those choosing post-secondary options; and (c) students develop- ing and participating in productive peer cultures for learning mathematics (Moses et al., 2008). The research reported here focuses on one of the sites from the large project; a small, urban community located in the midwestern United States. The APCM stu- dent cohort was comprised of 19 students in their first year of high school, the only high school in the community. The students, with parental or guardian consent, agreed to take two, 50-minute classes of mathematics each day with the same teacher for all 4 years of high school. Most of the students and their parents (or guardians) knew the APCM teacher as a member of their community prior to enter- ing high school. The APCM teacher is White, but she raised her bi-racial (Black) son, who was academically successful and a star on the football team, in the com- munity. Her son was about two years older than the APCM cohort students. How- ever, all of the children in the community who engaged in sports did so within the community leagues, and the majority of the male cohort students were also on the high school football team. The first author, on many occasions, observed students gravitating to the APCM teacher in times of need. Several of the Black male stu- dents referred to her using familial terms, such as “school mom” or “second moth- er.” The APCM teacher was observed reciprocating the students’ affections. For instance, she maintained a snack cabinet to feed hungry students; admonished poor decision making, in or out of school, while encouraging and expecting better in the future; and returned many unsolicited hugs. After her son graduated, the APCM teacher continued to participate in the community with students and to attend extra- curricular events. The state department of education designated 16 of the 19 cohort students as “Not Proficient” as freshmen based on a score received on the eighth-grade, state- mandated mathematics achievement test. The school and society at large, from our 7 Summer institutes were held each summer to provide students with opportunities to engage in mathematics and to develop leadership and other positive dispositions. The institute locations alter- nated between a large urban university and a moderately sized rural university, where the students lived on the campus of the hosting university for 2 weeks each year. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 98 perspective, underserved these students; our project endeavored to serve them dif- ferently with respect to developing mathematics literacy. Six Black male students from the cohort were purposefully selected for this study because they represented a sample cohort. The first consideration for selec- tion, and most obvious, was to represent the cohort demographically; the cohort was predominantly comprised of students who identified as Black (90%) and male (71%).8 A conscious decision to select only Black male students was made for sev- eral reasons: (a) the vast majority of cohort students identified as Black and male during the 4 years of high school; (b) in Year 4, there was only one student who identified as not Black and that student was new to the cohort; and (c) a personal interest of the first author to study Black students and their mathematics identity development. The other important consideration for selecting the sample was to re- strict selection to 4-year participants and to those participants with sufficient data. Thus, the small-group assessment videos (discussed later) were viewed to identify an initial list of eligible students, and two factors were used to cull that list: (a) the student was present for at least one small-group video segment for each of the 4 years;9 and (b) an equal number of students positioned by peers as leaders. Using these as criteria, six Black male students were selected. It is worth noting that no students who identified as female appeared in more than two years of the small- group assessment videos. Reasons for absences were twofold: either the student was absent on a particular group assessment day or there were technical challenges while videotaping. It was not uncommon for more students to be absent on assess- ment days, especially during the earlier years. We always announced research relat- ed data collection activities and allowed students to opt out without penalty, per the Institutional Review Board agreement. Moreover, the research team was responsi- ble for videotaping, and especially in the earlier years, unintended errors occurred such as failure to turn the camera on or uncharged batteries. The six Black male students selected included: (a) three students who were regularly positioned by peers as class leaders; (b) two students who were more out- spoken during class, one was positioned as a leader by peers and the other was not; and (c) two students who tended to be less vocal during class, one was positioned as a leader by peers and the other was not. Of the six students, only one was not an athlete, but all students engaged in extracurricular activities at school. A descriptive summary of the six students was compiled from the first author’s experiential knowledge and relationships with the students, class observations, and informal conversations with the APCM teacher (see Table 1). 8 The demographic percentage calculations represent averages calculated using cohort enrollment data over the 4 years of the APCM initiative. 9 There was one exception, Ray was not in a Year 1 video, but he appeared in two Year 2 videos, and one was used for his Year 1 assessment. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 99 The local university mathematician, mentioned earlier, was task developer and proctor for all of the small-group assessments. As the principal investigator and participant researcher, he participated in curriculum development with other math- ematicians from The Algebra Project, and supported the teacher as a local consult- ant for The Algebra Project curriculum during the 4 years she taught the APCM students. In that capacity, he became well known by the students because during their freshman year he regularly visited (2 to 4 days per week) and participated in mathematics instruction in collaboration with the teacher. Table 1 Student Descriptions Assessment Protocol The small-group assessment protocol had three components: (a) pose the problem, answering only questions related to task clarification; (b) provide no hints or validation during problem solving; and (c) encourage students to rely on peers for support. The mathematician, in most every instance, faithfully executed this Namea Leaderb Achievementc Summary of In-Class Persona Hal No Moderate A gregarious personality, a collaborative, confident, and enthusiastic mathematics en- gager; he identified passing the state test for mathematics as impactful Neo Yes High A hard worker, soft spoken student leader, logical defender of mathematical ideas, ready mathematics participant and collaborator; teacher calls him dependable Ray No Low A hard and persistent worker who puts forth great effort, encourages peer participation and focus; vocal in class, may have undiag- nosed learning disability Reg Yes High Confident, loyal, and success oriented, sup- portive of peers, focused and driven, and non-judgmental; recognized student leader Rex No Low A hard worker, willing to work with others, ready participant, may have undiagnosed learning disability Ted Yes High Class leader who led by example, willing to work with anyone, ready participant; passing the state test for mathematics was impactful a All names are pseudonyms. b Leader as positioned by peers. c Estimate of achievement as measured by state test scores and school grades. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 100 protocol over the 4 years, as evidenced by the video recordings. The purpose of the group assessments was to gain insights about: (a) The Algebra Project curriculum effectiveness in relation to students’ mathematics understanding, 10 and (b) the stu- dents’ sociocultural development for mathematics learning. We assert that the small-group, problem-solving assessment protocol and the established relationship between the mathematician and the students afforded a nar- row and fertile context for addressing the research questions. Over the years, the proctor and students built an amicable and trusting relationship, based on observa- tions by the first author. Gillen (2014), a long-time veteran teacher of The Algebra Project and social justice advocate, found from his extensive experience that an en- vironment where students have sufficient opportunity to engage mathematically within a receptive climate affords freedom for them to engage through a myriad of roles. All of the assessments were proctored using the same protocol and students were free to choose to work individually or collaboratively with no negative conse- quences. Because of these factors and the nature of an Algebra Project classroom as described by Gillen, we posit that the small-group assessment context minimized inherent power dynamics that exist in typical learning environments. The environ- ment afforded students opportunities to engage with limited or no barriers, and therefore afforded an unobstructed view of these students’ mathematics identity as it emerged and evolved. Data Collection Data collected for this study included six different tasks, captured using 15 video recordings of small-group assessments; the average length of the recordings was about 30 minutes (in total, approximately 450 minutes). The assessment proc- tor also created analytic field notes and collected document artifacts for each of the problems. The video recordings were captured using a stationary camera or non- professional videographer. The camera was placed or held near the small groups in order to record the participants’ discourses, interactions, and body language. The local university mathematician proctored all of the assessments and generated ana- lytic field notes about the group members and their work; these notes were used to add clarity to the video recordings. For example, if a student had a misconception about the mathematics, the proctor’s analytic notes may have described the miscon- ception; or if a student created a drawing and they were pointing out a particular 10 All of the problem-solving tasks, after the first problem, where designed using a context different from that used for instruction. This design component is significant because The Algebra Project curriculum situates mathematics learning within student shared experiences. The types of tasks de- veloped for the small-group assessments were not unique and could be characterized as worthwhile tasks (Stein et al., 2000). Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 101 aspect of it while talking, the proctor may have kept a copy of the drawing with his notes. The APCM curriculum addresses mathematics topics from the high school content standards outlined by the Common Core State Standards for Mathematical Practice (CCSS, 2010). The group assessment tasks used for this investigation tar- geted mathematics content from several CCSS high school content standards, in- cluding algebra, functions, and geometry. For example, one of the Year 1 tasks fo- cused on algebraic reasoning, a subset of the CCSS algebra standard. Reg, Rex, and Ted worked on that task; Reg and Ted were in the same group for their Year 1 as- sessment, they have the same Group #, while Rex worked on the same problem, but in a different group (see Table 2). Table 2 Mathematics Topics and Student Groupings by Year Reg Rex Ted Hal Neo Ray Year 1 Math Topics Algebraic reasoning Algebraic reasoning Algebraic reasoning Linear functions (continuous) Linear functions (continuous) Linear functions (piecewise) Group # 1 2 1 3 3 4a Year 2 Math Topics Linear functions (piecewise) Linear functions (piecewise) Linear functions (piecewise) Geometric construction (with paper) Geometric construction (with paper) Geometric construction (with paper) Group # 5 6 4 7 8 7 Year 3 Math Topics Area of composite shapes Area of composite shapes Area of composite shapes Area of composite shapes Area of composite shapes Area of composite shapes Group # 9 10 10 11 11 9 Year 4 Math Topics Direct variation (velocity com- parison) Direct variation (velocity com- parison) Direct variation (velocity com- parison) Direct variation (velocity com- parison) Direct variation (velocity com- parison) Direct variation (velocity com- parison) Group # 12 13 14 15 15 15 a Ray’s Year 1 task actually occurred during Year 2. Task selection for each year was done in a way to align the mathematics topics of his peers. Video Recording Data Analysis The analysis of video data was done using qualitative software (NVivo ver- sion 10) that allowed multiple researchers to analyze the same data, which simpli- fied comparative analysis of coding, and diminished the amount of transcription required. The protocol used for video analysis was as follows: (a) segment each video recording into time segments of about 1 to 2 minutes; (b) watch each seg- ment, and assign descriptive themes (i.e., nodes) that captured observed phenome- na, paying specific attention to each target student; and (c) for recordings with two Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 102 or more target students in the same small group, recordings were watched multiple times, at least once for each target student (counting re-watched video recordings, nearly 700 minutes of recordings were analyzed). The group assessment video recordings were coded and analyzed for recur- rent themes, allowing inferences to be made and supported (or disputed) via the da- ta—warranted claims were made (Wolcott, 2001). The analysis process was multi- level. The first-level analyses assigned thematic categories (i.e., nodes) to recording segments. The second-level used descriptive statistics and graphical representations of coded node frequencies to search for patterns. Heavily coded nodes (i.e., those with relatively high frequency counts) and patterns were used to draw inferences related to the research questions. Then the video data corpus was searched to find representative examples, evidence that confirmed or disputed inferences—the third- level analyses. The evidentiary data served to warrant the inferences from which claims were made (Erickson, 1986). A second researcher, a doctoral student whose worldview and biases differ from the first author’s, was recruited to work collabora- tively with the first author to improve the validity of findings while increasing the efficiency of the video analyses—a somewhat forth-level of analyses. Using multi- ple researchers to analyze the video data increased the trustworthiness of the anal- yses, which strengthens the validity of the findings (Lather, 1986). The first author used the video analysis protocol to code the first 2 years of video recordings after creating an initial codebook that defined thematic nodes based on the grounding literature. The nodes in the codebook were organized hier- archically beneath the primary constructs: agency, accountable for what, and ac- countable to whom. Analyzing the video segments led to defining emergent nodes during the analysis process to capture unanticipated phenomena observed that had not been included in the initial codebook; an approach described by Schwandt (1994) as an emic perspective. Additionally, mathematical work practices as an or- ganizing category was added late in the analysis process and after revisiting the lit- erature. The emergence of this category is described later in the Results section, as it was not part of the original analysis plan. Late during the analyses, we reor- ganized the a priori categories because of patterns in the data, which Lather (1986) described as face validity, which also strengthens qualitative research findings. The videos were watched several times by one or two researchers to establish and maintain an acceptable inter-rater reliability standard (Landis & Koch, 1977) with the Kappa coefficient > .70 and percent coding agreement > 85% throughout the coding process. To that end, the two researchers’ coded one video recording from Year 4; comparison analyses were run that showed coding did not meet the pre-established standard. The researchers met to review the coding and collabora- tively re-coded and refined the codebook definitions; refining the codebook during the analysis process ensured shared understandings for node definitions and con- sistency of coding between researchers. Collaborative coding continued until the Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 103 two researchers reached coding reliability agreement of > 85%. The two researchers then independently coded videos from Years 2 and 3; comparison analyses were run and the inter-rater reliability standard was met. Overall, the first author ana- lyzed video recordings for Years 1 through 3, and the second researcher analyzed video recordings for Year 4. Results The purpose of this study was to determine how mathematics identity devel- oped for six Black male students who choose to participate in the APCM initiative; students agreed to take two periods of mathematics taught by the same teacher for all 4 years of high school.11 The six Black male students participated in the APCM for all 4 years, passed the state’s graduation achievement test, and graduated high school in 4 years. According to exit interview data, these six students’ paths after graduation included college (2-year or 4-year), work, or military service. Their exit interviews also revealed that each student claimed mathematical readiness to pursue their planned path for the future. We turn our attention to the results from analyzing the video data. Mathematics Agency Three themes were evident from analyzing the 4 years of small-group, prob- lem-solving assessments: confidence, collaboration, and personal effort. The most heavily coded thematic nodes aggregated over the 4 years from the categories agency and accountable for what, the primary constructs for the analysis, are shown in Table 3. Interestingly, we noticed that several of the heavily coded thematic nodes might have been categorized as discourse practices. Manouchehri and St. John (2006) characterized discourse for mathematics learning as being comprised of both reflection and action for the purposes of gaining understanding of their peers’ perspectives and influencing them. The heavily coded nodes align with these characterizations and purposes: the discourses were reflective and action oriented for the purpose of gaining understanding of peer’s conceptions or garnering peer support. 11 Scheduling conflicts for credit attainment for graduating precluded students from taking the dou- ble periods of mathematics during their final year of high school. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 104 Table 3 The Predominant Codes Aggregated Over Year 1 to Year 4 One example of this type of discourse practice occurred as Hal worked to un- derstand how to fold a circular piece of paper in a way to construct parallel lines. As Hal grappled with the problem, he engaged in a mostly nonverbal way with a peer (Student 1, not a study participant) in his group: Hal: [Explains to the proctor why the lines he has constructed are not parallel. As the two other group members continue to work on folding their papers. He glances at Student 1’s folding a couple of times as he continues to contemplate his work.] Student 1: [After making several folds and examining his paper closely, he rotates the paper twice to examine the lines] “I don’t know if it’s right” [student giggles.] Hal: [Reaches over and picks up the paper his peer had been folding for closer examination.] It’s not bad, Dog. Student 1: [Nods in acknowledgement of Hal’s praise.] (Video recording, Year 2) In this interaction, there is little dialogue, but after taking Student 1’s paper, Hal goes on to explain why he believes Student 1’s constructed lines are parallel. This interaction clearly depicts a discourse practice in which Hal was reflective— by comparing his approach to Student 1’s—and led to action—explaining why Stu- dent 1’s lines were parallel. The purpose of the discourse was an example of Hal attempting to understand Student 1’s mathematics. A second example depicts Hal’s effort to garner peer support for an idea, us- ing the reflection/action discourse practice. This example occurred in Year 4 in a small group comprised of Hal, Neo, and Ray. In this episode, Hal and Ray are en- gaged in making sense of the problem involving two moving cars, A and B, travel- Thematic Nodes Hal Neo Ray Reg Rex Ted Totals Engaged in peer collaboration a 22 23 5 8 10 25 93 Engaged in individual process a 5 9 26 19 12 15 86 Collaborative sense making 17 16 2 10 3 14 62 Explaining ideas 13 12 7 10 7 12 61 Listening for understanding 9 13 6 3 7 22 60 Sharing ideas with peers 14 13 2 5 6 16 56 Consulting expert source 13 7 7 12 6 10 55 Listening to peer 11 13 8 3 7 10 52 Asking clarifying questions 7 7 6 4 7 13 44 Acknowledging contributions by others 7 6 2 5 1 14 35 a Work practices for mathematical engagement Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 105 ing at different rates, and their locations are described in terms of the other car and a referent red line at specific time intervals: Hal: There is no doubt, that it [car A] was one meter past the red line and car A was two meteres past the red line, and if they’re both exactly at four meters at three [sec], that means that it [car B] caught up, and passed, not passed it, but it just equaled up with it. So that it’s [car B] clearly faster. They’re going the same speed. Neo: Car B is at first, like already ahead of car A; and then car A tied it up. And car A is going faster. Hal: No, it says [referring to the problem], that it’s [car A] two meters past it [car B] already, so it’s [car A] in front of car B, it’s one meter past the red line. Neo: Oh, Ok. I thought… Hal: So, it’s kind of lined up like this [begins drawing and talking softly about his sketch.] (Video recording, Year 4) In this episode Hal again reflects on input received from a peer as he listens to Neo’s explanation, but then disputes Neo’s position using evidence from the text of the problem, which convinces Neo of Hal’s position. This reflective process leads to Hal taking action: depicting his thinking via a sketch. Hal’s purpose in the dis- course appears to be garnering peer support: Hal is seeking Neo’s support before investing in creating a pictorial representation. This action is representative of agency as articulated by Bandura (2005); Hal’s exercised agency was self-regulated and negotiated within the group’s social system. The two most heavily coded thematic nodes for participation represent two distinct student work practices for mathematics engagement, individual and collab- orative (Esmonde, 2009). The total coding for these nodes engaged in peer collabo- ration (93) and engaged in individual process (86) are much greater than the total for the next most heavily coded node, collaborative sense making (62; see Table 3). These two most heavily coded nodes were interpreted as students choosing to en- gage mathematically, as they did not opt to engage in off task behaviors or other- wise not participate. Close examination of this analysis led to two things: (a) reor- ganizing the nodes hierarchically around these two heavily coded nodes; and (b) examining how the students’ participation acts split across the two nodes. The qualitative analyses were done using software (NVivo version 10), which allowed for easily restructuring of nodes at any point within the analyses without disturbing prior analyses. The node restructuring led to additional analyses using this emergent perspective—looking at the students’ participation through the lens of their observed work practices. Immediately obvious, we found that some students mostly worked collaboratively (i.e., Hal, Neo, and Ted), while others opted to work independently (i.e., Ray and Reg), and one student (i.e., Rex) worked almost equal- ly across the two work practices (see Table 3). Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 106 When we reorganized the heavily coded thematic nodes by student work prac- tices for engagement, individual and collaborative, some nodes were combined be- neath an existing node or a new node was defined. In the end, the most heavily cod- ed nodes were found primarily under the collaborative work practice. Node descrip- tions are provided to make clear the meanings used for the coding process: catego- rizing what was observed (individual or collaborative) and students’ verbal utter- ances, actions, and gestures (see Table 4). Table 4 Definitions for Heavily Coded Thematic Nodes Organized by Categories Mathematical Identity Development Over 4 Years The reorganized codebook provided the foundation for our response to the primary research question: How did the mathematics identity of six Black male stu- dents participating in the APCM initiative develop over their 4 years of high school? We found the aggregate coding frequencies (i.e., totals for the six students by years) for the heavily coded nodes; doing so afforded an overall perspective of the students’ mathematics identity development across the 4 years. Summaries de- picting the evolution of the students’ identity are shown using summary line charts, using the following lenses: (a) students exercising individual problem-solving prac- tices, (b) students exercising collaborative problem-solving practices; and (c) for whom students were observed being accountable. Individual problem-solving practices. With respect to individual work prac- tices, there were only two heavily coded nodes asking clarifying questions and con- sulting expert source that emerged from the analyses (see Figure 1). These two par- ticipation behaviors were observed most during Years 2 and 3. Interestingly, stu- Heavily Coded Thematic Nodes Definitions of Nodes Category: Individual work practice Asking clarifying questions Independently initiates questioning, clarifying and probing questions Consulting expert source Seeks help and/or support from someone per- ceived as expert Category: Collaborative work practice Acknowledging contributions by others Making public recognition of a peer’s mathe- matical contribution Collaborative sense making Expressing efforts to understand while engaging with others Listening to peers Making public the effect of a peer’s verbal ex- pression Passive peer interactions Nonverbal response to another’s action Sharing ideas with peers Making public, verbally or by action, ideas, ex- planations, and artifacts Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 107 dents consulting a perceived expert source was highest in Year 2, and then declined for each year after. However, the students’ questioning increased from Year 1 and peaked in Year 3. We posit that this summary suggests that these students transi- tioned from reliance on a knowledgeable other to a greater reliance on self and col- laborations among peers. Two examples from the asking clarifying questions node from Years 2 and 3 are presented to illustrate this finding. In Year 2, Rex was observed asking more questions than he did in all of the other years combined. In the first example, stu- dents are given prices for purchasing Jelly bracelets from an online vendor. They are given a variety of information, such as the price for a specific number of brace- lets and volume shipping costs. The information, however, is not presented simplis- tically in a way that suggests a linear relationship. Rex asks several questions, such as: “Isn’t it [Jelly bracelet cost] going up?”; “What did you get?”; and “So, how much is it for one bracelet?” (Video recording, Year 2) These questions were asked of the group and are often focused on getting to the answer or seeking confirmation. Rex’s group members offered little in response to his questions, which led him to pose questions to the proctor about the given information. The proctor acknowledg- es that there is sufficient information to solve the problem, to which Rex replied, “Well, I didn’t find it.” (Video recording, Year 2); a response that suggests that Rex is done, unable to solve the problem and has no other options. Rex’s questioning was focused narrowly on getting support for finding the right answer to the prob- lem. The second example illustrates a different perspective that shows the evolu- tion of questioning among the students. In Year 3, Reg asked the most questions. The problem presented was about finding the linear measure, width, on each side of Figure 1: Summary of heavily coded nodes categorized by individual practice for all students by years. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 108 a rug centered in a room, given the room dimensions and the area of the rug. In ad- dition to asking fewer questions seeking support or confirmation, Reg poses ques- tions to the proctor: “Has any group solved this problem, yet?”; “Have we learned what we need to know in order to solve this question?”; and “Are we over thinking this?” (Video recording, Year 3) Reg’s questions appear to be seeking understand- ing about his preparedness to solve the problem. These questions are not searching for hints or support, but rather validation that he possesses all that is needed for success. What underlies these questions is an inherent trust that exists between the student and proctor given the student’s willingness to ask such questions and then to accept the response without question, even when he was faced with uncertainty about his solution. Reg persevered in problem solving after this exchange. Collaborative problem-solving practices. There were four heavily coded nodes for student collaborative work practices that emerged from the analyses (see Figure 2), three of which are aligned with discourse practices (e.g., Manouchehri & St. John, 2006): collaborative sense making, listening to peers, and sharing ideas. The shape of the lines that show a summary of coding for these three nodes are sim- ilar in shape, the lines are relatively flat between Years 1 and 2, peak in Year 3, and then fall in Year 4. These trends suggest that student identity development related to discourse practices followed an increasing trajectory and peaked in Year 3; howev- er, we hesitate to consider Year 4 because of previously stated reasons and the lack of observed participation in Year 4. Ted was observed as the most collaborative participant as measured by ob- served behaviors in this study. Therefore, we selected examples from video seg- ments featuring Ted to show a progression over the years as an illustrative case for Figure 2: Summary of heavily coded nodes categorized by collaborative practice for all students by years. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 109 all students (see Table 5). The Year 1 discourse practice is, for the most part, not collaborative, even though students are talking to one another. In Year 2, Ted gives up on the problem until he is lured back by a question posed by the proctor to Ray. In Year 3, Ted listens to peers and without invitation he alerts them of an error in their mathematics strategy for solving the problem. While he does not know the correct solution at the time, he was sufficiently engaged to recognize the potential pitfall and share his perspective with peers to redirect their trajectory. Each year, Ted’s level of mathematical engagement and discourse with peers seemed to gain in complexity in the sense that Year 1 was more collective than col- laborative—kids voicing ideas but not using them to improve mathematical under- standing—and by Year 3 discourses were unsolicited peer supportive for mathemat- ics learning. While Year 2 was clearly between the two extremes, greater persever- ance emerged and participation continued after claiming, “done.” Thus, over the years, the discourse frequency and complexity increased. Table 5 Ted’s Progression of Discourse Practice: An Illustrative Example of One Aspect of Mathematics Identity Development Year #: Problem/Context Description of Discourse Year 1: A trip is represented using a linear model Ted initiates conversation by sharing his idea; Reg responds, makes no comment about Ted’s idea, and shares his approach. Ted listens to Reg explain his idea more than once with little insight to further his solution; Reg disengages, “Can I work on my own?” Year 2: Find the total cost to pur- chase and ship Jelly bracelets, giv- en complex pricing information Ted declares himself done; the proctor poses a question to Ray, “Why are the price differences the same and then differ- ent?” Ted did not appear to be listening, but in response asks, “Which one?” Ted reengages with the problem. Year 3: Find the width around a rug centered in a room, given room dimensions and area of the rug Ted watches as Rex and the other group member discuss an idea; Ted recognizes an error in their logic. Ted takes Rex’s paper and by drawing on his paper, shows the group members the width they seek. The growing discourse practice may also explain the continual rise in students acknowledging contributions by others, the fourth heavily coded node, which should not be overlooked (see Figure 2). By far, Ted was the most observed engag- ing in this behavior, with the greatest number of instances observed in Year 4, mak- ing this node not as representative of the group, but Ted, the individual. Accountability. The aggregated coding of nodes from the thematic category accountable to whom is shown in Table 6. These nodes were coded less heavily in comparison to those coded for the participation nodes because interpreting whom one is accountable is not always transparent to an observer. Nonetheless, there were Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 110 patterns; notice that the students were observed as most accountable to peers when looking at aggregate totals across the years. However, the aggregate totals for ob- served accountability to expert and self are almost evenly split. Interestingly, Hal, Neo, Rex, and Ted were most accountable to peers, which is supported through analysis results that Hal, Neo, and Ted were observed being the most collaborative among the students. According to the APCM teacher, Reg mentored Ray and Ray modeled himself after Reg (Informal communication, Year 4), and interestingly, they were the only two among the sample with the least amount of coding for ac- countable to peers and with the most coding for accountable to expert and self. The descriptions used for this study for accountability to whom from the re- searchers’ perspective follow. Accountability to expert most often suggests that the learner is not sufficiently empowered and lacks autonomy for mathematics or mathematical understanding. There were many instances of this across the years where students requested validation from the proctor and their peers by asking questions such as: “So it took her 10 seconds to get to the end of the field?” (Hal, Video recording, Year 1); “What’s up with this, [Proctor]?” (Reg, Video recording, Year 2); “With your math knowledge is that correct?” (Hal, Video recording, Year 4) Table 6 Aggregated Coding of Nodes from One Category Showing Analyses of Small- Group, Problem-Solving Assessment Videos Over 4 Years Accountable to Whom Hal Neo Ray Reg Rex Ted Total Expert: seeks other to model or guide knowledge construction and validate 8 6 7 15 4 4 44 Peers: constructs understanding and validity collaboratively 17 22 2 5 7 18 71 Self: confident and autonomous constructs and validates inde- pendently 5 8 8 11 4 7 43 Accountability to self, means a level of learner confidence and belief with suf- ficient mathematical autonomy that he readily shares ideas with others so they may value, critique, or dispute them. From this stance, the proctor may be considered a peer at times; the proctor was positioned to share facts and not give hints or valida- tion. The proctor was observed not exercising his authority within the groups’ pow- er hierarchy. Accountability to peers is between accountability to expert or self if one viewed the three along a continuum. From a student perspective within a social system of a small group, being ac- countable to peers suggests sufficient confidence for overcoming the risk of being Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 111 wrong. The difference between peers and self is that for those who are accountable to peers, they are sufficiently free to rely upon others for support and adjust their ideas based on collaborations. On the other hand, those accountable to self may be more autonomous yet less open to “hear” critiques from others, which may be cog- nitively limiting, based on the situation, of course. From the summary line chart shown in Figure 3 we can see that observed in- stances of reliance on peers peaked in Year 3 and was maintained in Year 4. Ob- served instances of reliance on self grew steadily from Year 2 through Year 4 and over the same period observed instances of reliance on expert decreased steadily. These observations taken with the accountable to whom perspectives, we conclude that students made a shift from relying on knowledgeable others to relying on them- selves and/or peers and they were sufficiently confident to risk being wrong, yet free enough to be influenced by collaborations. Summary of Findings: Mathematics Identity Development The first point to make is that the APCM initiative created a receptive climate and we posit the freedom and nurturing was fertile ground for the six Black male students’ mathematics identity development during high school. The students ex- pressed, their teacher described, and the researchers observed students’ mathemat- ics confidence. One student described his mathematics confidence as, “I can do an- ything that I put my mind to” (Hal, Interview, Year 4). This simple statement is emblematic of the way these students saw themselves mathematically (i.e., their mathematics identity), and it suggests that their confidence was connected to per- sonal effort. What is not captured by this particular statement is the value the stu- Figure 3: Summary of heavily coded nodes categorized by accountable to whom for all students by years. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 112 dents placed on their peers and collaborations for learning mathematics, an aspect that was evident across the observations. Another key aspect of these students mathematical identity was their inten- tional choice to engage mathematically. The students opted to engage in individual and/or collaborative practices; however, even for those who favored individual practices they were observed in discourse through asking questions, which often led to verbal discourses with others. Overall, we observed students engaged in rich dis- course practices that were reflective and action oriented for the purpose of gaining understanding or garnering peer support. The evolution of these six Black male stu- dents included transitioning from reliance on knowledgeable others to reliance on self and peer collaborations or mathematical participation, which reified the ob- served increase in the number of discourses and their complexity within their small- group, problem-solving assessments. Therefore, it was not surprising to realize that confidence appears to be the foundation for our students’ mathematics identity de- velopment when viewed through the lens of mathematical agency, a relationship established by Bandura (2002). Discussion and Conclusion Research on reform-based mathematics calls for classrooms environments where mathematical autonomy and freedom abound and are available for all learn- ers (Carpenter, Fennema, Franke, Levi, & Empson, 2000; Carpenter & Romberg, 2004; Hiebert et al., 1997; West & Staub, 2003). Such autonomy and freedom, however, cannot be taken for granted, especially from those students (urban and rural) who are underserved (Gillen, 2014). We agree that pedagogical content knowledge and mathematical knowledge for teaching (e.g., Ball & Bass, 2003; Ball, Hill, & Bass, 2005; Hiebert et al., 1997; Hufferd-Ackles, Fuson, & Sherin, 2004) are necessary conditions in creating effective learning environments; howev- er, these knowledges are not sufficient conditions for creating equitable learning environments for all children (e.g., Martin & Herrera, 2007; NCTM 2000, 2014). Although NCTM (2000) established the Equity Principle long ago, equity continues to elude many mathematics classrooms, especially those with large numbers of ra- cial and ethnic minority students (Berry, 2008; Martin, 2008). Further research is needed to understand the full impact of practices and poli- cies on student mathematics identity development, and to articulate specific reforms needed to free our children and our classrooms from those practices and policies that inhibit mathematics literacy, leadership, and freedom (Hope et al., 2015). We must serve underserved students differently so that they are afforded opportunities to choose mathematics literacy. As the research reported here demonstrates, histori- cally underserved students develop different mathematics identities when provided access to classroom environments that are not reliant on traditional remediation ap- Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 113 proaches. We know that mathematics efficacy and confidence are essential disposi- tions for exercising mathematics agency (Bandura, 2002), which supports our find- ing that the students’ confidence played a role in the ways they engaged in the small-group problem solving. We also posit that the receptive climate and freedom in the groups contributed to the ways their participation manifested in relation to their mathematics identities. These findings were not unanticipated. In another study about APCM students, Grant (2014) examined students’ verbal and written reflections about the ways they interact with peers while learning mathematics. She found that APCM students from two cohorts, one urban and the other rural, de- scribed productive classroom culture for mathematics learning as students getting along with peers, working hard, and supporting one another. In the end, the study reported illustrates the development of mathematics identity as participation of six Black male students; the findings add to the literature extolling the virtues of Black learners. The documented participatory freedom exer- cised by the students may be useful for those looking for new approaches in trans- forming the culture of participation and agency in mathematics classrooms. The study, however, is limited in the sense that it looked closely at only six Black male students in one mathematics teacher’s classroom, over a 4-year time period. More- over, the structure of the APCM initiative—students studying with one teacher for all 4 years—worked for the students and teacher reported here. It is important to note, however, that there were instances with other APCM cohorts where that was not the case. Some teacher–student relationships were not synergistic, and did not promote effective mathematics learning. Nevertheless, one implication of this study is that the APCM initiative pro- vides guidance for those interested in creating equitable and receptive environments for underserved students generally, and for Black male students specifically. Policy makers and other stakeholders have claimed interests aligned with equity as evi- denced by names such as “No Child Left Behind” and “Race to the Top.” These and other initiatives funded through public and private organizations are all inno- cently labeled as accountability measures. These labels, however, are misleading and have been far reaching with many unintended negative consequences for U.S. schools and mathematics classrooms. These mandates manifest in education sys- tems as hierarchies where teachers and students are at the bottom, with little or no choice or autonomy (Gillen, 2014). Gillen and others (e.g., Leonard & Martin, 2013) argue, and we concur, that students need different learning environments and opportunities if they are to develop the types of positive mathematics identities de- scribed here. Acknowledgments A special thank you is extended to the APCM cohort students, Mrs. Amanda Tridico Clawson, the APCM teacher, and Dr. Lee McEwan, the mathematician and proctor for the group assessments, Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 114 without whom this project would not have been possible. The APCM initiative is based on research and development supported by the National Science Foundation DRK-12 under Award No. 0822175. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation or the The Algebra Project, Inc. References Anyon, J. (2006). Social class, school knowledge, and the hidden curriculum: Retheorizing reproduction. In L. Weis, C. McCarthy, & G. Dimitriadis (Eds.), Ideology, curriculum, and the new sociology of education: Revisiting the work of Michael Apple (pp. 37–46). New York, NY: Routledge, Taylor and Francis Group. Ares, N. (2006). Political aims and classroom synamics: Generative processes in classroom communities. Radical Pedagogy, 8(2), 12–20. Balfanz, R., & Legters, N. (2004). Locating the dropout crisis. Which high schools produce the nation’s dropouts? Where are they located? Who attends them? Report 70. Center for Research on the Education of Students Placed at Risk CRESPAR. Baltimore, MD: John Hopkins University. Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmet (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 3–14). Edmonton, Canada: CMESG/GCEDM. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14–46. Bandura, A. (2002). Social cognitive theory in cultural context. Applied Psychology: An International Review, 51(2), 269–290. Bandura, A. (2005). The evolution of social cognitive theory. In K. G. Smith & M. A. Hitt (Eds.), Great minds in management: The process of theory development (pp. 9–35). Oxford, United Kingdom: Oxford University Press. Bandura, A. (2006). Toward a psychology of human agency. Association for Psychological Science, 1(2), 164–180. Berry, R. Q., III. (2008). Access to upper-level mathematics: The stories of successful African American middle school boys. Journal for Research in Mathematics Education, 39(5), 464– 488. Berry, R. Q., III., Ellis, M., & Hughes, S. (2014). Examining a history of failed reforms and recent stories of success: Mathematics education and Black learners of mathematics in the United States. Race, Ethnicity & Education, 17(4), 540–568. Berry, R. Q., III., Pinter, H., & McClain, O. (2013). A critical review of American K–12 mathematics education, 1954–Present: Implications for the experiences and achievement of Black children. In J. Leonard & D. B. Martin (Eds.), Beyond the numbers and toward new discourse: The brilliance of Black children in mathematics (pp. 23–53). Charlotte, NC: Information Age. Booker, K., & Mitchell, A. (2011). Patterns in recidivism and discretionary placement in disciplinary alternative education: The impact of gender, ethnicity, age, and special education status. Education and Treatment of Children, 34(2), 193–208. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 115 Bracey, J. M. (2013). The culture of learning environments: Black student engagment and cognition in math. In J. Leonard & D. B. Martin (Eds.), The brilliance of Black children in mathematics: Beyond the numbers and toward new discourse (pp. 171–195). Charlotte, NC: Information Age. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2000). Cognitively guided instruction: A research based teacher professional development program for elementary school mathematics (Report No. WCER-003). Washington, DC: National Science Foundation. Carpenter, T. P., & Romberg, T. A. (2004). Powerful practices in mathematics & science: Research- based practices for teaching and learning. Madsion, WI: University of Wisconsin. Retrieved from http://ncisla.wceruw.org/research/powerful.html Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education, 40(1), 40–68. Cobb, P., & Hodge, L. L. (2002). A relational perspective on issues of cultural diversity and equity as they play out in the mathematics classroom. Mathematical Thinking & Learning, 4(2/3), 249–284. Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Doerr, H. M. (2006). Examining the tasks of teaching when using students’ mathematical thinking. Educational Studies in Mathematics, 62(1), 3–24. Durbin, K. (2012, May 23). We’re trying to make sure students know a college education is possible. Mansfield News Journal. Retrieved from http://www.mansfieldnewsjournal.com/ Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 119–161). New York, NY: Macmillan. Esmonde, I. (2009). Mathematics learning in groups: Analyzing equity in two cooperative activity structures. The Journal of the Learning Sciences, 18(2), 247–284. Esmonde, I., Brodie, K., Dookie, L., & Takeuchi, M. (2009). Social identities and opportunities to learn: Student perspectives on group work in an urban mathematics classroom. Journal of Urban Mathematics Education, 2(2), 18–45. Retrieved from http://ed- osprey.gsu.edu/ojs/index.php/JUME/article/viewFile/46/35 Esmonde, I., & Langer-Osuna, J. M. (2013). Power in numbers: Student participation in mathematical discussions in heterogeneous spaces. Journal for Research in Mathematics Education, 44(1), 288–315. Franke, M. L., Kazemi, E., & Battey, D. (2007). Understanding teaching and classroom practice in mathematics. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 225–256). Charlotte, NC: Information Age. Fullilove, R. E., & Treisman, P. U. (1990). Mathematics achievement among African American undergraduates at the University of California, Berkeley: An evaluation of the mathematics workshop program. Journal of Negro Education, 59(3), 463–478. Gee, J. P. (2000). Identity as an analytic lens for research in education. Review of Research in Education, 25, 99–125. Gibson, P. A., Wilson, R., Haight, W., Kayama, M., & Marshall, J. M. (2014). The role of race in the out-of-school suspensions of Black students: The perspectives of students with suspensions, their parents and educators. Children and Youth Services Review, 47(3), 274–282. Gillen, J. (2014). Educating for insurgency: The roles of young people in schools of poverty. Oakland, CA: AK Press. Gilpin, L. S. (2006). Postpositivist realist theory: Identity and representation revisited. Multicultural Perspectives, 8(4), 10–16. http://ncisla.wceruw.org/research/powerful.html http://www.mansfieldnewsjournal.com/ http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/viewFile/46/35 http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/viewFile/46/35 Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 116 Grant, M. R. (2014). Urban and rural high school students’ perspectives of productive peer culture for mathematics learning. Journal of Urban Learning, Teaching, & Research, 10, 39–49. Greeno, J. G. (1997). Theories and practices of thinking and learning to think. Middle-School Mathematics through Applications Project, 106(1), 85–126. Gregory, A., Skiba, R. J., & Noguera, P. A. (2010). The achievement gap and the discipline gap two sides of the same coin? Educational Researcher, 39(1), 59–68. Gresalfi, M. (2009). Taking up opportunities to learn: Constructing dispositions in mathematics classrooms. The Journal of the Learning Sciences, 18(3), 327–369. Gresalfi, M., Taylor, M., Hand, V., & Greeno, J. G. (2009). Constructing competence: An analysis of student participation in the activity systems of mathematics classrooms. Educational Studies in Mathematics, 70(1), 49–70. Gutstein, E. (2007). “And that’s just how it starts”: Teaching mathematics and developing student agency. Teachers College Record, 109(2), 420–448. Haberman, M. (2010). The pedagogy of poverty versus good teaching. Phi Delta Kappan, 92(2), 81– 87. (Orginal work published 1991) Hand, V. (2010). The co-construction of opposition in a low-track mathematics classroom. American Educational Research Journal, 47(1), 97–132. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Hope, E. C., Skoog, A. B., & Jagers, R. J. (2015). “It’ll never be the white kids, it’ll always be us”: Black high school students’ evolving critical analysis of racial discrimination and inequity in schools. Journal of Adolescent Research, 30(1), 83–112. Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81– 116. Jett, C. C. (2010). “Many are called, but few are chosen”: The role of spirituality and religion in the educational outcomes of “chosen” African American male mathematics majors. The Journal of Negro Education, 79(3), 324–334. Kolb, D. A. (1984). Experiential learning: Experience as the source of learning and development. Englewood Cliffs, NJ: Prentice Hall. Ladson-Billings, G. (1994). The dreamkeepers: Successful teachers of African American children. San Francisco, CA: Jossey-Bass. Ladson-Billings, G. (1997). It doesn’t add up: African American students’ mathematics achievement. Journal for Research in Mathematics Education, 28(6), 697–708. Ladson-Billings, G. (2006). From the achievement gap to the education debt: Understanding achievement in US schools. Educational Researcher, 35(7), 3–12. Lampert, M. (2004). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in mathematics education research (pp. 153–171). Reston, VA: National Council of Teachers of Education. (Original work published 1990) Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33(1), 159–174. Lather, P. (1986). Issues of validity in openly ideological research: Between a rock and a soft place. Interchange, 17(4), 63–68. Leonard, J., & Martin, D. B. (Eds.). (2013). The brilliance of Black children in mathematics: Beyond the numbers and toward new discourse. Charlotte, NC: Information Age. Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 117 Love, A., & Kruger, A. C. (2005). Teacher beliefs and student achievement in urban schools serving African American students. Journal of Educational Research, 99(2), 87–98. Manouchehri, A., & St. John, D. (2006). From classroom discussions to group discourse. Mathematics Teacher, 99(8), 544–551. Marrus, E. (2015). Education in Black America: Is it the new Jim Crow. Arkansas Law Review, 68, 27–54. Martin, D. B. (2008). E(race)ing race from a national conversation on mathematics teaching and learning: The national mathematics advisory panel as White institutional space. Montana Mathematics Enthusiast, 5(2/3), 387–397. Martin, D. B. (2009). Does race matter? Teaching Children Mathematics, 16(3), 134–139. Martin, D. B. (2012). Learning mathematics while Black. Educational Foundations, 26(1/2), 47–66. Martin, D. B. (2013). Race, racial projects, and mathematics education. Journal for Research in Mathematics Education, 44(1), 316–333. Martin, D. B. (2015). The collective Black and Principles to Actions. Journal of Urban Mathematics Education, 8(1), 17–23. Retrieved from http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169 Martin, T. S., & Herrera, T. (2007). Mathematics teaching today: Improving practice, improving student learning. Reston, VA: National Council of Teachers of Mathematics. Maton, K. I., Hrabowski, F. A., III., & Greif, G. L. (1998). Preparing the way: A qualitative study of high-achieving African American males and the role of the family. American Journal of Community Psychology, 26(4), 639–668. Matthews, L. E., Jones, S. M., & Parker, Y. A. (2013). Advancing a framework for culturally relevant, cognitively demanding mathematics tasks. In J. Leonard & D. B. Martin (Eds.), The brilliance of Black children in mathematics: Beyond the numbers and toward new discourse (pp. 123–150). Charlotte, NC: Information Age. McGee, E. O. (2013a). Threatened and placed at risk: High achieving African American Males in urban high schools. The Urban Review, 45(4), 448–471. McGee, E. O. (2013b). Young, Black, mathematically gifted, and stereotyped. The High School Journal, 96(3), 253–263. McGee, E. O., & Martin, D. B. (2011a). From the hood to being hooded: A case study of a Black male PhD. Journal of African American Males in Education, 2(1), 46–65. McGee, E. O., & Martin, D. B. (2011b). “You would not believe what I have to go through to prove my intellectual value!” Stereotype management among academically successful Black mathematics and engineering students. American Educational Research Journal, 48(6), 1347– 1389. Moses, R. P., & Cobb, C. E., Jr. (2001). Radical equations: Civil rights from Mississippi to The Algebra Project. Boston, MA: Beacon Press. Moses, R. P., Dubinsky, E., Henderson, D. W., & West, M. (2008). The development of student cohorts for the enhancement of mathematical literacy in under served populations: National Science Foundation Award 0822175. Nasir, N. S., & Hand, V. M. (2006). Exploring sociocultural perspectives on race, culture, and learning. Review of Educational Research, 76(4), 449–475. Nasir, N. S., McLaughlin, M. W., & Jones, A. (2009). What does it mean to be African American? Constructions of race and academic identity in an urban public high school. American Educational Research Journal, 46(1), 73–114. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169 Grant et al. Identity as Participation Journal of Urban Mathematics Education Vol. 8, No. 2 118 National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics. National Research Council. (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Mathematics learning study committee, center for education, division of behavioral and social sciences and education. Washington, DC: National Academy Press. Nzuki, F. M. (2010). Exploring the nexus of African American students’ identity and mathematics achievement. Journal of Urban Mathematics Education, 3(2), 77–115. Retrieved from http://ed- osprey.gsu.edu/ojs/index.php/JUME/article/download/45/68 Patterson, G. A. (2014). Boys from the ‘hood—often misunderstood. Phi Delta Kappan, 96(2), 31– 36. Schwandt, T. A. (1994). Constructivist, interpretivist approaches to human inquiry. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 118–137). Thousand Oaks, CA: Sage. Steele, C. M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist, 52(6), 613–629. Steele, C. M. (2006). Stereotype threat and African-American student achievement. In D. Grusky & S. Szwlwnyi (Eds.), The inequity reader (pp. 252–257). Boulder, CO: Westview Press. Steele, C. M., Spencer, S. J., & Aronson, J. (2002). Contending with group image: The psychology of stereotype and social identity threat. Advances in Experimental Social Psychology, 34, 379– 440. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press. Stinson, D. W. (2006). African American male adolescents, schooling (and mathematics): Deficiency, rejection, and achievement. Review of Educational Research, 76(4), 477–506. Stinson, D. W. (2008). Negotiating sociocultural discourses: The counter-storytelling of academically (and mathematically) successful African American male students. American Educational Research Journal, 45(4), 975–1010. Stinson, D. W. (2013). Negotiating the “White male math myth”: African American male students and success in school mathematics. Journal for Research in Mathematics Education 44(1), 69–99. Stinson, D. W., Jett, C. C., & Williams, B. A. (2013). Counterstories from mathematically successful African American male students: Implications for mathematics teachers and teacher educators. In J. Leonard & D. B. Martin (Eds.), The brilliance of Black children in mathematics: Beyond the numbers and toward new discourse (pp. 221–245). Charlotte, NC: Information Age. Tyner-Mullings, A. R. (2012). Central Park East Secondary School: Teaching and learning through Freire. Schools: Studies in Education, 9(2), 227–245. Varelas, M., Martin, D. B., & Kane, J. M. (2013). Content learning and identity construction: A framework to strengthen African American students’ mathematics and science learning in urban elementary schools. Human Development, 55(5/6), 319–339. Walker, E. N. (2006). Urban high school students’ academic communities and their effects on mathematics success. American Educational Research Journal, 43(1), 47–73. Watson, D. (2012). Norming suburban: How teachers talk about race without using race words. Urban Education, 47(5), 983–1004. West, L., & Staub, F. C. (2003). Content-focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann. Wolcott, H. F. (2001). Writing up qualitative research. Thousand Oaks, CA: Sage. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/download/45/68 http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/download/45/68