 Race, Ideology, and Academic Ability: A Relational Analysis of Racial Narratives in Mathematicsby Niral Shah - 2017 Background/Context: There is evidence that race affects students’ learning experiences in mathematics, a subject typically thought of as “race-neutral” and “culture-free.” Research in psychology and sociology has shown that racial narratives (e.g., “Asians are good at math”) are pervasive in U.S. culture and play a critical role in shaping people’s lived experiences. However, racial narratives have received little explicit attention in the mathematics education literature. Purpose: The purpose of this study was to examine the racial ideological context of mathematics education, specifically in terms of how students made sense of racial narratives about academic ability. Participants: Thirty-five students identifying as Asian, Black or African American, Latinx, Polynesian, White, and mixed race were interviewed. These students were recruited from four mathematics classrooms observed by the author at a racially diverse high school in Northern California. Research Design: This qualitative study employed an ethnographic research design to gather data on the meanings students constructed around issues of race in the context of mathematics. Data Collection and Analysis: A semistructured interview protocol was used to conduct individual interviews with each student participant. Field notes were taken during 130 hours of participant observation over the course of a school year. Interview transcripts and field notes were analyzed for instances in which participants invoked racial narratives. Each of these narratives was first coded by topic and by the racial group to which the narrative referred. Narrative clusters were then identified and analyzed in order to understand how the narratives were related to each other. Findings: Students invoked a variety of racial narratives about both mathematical and nonmathematical topics (e.g., athletic ability, general intelligence, parenting practices). Importantly, students did not invoke these narratives in isolation. Instead, nearly all of these narratives were invoked in conjunction with at least one other narrative. This relationality among racial narratives shows how the academic abilities of learners from diverse racial backgrounds are constructed in relation to each other, often in ways that position non-Asian students of color as inferior in mathematics. Conclusions/Recommendations: This article suggests the need for study designs and analytical approaches that theorize race as a relational construct that transcends the Black-White paradigm. Further, this article challenges researchers and practitioners to reconsider boundaries between what is deemed “mathematical” and “nonmathematical” in classroom discourse, specifically with respect to sociopolitical phenomena like race. Two African American boys, Will and Derrick, wait patiently as their math teacher circulates the room. She is passing back the results of the latest math test. Will receives his grade: He earned an A. Derrick sees Wills grade and exclaims: Oh youre hecka smart in math! You must have some Asian in you! Research on race in mathematics education has increasingly focused on students racialized experiences as mathematics learners (Berry, 2008; Martin, 2006, 2009; Stinson, 2008). Those experiences are shaped by widely circulating racial narratives related to mathematical ability (McGee & Martin, 2011; Nasir & Shah, 2011; Steele, 1997). Racial narratives can be thought of as stories that circulate in society about the supposed traits and behaviors of particular racial groups, and also about whether and how race matters in a given context (Bonilla-Silva, 2003; Pride, 2002). Although all human beings have the capacity to do and learn mathematics (Devlin, 2000), certain racial narratives propagate the false, hierarchical view that certain groups possess an inherently greater mathematical ability than other groups (Nasir & Shah, 2011; Stinson, 2008). Understanding the full scope of these narrativesand the relations among themis critically important to the project of designing racially equitable classrooms in mathematics. To date, though, our understanding of these narratives is limited in two ways. First, the racial narratives that have been identified as relevant to mathematics learning comprise but a subset of the broader racial ideology. Much attention has been given to racial narratives about Black mathematics learners, and how those narratives position Black students as mathematically inferior relative to White students (McGee & Martin, 2011; Steele, 1997; Stinson, 2008). However, far less is known about narratives that refer to the mathematical ability of other racial groups. For example, the Asians are good at math narrative is pervasive in the U.S. context (Aronson et al., 1999; Cvencek, Nasir, O'Connor, Wischnia, & Meltzoff, 2014; Shah, 2009). And yet, few studies have considered how students make sense of this particular narrative or how it gets taken up in classrooms. Expanding beyond the Black-White paradigm is essential to developing a comprehensive theory of how race operates in mathematics learning contexts for students of all racial backgrounds. A second limitation is that extant research has not considered linkages among racial narratives pertaining to mathematical ability. Consider the example of a racial narrative that positions Polynesians as mathematically inferior. By itself, this narrative is certainly consequential for Polynesian learners experiences and positionalities in mathematics. But this narrative does not exist in isolationit is linked to racial-mathematical narratives about Whites, Asians, Latinx or Latinxs,1 and other racial groups. Here I use the term relationality to refer to linkages among racial narratives. Acknowledging relationality can reframe the kinds of questions asked in studies of race in mathematics education. Typically, researchers have framed questions as follows: How does being Polynesian affect a students experience as a learner in mathematics? Through a relational frame, however, the question becomes: What does it mean to be a Polynesian mathematics learner in the context of how other racial groups are positioned in mathematics? The notion of relationality can also be generative in conceptualizing linkages among racial-mathematical narratives and racial narratives that do not explicitly refer to mathematics. Racial narratives about topics such as general intellectual capacity, athletic ability, and parenting practices also circulate in societyhow might they relate to racial narratives that explicitly reference mathematical ability? The role of nonmathematical racial narratives in positioning students with racialized mathematical identities has received little attention in research on race in mathematics education. To illustrate why these gaps in the current literature are consequential, consider the racial exchange presented at the beginning of this article. The student, Will, reported this exchange while being interviewed as part of a prior study (see Nasir & Shah, 2011). Why does a racial-mathematical narrative about Asians come up in a conversation between two African American students? What is the significance of Derricks allusion to Will being smart in math? How do the multiple racial narratives invoked in this exchange interact to position Will and Derrick as mathematics learners and as racialized individuals? Conceptual frameworks currently available in the literature offer only partial answers to these questions. In this article, I use ethnographic techniques to investigate students perspectives on and subjective experiences with racial narratives related to mathematical ability. Building on theoretical perspectives that take race to be a relational construct (Friedman, 1995; Kim, 1999; Leonardo, 2013; Lucal, 1996), I document the various types of racial narratives that students invoke while making sense of issues of race in mathematics education. Further, I map the explicit and implicit linkages students draw among racial narratives. Analysis reveals how perceptions of mathematical ability are constructed through a complex web of multiple, interconnected racial narratives. To parse relations among these narratives, I introduce two constructs: cross-group relationality, or linkages among narratives about the same topic but different racial groups (e.g., how Asians are good at math mutually constitutes Polynesians are bad at math); and within-group relationality, or linkages among narratives about different topics but the same racial group (e.g., how Latinx or Latinxs are bad at math mutually constitutes Latinx or Latinxs are less intelligent, Latinx or Latinxs are manual laborers, and other societal narratives about Latinx or Latinxs). I argue that these types of relationality contribute to a more comprehensive understanding of what I call racial-mathematical ideology, and the processes by which students come to be racialized as learners in mathematics. CONCEPTUAL FRAMEWORK I begin by conceptualizing race and racial ideology, focusing on several of their defining theoretical attributes. This is followed by a treatment of racial narratives, which I characterize as elements of racial ideology. Grounded in concepts from the literature, I then offer a working definition of racial narrative for the purpose of this article. Finally, I consider my conceptual framework in light of extant literature on racial narratives in mathematics education. RACE AND RACIAL IDEOLOGY AS SOCIOPOLITICAL AND RELATIONAL Race is a fluid, sociopolitical concept invented during the 16th century by White Europeans to categorize human beings. Race has no biological basis, but racial categories are often viewed as a neutral way of characterizing human difference (Gould, 1996; Long, 2004). This view belies the fact that since its invention, individuals and institutions have deployed race to facilitate particular ideological and economic interests (Mills, 1997; Omi & Winant, 1994). Ethnicity, nationality, and culture are related constructs that have also been used to serve political ends. However, what distinguishes race is its concern for matters related to the physical body. That is, race-based claims assume correlations among somatic traits (e.g., skin color, hair type) and peoples perceived characteristics, including intellectual capacity, physical abilities, and moral tendencies (Goldberg, 1993). From a racial standpoint, phenotype assumes an ontological meaning as the body is treated as a window onto how a group of people is believed to think and act. Race and race-related phenomena (e.g., racism) have ideological and material dimensions (Goldberg, 1993; Leonardo, 2013). Ideology is about more than personal beliefs; it refers to a value-laden framework of interpretation that allows people to make sense of themselves and the world around them (Hall, 1996). These interpretive frameworks are often fragmented and fluid because they are highly context dependent (Hall, 1996; Philip, 2011). Local circumstances, historical context, and individual positionality all affect how a person interprets complex phenomena like race. And yet, this does not mean that each person constructs racial meanings in entirely idiosyncratic ways. Common themes across individual ideologies point to the existence of broader racial ideologies that are collectively shared and reproduced throughout society. As individuals express their racial ideologies through discourse (Leonardo, 2003), they shed light on the content and structure of these society-wide racial ideologies. I elaborate on this point later when I argue for the utility of analyzing racial discourse as a way of researching racial ideology. By focusing on racial ideology in this article, I do not mean to dismiss the materiality of racialized experience (Essed, 1991; Omi & Winant, 1994). People are denied loans, prevented from living in certain neighborhoods, and targeted by police because of their race. Race and racism are not purely mental phenomena. Still, racial ideologies undergird and are reproduced by the social practices of race (Goldberg, 1993). It is more useful to think of ideology and materiality as inextricably linked (Althusser, 1971). A key point about racial ideologies is that they are relational. That is, race has always functioned as a framework for organizing groups of human beings in relation to each other (Leonardo, 2013; Lucal, 1996). Although race relations are sometimes conceptualized in terms of single binaries (e.g., the Black-White binary), racial positionality is triangulated within a broader network of multiple race relations (Friedman, 1995; Kim, 1999). For example, what it means to be Black cannot be decoupled from what it means to be White, Asian, Polynesian, and so on, as defined in a particular historical moment and geographical context. The racial meanings that circulate in a social discourse about various groups are mutually contingent and constitutive. Historically, racial positionalities have not been equal. Instead, relations among racial groups have been hierarchical, as some racial groupstypically Whites and sometimes Asians in the U.S. contextare considered superior to other groups (Bonilla-Silva, 2003; Goldberg, 1993; Said, 1979). The hierarchical ordering of racial groups has both ideological ramifications (Du Bois, 1903/1965; Fanon, 1967) and material effects on the distribution of resources in society (Oliver & Shapiro, 2006; Roithmayr, 2014). Thus, analyses of racialized experiences and positionalities must account for relationality among racial groups. THEORETICAL PERSPECTIVES ON RACIAL NARRATIVES Racial ideology is realized through racial discourse (Leonardo, 2003). Racial narratives are a form of discourse central to racial talk. Racial narratives have been conceptualized in a variety of ways using a range of terminology across multiple literatures. While a full review of the concept is beyond the scope of this article, here I draw on relevant literature to discuss some of its core ideas, as a way of building toward a working definition of racial narrative for the purpose of this article. Narratives and the act of storytelling are central to how people explain social phenomena and bring coherence and meaning to their lives (Delgado, 1989; McAdams, 2013). In particular, racial narratives play a critical role in how people experience and make sense of race while engaging in everyday social interaction (Bonilla-Silva, 2003; Friedman, 1995; Pride, 2002). In his seminal study of color-blind racial ideology, Bonilla-Silva (2003) analyzed how and for what purposes White participants invoked racial story lines while discussing various racial topics, such as affirmative action and reparations. According to Bonilla-Silva, racial story lines are socially shared tales that are fable-like and incorporate a common scheme and wording (p. 76). He notes that racial story lines are often devoid of specific narrative content, instead manifesting as generic racial claims involving one-dimensional stock characters. For example, a prominent racial story line that emerged during the Reagan era was that of the welfare queen. Although initially this phrase referred to a particular person and particular events (i.e., a story in the traditional sense of the word), it eventually came to signify a broader discourse or metanarrative about African Americans as unmotivated and dependent. This example also captures Bonilla-Silvas point about racial story lines as fable-like, in that the welfare queen constitutes a kind of mythological figure positioned as representative of the proclivities and pathologies of an entire racial group. Building on Bonilla-Silvas construct, I note that racial narratives can come in multiple forms with varying degrees of specificity. On the one hand, racial narratives can manifest as brief statements that explicitly refer to particular groups and their supposed traits or behaviors (e.g., Latinx or Latinxs are less intelligent or Polynesians are good at sports). In fields such as sociology, cognitive science, and social psychology, such statements are typically known as stereotypes. Much important work has been done to investigate how racial stereotypes shape individuals attitudes (Bobo, 2001; Katz & Braly, 1933), and affect individuals performance under certain conditions (Aronson et al., 1999; Steele, 1997, 2010). And despite their connotation as in the head phenomena, stereotypes are very much socially constructed artifacts (Haslam, Turner, Oakes, Reynolds, & Doosje, 2002; Murphy & Walton, 2013). Indeed, they can be understood as stories about racial groups that circulate in society; racial stereotypes are a type of racial narrative. The concept of racial narrative as I use it here, though, goes beyond stereotypical statements. In addition to referring to racial groups traits, racial narratives can also express general perspectives on the nature of race, how it functions, and its significance (or lack of significance) in society and in a given domain like mathematics. These racial narratives constitute metastatements about race. Examples of racial narratives of this type include: racism is a thing of the past and I never owned slaves (see Bonilla-Silva, 2003). The absence or suppression of racial talk (Pollock, 2004), and the suggestion that race is irrelevant in a given space (Lewis, 2001), are themselves racial narratives. Compared with stereotypes, this kind of racial narrative makes more sweeping claims about race by operating at a larger grain size. Members of all racial groups invoke and deploy racial narratives, albeit for different purposes. Whereas dominant narratives preserve and perpetuate the status quo of power relations (Bonilla-Silva, 2003; Delgado, 1989), counternarratives aim to destabilize what the dominant group takes to be normal. In the context of race studies in education, counterstorytelling constitutes the methodological core of the domain of Critical Race Theory (Bell, 1992; Solórzano & Yosso, 2002; Yosso, 2006). According to Yosso (2006), counterstories offer a challenge to majoritarian narratives, which tend to frame people of color as inferior. Counterstories straddle the line between personal and societal narratives, in that a counterstory can represent a persons individual narrative about race while also making reference to societal narratives about race. Further, in providing a retort to majoritarian or dominant racial narratives, counterstories can come to constitute a new societal narrative that challenges dominant views about race. Overall, societal narratives about topics like race matter because of how they can become consequential in positioning. Positioning refers to the processes through which individuals become discursively constituted as being certain types of people, with particular traits, capacities, dispositions, and perceived tendencies. (Davies & Harré, 1990). With respect to how individuals understand and position themselves, racial narratives can serve as building blocks in peoples personal narratives (Pride, 2002) or testimonies (Bonilla-Silva, 2003) around the role of race in their lives. Racial narratives can be deployed to position others as well (Nasir & Shah, 2011). In the story at the beginning of this article, Derrick invokes the Asians are good at math narrative in a way that positions Will as someone who succeeds in mathematics because of an innate ability, rather than because of other factors like Wills strong work ethic. Further, Derrick positions himself and African Americans more generally as being less capable in mathematics because they are not Asian. As I aim to show in this article, the racial narratives we invoke about ourselves, as well as the narratives we invoke about others, can affect whether people have access to social positions of dominance or marginality (cf. Sfard & Prusak, 2005). CONCEPTUALIZING RACIAL NARRATIVES AS ELEMENTS OF RACIAL IDEOLOGY For the purpose of this article, I define racial narratives as stories that circulate in society about the supposed traits and behaviors of particular racial groups, and also about whether and how race matters in a given context. Racial narratives can be thought of as elements of racial ideology. If racial ideology is an interpretive framework for making sense of the world in racial terms, then racial narratives are constituent elements of that framework. When articulated in discourse, racial narratives become tools for bringing order and meaning to social phenomena. Figure 1. A general schematic of racial narratives as linked elements of racial ideology
The schematic in Figure 1 is a general representation of the content and structure of a hypothetical racial ideology. It is intended to visually depict the key ideas I have discussed about racial narratives, racial ideology, and the relationship between them. The specific boxes and arrows in the figure are hypothetical and meant for illustration purposes only. As I have discussed, a variety of narratives about different racial groups exist in society. The individual boxes in Figure 1 (e.g., Blacks are&) represent the content of these narratives (i.e., the racial groups and topics to which they refer). The topics of the narratives are left unspecified in this general schematic because they are domain-dependent. That is, racial narratives in mathematics may differ from racial narratives in other domains, although they may also overlap. Further, given that race is a relational construct, we should expect these racial narratives to be linked. The linkages among narrativesrepresented by the hypothetical arrows connecting the boxesconstitute the structure of a racial ideology. Finally, the existence of particular narratives in a given ideology is not coincidental. Each narrative within a network of narratives serves a particular function. This means that an analysis of racial ideology should ask the following questions: Why do certain narratives comprise the racial ideology under consideration? What purpose do they serve? A contribution of this article is that later I present a refined version of the schematic in Figure 1 specific to the case of racial-mathematical ideology, which will serve to illuminate the particular ways in which race operates in the context of mathematics. Having articulated the conceptual framework that informed the study, next I review the extant literature on racial narratives in mathematics education. EXTANT RESEARCH ON RACIAL NARRATIVES IN MATHEMATICS EDUCATION In mathematics education, much research remains rooted in theoretical frameworks that consider race to be solely a demographic variable (Gutiérrez, 2008; Martin, 2009; Parks & Schmeichel, 2012). However, a growing body of work treats race as an experiential phenomenon connected to issues of identity. For example, Martin (2006) analyzed the racialized experiences of successful African American mathematics learners through the lens of mathematics identity, which he defined in terms of peoples beliefs about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives (p. 206). An important contribution of this work was to highlight a relationship between mathematics identities and racial identities. Many of Martins participants reported facing racial obstacles in their educational trajectoriesfor example, being denied access to advanced mathematics courses because of their race. For these individuals, being a mathematics learner was inseparable from being African American. Subsequent studies involving African American learners have corroborated these findings (see Berry, 2008; Moody, 2004; Spencer, 2009; Stinson, 2008). And yet, although it is implicit in much of this work, few studies in mathematics education have directly engaged the topic of racial narratives. An exception is research by McGee and Martin (2011), which focused on identifying specific tactics that African American males use to repurpose and resist pejorative narratives. However, the narratives themselves were not objects of inquiry in that research. In his work, Stinson (2008) has focused more directly on students sense making about the narratives themselves. In particular, Stinson identified several racial narratives that were salient to his participants, including narratives about African Americans being apathetic and prone to criminality. He argued that those narratives exist in opposition to a White male math myth, which claims mathematical ability as the exclusive property of White males (cf. Ernest, 1991; Ladson-Billings, 1997). An important contribution of Stinsons work is that it demonstrates how racial narratives become consequential for students positioning as learners in mathematics. He argues that students positionality is constituted through the interplay of multiple narratives and broader discourses (cf. Hall, 1996). Again, this underscores the importance of researching racial narratives, as they represent identity statements about groups of people: who they are thought to be, how they are thought to act, and what they are thought to value. Reflecting on the mathematics education literature as a whole, it is evident that most studies have focused exclusively on Black learners. In and of itself, this is not problematic. Given that prior research had considered Black students primarily through deficit perspectives that compare them to White students (Gutiérrez, 2008; Martin, 2009), it is reasonable and necessary to study Black students experiences in their own right. Such research has illuminated the brilliance of Black children independent of the need for comparison groups (Martin, 2013; Perry, Steele, & Hilliard, 2003). And yet, beyond Black learners, little is known about how the positionalities and experiences of other racial groups contribute to the larger racial context of mathematics education. For example, there is evidence that students across the K-12 mathematics pipeline are aware of the Asians are good at math narrative (Cvencek et al., 2014; Shah, 2009), and that this narrative can affect mathematical performance in older students (Aronson et al., 1999; Shih, Pittinsky, & Ambady, 1999). However, there is little research on how students make sense of this narrative or how it is invoked in mathematics classrooms. Further, the Asians are good at math narrative exists alongside narratives about other topics that position Asians in the U.S. as a model minority (Lee, 1996; Wu, 2003). How do narratives about Asian intelligence or body type or the mythology around Asian parenting (see Chua, 2011) relate to racial narratives about Asians mathematical ability? Linkages of this kind among narratives about the same racial groupand the implications for all racial groups in mathematics educationhave not been explored. Beyond Asian and Black students, it is also the case that little is known about the racialized mathematics learning experiences of learners of other racial backgrounds. For instance, while Polynesians are sometimes categorized as Asian, the ways in which they are racialized in schools often do not conform to the model minority discourse (Vaught, 2011). How are Polynesian students racialized as learners in mathematics? One of this articles contributions is to highlight the voices and experiences of students from multiple racial backgrounds in an effort to theorize racial ideology in mathematics education beyond the BlackWhite paradigm. To be clear, I am not suggesting that researchers open separate lines of inquiry for every racial group (e.g., Asians in math, Latinx or Latinxs in math, Indigenous people in math). Rather, what is needed is a theoretical perspective robust enough to simultaneously conceptualize the positionalities of all racial groups in terms of a dynamic set of relations among those groups. Martin (2009) has proposed the notion of a racial hierarchy of mathematical ability in which students who are identified as Asian and White are placed at the top, and students identified as African American, Native American, and Latino are assigned to the bottom (p. 315). This is a useful first step in considering relationality among different racial groups of mathematics learners. What is needed is a systematic empirical inquiry into the nature of this hierarchy, and how it is constructed and supported by linkages among racial narratives in and out of mathematics education. RESEARCH QUESTIONS To summarize, the current literature documents the existence of racial-mathematical narratives, but little is known about the connections students perceive among them. Further, more insight is needed into the racial narratives about topics beyond mathematics that students find relevant to mathematics learning. Altogether, these narratives comprise a key part of the racial ideological context of mathematics education, and the goal of this research was to understand how linkages among narratives shed light on students racialized experiences and positionalities. To that end, the following research questions guided the study: 1. What racial narratives do students at a racially diverse high school invoke while making sense of race in the specific context of mathematics learning? 2. 3. How are those narratives organized in relation to each other, and what do those relations reveal about how students are positioned as mathematics learners with respect to race? METHOD The data presented here were collected during a yearlong ethnographic study of processes of racialization in mathematics classrooms (see Shah, 2013). Mathematics was chosen as a context for the research because of the apparently conflicting ways in which it is perceived with respect to issues of race. On the one hand, mathematics is often viewed as neutral and culture-free. Unlike other academic subjects like English and social studies, mathematics is perceived to be pure and independent of the messy realities of everyday life. On the other hand, there is evidence that mathematics classrooms are not immune to the problems of race and racism that persist in U.S. society more generally (Ladson-Billings, 1997; Martin, 2006; Oakes, 2005; Stinson, 2008). It was this tensionthe coexisting perceptions of mathematics learning as simultaneously neutral and a racialized form of experience (Martin, 2006)that made mathematics an interesting and important context for this work. This article focuses on analyzing the content and structure of racial ideology in mathematics. In the remainder of this section I elaborate how a qualitative approach was organized to conduct this analysis. RESEARCH SETTING Research was conducted during the 20102011 school year at Eastwood High (pseudonym), a comprehensive high school located in a working-class area of Northern California. In 1960, the city of Eastwood was 98.5% White. However, over the next three decades Eastwood experienced a dramatic demographic shift. Mexican farm workers had been coming since the 1940s for seasonal work in agricultural industries based in the city, and eventually these workers and their families established permanent residency. At the same time, White families began to move to outlying suburbs as new groups of immigrants from various parts of Asia and Latin America arrived. In 2010 Eastwood had a Latinx or Latinxs plurality and was one of the most diverse cities in California with respect to race, ethnicity, and language. Table 1 shows the racial and ethnic demographics of the school at the time of the study. Eastwood High was one of three high schools in the Eastwood Unified School District. Compared with these other schools, Eastwood was known within the community as the diverse school. Following other researchers who have studied race in schools (see Park, 2011; Van Ausdale & Feagin, 2001), a racially diverse research setting was sought under the assumption that racial talk and racial sense making might be more explicitly prevalent in a place where students are constantly confronted by various forms of difference.
Data collection took place in four focal classrooms: Geometry, Precalculus, Algebra 2, and AP Calculus. Similar to many large high schools, Eastwood High had two separate tracks for 9th grade students in mathematics, which affected students access to AP Calculus by the 12th grade. Although prior research shows that such tracking mechanisms tend to correlate with racial disparities in course-taking patterns (Darling-Hammond, 2010; Oakes, 2005), this was the not the case in the focal classrooms, where the racial demographics were nearly identical to those of Eastwood High overall. Despite proportional representation on this level, though, the data will show that race was still part of everyday life at Eastwood (cf. Gutiérrez, 2008). DATA COLLECTION AND PARTICIPANTS Data were collected using an ethnographic approach, which privileges the social context and subjectivity of human experience in an effort to document the meanings that individuals construct about everyday life (Eisenhart, 1988). A number of educational studies have used ethnographic methods to investigate students racial ideologies and racialized experiences (see Lewis, 2001; McGee & Martin, 2011; Nasir, McLaughlin, & Jones, 2009; Park, 2011; Pollock, 2004; Schaffer & Skinner, 2009; Trainor, 2005; Van Ausdale & Feagin, 2001). The findings presented here rely on analysis of two data sources: field notes based on classroom observations, and transcripts of interviews with students from the focal classrooms. I conducted participant observation two to three times per week in the four mathematics classrooms. Over a 7-month period, 93 class sessions were observed for approximately 130 hours of total observation time. I also spent many additional hours interacting with teachers and students outside of formal class time during lunch, before school, and in between classes during passing periods. All observations were documented in field notes. In addition to my role as a researcher, I served as a classroom tutor. In a small way, this service afforded me an opportunity to reciprocate the time and access that teachers and students at Eastwood were providing me by participating in the study. In another sense, tutoring allowed me to build deeper relationships with students, which may have facilitated students willingness to share their experiences and perspectives during the interview component of the study. All students in the focal classrooms were invited to participate in individual interviews. Of those who gave consent, 35 students (19 boys and 16 girls, as classified by official school records) were chosen (see Table 1). Rather than trying to match the exact racial demographics of the school, the panel of participants was purposefully assembled using maximum variation sampling in order to ensure representation from all of the racial groups at Eastwood High (Glesne, 1999, cited in Park, 2011). For this reason, the relative overrepresentation of certain groups did not pose an obstacle to investigating the research questions. In fact, this sampling approach is consistent with the theoretical concerns detailed in the conceptual frame, as a primary goal here was to analyze how students from different racial backgrounds were positioned in relation to each other. Further, lower numbers of participants in certain racial groups was not problematic because it was not the goal of the study to make claims about the racial ideologies of particular racial groups. Racial demographics of the interview participants shown in Table 1 were self-reported by the students interviewed. Interviews followed a semistructured protocol, which was implemented in a fluid, conversational style. The semistructured approach affords the flexibility to follow up on unexpected responses while still allowing for comparison of responses across interview participants (Merriam, 2009; Trainor, 2005). Each interview was audiotaped and lasted approximately 30 minutes, with several interviews lasting 90 minutes. Students were told that the general purpose of the interviews was to learn about their experiences learning mathematics in school, as well as their perceptions of mathematics itself and what it takes to succeed in mathematics. The first half of the interview protocol focused on students beliefs about the nature of mathematics and mathematics learning, as well as their perceptions of themselves as mathematics learners. This part of the interview included a task where students were asked to react to a cartoon drawing. The cartoon showed a student telling a math teacher: I had my doctor do a DNA blood analysis. As I suspected, Im missing the math gene. Given that perceptions of mathematical ability as innate are pervasive (Devlin, 2000; Ernest, 1991), I conjectured that this task might indirectly shed light on students racialized perceptions of mathematical ability. The second half of the interview protocol more directly probed students knowledge, perceptions, and sense making about issues of race in relation to mathematics learning. To prompt this part of the interview, students were asked: Have you heard people say that some groups are better than others at math? Extensive piloting suggested that this particular questionwhile latently race-neutralwould be likely to trigger racial thinking without having to explicitly cue race. Further, pilot data revealed a tendency for students to respond to this question by citing the Asians are good at math narrative. For this reason, initial questions in this part of the interview protocol included prompts about the mathematical, academic, and racial positionality of Asians in the United States and globally. However, conversation about these topics often segued into discussions about the experiences of other racial groups in mathematics. For example, non-Asian participants were asked to comment on how their own racial group is perceived in mathematics, and whether their experiences in mathematics would have been different had they been Asian (as opposed to Black, Latinx, White, or Polynesianthe primary racial groups represented at Eastwood High). These questions were specifically designed to investigate students perceptions of how racial groups in mathematics are positioned in relation to each other. It is important to note that the primary objective of this study was to gauge students awareness and sense making about race, not their endorsement. Rather than attempting to determine students personal beliefs, I was interested in mapping the racial ideological context of mathematics education. This is the difference between, for example, a student reporting having heard other people say that Black students are less capable of learning math, and that same student personally believing that to be true. As a result, interview questions were intentionally framed in a de-centered way (i.e., what people say, rather than what students themselves believe). This design decision may have contributed to students feeling more comfortable in the interviews to discuss a sensitive topic and vocalize potentially problematic racial statements. ANALYTICAL APPROACH Building on theoretical assumptions put forth by Volosinov (1976) and Fairclough (2003), Bonilla-Silva (2003) notes that &ideology, racial or not, is produced and reproduced in communicative interaction (p. 11). In other words, ideology is realized through discourse (Leonardo, 2003). This suggests that in the analysis of racial ideology, racial language used by participants represents a critical object of inquiry. Within the scope of racial language, racial narratives constitute a productive unit of analysis because they are central to how people speak about race. Thus, racial narratives represent fundamental building blocks of racial ideology. Although racial language does not always involve the invocation of racial narratives, such narratives do tend to emerge in racial talk. Analysis proceeded in two phases. The first phase focused on cataloging all of the racial narratives invoked by participants in real-time interactions and in interviews. In other words, which narratives comprise the content of racial-mathematical ideology? To which racial groups do the narratives refer? What topics (e.g., mathematics, intelligence, parenting) are the narratives about? The second phase focused on analyzing relations among the narratives. In other words, what are the characteristics of the structure of racial-mathematical ideology? What do relations among racial narratives reveal about the complex identities with which mathematics learners come to be positioned? To catalog the racial narratives invoked by participants, I developed an operational definition of racial narrative based on the conceptual framework discussed earlier. In its most basic form, any instance in the data of an utterance linking a racial group (e.g., Asian, White) to a particular trait or behavior was coded as a racial narrative. To illustrate the coding procedure, consider the following statement made by one of the participants interviewed for the study: They think that just because Polynesians are so big that we dont know how to do math. In this case, a particular racial group (Polynesians) is mentioned in reference to two traits: body size and mathematical ability. Thus, two racial narratives would be coded as being invoked in this excerpt of transcript: Polynesians are physically large and Polynesians are weak at math. More general statements about race (i.e., those that did not refer to particular racial groups) that refer to the significance of race in society or in relation to mathematical ability were also identified in the data. Examples of such statements include: I dont think race really matters, and I think every race is smart. How and for what reason participants invoked narratives of this type revealed an additional dimension of the nature of racial-mathematical ideology. Using this operational definition, both the observational data and interview data were analyzed for instances where participants invoked racial narratives. All field notes were searched for evidence of race-related language. This yielded a subset of field notes of racialized classroom interactions, which were entered into a database using Filemaker Pro software. This subset of observational data was further analyzed to identify those interactions where a racial narrative was invoked, as well as to identify the particular narratives that were invoked. With respect to the interview data, all 35 interviews were transcribed in full, and the same coding procedure was used to identify racial narratives in the interview transcripts. All of the racial narratives identified in both data sets were entered into a Microsoft Excel spreadsheet. In total, 98 unique racial narratives that cited particular racial groups and their perceived traits or behaviors were identified. Each of these narratives was then tagged in two ways: by the racial group to which it referred and by topic (e.g., mathematical ability, body type). A preliminary analytical pass through the topics was used to develop initial conceptual categories (Harry, Sturges, & Klingner, 2005), which were then refined through an iterative process of comparing, contrasting, and collapsing topics. Through this process seven topical categories of racial narratives emerged: 1) math/STEM (science, technology, engineering, mathematics) ability; 2) intelligence; 3) general academic performance; 4) personality traits; 5) body type or athletic ability; 6) family life or cultural practices; and 7) career paths. These categories facilitated investigation of the first research question regarding the types of racial narratives students perceive to be relevant to mathematics education, as well as the frequency with which participants invoked these narratives. In the case of racial narratives in the interview data, multiple invocations of a given narrative by the same student were counted only once, so as not to skew the relative frequency of the narratives. Overall, Phase 1 of the analysis shed light on the content of racial-mathematical ideology. Phase 2 of the analysis focused on how the racial narratives identified in Phase 1 were related to each other (i.e., the structure of racial-mathematical ideology). Investigating relations among the narratives required a shift in the unit of analysis from the individual narratives to what I call narrative clusters, or segments of discourse where two or more narratives were invoked in an uninterrupted sequence of statements. The following excerpt from one of the student interviews is an example of a narrative cluster: I heard that Asians, specifically, are probably on the top, but I think Asians are good at everything, so I don't think it's just math or science or nothing like that. Talking about that: me and Lauren are the only African Americans in the Precalculus class, so it seems like the numbers show that, but I don't think that's necessarily the truth. Multiple racial narratives are being invoked in this excerpt. The statement that Asians are good at everything would be coded as an instance of a narrative about Asians general academic performance. The statement that I dont think its just math or science would be coded as an instance of a narrative about Asians STEM ability. Finally, the statement about there being few African Americans in the Precalculus class confirming the numbers would be coded as an instance of a narrative about African Americans mathematical ability. Using this coding procedure, 96 narrative clusters were identified in the data and entered into a Microsoft Excel spreadsheet. Each cluster was then analyzed to determine how the narratives within a given cluster were related to each other. This analysis revealed two types of relationalitycross-group and within-group relationalitythat characterize the structure of racial-mathematical ideology. These concepts, as well as their implications for students racial positionalities as mathematics learners, are elaborated in the findings. RESEARCHER POSITIONALITY AND ETHICAL ISSUES Research is neither a neutral nor objective endeavor; a researchers positionality affects all aspects of the research process (Fine, 1994; Foote & Bartell, 2011; Villenas, 1996). In considering this issue, I avoid a confessional approach that views positionality as solely a threat to validity (Maxwell, 2013; Pillow, 2003). Like all human beings, researchers simultaneously negotiate multiple identities that constitute their positionality. However, given the topic of the present study, here I reflect primarily on issues of racial positionality in relation to data collection and analysis. Based on my light-brown skin, black hair, and South Asian first and last name, I am frequently identified as Asian, South Asian, Indian, or even sometimes Arab. What are the implications of this racial positionality for a study of racial ideology in mathematics, where Asians are prominently positioned as superior in mathematics? In and of itself, my physical presence in the focal classrooms and during interviews constituted a potential signifier of the Asians are good at math narrative. Many students were also aware of my background as a former high school mathematics teacher and as a university researcher studying mathematics education. It is possible that this positionality cued students to invoke narratives about Asians in the interviews. Further, because most of those narratives are superficially positivein that they align with the model minority discourse about Asians in the U.S.it is possible that students felt more comfortable in talking about race with an Asian-identified person. That is, talking about race in a positive way may have posed less risk to participants of appearing racist, therefore making them more open to discussing race overall. A Black, Latinx, White, or Polynesian researcher may have elicited fewer invocations of racial narratives and in different relative proportion by racial group and topic. My positionality as a person of color from a model group may have affected the kind of access I was afforded to participants experiences and sense making about race in mathematics education. I also note an ethical issue associated with researching a topic like race. In discussing racial narratives throughout this article, I recognize the danger in inadvertently reifying these false perceptions about the inherent capacities and proclivities of different racial groups, as well as the socially constructed concept of race itself. It can also be painful for the reader to readand for me as a researcher to write aboutovertly racist statements documented in the study. And yet, if the alternative is to avoid research of this type altogether, then the potential risks are worthwhile on net. Undermining false racial hierarchies requires an understanding of the racial ideologies that support them. FINDINGS The findings from this study are presented in three sections. First, I provide an overview of the types of racial narratives that students invoked. Second, I describe the linkages that students drew among the narratives. Analysis focuses on two types of linkages: cross-group relationality and within-group relationality. In the third and final section, I explain how cross-group and within-group relationality function simultaneously to position students as mathematics learners with varying degrees of potential to succeed in mathematics. When reading the data, it is important to keep in mind the distinction between invoking a narrative and endorsing a narrative. The students in this study reported hearing numerous racial narratives at school, at home, and in the media. Some students explicitly endorsed these narratives, but most students invoked narratives while rejecting their validity. Further, participants tended to describe their perceptions of what other people believe, rather than their own beliefs. It is worth noting that even participants stated beliefs are themselves partially appropriated from pre-existing racial discourses that are based on pre-existing racial ideologies. In other words, participants are not the sole authors of the discourses they speak. Thus, the data should be understood less as reflecting the personal ideologies of individuals, and instead a window onto the broader racial ideologies that circulate in society. RACIAL NARRATIVES ABOUT MULTIPLE RACIAL GROUPS AND MULTIPLE TOPICS Overall, 98 different racial narratives invoked by students were documented across the data corpus. Table 2 shows the frequency with which students invoked the different categories of narratives. Narratives about multiple racial groups and multiple topics were invoked a total of 267 times, nearly all of which occurred during the 35 student interviews. This is remarkable given that students were not told in advance that the interviews would engage issues of race. All students invoked at least one racial narrative, with a median of 7 narratives per student.
Two key trends in Table 2 are worth noting. The first trend is that over half of all invocations (140 of 267) involved a racial narrative about Asians.2 This is likely due to the fact that nearly all students initially invoked the Asians are good at math narrative, which prompted follow-up questions related to Asians that, in turn, led to more opportunities for students to invoke Asian-related narratives. The next largest number of invocations involved racial narratives about Blacks. Narratives about the other racial groups at Eastwood High (Latin@s, Polynesians, and Whites) were also invoked but with less frequency. The second trend is that 79% of the narratives invoked (212 of 267) were not explicitly about mathematics or STEM. Despite the interviews focusing on students personal histories learning mathematics, and the observations being conducted in mathematics classrooms, the vast majority of racial narratives pertained to nonmathematical topics. This finding prompts the following questions: How are narratives about parenting style or intelligence related to narratives about Asians mathematical ability? What do perceptions of Black athleticism have to do with perceptions of Black students potential in mathematics? I take up these types of questions later in the article. Overall, these findings suggest that students in the study perceived race to be relevant to issues of mathematics education. Of course, this does not mean that race was salient to all students in the same way or to the same degree. While some students reported that race and racism were central to their learning experiences, for other students race was far less personal. The common thread across the data was that all students were aware of connections among narratives about race and narratives related to mathematical ability. In the next section, I analyze specific statements made by students to illuminate the linkages they drew among the racial narratives they invoked. RELATIONALITY AMONG RACIAL NARRATIVES Isolated invocations of racial narratives were rare in the data. In fact, 90% of all racial narratives in the data (241 of 267) were invoked in conjunction with other narratives. This suggests a high degree of relationality, and it is these narrative clusters that provide insight into how narratives about multiple racial groups and multiple topics are linked. Here I use two concepts to help analyze those relationships: cross-group relationality and within-group relationality. Cross-group relationality refers to linkages among racial narratives about different racial groups (e.g., the relationship between Asians are good at math and Blacks are bad at math), while within-group relationality refers to linkages among narratives about a single racial group (e.g., the relationships among Polynesians are good at sports, Polynesians are not smart, and Polynesians are bad at math). Of the 96 narrative clusters that were identified in the data: 19% were instances of cross-group relationality; 69% were instances of within-group relationality; and 12% involved both types of relationality. As I discuss each in turn, I show how these two types of relationality help illuminate the broader structure of racial ideology in mathematics. Cross-group Relationality For each topical category in Table 2, nearly all of the racial narratives invoked about Asians were superficially favorable, while nearly all of the racial narratives invoked about non-Asians of color were pejorative. Narratives invoked about Whites were mixed. In this section, I argue that linkages among narratives implied hierarchical relations across racial groups. To support this claim, I begin by focusing on cross-group relationality among narratives about mathematical ability, which is followed by an analysis of cross-group relationality for the other topical categories of narratives in the data. All 35 students interviewed reported awareness of the Asians are good at math narrative. Conceptually speaking, a narrative that deems Asians to be good at math implies that there is at least one non-Asian group that is bad or less good at math. In other words, the linguistic structure of the narrative suggests a hierarchical, cross-group relationality. There is empirical evidence to support this idea, as 71% of the students interviewed (25 of 35) invoked a superficially favorable racial-mathematical narrative about one racial group in relation to a pejorative racial-mathematical narrative about another group. That is, most students made sense of racial-mathematical narratives in the context of a hierarchical dynamic. In the following representative transcript, Rachel (White, 12th grade) articulates this perspective as she explains how different racial groups are compared with respect to mathematical ability: Well I think everybody had the list in their head: "Oh it's Asians, and then it's White people, and then&Blacks and Mexicans are tied" (laughs). But nowadays it's just an individual thing. It's like, "Okay, the Asian girl is at the top of the classno surprise!" But I think now because students are doing so varyingly in their own race in all subjectsbut especially mathit just kind of depends. So if you like math and you do well and you happen to be Mexican, well then okay, you're Mexican and you like math. I think it just depends on the person. With Asians at the top and two non-Asian groups of color at the bottom, Rachels list echoes Martins (2009) notion of a racial hierarchy of mathematical ability. Rachel describes how race informs peoples expectations about mathematical ability. In her words, it is no surprise when the highest performer in a class is an Asian student. But Rachel rejects this hierarchical ideology as a relic of the past that is no longer relevant. Instead, she argues that, &nowadays it's just an individual thing. Rachels framing of mathematical performance as an individual thing is noteworthy but ambiguous. On the one hand, it echoes an ideology of individual liberalism (Moore & Pierce, 2007), which diminishes the ongoing significance of structural racism through a focus on individual actions and motives. On the other hand, Rachels subsequent comment (if you like math and you do well and you happen to be Mexican, well then okay, you're Mexican and you like math) suggests that she views any connection between race and interest or performance in mathematics as coincidental. Indeed, her final statement (I think it just depends on the person) can be interpreted as Rachel emphasizing variation in individual ability to disrupt a racial ideology that claims a deterministic relation between mathematical ability and race. Many other students in the study made similar arguments to challenge the dominant hierarchical ideology. For some students the racial hierarchy of mathematical ability was more than a theoretical construct. In the following representative transcripts, two students from groups positioned at the bottom of the hierarchy describe how that positionality can have a material impact on everyday experiences in mathematics classrooms: Like I'm Black but I'm good at math. So are you not going to ask me for help because I'm not Asian? So I don't really agree with the stereotype. - Monet (Black, 12th grade) Yeah, so say a substitute teacher would come in [to class] and she'll see the Indian kid and think, "Oh he must be the best one here in math," and she'll look at me and think, "How did he get into this class? What the heck is he doing here?" - Troi (Samoan, 12th grade) In the first excerpt, Monet rejects the Asians are good at math narrative by citing herself as an example of a non-Asian student that excels in mathematics. In doing so, she implies the existence of a narrative that positions Black learners as mathematically less capable: Im Black but Im good at math (emphasis added). Monet suggests that not being Asian could cause classmates to ignore her when they are looking for a competent student to help them. This is significant because the act of asking a person for help positions that person as competent, whereas students not asked for help are not afforded these opportunities. For Monet, then, racial-mathematical narratives about Asians and Blacks are mutually constitutive, and it is through this relation that she is positioned with respect to mathematical ability. In the second excerpt, Troi describes another way in which cross-group relationality can become consequential for students. At the time of the study, Troi was a higher performing student in Precalculus, an advanced course in which few students at Eastwood High were enrolled. He describes a scenario where racial-mathematical narratives can influence teachers first impressions of students. Whereas a teacher would automatically assume that the Indian kid is the best one here in math, Troi asserts that his very enrollment in the class would be called into question, presumably because of an existing narrative that Samoans are not good at math. Again, racial-mathematical narratives about multiple groups are invoked in conjunction with each other. Troi understands his positionality in mathematics as a function of how both Samoans and Asians are perceived in mathematics. Cross-group relationality also informed how students made sense of racial narratives about topics less explicitly related to mathematics. In addition to mathematics, students also articulated racial hierarchies about general academic performance independent of subject area. In the following transcript, James (African American, 12th grade) recalls how four years prior his mother told him to look up racial patterns in graduation rates at Eastwood High before deciding to enroll there for high school: If you go on the district website, it shows a little circle graph where it shows the percentage of people who graduate. My mom made me look at that before I wanted to come to this school, and it was like 90% of Asians graduated, and for African Americans it was like 50-something. So it was kind of&it was kind of disturbing, but you try and use that as fuel so you can get into that good 50%. You don't want to fall behind and help out the stereotype or anything like that, you just try and fight against that. That's how my mom taught me. James interprets the low graduation rate of African American students at Eastwood as signifying a narrative about African Americans not doing well in school. He describes how his mother taught him to fight against that narrative, and to use the statistic as fuel so he can be a part of the good 50%. Importantly, James does not consider African American graduation rates in isolation. He compares African American graduation rates to Asian graduation rates, which imply a contrasting narrative about Asians doing well in school. The cross-group relationality among racial-academic narratives implied in Jamess statement was representative of how other students talked about racial-academic narratives. And similar to the racial-mathematical narratives that emerged in the data, the racial-academic narratives students invoked tended to position Asians at the top, Whites in between, and non-Asians of color at the bottom of the academic hierarchy. It is also noteworthy that students invoked racial narratives about intelligence in ways that implied a similar hierarchy. Whereas Asians were often talked about as smart and as geniuses, students noted that non-Asians were often seen as dumb and stupid. I discuss racial-intellectual narratives in greater depth in the next section. The final category of racial narrative where there was evidence of cross-group relationality was racial narratives about family life and cultural practices. In particular, students compared the parenting styles of various racial groups. To illustrate, consider the following transcript from an interview with a 12th grade Tongan student named Isaac: If I were Asian my parents would have been stricter. They would have taken things away from me, you know: having fun. Like being Tongan, my dad gave me freedom, but not too much freedom. Like when I would go to my cousin's house, my dad would let me sleepover. But how I'm thinking is that an Asian family wouldn't let him. They'd make him come right back home to do your homework. When my dad is too tired, he wouldn't tell me to do my homeworkhe'll expect me to do it. From Isaacs perspective, Asian parents are stricter, which he compares with his Tongan dad who gave me freedom, but not too much freedom. Isaac is not saying that Tongan parents are indifferent to their childrens education. Indeed, he notes that even though his dad might not explicitly tell Isaac to do his homework, he still expects Isaac to do it. For Isaac, both Asian and Tongan parents are strict to a point, but he perceives Asian parents to be relatively stricter. Racial narratives about relative differences in parenting styles related to education were the second most frequently invoked type of narrative. For categories of narratives that were less frequently invoked (e.g., body type/athletic ability, personality traits), there was less evidence of students invoking narratives about different groups in the same turn of speech. However, comparing isolated invocations of these narratives suggests linkages among them. For example, apart from narratives about Asians, the most frequently invoked category of racial narrative was about Blacks being perceived as good at sports. Polynesians and Latin@s were also viewed as athletically proficient. In contrast, narratives invoked about Asians positioned them as physically small and athletically deficient. Similarly, narratives invoked about personality traits positioned Blacks, Latin@s, and Polynesians as angry, violent, and rebellious, compared with Asians who were positioned in an opposite way as quiet, disciplined, and patient. But what do these types of narratives have to do with mathematics learning? In the next section, I elaborate on the relationship among nonmathematical racial narratives and racial narratives explicitly about mathematical ability. Within-group Relationality The majority of narrative clusters (78 of 96) in the data involved within-group relationality. I argue that the narratives in these clusterseven those that appear unrelated to mathematicsconverge to position students as mathematics learners. In other words, what it means to be a Latin@ mathematics learner or an Asian mathematics learner is constituted through relations among racial narratives about a range of topics, not only those that explicitly refer to mathematical ability. To support this claim, in this section I conduct an in-depth analysis of three representative examples from the data. These exemplars were chosen because in each of them, students invoked four or more racial narratives in the same turn of speech, which make these data useful for illustrating the dynamics of within-group relationality. Further, the exemplars involve three different racial groups, which makes it possible to compare and contrast how within-group relationality operates for different groups. The first data exemplar involves Carlos, a 12th grade student who identified as Mexican. During elementary school, Carlos reported that he had little interest in school and gave little effort in math class. As he entered middle school, though, Carlos increasingly became the target of racist remarks from classmates that belittled his mathematical ability and intellectual potential. In the following transcript, he describes how the convergence of multiple racial narratives about Mexicans caused him to reorient his approach to learning mathematics: I've been biased against because I'm Mexican. I didn't want people to say about me, "Oh look, there's just another Mexican that doesn't know how to do math." So that was one of the main reasons I wanted to do the best, because I wanted to show people that Mexicans are smart, and that they don't just work and do gardens all the time. In this excerpt, Carlos invokes multiple racial narratives about his own group in close succession. Initially, a narrative positioning Mexicans as mathematically inferior becomes a source of motivation for Carlos. However, disproving this racial-mathematical narrative is about more than proving his mathematical potential. Carlos views success in mathematics as a way to show people that Mexicans are smart: the racial-mathematical narrative and the racial-intellectual narrative are linked. This reflects a more general tendency for people to perceive mathematical ability as indicative of a persons intellectual capacity (Ernest, 1991). To be sure, one of the most common types of narrative clusters in the data involved racial narratives about mathematical ability and intelligence being invoked in conjunction with each other. Another connection Carlos makes is between the racial-intellectual narrative and a racial narrative about the type of jobs Mexicans are perceived to do. I argue that it is not coincidental that he invokes a narrative that Mexicans do gardens all the time in the context of talking about mathematics and intelligence. Gardening is not just any type of laborit is manual labor. The subtext of this narrative is that Mexicans are thought to be physically capable but not mentally capable. By extension, this explains why Carlos invokes a racial-mathematical narrative alongside these other two narratives. Mathematics provides Carlos a way of challenging narratives that position Mexicans as unintelligentsuch as those about gardeningbecause doing well in mathematics signifies intelligence in the U.S. context. Overall, analyzing within-group relationality here reveals unexpected linkages among seemingly unrelated narratives. In this case, racial narratives about career and about physicality were related to mathematical ability, and narratives about intelligence served as the connective tissue linking them together. The second data exemplar involves a similar dynamic. Polynesian students at Eastwood High were a small but prominent population on campus. Samoan and Tongan cultural practices were well represented in school events, and the Polynesian male students in particular were known for their participation in contact sports, such as football and rugby. Several of the faculty I spoke with viewed them as troublemakers and found them difficult to manage in their classes. In the excerpt below, Troi (Samoan, 12th grade) elaborates on how perceptions of Polynesian bodies and personalities contributed to their being positioned as mathematically, academically, and intellectually inferior: Other students just see me as big and mean&and here [at Eastwood High], the Polynesian kids are seen as like we're big, that we do whatever we want. Like we're not very intellectual, and like we're not smart. But once they meet me they'll know that I'm actually very intelligent, and I can do math, I know how to do English, I can do science&all that kind of stuff. I think when I come in they just see me as someone who's going to hurt them or beat them up or someone who freaking wants to kill. They're not going to take time out to sit and talk with me, and actually greet me and actually get to know me. In this excerpt, Troi draws connections among multiple categories of racial narratives. Initially, he connects a narrative about Polynesians being big to narratives about Polynesians being seen as mean and someone whos going to hurt them or beat them up or someone who freaking wants to kill. The relations among these narratives evoke an image of Polynesian students as angry and violent people that others should fear. Indeed, Troi implies that classmates tend to avoid him, and do not attempt to greet me and actually get to know me. But Troi perceives these narratives to be consequential in ways that go beyond his social standing. They also matter for how Troi is positioned from an intellectual standpoint. After invoking narratives about Polynesians body type and personality, he notes that people view Polynesians as being not very intellectual. Other Polynesian students in the study also indicated that their group was perceived as unintelligent, which they linked to their participation in football. In their view, being stereotyped as athletically superior meant that they were viewed as intellectually inferior. This reflects the notion of the dumb jock, a common trope in the U.S. where athletic ability and intelligence are incorrectly viewed as inversely related. Further, narratives about body size and personality are relevant here because Polynesians are associated with football, a sport defined by physicality and brute force. Overall, Trois articulation of how Polynesiansand Polynesian males in particularare positioned echoes a longstanding racist discourse about men of color being physically imposing, aggressive, and violent (see Goff, Eberhardt, Williams, & Jackson, 2008). The within-group relationality among the narratives invoked by Troi about Polynesians parallels the within-group relationality among the narratives invoked by Carlos about Mexicans. Both students linked the body to the mind (i.e., intellectual capacity). For Carlos, being stereotyped as performing manual labor jobs contributes to Mexicans being perceived as unintelligent. For Troi and his Polynesian classmates, being stereotyped as excelling in a highly physical sport contributes to Polynesians being perceived as unintelligent. This matters because, as Troi puts it, narratives about intelligence relate to whether people assume he has the capacity to succeed in mathematics and other academic subjects. Once again, the racial-intellectual narrative serves as a bridge between narratives about the body and narratives about mathematical ability and general academic ability. The final data exemplar focuses on Asians, the racial group most consistently perceived as dominant in mathematics by students in the study. Whereas in the previous two pieces of data students reflected on narratives about their own groups, here a Salvadorian student named Isabel (12th grade) invokes multiple racial narratives to articulate how she perceives Asians mathematical ability in relation with other racial narratives about Asians: Not to be racist, but I know that a lot of Asians are really good at technology, and that has a lot to do with math. You hear it all the time over there in China, over there in Philippines&they're ten years more advanced in technology than we are, and it's probably true because they're really smart. It's the way that they focus and the way that their parents have taught their child, and how their child has taught their children&to be focused and to be well-trained and to not get distracted, versus other people who get distracted easily. I don't know, just the way that they learn; they're always focused, always determined. Isabel begins by stating that a lot of Asians are really good at technology, a phenomenon she attributes in part to Asians being good at mathematics. Elaborating on her assertion, Isabel draws a racialized international comparison between two Asian countries and the U.S. (theyre ten years more advanced in technology than we are). And like Carlos and Troi, racial-intellectual narratives also figure in Isabels sense making (its probably true because theyre really smart). According to Isabel, mathematical ability and general intellectual ability are both contributing factors to Asians being really good at technology. Similar to Carlos and Troi, Isabel invokes racial-intellectual narratives alongside other racial narratives about Asians. These other narratives concern Asians personality traits, such as the way that they focus and how they are always determined. Isabel explains how these traits are a byproduct of Asian parenting, which in her view has inculcated Asian children with certain habits and a particular mentality. One way of interpreting Isabels statement is that these habits and mentality make Asians really smart, which in turn makes them really good at technology. Looking across all three students, there is a key difference in how students perceived Asians as opposed to other students: narratives about the body were more likely to be invoked in discourse about non-Asians than Asians. Of the 30 unique times when racial narratives about body type or athletic ability were invoked, only 5 (17%) pertained to Asians (see Table 2). On the few occasions when narratives about Asian bodies were invoked, they did not denigrate Asian intelligence in the way that Carlos and Troi articulated for their groups. For example, on one occasion a student bragged to a teacher that he played ping pong like an Asian girl (Field Notes, 2/1/11). Unlike football and manual labor, ping pong is not an activity associated with extreme physical exertion. Indeed, most of the narratives invoked about Asian bodies positioned them as weak and not athletic. And in fact, one might argue that these narratives actually bolster perceptions of Asians as intelligent and mathematically gifted because they align with stereotypical images of mathematicians as diminutive and athletically challenged. Overall, within-group relationality illuminates how a students positionality in mathematics is constituted through relations among multiple, seemingly unrelated narratives about that students racial group. In the next section, I synthesize the findings of this study by discussing how cross-group relationality and within-group relationality operate simultaneously to position students with respect to race and mathematical ability. CROSS-GROUP & WITHIN-GROUP RELATIONALITY AS SIMULTANEOUS FORCES OF POSITIONING Earlier I presented Figure 1 as a way of representing the idea that racial narratives can be thought of as constituent elements of racial ideology. Based on the findings of this study, that representation can now be updated to reflect the specific content and structure of racial ideology in mathematics (see Figure 2). The findings indicate that racial ideology in mathematics consists of narratives that span a wide breadth of topics and that refer to all racial groups. However, I note that the diagram in Figure 2 shows only three groups: Asians, non-Asians of color, and Whites. This multiracial, hierarchical model reflects both the quality of the narratives and how they clustered in the data, as Asians were clearly positioned as dominant in mathematics, non-Asians of color were clearly positioned as inferior in mathematics, and narratives about Whites were mixed. As was discussed earlier, these groups positionalities are mutually constitutive (i.e., cross-group relationality). And within each of these groups, the racial narratives about a particular group are also linked (i.e., within-group relationality). Figure 2. A schematic of racial-mathematical ideology
Overall, Figure 2 highlights an important point: the positioning effected by both cross-group and within-group relations among racial narratives occurs simultaneously. To illustrate, recall the data presented earlier about Carlos. Carlos was aware of numerous pejorative narratives about Mexicans that position him as mathematically, intellectually, and culturally inferior. These narratives comprise the racial context of Carloss everyday efforts to learn mathematics. At the same time, though, Carlos must also reckon with the fact that he is not a member of a model racial group that possesses all of the capacities and dispositions that Carloss group is thought to lack. The comparison between Mexicans and more favorably positioned groups reinforces Carloss marginal positionality in mathematics. The situation is identical for dominant groups in mathematics but in the opposite direction. What it means to be an Asian mathematics learner, for example, is certainly constituted through linkages among the many narratives about Asian intelligence, parenting, and other topics. At the same time, though, being Asian in mathematics is bolstered by not being Black in mathematics, and by not being Polynesian in mathematics, and even by not being White in mathematics. In other words, cross-group relationality and within-group relationality can be understood as simultaneously converging forces of positioning, and it is within this complex ideological context that mathematics learning is situated. DISCUSSION Despite prevailing views about mathematics as a race-neutral and culture-free domain, a growing body of research has shown mathematics education to be a highly racialized domain (Battey, 2013; Chazan, Brantlinger, Clark, & Edwards, 2013; Faulkner, Stiff, Marshall, Nietfeld, & Crossland, 2014; Martin, 2006, 2009; McGee, 2014; Stinson, 2008). The present study builds on this literature by illuminating the central role of racial narratives in processes of racialization in mathematics education. Here I discuss some of the theoretical and methodological issues and questions that follow from this work. A major theoretical contribution of this research is that it suggests the need for a more expansive view on which racial groups merit attention in studies of race in mathematics education. Whereas prior work has tended to focus on the BlackWhite paradigm, a key finding in this study is that narratives about Asians were most salient to students. Indeed, the importance of Asians as the nucleus of racial-mathematical ideology has been underestimated in the extant literature. Beyond Asians, though, this research also brings attention to the mathematics learning experiences of Polynesian students, a marginalized population in the U.S. that has received little attention in the educational research writ large (Vaught, 2011 is an exception). Overall, this study reveals some of the limitations associated with relying on the BlackWhite paradigm. Conceptual frameworks are needed that can account for the ways in which racial ideologies implicate learners of all racial backgrounds. This study also problematizes the positionality of White students in mathematics. Typically, Whites are grouped alongside Asians at the top of the racial hierarchy of mathematical ability, with non-Asians of color positioned at the bottom (Martin, 2009). However, the data here complicate this binary racial model. No other racial group in this study had a more ambiguous positionality in mathematics than White students. Although there were those who viewed Whites as being good at math, students were just as likely to perceive Whites as being bad at math. This may depend on the racial groups against which Whites are compared. Relative to Black students and other non-Asian students of color, the White male math myth may apply (Stinson, 2008). However, relative to Asian students, there is evidence that even Whites consider themselves to be mathematically inferior (Aronson et al., 1999). In lieu of a binary hierarchy, here I have proposed a multiracial model in which Asians, Whites, and non-Asians of color are viewed as occupying distinct positions on the racial-mathematical order. Although Whites remain dominant in most sectors of U.S. life, I argue that mathematics is currently a domain where Asians are positioned as the dominant group. What are the implications of this dynamic? To what extent does this positioning benefit Asians, and in what ways might it create problems for them and for non-Asian students of color? Further, what interests do Whites occupying an ambiguous middle position serve? Research into these questions would lead to a more nuanced understanding of all students racialized experiences in mathematics. Relative to narratives about Asians and non-Asians of color, it is also the case that narratives about Whites were less frequently invoked. One possible interpretation of this finding is that Whites were a minority population at Eastwood High, and thus were less salient to students in the study. However, this seems implausible: there were even fewer Asian students at the school, but narratives about Asians were pervasive in the data. Another interpretation is that the relative absence of Whites in students racial discourse reflects the ways in which Whiteness operates more broadly as an ideology, attempting to erase the role of Whites in the broader historical narrative of race (Leonardo, 2009). That is, Whiteness frames race and racism as issues concerning people of color but not Whites. The positionality of Whites in mathematics and its implications for the broader politics of race in mathematics education are issues worthy of further inquiry. Another contribution of this work is that it demonstrates the utility of conceptualizing race in relational terms. I offered the notion of cross-group relationality as a way of conceptualizing how the positionalities of students from different racial backgrounds are fundamentally linked and mutually constitutive in a hierarchical fashion. This relationality means that a students racialized experiences cannot be fully understood without considering them in light of the racialized experiences of students from other racial backgrounds. In short, hierarchy is the beating heart of racial ideology. From a methodological standpoint, research designs involving racially diverse groups of participants may be better positioned to study peoples racialized experiences in particular local settings. To be clear, I am not suggesting that single-race studies are without merit. As was discussed earlier, over a decade of research focused on the experiences of Black learners in mathematics has provided valuable insight into the form and function of race and racism in mathematics education. The experiences of students of color, in particular, can and should be studied in their own right. And yet, learners racialized experiences do not exist in a vacuum. A study of what it means to be Black while learning mathematics in a given context would be enhanced by also investigating how Whiteness, Asianness, Latin@ness and Polynesianness, for example, are configured in that same context. A concrete way of accounting for this issue is to build interview protocols that include questions about learners of other racial backgrounds. For example, White learners might be explicitly asked to reflect on the experiences of Asian and Latin@ students in mathematics, and vice versa. Through this act of projection (i.e., asking students to consider the positionality of other racial groups), researchers would gain deeper insight into how students make sense of the experience of being members of their own racial groups. The idea is to study racialized experience within the context of broader race relations. It is also noteworthy that so many of the racial narratives that students invoked were about topics other than mathematics. Narratives about the relative intellectual capacities of different racial groups figured prominently in students sense making. And given that mathematical ability is often taken to be a signifier of general intelligence (Ernest, 1991; Martin, 2009), the prevalence of racial-intellectual narratives in conversations about mathematics was not surprising. However, students also invoked narratives about topics seemingly unrelated to mathematics, such as parenting practices and athletic ability. What do these categories of narratives have to do with students mathematical abilities? In considering that question, I offered the notion of within-group relationality as a way of conceptualizing networks of linkages among mathematical and nonmathematical racial narratives. A key finding was that these narratives are closely interrelated, such that students are racially constituted as being certain types of mathematics learners through relations among racial narratives about their groups ability in mathematics and racial narratives about their groups capacities and proclivities outside mathematics. This explains how narratives that otherwise seem like strange bedfellows can converge to position students. For example, I showed how students perceive racial narratives about Mexicans being gardeners and Polynesians being physically large as consequential to perceptions of mathematical ability. Further, it is not coincidental that how the racial narratives clustered aligns with broader societal discourses about particular racial groups. That is, narratives about Asians were consistent with the longstanding discourse in the U.S. about Asians as the model minority (Lee, 1996; Wu, 2003), and narratives about non-Asians of color were consistent with the longstanding discourse about the culture of poverty. This shows that mathematics classrooms are not impervious to the racial discourses that circulate in society. Students are constituted as racial subjects in mathematics, and they continue to be racial subjects when the bell rings and they leave the school grounds. This shows that mathematics is a discursive space where societal discourses of race are invoked and perpetuated. Racial hierarchies of mathematical ability constituteand are constituted byracial hierarchies in other domains of everyday life. A broader implication of this point is that it calls into question the line between what is deemed mathematical and nonmathematical. There has been much debate in the field about what should count as mathematics education research (see Battista, 2010; Confrey, 2010; Heid, 2010; Martin, Gholson, & Leonard, 2010). This study raises questions about how to interpret nonmathematical talk in mathematics classrooms. If some students are deemed the smart ones and other students are labeled as the jocks, to what extent might this positioning affect perceptions of these students potential to succeed in mathematics? A broader lens on what counts as mathematical can open researchers and practitioners to phenomena that initially seem unrelated, but that may be highly consequential for students opportunities to learn and to be identified as capable in mathematics. This study represents an initial effort to explore these issues; more research is needed to understand relationships among mathematics and racial discourses about the body, the mind, personality traits, and cultural practices like parenting. In summary, this study shows that the dominant racial ideology in mathematics education is organized in a way that generates and reproduces a hierarchy of mathematical ability. Students of all racial backgrounds in the study were aware of a diverse set of racial narratives that comprise this racial ideology. And in fact, given that students were not asked directly to comment on the particular narratives that emerged in the data, the findings may actually underestimate students awareness of racial narratives that were less frequently invoked, such as those related to non-Asian groups. Further, students of all racial backgrounds in the study leveraged these societal narratives as building blocks in the personal narratives they constructed about their own lives and learning experiences in mathematics, as well as in the local stories they constructed about the mathematical capacities of their classmates. It is an open question whether the same narratives invoked by the students here would hold similar salience for students at schools with a different racial makeup or in a different geographical location. Building on the present research, which was carried out in a racially diverse setting, research on racial ideology in racially homogenous settings is needed as well. Intersectional analyses also offer a promising direction for future work, as some of the data presented here reveal how racial narratives can intersect with and be inflected by narratives related to other social markers, such as gender and class. Ultimately, for many students issues of race in education come down to a matter of recognition. Do all students have the opportunity to be seen for what they are truly capable of doing in a classroom? In this article, many students of color expressed pain and frustration with having their capabilities ignored or misread through a racial lens. And this goes beyond personal feelings, as students that are not seen as competent may be given fewer opportunities to participate in class, or might be passed over for placement in advanced courses. In other words, the ways in which racial narratives position students can have material consequences for their educational trajectories. CONCLUSION Just as race continues to matter in society, race continues to matter in mathematics education. This article has sought to illuminate the complex web of racial narratives that position students in mathematics as learners with varying levels of ability. I have argued that this web of racial narratives comprises a key part of the racial context of mathematics education. Of course, these narratives are not deterministic. Students can and do deploy counternarratives in resisting the problematic ways in which dominant racial narratives position them. And yet, the persistence of racial performance, participation, and opportunity-to-learn gaps means that too many students still do not have access to the personal, social, and institutional resources needed to succeed in subjects like mathematics. Interventions that attend to both the ideological and material conditions that students face stand a better chance of producing more equitable outcomes, particularly for students from historically marginalized racial groups. Acknowledgment This research was supported in part by a dissertation grant from the National Academy of Education and Spencer Foundation, and the Institute of Education Sciences under grant R305B090026. The opinions expressed are those of the author and do not represent views of the Spencer Foundation, the National Academy of Education, or the Institute of Education Sciences. I would like to thank Alan Schoenfeld (and the Functions Group), Nailah Nasir (and CPLD), Zeus Leonardo, and Michael Omi for their support in this research. I would also like to thank Katie Lewis, Amy Parks, and the two anonymous reviewers for their thoughtful comments on earlier version of this manuscript. Finally, I deeply appreciate the teachers and students of Eastwood High School for their time and openness. Notes 1. With the term Latinx, I follow Gutiérrez (2013) and others in using gender-fair language to represent Latina students, Latino students, Latin@ students, and students who identify with more fluid interpretations of gender or reject the concept of gender altogether. 2. The use of the term Asian here reflects the fact that most students in the study did not differentiate among Asian subgroups and their particular histories and traditions. 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