Centering Children in Mathematics Education Classroom Research

Conventions in Describing Instructional Episodes in Mathematics Classrooms

The majority of classroom interactions described in mathematics education research occur in whole-class settings, and the ways children’s contributions are typically reported in these settings make it difficult to make sense of the varied ways that children participate in whole-class activities. For example, little contextual information is typically provided about how many children in a classroom contribute to the discussions reported in articles or about whether the children whose contributions are reported are representative of the class in terms of mathematical knowledge, temperament, racial and ethnic identities, gender, or social class positioning.

For example, the following transcript is an excerpt from a study of how teachers use pauses in mathematics discussions:

167 TEA: [so what was the most popular?]

As in many studies, the children are not identified with pseudonyms, making it difficult to track the experience of individuals across the text, and some of the children’s contributions to the discussion are reported chorally, making it difficult to assess the extent of participation within the class. Children who did not participate in the choral response are not described, and children whose bodies might indicate a lack of attention to the conversation are also not attended to. In these ways, the way whole-group conversations are represented in transcripts masks variation in individual children’s participation in these conversations.

These sorts of gaps in contextual details are often present in summaries of classroom episodes as well. For example, in reporting a whole-class conversation, Hewitt (2014) wrote, “[T]his was worked on successfully by the class as a whole with one student making the movements on the interactive white board (IWB) and the rest of the class calling out directions” (p. 13). Although efficient in terms of writing, descriptions such as these convey the idea that all students in a classroom are participating equally and similarly. These descriptions do not account for students who may be staring out the window, engaged in a side conversation, or merely sitting silently while classmates’ answer. Yet these sorts of differences in participation may do much to explain differential success in mathematics.

Children’s conversations with each other and their embodied interactions are more often described in studies that explicitly focus on small-group interactions in mathematics (e.g., Bishop, 2012, Esmonde & Langer-Osuna, 2013; M. Wood, 2013). For example, M. Wood (2013), in describing the ways that a diverse group of children solved an area task, showed the ways that children’s mathematical knowledge, racial identity, gender, and temperament all contributed to differences in their experiences working on the task, while Esmonde and Langer-Osuna (2013) described the ways the youth in their study used embodied interactions to communicate. Still, studies that report children’s interactions in small groups have tended to describe participation in structured group tasks, rather than that the sorts of conversations that occur frequently among children as they work with each other in a variety of settings, such as stations, games, or drill and practice sessions.

Children’s experiences doing work independently tend to be described even more rarely than their interactions in small groups. This may be in part because spoken language is more difficult to capture during independent work and as a community, we have not made a practice of reporting embodied interactions. Roth and Maheux (2014) offered one model for describing independent work by using photographs to show how a young girl approached a geometric task, demonstrating how her vocalizations during this task provided insight into her mathematical thinking that would not have been possible to identify by looking at her finished work or her contributions to a whole-class conversation. Lambert (2015, p. 10) provided another rare glimpse into interactions during independent work in reporting the following conversation:

In the second semester, two boys sitting side by side, working feverishly to complete a worksheet on addition and subtraction with integers, had the following discussion,

Interactions by individual children during independent work such as described by Roth and Maheux and with peers such as described by Lampert are likely occurring frequently in nearly all mathematics classrooms and are likely to be playing a significant role in students’ learning of mathematics and in their conceptions of themselves as mathematical learners. However, they are rarely reported in research.

Given that the vast majority of descriptions of interactions in mathematics classrooms focus on children’s interactions with the teacher, we do not, as yet, have as rich an understanding of peer-to-peer interactions during both small group and independent work settings as we do of whole-class conversations. In addition, we know much less about the ways that children who participate infrequently in whole-class conversations (either in individual lessons or over time) experience mathematics than we do about the experiences of students who participate frequently.

Context and Children’s Identities in Mathematics Education

There is relatively little research in mathematics education that describes schooling in rural settings (Silver, 2003), and even less that describes the experiences of African American ¹ children in those settings. Yet children’s schooling experiences in mathematics and other content areas are always saturated in broader identity discourses, such as those around race, gender, and geographic context (Gholson & Martin, 2014; Gutiérrez, 2013). One goal of this study is to illustrate some of the ways these broader identity discourses overlapped to shape children’s mathematical experiences.

The rural character of the school was one contextual feature in this study that operated in subtle but important ways. Although there is little research that attends specifically to mathematics education in rural schools, broadly much of the research on rural contexts has emphasized challenges. For example, Semke and Sheridan (2012, p. 23) described rural schools as “geographically isolated” and went on to say that they “tend to be hard to staff with high teacher turnover, a high percentage of inexperienced or poorly prepared teachers, inadequate resources, and poor facilities.” In high schools, these issues sometimes show up in difficulties in hiring teachers with appropriate content area backgrounds, although these issues are generally less problematic in the elementary grades.

In the study reported here, the rural context of the school shaped two important features of the children’s mathematics experiences. First, the small size of the school (approximately 300 students from prekindergarten to Grade 12) meant that all students in a grade-level cohort—with the exceptions of students who moved away from or into the school—experienced 13 years of schooling together, with few or no changes in who was in the classroom from year to year. This meant that unlike in schools where children are redistributed across three to four classrooms each year, children’s relationships with each other had more opportunities to become entrenched over time.

Second, the small size of each grade-level cohort in the school led to large variations in the numbers of boys and girls in each cohort. Thus, one outcome of the rural context of this study was that my cohort of child participants had far more boys than girls—in the final year of the study, 5 girls and 12 boys. The cohort a year ahead of this one was similarly imbalanced, but in favor of girls, although across the 13 grade levels of the school, the distribution of boys and girls was approximately equal. These expected statistical variations worked to create particular gendered contexts for children’s experiences in small schools that are much less likely to occur in urban schools that serve larger populations. The girls in the cohort I studied will experience all 13 years of their schooling in classrooms where they are vastly outnumbered by boys, and it is necessary to understand their participation in the classroom as shaped by this experience.

In pointing to gender here, I draw on work in line with Walshaw’s (2001) discussion of gender and mathematics education, which emphasizes the ways that gender differences are produced through social interactions rather than linked to biological differences in human beings. While gender work in mathematics education has been less prominent in recent years, perhaps in part because of narrowing gender gaps related to achievement, course-taking, and pursuit of STEM (science, technology, engineering, and mathematics)–related majors in college (Corbett, Hill, & St. Rose, 2008; Lindberg, Hyde, Petersen, & Linn, 2010), researchers have continued to point to some gendered ways that students experience mathematics, including higher levels of mathematics anxiety among girls (Devine, Fawcett, Szűcs, & Dowker, 2012) and the circulation of gendered stereotypes about who can do mathematics, which U.S. children report as early as second grade (Cvencek, Meltzoff, & Greenwald, 2011).

In addition, some studies that have not named gender specifically as an area of inquiry have pointed to how gendered ways of interacting shape mathematical experiences. For example, Esmonde and Langer-Osuna (2013) showed how two high school girls used their friendship and social skills to create openings for interacting around mathematics in a problem-solving activity with a high school boy. Similarly, research outside of mathematics education has discussed the gendered aspects of schooling experiences. In particular, researchers have documented educators’ tendencies to read girls as docile and boys as challenging (Sadker & Sadker, 2010). In an interview study with British teachers, Jones and Myhill (2004, p. 553) found that teachers routinely positioned girls as “compliant” and boys as “disruptive” or “immature,” and expected girls to typically do well in school and boys to struggle. Walkerdine (1998) connected these gendered beliefs about behavior to teachers’ conceptions of who was good at mathematics, arguing that educators’ perceptions of girls possessing procedural rather than conceptual understandings were related to seeing them as rule followers. These gendered discourses around classroom behavior were very present in the context of this study.

In addition, discourses around race were constantly circulating in interactions at the site of the study, where over the course of 3 years, White teachers taught mostly Black students. In discussing the race of the participants, my thinking is framed by Omi and Winant’s (2004, p. 9) discussion of race as both socially constructed and socially real and their call to attend to the effects of “race-thinking (and race acting)” in human interactions. In pointing to the ways in which the focal children’s experiences were racialized, I want to explicitly highlight Martin’s (2009) suggestion that researchers discuss race outside of comparative frameworks—for example, only understanding African American children’s experiences in relation to White children’s experiences. As Berry, Ellis, and Hughes (2014, p. 551) point out, research that attends to race only for purposes of comparison suggests that “Black children are not worth studying in their own right.” (A similar argument might be made for attending to rural contexts outside of comparisons to urban ones.)

A growing body of scholarship has characterized a number of features of the experiences of Black children and youth in mathematics classrooms, although much of this scholarship has focused on children older than the participants in the current study (e.g., McGee & Martin, 2011; Nasir & Shah, 2011). For example, in interviews with parents, Martin (2009) documented African American parents’ perceptions of ways their children were overlooked in schools as mathematics learners, while Stinson (2013) reported that the African American boys in his study had to negotiate damaging stereotypes and media images in defining themselves as successful mathematics learners. Although they look beyond mathematics, Wright and Counsell (2018) describe ways in which even very young Black boys must navigate anti-Blackness in schools through unfounded assumptions about their dangerousness, lack of representations of children like themselves in literature, and lack of faith by educators in their capabilities. Similarly, researchers have found that both Black boys and girls are perceived through a lens of “adultification,” wherein Black boys are perceived as more dangerous and in more need of harsh punishment, and Black girls are perceived as needing less support and as being more independent (Epstein, Blake, & González, 2017; Morris, 2007; Wright & Counsell, 2018).

In addition, researchers have documented the ways in which urban and minoritized children are more likely to encounter back-to-basics pedagogies (e.g., emphasis on test preparation, repeated practice, and production of correct answers) and frequent assessments (Berry et al., 2014; Parks & Bridges-Rhoads, 2012). This focus on skill mastery for test performance was heavily present at the site of the study, where children took a state-mandated, high-stakes test at the end of first grade.

Of course, identity discourses around ruralness, gender, and race do not exist in isolation from each other. In a study of third-graders, Gholson and Martin (2014) found that discourses around gender (e.g., valuing of slender bodies and feminized clothing) intersected with discourses of race (e.g., valuing of lighter skin tones) to shape girls’ positions in the classroom as mathematically competent. In addition, research outside of mathematics education around Black boys in elementary schools has shown that Black boys are more likely to be targeted for more severe disciplinary reactions from White teachers (Ferguson, 2010; Love, 2013; Smolkowski, Girvan, McIntosh, Nese, & Horner, 2016) and that their behavior is more likely to be interpreted as challenging. In contrast, teachers and children are more likely to hold positive views of the behavior and expected achievement of African American girls (Hudley & Graham, 2001; D. Wood, Kaplan, & McLoyd, 2007). Although for some time, research has shown that teachers’ tend to emphasize social skills over academic strengths when describing Black girls, praising them for their work habits or the lack of attention they require, while also calling them average students (Grant, 1984). This emphasis by teachers on social rather than academic strengths may contribute to the effect Gholson (2016) describes in her analysis of scholarship about Black girls in mathematics, showing that Black girls and women develop less positive relationships with mathematics over time and do not enter STEM fields at the same rates as Black boys and men. Gholson (2016) urged researchers to attend to entanglements of gender and race in ways that do not make Black girls invisible by overemphasizing the experiences of Black boys and White girls.

Thickening Identities Through Schooling

To analyze two children’s participation in mathematics over 3 years, this article draws on theories that see identity as a construct that emerges and shifts in relation to interactions with others and with the material world (e.g., Bakhtin, 1981; Wortham, 2001, 2006). From this perspective, identity can be seen as interactive, emerging “through cultural practices—as people ‘do’ life” (Nasir & Saxe, 2003, p. 14). This interactive conception of identity—whether called “dialogic” (Bakhtin, 1981, p. 279), “positional,” (Holland, Lachicotte, Skinner, & Cain, 1998, p. 125) or “narrative” (Wortham, 2001, p. xi)—draws attention to the ways that interactions with others shape one’s identity. Broadly, from this theoretical perspective, identity is defined as a way of being in a particular social context that allows people to see themselves—and others to see them—in ways relevant to that context—such as being a “good” or “troublesome” student or a “capable” or “incapable” mathematics learner. In particular, a mathematical identity is a way of being in relation to mathematics that emerges “through the claims people make about themselves and others in mathematical spaces” (Langer-Osuna & Esmonde, 2017, p. 639).

Social interactions are understood to impact children’s identities in relation to the learning of academic disciplines, such as when children learn what it means to be a person who knows or does mathematics. In the primary grades, this might mean being someone who offers right or wrong answers, is fast or slow, or gives or offers help. But also, as Dyson (1997) points out, social interactions teach children about who they are in the broader world. Through classroom interactions, young children learn not only literacy and mathematics but also about “the words available in certain situations to a boy or a girl, to a person of a particular age, ethnicity, race, class, religion, and so on” (Dyson, 1997, p. 4). This identity learning is often not the result of explicit statements about who a child is, but is constructed through observations of the ways peers and teachers respond to one’s speech and silence, stillness and movement, and completed and uncompleted written work.

In addition, interactive perspectives on identity also highlight the ways in which the material world, including bodies and objects, does identity work. Haraway (1991) writes about bodies as “maps of power and identity” (p. 180), asking readers to consider how bodies are read and responded to in social situations. As discussed in the previous section, common identity markers encoded in bodies, such as race, gender, and age, can be made more or less salient through social interactions in the classroom and can shape the ways a person understands the social engagements of others. This means that it is impossible to consider a spoken exchange in a classroom without also considering the bodies of the speakers. In other words, a particular statement made by a White adult woman to a young, Black girl will do a particular kind of identity work. The focus on the material pushes researchers to consider data beyond spoken language. The ways children and adults sit, move their bodies, or make facial expressions all contribute to identity work in the classroom.

Finally, when looking at identity from this perspective, it’s important to consider ways that certain aspects of identity may be made salient in particular moments and then recede in others, while other aspects of identity may be reinforced again and again until they become easily recognized and more commonly taken up by the individual (Langer-Osuna & Esmonde, 2017; Wortham, 2006). “[T]his thickening of a particular identity—or sense of self—becomes a significant source from which a person draws as she interacts with others and the world around her” (Jones, 2012, p. 446). Thus, through repeated interactions, children come to understand themselves as particular kinds of people.

The focus on this article is on the ways that interactions with the social and material worlds over 3 years shaped the identities of two children primarily in relation to mathematics.

Methodology

This study was situated within a larger interpretive (Erickson, 1986) project that sought to examine the mathematical learning of young Black children in a rural school over 3 years. The larger project sought to document the ways that context affected young children’s mathematical experiences by collecting data in and out of school, during play, formal instruction, and assessment interviews, and in three different classrooms with three different teachers. Ethnographic methods (e.g., Dyson & Genishi, 2005; Geertz, 1973), which emphasize long-term immersion in particular places and cultures and understanding of participants’ perspectives through participation in local practices, framed every aspect of the study.

The part of the study reported here was a longitudinal multiple case study (Stake, 2005) that aimed to follow two children over 3 years of schooling, using similarities and differences in the two children’s participation in typical mathematics classrooms to understand how their mathematical identities were constructed over time. I crafted case studies for this part of the project because they allowed me to closely examine the role of social context in the focal children’s mathematical participation and to draw together multiple sources of evidence. The longitudinal nature of the study allowed me to understand the thickening of children’s mathematical identities. In addition, I drew on ethnographic tools for rich description (Emerson, Fretz, & Shaw, 1995) to make the affective, embodied aspects of children’s experiences more salient. These tools included a focus describing the body (e.g., where bodies were placed in classrooms, ways that children moved their bodies) and attention to affect (e.g., noting affective responses to children myself and drawing on my 3 years of engagement with these children to interpret their own displays of emotion).

In addition to supporting an analysis over time, focusing my analysis on two children allowed me to create a rich sensory portrait of their engagements in mathematics, looking at, for example, the content of the mathematics, the teacher’s pedagogy, the child’s placement in the room, the material tools used during the experience, the child’s bodily engagements and facial expressions, the language used, and the context of the child’s social standing in the classroom.

In examining identity in an ethnographic study, I privileged data that it was possible to observe—such raising or not raising hands, speaking or not, completing written assignments or not. Because I did not have access to children’s thoughts throughout the lessons I observed, it was not always possible to know which aspects of classroom life they found salient in relation to their own mathematical identities. As a result, some aspects of children’s mathematical identities—particularly those that may have been developed through the observations of the interactions of others rather than through their own participation—may be underrecognized in this study.

Participants

In this project, the two focal children chosen were both atypical (in different ways) from many children portrayed in transcripts of classroom interactions and from the other children in the cohort I studied. They were chosen intentionally because they presented “telling” cases (Mitchell, 1984, p. 239) that allowed me to examine how mathematical identities could be developed in different ways in the same classrooms. For example, given that I was interested in the ways that children who speak rarely in whole-class discussions experienced mathematics, it would not have been analytically useful to have chosen two focal participants who were called on frequently by the teacher. Similarly, because I was interested in ways that children engaged in classrooms in unexpected and unsanctioned ways, it was analytically useful to include a participant who frequently challenged classroom norms for behavior and participation.

The two focal children described here (chosen out of a cohort that varied from 16 to 20 children across the 3 years of the larger project) were Black and from low-income families, as were nearly all the children in the cohort. The chart below identifies all the children quoted by name in the article as a reference for readers. The race and gender identifiers used in the chart were those used most frequently by the children’s caregivers on forms or by the children themselves.

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Table 1

Demographic Information About Child Participants

Name	Race/Ethnicity	Gender	Years in Study
Aisha	Black	Girl	3
Jeremiah	Black	Boy	3
Trevion	Black	Boy	3
Kashon	African American	Boy	3
Clay	African American	Boy	3
Dahlia	Latina	Girl	3
Dhruv	Indian	Boy	3
Dave	Black	Boy	3
Jada	Black	Girl	3

Although there was some mobility across the 3 years, the makeup of the children in the cohort was relatively similar across the study, which in first grade included 17 children, 14 of whom identified as African American or Black, 1 as White, 1 as Latina, and 1 as Indian.

For the project, I conducted problem-solving interviews with the students in the beginning of prekindergarten and at the end of each academic year. In the fall prekindergarten interviews, the two focal children—Aisha and Jeremiah—demonstrated roughly the same skills, including rote counting, counting with one-to-one correspondence to 10, and identifying shapes; however, by the end of first grade, Jeremiah was able to solve addition, subtraction, and multiplication story problems in a variety of formats with greater accuracy and efficiency than Aisha. This contrast was one of the reasons I chose to construct cases about these two children.

In the classroom, Jeremiah was frequently corrected for his behavior. His participation in mathematics, particularly in prekindergarten and kindergarten, was often limited by management practices, such as seating him separately with a paraprofessional or scolding him. These negative interactions sometimes resulted in emotional outbursts from Jeremiah, which prompted the prekindergarten and kindergarten teacher to send him out of the room or even to call his mother to request that he be picked up from school. Despite these tensions, Jeremiah was often happy in the classroom, playing with classmates, telling jokes, and taking particular care with other children who were smaller than their classmates or emotionally fragile in some way. In addition, all three teachers commented on Jeremiah’s strengths in mathematics, noting that he could be counted on to answer questions correctly and always did well on assessments. Jeremiah was a frequent participator in whole-class conversations, in both sanctioned and unsanctioned ways.

Aisha was identified as a “model” or “ideal” student by all three teachers, although she rarely participated in whole-class conversations in mathematics. Across the 15 lessons examined over the 3 years, Aisha was seated in the back row in 13 of the lessons, both when the children were gathered together on a carpet and when they were seated at desks. Although the teachers never addressed this explicitly in interviews, her physical positioning was almost certainly in part because she could be trusted not to disrupt the lesson no matter where she was seated. (In contrast, Jeremiah was frequently placed near the teacher.) She was also praised for her behavior in all three classrooms much more frequently than she was praised for her engagement with mathematics. Aisha was sought as a playmate while on the playground by both boys and girls and was identified as a “friend” by the other four girls in the classroom during interviews at the end of first grade.

All three of the teachers in the study were White, middle-class women. The prekindergarten teacher had more than 20 years of experience, had taught some of the parents of children in Jeremiah’s and Aisha’s cohort, and lived in a nearby town. The first-grade teacher had only slightly less experience and lived and grew up on a farm in the rural community where the school was located. At the time of the study, she was pursuing a master’s degree in early childhood education. The kindergarten teacher was starting her second year of teaching at the time of data collection in her room and lived farthest from the school, commuting approximately 40 minutes.

As the researcher, I am also a White woman, who was an elementary teacher before becoming a professor. In addition, I grew up in a different geographic region of the United States than all the other participants in my study, which marked me as an outsider as much as my race and social class in many interactions. To overcome some of this outsider status, I led 2 years of professional development at the school before the research project began, organized community events, and conducted observational research in the prekindergarten classroom. By the end of the project, I had spent 7 years visiting the school, which built a certain amount of familiarity with the people, the place, and the routines.

In engaging with the children in the classroom, I attempted to emulate early childhood ethnographer Dyson (1997, p. 25) by taking on the role of a “curious, rather ignorant but very non-threatening person, who wishes to witness their goings on.” For example, children learned that I could not be relied on to provide answers about what they were allowed to do in the classroom or to resolve arguments, although I would get tissues, sharpen pencils, and tie shoes. I tried to call on Dyson’s strategy of being “very busy” with my work to avoid being called on to help with assignments, but there were times when the classroom teacher was overwhelmed or when children were particularly frustrated that I did provide guidance on mathematical questions. In addition, I frequently asked children to describe for me what they had done in mathematics, and, as my relationships with the children grew more established, I occasionally intervened in times of emotional upset, as described in the opening of the article. These interventions felt important as ways of humanizing my relationships with the children in the study and as ways of becoming a more full participant in the context I was studying.

Data Collection

A variety of data were collected, including weekly video of 67 classroom lessons across prekindergarten, kindergarten and first grade, fieldnotes of the lessons that focused on focal children’s engagements, video of yearly mathematics assessment interviews of all children, audio files and transcripts of at-home interviews with parents and focus groups conducted at school, audio files and transcripts of interviews with teachers, photographs, and student work (Parks & Schmeichel, 2014). Table 2 summarizes the quantity and types of data collected.

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Table 2

Summary of Data Collected

Year	Type of Data	Quantity
Prekindergarten	Mathematics lessons (videos, artifacts)	22 days across year
	Individual child interviews (videos)	15 in fall; 15 in spring
	Family focus groups (audio, transcripts)	2, approximately 45 minutes each
	Family math night (video)	1, approximately 1 hour
Kindergarten	Mathematics lessons (videos, artifacts)	25 days across year
	Individual child interviews (videos)	14 in spring
	Take home activities (audio, pictures)	8 families
Grade 1	Mathematics lessons (videos, artifacts)	20 days across year
	Individual child interviews (video)	16 in spring
	Family focus group (audio, transcript)	1, approximately 45 minutes
	Individual parent interviews (audio, transcripts)	6, approximately 60 minutes Each

I relied on graduate research assistants to do much of the camera work, which allowed me to engage in the classroom as an ethnographic participant observer (Emerson et al., 1995; Geertz, 1973). For me, this meant sitting as close as possible to the focal child I was observing for the day to see the classroom from his or her perspective. This meant that often the comments and actions of the children sitting closest to the focal child were more salient in my fieldnotes than the comments and actions of the teacher. My data collection was also heavily informed by current work on the body (MacLure, Holmes, MacRae, & Jones, 2010; Pink, 2015), which resists reducing all experiences to the words of participants. Although interviews, both formal and informal, were conducted each year, the words said in those interviews were not privileged as data above the daily, lived engagements observed in the classrooms. Children’s facial expressions, postures, and other embodied expressions were seen as data that could inform analysis.

Data Analysis

All data—including the 67 mathematics lessons, the child interviews, and the teacher interviews—was labeled with indexical codes using qualitative analysis software by a member of the research team. The codes included grade level, child’s or teacher’s name, mathematical content, and participation structure (e.g., whole class, small group, independent). The purpose of these codes was both to perform an initial analysis and to make it easier to retrieve data later.

For the current study, I wanted to look deeply at the ways in which two children engaged in the classrooms. This required examining each lesson from multiple perspectives. To make this close examination possible, I identified five representative lessons from each year for multiple additional rounds of coding (15 lessons total). These lessons were chosen because they had footage of Jeremiah and Aisha, portrayed lessons across the academic year, and varied in mathematical topics. These lessons did not differ in any significant way in terms of either topic or structure from most other lessons observed in each of the three classrooms (as determined by comparing the indexical codes for the lessons in the total data set to the indexical codes for the lessons chosen for close analysis). The research team open coded each of the selected lessons to identify the major instructional experiences in each lesson and to document social interactions (e.g., praise, correction, emotional tone) as well as opportunities for mathematical learning (e.g., types of questions, time on task). Table 3 shows an example of this initial coding.

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Table 3

Example of Coding

Episode From Fieldnotes Supported With Video	Codes	Key Related Themes
The first-grade teacher asked the children to look at two fact families arranged in triangles as shown below: The teacher asked the children whether both fact families were correct. Several children shook their heads and raised their hands, and she called on Jeremiah. Jeremiah said that B was wrong and she asked him how he knew.Jeremiah held up four fingers, saying “If I held up four fingers, and I hold up two more.” Jeremiah held up two fingers on his other hand. “One, two. I know four plus two is not seven. The answer is six.”The teacher said, “So, if you had to change one of those numbers to make it a fact family, what would it be?”Jeremiah said, “I would change the seven to a six.”The teacher responded, “Good job, Jeremiah. Did y’all here what he said?”During this exchange, Aisha sat silently and her gaze alternated between Jeremiah and the teacher and the opposite side of the room.	Indexical codes: First grade, Jeremiah, Aisha, whole-class, addition-subtractionSecond-level codes: Fact-families, constructive Jeremiah’s confidence, whole-class conversation, analysis of error, Jeremiah public explanation, teacher follow up, Aisha quiet, calm, praise, mathematics Jeremiah, Aisha silent, gaze averted	The task allowed Jeremiah (and some other children) to explain his thinking and identify errors in mathematics problems.The exchange seemed to move Jeremiah toward a more conceptual understanding of mathematics and to assume a position as expert.In contrast, this exchange seemed to move Aisha toward a continuing passive experience of mathematics in the classroom.Aisha seemed neither excited nor upset by the interaction.

In seeking to understand the classroom interactions from as many perspectives as possible, I worked through many cycles of coding (Saldaña, 2016). These included codes that aimed to describe social interactions and opportunities for mathematical learning (as described above), open coding that focused on movement, gaze, and emotion (Pink, 2015), and coding drawn from a priori frameworks, including documenting the cognitive demand of the mathematical tasks (Henningsen & Stein, 1997; Stein & Smith, 1998) and the level of cognitive engagement of the pedagogical interactions (Chi & Wylie, 2014). In addition to these different coding frameworks, I also coded each video with a focus on each focal child and on the teacher. For each lesson, I created data displays that showed the action of the classroom in three columns: the class as a whole, Jeremiah, and Aisha. Table 4 shows an excerpt of these tables, focused on just one instructional activity in one lesson. For the lesson shown, there were two other instructional activities that day. A similar table was made for all instructional activities in the 15 focal lessons. The goal of these tables was to further reduce the data from the open coding shown in Table 3. These tables allowed me to identify patterns in the data, which I could then use to read transcripts with more analytic power.

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Table 4

Excerpted Table of Coding Cycles (Showing One Activity, in One Lesson)

Date	Grade Level	Instructional Activity	Class	Aisha	Jeremiah
10–10	1	Whole class: Students take turns modeling problems on the number line. (10 minutes)	ICAP: ActiveCogDem: Procedures w/connectionsTeacher questions: (22)Open (8)Closed (14)Choral (12)Directed (10)Teacher praise (2)Teacher correction (4)Body: At Smartboard, standingClass: In desks, sittingTeacher expression: Calm	ICAP: PassiveCogDem: Procedures w/connectionsParticipation:Calling out (0)Teacher-initiated (0)Praised (0)Corrected (0)Body: Back row, sitting, following with eyes, pointing to number lineFacial expression: Calm	ICAP: ActiveCogDem: Procedures w/connectionsParticipation:Calling out (2)Teacher-initiated (2)Praised (0)Corrected (0)Body: Isolated at desk in front, sitting and standingFacial expression: Excited

Note. ICAP = interactive, constructive, active, and passive.

While it was not feasible to bring this level of analysis to all 67 lessons, I did use the indexical codes to locate all videos that showed Aisha or Jeremiah after the analysis and watched them to determine if the interactions in the closely analyzed videos were markedly different from the interactions in other lessons. Although I did not do multiple cycles of coding with all these videos, Aisha and Jeremiah did not appear to be participating in significantly different ways in the lessons in the broader data set.

The analysis presented in the article is drawn from work across all these coding cycles. Table 3 shows an excerpt of fieldnotes along with the codes assigned at various stages of the analysis process and excerpts from analytic memos describing key themes closely connected to the excerpt portrayed. Throughout the project, I wrote analytic memos (Emerson et al., 1995) to make connections across different kinds of codes and to identify and then later clarify major themes.

After the multiple cycles of coding, I created written case studies for both Jeremiah and Aisha, describing their participation in the classrooms across the 3 years in the three primary activity settings: whole-class, small group, and individual work. I then returned to the transcripts of the teacher and parent interviews to extend the case studies by including descriptions of the two children from the perspectives of their parents and teachers. Finally, I reviewed the videos and written records from the yearly assessments I conducted with both children.

The Three Classroom Contexts

The character of instruction across the three classrooms differed in some significant ways. As might be expected, the number of instructional activities in each lesson increased across the 3 years, with the prekindergarten teacher typically implementing two instructional activities in each mathematics lesson, and the first-grade teacher typically implementing three different activities. The majority of instructional activities in all three classrooms (approximately 2/3) were conducted in whole-group settings. The prekindergarten children engaged in only one independent activity in the five lessons observed, while in kindergarten, the children engaged in two independent activities, and in first grade, the children engaged in three independent activities across the five lessons. Finally, the first-grade teacher implemented more experiences that actively engaged children with materials or in conversations with their classmates than the other two teachers. The first-grade teacher was also the only teacher of the three to design experiences that asked children to find and correct errors in problems, solve nonroutine problems, or to discover and discuss mathematical rules. In contrast to the first-grade classroom, most instructional experiences in the prekindergarten classroom involved the teacher focusing children’s attention on mathematics, but not asking them to generate understandings for themselves. For example, the most common experiences were counting with the support of the teacher and drawing a shape from a model. The kindergarten teacher designed more passive whole group experiences than the other two teachers, such as by reading a picture book aloud without encouraging children to discuss the book or answer questions, which the prekindergarten teacher did routinely, such as by asking children to count objects or to identify shapes in the story.

In addition to these differences in the structure of the mathematics lessons, the three classrooms also differed in terms of emotional climate. The preschool teacher engaged in the most positive interactions with children across the five lessons observed, with almost half of her interactions with children involving praise, often supported by large smiles and an enthusiastic tone of voice. In contrast, about a third of the kindergarten teacher’s interactions contained praise, the smallest percentage of the three. She also engaged in the most corrective language. Although the first-grade teacher engaged in more praise and corrective interactions than the kindergarten teacher, her tone in both was quite low-affect, frequently whispering “good” to individual children or correcting behavior with a stern look.

The Average Experience of the Average Child

In order to understand the ways in which Aisha’s and Jeremiah’s experiences differed from the typical and to illustrate ways that presenting observational data that centers teachers, portrays children’s experiences as relatively monolithic, I begin by describing a typical instructional episode in two of the focal classrooms.

In the prekindergarten classroom, the teacher elicited the highest levels of participation from the children; however, the mathematics addressed, which included reciting the number sequence, number recognition, and shape identification, was often low cognitive demand. For example, in the following excerpt, taken from a conversation focused on shape recognition, one child participates individually and many participate chorally. However, the teacher did not emphasize the relationship between the number of sides and the shape and provided a nonmathematical cue (the starting sound of the shape name) when a child struggled to correctly answer her question.

In the lesson, the teacher began by drawing an equilateral triangle on the board with the base parallel to the bottom of the board and asked children to provide its name and guided the class in collectively counting each side. Then, she drew a square.

The teacher then repeated this routine with a rectangle and a circle, calling on different children to answer individually and leading the class in a choral count.

Reporting the episode in this way emphasizes the fact-based questions the teacher asked as well as the way she took on the role of evaluating whether children’s responses were mathematically correct. Mathematical participation in this classroom—if replicated over many similar lessons over the course of the year—would move children toward understanding mathematics as the production of correct answers and away from conceptions of mathematics that include argument and justification. My argument is not that this way of understanding the classroom is untrue, but that it obscures ways that individual children’s identities may develop as they participate in these interactions in ways that are unique and personal. For example, during the excerpt above, Aisha was sitting in the back row with her hands in her lap and did not participate in any of the choral responses. Meanwhile, Jeremiah was moving about the carpet, looking at toys on shelves and bouncing up and down, until he was scolded by the paraprofessional and pulled onto her lap. Thus, although Aisha and Jeremiah, along with the other children in the class, had a particular relationship to mathematics reinforced for them (i.e., valuing correct answers), they also had other aspects of their identities made salient in the lesson.

In contrast to the prekindergarten classroom, children in first grade frequently had opportunities to solve problems, present their work, and answer questions about their thinking. In one lesson, the first-grade teacher assigned some problems where students were asked to fill in missing numbers in sequences that had different starting points, but all counted up by tens. After filling in blank boxes, children explained in writing their thinking.

In this episode, the teacher assigned problems that asked children to attend to place value structure and to explain their thinking in writing. After asking Clay to share his thinking publicly, the teacher connected his ideas to something that she had seen Jada do. She then went on to ask other children about their work in ways that highlighted place value relationships. Neither Jeremiah nor Aisha participated in this conversation. Jeremiah was focused on items in his desk throughout, while Aisha sat quietly watching, but relied on her tablemates to complete the assignment.

Because Aisha’s and Jeremiah’s ways of participating were not central to the main story of either of the lessons described above, their actions were not included in the transcript above. Thus, reporting the transcript in this way did not make Aisha’s and Jeremiah’s participation explicit for analysis. As with the transcript of the prekindergarten lesson above, the way this classroom interaction is reported makes the teacher’s practice central. It is easy to notice that this teacher is engaging the children with more significant mathematics problems and that she is asking questions that require them to communicate their reasoning. What is less apparent is how widely the practice of engaging in public reasoning is in this classroom and which children are taking up the opportunities to engage with mathematics in this way.

In the rest of this article, the focus in the reporting of classroom episodes will be on Aisha and Jeremiah, rather than the teachers. The goal is to understand how children who are not part of the dominant story line of the lesson developed individual mathematical identities within the same classrooms and to draw attention to how the ways that classroom episodes are reported in research shapes our supports our understanding of the ways that multiple mathematical identities develop. It is important to note that while space limitations support a close up focus on only two children, it is likely that many children in each classroom had experiences that were different in some meaningful ways from those of the children who appear most frequently in transcripts of the teacher’s public interactions with students. Focusing closely on two children not only illuminates their own experiences but also the ways in which the choices we make in representing classroom episodes put different people at the center.

Aisha: Mathematics From the Sidelines

Aisha’s peripheral participation in mathematics was most noticeable during whole-class episodes, which were the dominant instructional format in all three classrooms in that they occurred most frequently and took up most of the time devoted to mathematics in the 15 episodes analyzed. Aisha was frequently silent, regardless of differences in grade level, quality of the instructional tasks, or teachers’ questioning styles. This was also reflected in her physical positioning during whole-class discussions in all three classrooms. In 13 of the 15 lessons observed across the 3 years, Aisha was seated in the back of the group, either on the carpet or in desks. Figure 1 shows Aisha in a typical pose during a prekindergarten lesson, seated at the edge of the carpet farthest away from her teacher with her chin in her hands.

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Figure 1.

Aisha in whole group lesson.

Neither her expression nor posture reveals joy or excitement in the lesson, although it is possible she is feeling these emotions.

Often in whole-class settings, Aisha remained quiet, and typically, her attempts to enter conversations were unsuccessful. For example, during a lesson in prekindergarten on shapes, the teacher asked the children to name foods that resembled a triangle. After a few moments, many children started calling out ideas without raising their hands, including Jeremiah, and leaning forward on their knees to capture the teacher’s attention. During this time, Aisha sat quietly in the back row, raising her hand. The teacher then praised Aisha to redirect the children.

During this lesson, the teacher never did return to Aisha or ask her to make a mathematical contribution. Aisha kept her hand in the air for a few more moments after the teacher spoke to Kashon, but then she put it down and did not raise her hand throughout the rest of the lesson. She continued to sit still and silently, largely watching the teacher, and never spoke, even to a neighbor. Aisha received this sort of praise for her behavior in all three classrooms, being commended for putting her finger in the correct place on worksheets, sitting quietly, and keeping her hands in her lap. During this lesson, children took 20 mathematical turns in the conversation, with eight of those turns being taken by just two boys and another three turns taken by Jada, the only girl who spoke multiple times. Of all the children in the class, Aisha received the most frequent praise for her behavior in this lesson, as well as across the 15 lessons analyzed. (Her classmate Clay received the most frequent praise for his mathematical thinking.) Interactions like these, repeated over time, worked to make Aisha’s identity as a well-behaved student more salient for her (and her classmates) than her identity as a mathematical learner.

Aisha’s “good student” identity was also reinforced outside of mathematics lessons. Each year, Aisha was one of the first children chosen as a student of the month. Her name regularly remained on “green” on the classroom behavior charts in kindergarten and first grade, and she was always featured on the quarterly recognition board in the hallway for “attitude, attendance, and academics.” In addition, each year, the Grade 1 to Grade 12 classrooms were asked to nominate a couple to the homecoming court. Aisha was chosen by the class as the first-grade girl representative, a sign of recognized good behavior and social standing.

Aisha’s identity as a well-behaved child was also reinforced by her mother and by Aisha herself. In an interview in her home during the Grade 1 year, Aisha’s mother, who called Aisha an “angel at school,” said, “she the one keeping the other ones in order. She never in trouble at school. She always good at school.” My observations confirmed Aisha’s mother’s perception that Aisha took on a caretaker role with other children. Aisha occasionally corrected other children, often with a look or a quiet reminder. After art or other messy activities, she was often one of the first to grab the broom to sweep up. In prekindergarten she spent much of her free time playing in the kitchen and with dolls, assuming a caregiver role with her peers and with the toys. One of Aisha’s teachers, reinforced this perception of her, calling Aisha a “little mother.” However, during the at-home interview, Aisha revealed a more complicated picture of herself as a caretaker. In talking about what she would do when she grew up, Aisha said she would find an apartment and a husband, and “probably have kids, but I don’t know if I want them, running around the house, getting my nerves and everything, just like what I do with you.” Similarly, she seemed to occasionally find the behavior of her classmates frustrating, asking them, for example, to “stop playing” when they were supposed to be completing an assignment. Thus, while Aisha’s identity as a good student and a caretaker was reinforced in multiple contexts and over years, she also expressed some desire to carve out an alternative conception of herself, although she often embraced feminized aspects of her identity. When asked if there was anything she would change about school, she said, “change me to a princess.”

Aisha’s identity as a “good” student likely contributed to the few opportunities she was given to speak publicly in the mathematics classrooms. Across the 3 years, Aisha experienced only one extended interaction (more than one turn) in a whole-class setting. Most commonly, she remained on the periphery of lessons, both physically and in the conversation, as in the following kindergarten interaction. During a whole-class discussion in kindergarten, the children were asked to compare coins. Some verbally expressed confusion about whether a dime or a penny was larger in size. The teacher chose to engage one child in an extended interaction to explore this idea.

This interaction went on for several more turns as the teacher asked Clay to explore the two coins in a variety of ways so he could see that the penny was larger. During this time, the teacher’s voice grew quieter, and many of the other children in the class started to examine the coins in front of them and to talk to their neighbors about which they thought was larger (or about unrelated topics). Even as the teacher’s voice grew quieter and her interactions more and more focused on Clay, Aisha sat silently through the entire interaction with her coins on the table in front of her and in her hands in her pockets, as the teacher had initially directed. Many other children either called out to engage in the exchange with the teacher (as Dahila and Dhruv did) or independently explored differences in the coins in front of them (as Jeremiah did). Aisha, following the teacher’s directions, did not have a mathematical engagement during this exchange because she was too far away to clearly hear the teacher and she did not use the time to investigate the coins in front of her. Rather, she followed the directions given to her by the teacher to sit quietly with her hands in her lap.

In the first-grade classroom, Aisha was called on in whole-class settings even more rarely than in prekindergarten or kindergarten. She sat silently in two of the five lessons. When the teacher called for active participation by asking children to execute algorithms mentally or by asking children to analyze and explain errors in fact families, Aisha remained still and quiet, her face largely impassive.

The first-grade teacher often provided opportunities for using procedures in meaningful ways or occasionally for mathematical reasoning, such as by asking children to describe what they learned or to make connections to the ideas of others. For example, when asked to describe what was learned in a lesson on fact families, Jeremiah said, “I learned that we can subtract . . . when you are doing a subtracting problem you can look at the other problems and figure out what the answer will be.” However, Aisha was almost never called on to publicly share what she had learned in this way.

In considering Aisha’s participation in the episodes described, as well as others not described here, it is clear that Aisha was developing an identity as someone who engaged with mathematics as a primarily a passive activity, where one quietly watches the engagements of others. Importantly, while the kindergarten and first-grade teacher, unlike the prekindergarten teacher, did not quickly correct children provide them with “hints” so they would immediately produce correct answers, the differences in questioning practices made far less difference to Aisha than to other children in the development of her identity in relation to mathematics because she was not actively engaged in such questioning practices. Similarly, the first-grade teacher’s practice of asking children to explain their thinking was much less relevant to Aisha than to children who answered such questions frequently. Because Aisha was engaged by her teachers with questions so rarely, differences in the kinds of questions being asked across the three classrooms did not significantly change her identity as a mathematical learner, which is not to say that these questioning practices had no impact at all. Observing her classmates engage with mathematics in this way may have changed how Aisha conceptualized mathematics and therefore her own understanding of her mathematical identity; however, it seems unlikely that these questioning practices had as powerful an impact on Aisha’s identity as on those of the classmates who engaged more actively with these practices. This dynamic demonstrates ways that pedagogical practices used widely in classrooms may develop different students’ identities in divergent ways.

Small-group activities and independent work, which were less public and less regulated by the teachers, provided potential contexts for Aisha to develop her identity as a mathematics learner, outside of her identity as a good student. Across the 3 years, Aisha spoke more and engaged with more materials in small-group settings than in whole-class discussions. Although because these settings occurred much less often than whole-class interactions, these more active engagements were a much smaller set of Aisha’s experiences in mathematics, which means she had fewer opportunities to reinforce her identity as a mathematical learner in this way. The increased engagement in small-group settings resulted from Aisha being able to manipulate materials (such as bingo counters and tens frames) even when she was not engaged by the teacher and from the teachers calling on Aisha in these settings more often than they did in whole-class discussions.

However, Aisha sometimes fell into an observer role in small-group settings as well. For example, in one lesson, the children were asked to work with a partner to solve nine story problems using any strategy that made sense to them. Aisha worked with her tablemate, Dave, on the task. For the first problem—“There are eight marbles outside the bag. There are 15 marbles inside the bag. How many marbles are there all together?”—she and Dave both began by drawing 15 circles in a top row and 8 circles in a second row on their whiteboards, a strategy that had been frequently used in the class. Aisha counted aloud as she drew, looking back and forth between her board and her partner’s. Dave counted his total, whispering each number and announcing “23” at the end. He wrote “15 + 8 = 23” on his worksheet. Aisha, after counting the circles in her top row to get 15, stopped and looked at Dave’s paper. She then wrote “23” on her worksheet, asking Dave, “Is the answer 23?” He nodded.

For the next problem, Aisha watched Dave draw a row of 14 circles and then a row of 7 circles before writing “14 + 7 = 21” on his paper. She then wrote the same answer on her paper without drawing anything on her whiteboard. Throughout the worksheet, Dave continued to work independently on the problems, never asking Aisha a question. He answered hers and occasionally volunteered a suggestion: “That one’s subtraction. Don’t make two rows.”

Dave experienced this activity as an opportunity to strengthen his identity as a flexible problem solver. He created pictorial representations of the problems and connected these to numeric representations to get the answers, but Aisha’s experience involved counting objects and taking notes on what Dave did, rather than constructing her own understandings or truly engaging with his ideas through conversation. As a result, many of Aisha’s small-group experiences continued to move her toward a spectator role in relation to mathematics, even while other children (like Dave) experienced the same task in ways that were more active. Of course, Aisha’s identity as a mathematics learner was developing along with her identity as a Black girl. In working with Dave, a Black boy, it is likely that Aisha felt comfortable deferring to him and that Dave felt comfortable supporting her because this kind of interaction reinforced common gendered ideas about how people interact that both Dave and Aisha experienced in their lives and in the media they consumed.

In fact, when Aisha was partnered with Jada, another Black girl, on a similar problem, she did not ask Jada for help but sought out support from adults in the room. Because the paraprofessional and teacher were busy with other children, I chose to respond to Aisha’s request for help. As I talked Aisha through the problem, I had to ask Jada to let Aisha to answer for herself because Jada kept jumping in for her. When Aisha reached the same answer Jada had announced earlier, Jada said, “I told you,” but Aisha said, “I did it,” revealing pride and pleasure in having solved the problem herself. Sometimes the classroom teachers or the paraprofessionals were able to assist Aisha in a similar way, but in the busy classroom, Aisha was often not successful in getting the attention of adults, who were frequently interacting with more vocal children.

Of all the activity settings in the classrooms, independent work times provided Aisha with the most opportunities to develop her identity as a mathematical learner; however, these opportunities were few and far between. In the 15 lessons closely analyzed for this study, there were three in first grade and two in kindergarten, and the time spent on these experiences was relatively short; however, because she was expected to work on her own, Aisha was often, although not always, able to engage actively with mathematics in independent activity settings. For example, in kindergarten, the children were asked to measure a construction paper foot with three different nonstandard units. In this activity, Aisha worked on her own without looking at what her tablemates were doing. First, she snapped together a row of unifix cubes and lined it up next to the foot break it off at the longest point of the construction paper foot. She wrote “11 unifix cubes” on the foot. Next, she lined up teddy bears across the foot in a straight line, being careful not to leave spaces between them and to choose bears that were the same size, but not necessarily the same color, demonstrating that she recognized that size was a more important attribute in this activity. After counting, she wrote “15 bears” on the foot. Finally, she picked up plastic dinosaurs, which were visibly longer than the bears. After putting two down, she whispered, “This is going to be less.” When she finished counting, she wrote “7 dinosaurs” on the foot. During this activity, Aisha was involved in measuring, and it would seem from her vocalizing, constructing a relationship between the size of the unit and the final measure. This active experience stands in contrast to her passive engagement during the conversation that preceded the activity, in which she was never called on and never raised her hand to speak.

However, it is worth noting that not all Aisha’s independent work times were as productive. In a first-grade lesson, Aisha, like the other children in the class, was instructed to work on her own to solve addition and subtraction problems; however, she did not solve any problems independently, but waited for her tablemate Dave to solve each problem before leaning over and copying his answer (similar to the way she relied on Dave in the small group episode described in the previous section). Although her motions were not particularly subtle, her use of Dave as a support was never noticed by the teacher. In fact, when the first-grade teacher viewed a video clip of a similar interaction, although from a different lesson, she expressed surprise and said she had never seen Aisha copy from a neighbor. Thus, while independent activities provided a chance for Aisha to broaden her identity as a mathematics learner, the ways of being she enacted during whole class and small-group activity settings bled over into these spaces. And because, whole group and small-group activities were so much more frequent, Aisha had many more opportunities to reinforce her identity as a passive spectator of mathematics than to develop her identity as an active learner.

Looking across Aisha’s experiences in all three classrooms in different instructional formats, it is apparent that although the teachers’ practices varied in some important ways over the 3 years—cognitive demand of tasks, opportunities to work independently, questioning styles—Aisha’s experiences in mathematics remained relatively consistent. She was rarely engaged publicly, never asked to explain or defend her reasoning, and was allowed to rely on others to mask her own lack of participation in mathematics. Through interactions like these over the 3 years, Aisha’s identity as an observer of mathematics became more and more deeply inscribed.

This aspect of her identity likely shaped her engagement in an interview I did with her at the end of first grade. She demonstrated confidence and competence with counting, adding, subtracting and shape identification, when the questions were specific and focused on procedures. However, she expressed uncertainty with problem solving tasks. For example, in my first-grade interview with her, I asked her to solve the doughnut problem her teacher had assigned a few months before. (“There are 18 doughnuts. Six are chocolate. How many are not chocolate?”) In response, Aisha drew 18 doughnuts in a box and six outside and counted them all. In solving a novel story problem—“My daughter had 12 stickers. She gave away 8. How many does she have now?”—Aisha drew 12 stickers and said the answer was 12. When questioned again, she said “8.” Her responses are not surprising, given how rarely she solved problems like these on her own in the class or verbally expressed her reasoning. While her experiences in the classroom gave her opportunities to encounter story problems like these, she did not have opportunities to make sense of them herself or to develop an identity as someone who could solve such problems.

In this interview, Aisha said that she liked math. When asked why she said, “You can learn about things like math. You can learn your numbers and how to subtract and to put together.” It is perhaps worth noting that in answering she did not discuss things that she liked to do in math, but instead focused on her idea of what math was. When asked how she was good at math, Aisha said, “I do my homework,” suggesting that her conception of a “good mathematics student,” relied heavily on the idea of doing what is expected. In addition, when asked later in the interview who in her class was good at math, she did not name herself – or any of the other girls. Instead she named Clay and Kashon, two boys who were frequently recognized for their whole-class contributions. When I asked how she knew, she said they were good at problem solving and at a game they played recently, focusing less on their willingness to do what was asked (complete in-class assignments and homework) and more on what they did with mathematics.

Broadly, it is possible that her interactions in mathematics over the 3 years contributed to a gendered conception of what it meant to be good at mathematics because boys in her class were for frequently called on to do mathematics publicly and more frequently publicly praised for problem solving. In considering Aisha in particular, it is clear her identity thickened over her first 3 years of schooling to center following directions, being quiet, and completing tasks with a minimum of disruption, rather than on understanding mathematics, publicly explaining her thinking, or independently solving problems.

Jeremiah: Mathematics Under Siege

Unlike Aisha, whose participation in mathematics remained relatively consistent from day to day, Jeremiah’s participation in mathematics varied widely from high levels of engagement in cognitively demanding tasks to periods of exclusion and isolation. However, these variations occurred primarily across individual lessons, rather than across years. For example, in all three classrooms Jeremiah frequently spoke during whole-class settings and was seated near the teacher in about half the lessons observed. This physical placement allowed him to both contribute frequently to mathematical conversations and to be in a position where his behavior was under constant observation by the teachers.

In whole-class conversations, Jeremiah, especially in kindergarten and first grade, engaged in a lot of talk, both solicited and unsolicited, as well as a great deal of movement. Although these behaviors drew negative attention from the teachers, they also created opportunities for him to participate in mathematics in productive ways. For example, during the first-grade lesson on even and odd numbers, Jeremiah, whose desk was in the front of the room, participated actively both when he was called on and when he was not.

In this episode, Jeremiah is corrected twice, once explicitly and once implicitly, by the teacher; however, he is also actively engaged in responding to the teacher.

By participating in ways that violated classroom rules and norms, Jeremiah was able to maximize his engagement with the mathematics, even sometimes beyond the intention of his teachers. For example, during one kindergarten lesson, the teacher was reading a book about money to the children with the expectation that they would silently listen to the story. However, twice during the story, Jeremiah got out of his seat without permission to interact with the text, once looking closely at the pictures to see the difference between two coins and once touching his finger to a set of pennies to count them, turning his own engagement active. Through repeated interactions such as these, Jeremiah deepened his identity as a learner who could take an active role in making sense of mathematics.

Similarly, in a measurement lesson also in kindergarten, Jeremiah approached the balance without permission to observe closely and touch the materials, as shown in Figure 2 (the face of another child in the class is blurred).

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Figure 2.

Jeremiah in whole group lesson.

Jeremiah frequently took up a lot of space in the classroom with his body and interacted with mathematical materials, whether or not he was invited to do so by the teacher. As in the case of this measurement lesson, Jeremiah’s engagements sometimes resulted in him being shouted at or disciplined by having his clip moved down on the behavior chart. In this lesson (as was frequently the case), this resulted in him sobbing loudly and being sent to calm down in the hallway, where he missed the remainder of the mathematics lesson. Other children in the classroom did not seem to be troubled by Jeremiah’s movement; however, his emotional reactions to teachers’ reprimands often drew attention from other children. These repeated interactions where Jeremiah’s active participation in making sense of mathematics was linked to reprimands from teachers in whole-class settings operated to shape Jeremiah’s identity as a successful mathematics student, who was simultaneously perceived as problematic by his teachers.

As shown in Figure 3, Jeremiah received far more corrections from his teachers in all 3 years than any other child in the classroom (although only the children referred to by name in this article are shown on the chart.) His mother said he was sent home “almost every week” in prekindergarten and occasionally in kindergarten. These frequent interactions around Jeremiah’s behavior shaped not only his life at school but also his interactions with his family. In an interview, his mother said, “I was so tired. They just kept calling me. There was no need to have him in school if they were just going to keep calling and telling me to come get him.” Jeremiah went on to describe the multiple interventions she tried with Jeremiah, including conversations, rewards, and punishments. Ultimately, she was able to mobilize the school’s special education system to put an agreement into place that Jeremiah would be allowed to recover in the guidance counselor’s office rather than being sent home. His mother reported that Jeremiah had not been sent home throughout first grade. Jeremiah’s first-grade teacher agreed that Jeremiah’s behavior had significantly improved from what she had seen in kindergarten, but said he still received frequent reminders about staying on task and in his seat. These frequent conflicts with his teachers took a toll. Jeremiah, in our end-of-the year interview in first grade, stated, “None of my teachers liked me.”

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Figure 3.

Behavior corrections in whole-class lessons over 3 years.

As for Aisha, Jeremiah’s experiences in small-group activities and independent work provided more flexibility in the development of his identity as a mathematics learner. In the same ways that the reduced regulations for behavior during small group and independent work allowed Aisha to worry less about following directions, this openness allowed Jeremiah to solve problems and take on leadership roles without constant corrections for his behavior. Thus, these settings were more likely to reinforce a positive mathematics identity for Jeremiah.

For example, when working with a group to solve the problem —“There are 18 doughnuts in a box and 6 are chocolate. How many are not chocolate?”—Jeremiah quickly drew a box of 18 doughnuts, colored in 6 and counted the remaining ones before writing in the answer. When his partners, Dahlia and Trevion, touched his arm to ask “Is this adding?” about the first problem, he said that it was, and then went back to his own worksheet. When asked how to do the doughnut problem, he replied: “Draw like this,” and showed his whiteboard. By the end of the worksheet, however, he began to get frustrated. When Dahlia asked again, “Jeremiah, can you help me?” he snapped “Why you looking at my paper?” Like Dave, Jeremiah experienced this activity as one where he could solve problems through a variety of solution strategies and choose to offer (or withhold) help from his partners. At the same time, the frequent requests for help from his classmates, which took him away from his own work, often caused Jeremiah to grow irritable, which occasionally resulted in further corrections from his teachers for his language or tone in talking to classmates, and further contributed to the link between Jeremiah’s identity as a successful mathematics learner and a problematic student.

Like Aisha, Jeremiah was able to use independent work times in mathematically productive ways on most occasions. Although Jeremiah frequently stood at his desk while working in both kindergarten and first grade (sometimes hopping from foot to foot), he was generally very focused, working quickly through the assigned problems and rarely calling out to the teacher, as he did in other instructional formats. Often, he finished early. Sometimes, this led to off-task behavior, but sometimes he used this time to help his classmates. Dahlia frequently asked him to check her work, and while sometimes he grew impatient, occasionally he would support her by asking questions—“So how many sheep should you draw? Point for me.” These experiences provided additional positive experiences of mathematics for Jeremiah, who would occasionally show joy by clapping for Dahlia when she correctly solved a problem.

Sometimes, Jeremiah created independent work opportunities by working ahead the teacher’s directions. For example, in the subtraction with number lines lessons, the first-grade teacher took the class step-by-step through the first three problems asking children to put their fingers on particular numbers and then count back. During this lesson, Aisha followed along by moving her finger, but waited until the teacher wrote an answer on the interactive whiteboard before writing an answer on her own paper. In contrast, Jeremiah worked through all the subtraction problems on the first side, using his fingers to count back, before the class completed the three example problems. In doing so, he turned whole-class problem solving led by the teacher into independent problem solving.

Jeremiah’s experiences in mathematics over the 3 years supported him in developing an identity as an active learner of mathematics. In whole group, small group, and independent settings, Jeremiah displayed confidence that he could solve problems and explain his thinking. He also demonstrated agency in relation to his mathematics learning by offering answers, asking questions, and engaging with materials in ways that supported his understanding, even when he had to act in opposition to the wishes of his teachers to achieve these understandings, such as when he got up to interact with the balance or to count coins.

Jeremiah’s confidence as an active learner of mathematics was demonstrated in his problem-solving interview at the end of first grade. In response to the doughnut problem (“There are 18 doughnuts in a box and six are chocolate. How many are not chocolate?”), Jeremiah solved it the same way he had in class months before. In solving the problem, “My daughter had 12 stickers. She gave away 8. How many does she have now?” Jeremiah counted up from 8 to get four. He solved each of the problems I gave him during the interview, drawing on a variety of strategies, including counting up, mental addition, and making physical models.

However, Jeremiah’s affect during the first-grade interview was noticeably different from that during the prekindergarten interview I did with him. In the prekindergarten interview, Jeremiah laughed, played with materials, asked me questions, and clapped for himself when he solved particularly tricky problems. In first grade, he occasionally put his chin on the table, smiled and laughed rarely, and exhibited little enthusiasm. On the day of the interview, he had been having a particularly challenging day and had had his “clip” on the behavior chart moved down to red, which he did not like. However, the differences in affect across these two interviews also show the ways in which his identity was developing as a student. He was a confident mathematics learner, but did not seem to be developing an identity as someone who enjoyed school or who felt like a “good student.” When I asked why “none” of his teachers liked him, in response to his claim, he said, “I used to get in trouble.” In addition, he expressed little enthusiasm about school in general, as in the following exchange:

Jeremiah’s amazement that I had been in the prekindergarten classroom was the only real excitement he displayed in the first-grade interview. He worked competently through all of the questions I asked, but displayed no real joy. All this suggests that while Jeremiah’s interactions in the three classrooms allowed him to develop an identity as a capable mathematics learner, the interactions also shaped his conception of himself in less productive ways. The frequent corrections to his behavior and his early experiences being sent home from school made it difficult for Jeremiah to develop an identity as a good, or even well-liked, student.

As with Aisha, Jeremiah’s identities as a math learner and a student were also being shaped in relation to his identities as a raced and gendered human being. All Jeremiah’s teachers were White women, as such, their conceptions of what behavior was in need of correction was almost certainly shaped by their engagement with Jeremiah across lines of race and class difference and problematic conceptions of Blackness drawn from the broader discourse. Jeremiah’s mother alluded to this obliquely by saying her first thought when she began getting phone calls from teachers was, “He’s four years old. Handle it!” Although she was reluctant to criticize teachers, and, in fact, said she appreciated how much time they had devoted to her son, she did express frustration that what she (and I) saw as typical behavior for a young child was treated as serious enough to cause her son to be sent home from school. For Jeremiah, these difficult interactions over 3 years reinforced his identity as a student who was frequently framed in opposition to his teachers, even while he came to see himself as a competent mathematical learner.

Discussion

What is striking in relation to both their mathematical and social worlds is the ways that Jeremiah and Aisha’s unique identities were thickened in each year of the study, although the teachers’ pedagogical practices in the three classrooms differed in significant ways. The three classrooms varied in relation to the cognitive demand of the tasks presented, the questioning practices of the teachers, and the behavior management strategies used; however, despite these differences, Jeremiah and Aisha continued to have similar aspects of their identities reinforced over the 3 years. This finding raises questions about the ways that global changes in teacher practice might affect individual children. Even if teachers, through professional development, curricula, or preservice education, come to ask more meaningful questions or to choose richer tasks, we cannot assume that these changes in practice impact all children in the same ways. This finding has implications for both practitioners and researchers.

For practitioners, this finding suggests the importance of developing strategies for monitoring the participation of all children in the classroom, and particularly the criticality of attending to patterns in participation. As Ing et al. (2015) found, differences in the ways that teachers interact with individual children impact children’s achievement in mathematics, regardless of the nature of the pedagogical practices used broadly in the classroom. Similarly, both Jeremiah and Aisha’s cases point to the importance of considering the ways that interactions around behavior shape children’s experiences in mathematics. For Aisha, continual positive attention for her stillness and silence, rather than her mathematical contributions, reinforced her identity as a passive mathematics learner, while the frequent corrections Jeremiah experienced deepened his identity as a student who his teachers perceived as troublesome. In addition, this research supports previous calls for work with educators around explicit and implicit bias. If White teachers’ interactions with young Black children are immersed in discourses of adultification (Epstein et al., 2017), then Black children are unlikely to get needed opportunities to engage with new pedagogical practices in ways that develop their identities as mathematical learners. Unpacking assumptions about Black girls’ assumed maturity and independence and Black boys’ assumed problematic behavior needs to be done alongside efforts to improve questioning practices or promote the assignment of open-ended problems. Of course, professional development is also needed to explore and dismantle problematic discourses related to other minoritized groups so that teachers ensure that they are creating classrooms where children’s social identities contribute to, rather than work against, their developing identities as mathematics learners in their classrooms.

For researchers, this study suggests that descriptions of classrooms that focus on the teachers’ interactions with multiple children will not always adequately convey the quality of mathematics instruction experienced by individual children. In addition, large-scale studies that look at average effects of curricula and pedagogies may mask differences in individual experiences. It will be important for some analyses that focus on the quality of questions teachers ask or of the mathematical complexity of tasks posed to document whether all children in the classroom have equal access to these questions and tasks and to report data in ways that make clear the breadth of participation in whole-class discussions, verbal and embodied ways of participating, and ways that children’s participation is shaped by their gendered, raced, classed, and other social identities.

Thinking about Aisha’s experiences in the context of discourses around Black girlhood (Gholson & Martin, 2014; Grant, 1984) raises questions about how notions of femininity, particularly Black femininity, played into teachers’ engagements with her. Although not a focal child in this study, Jada, another Black girl in the class, received far more attention from teachers than Aisha, but was also chided for her behavior in ways Aisha was not. Aisha and Jada’s varied interactions with teachers in the three classrooms may have been located in ideas about the “proper” way for Black girls to behave. Praising Aisha for her silence and stillness and correcting Jada for her exuberance both worked to maintain a normative view of Black girl femininity, which, as documented above, interfered with Aisha’s ability to develop her identity as a mathematics learner.

Jeremiah’s interactions with teachers and classmates were also located amid discourses of Black boyhood. Many scholars have written about the ways in which White teachers tend to perceive even young Black boys as dangerous (e.g., Ferguson, 2010; Love, 2013). Some of Jeremiah’s positioning as disciplinary problem needs to be understood in relation to these discourses, where his movement and verbal interruptions were treated as such significant problems that at the end of his first 3 years of schooling he said that none of his teachers have liked him. Thus, even though his teachers, and Jeremiah himself, thought of him as a capable mathematical learner, the constant negative pressure around his behavior worked over 3 years to create an identity that excluded Jeremiah from the category of good student. It is impossible to know what the long-term impact of this identity work will be for Jeremiah, but given the density of the negative interactions in the first 3 years, it seems reasonable to assume that this is an identity that will be hard to disrupt.

One possibility for such disruptions may lie in the possibilities of independent work settings for more flexible identity development. For both Aisha and Jeremiah, independent tasks provided more opportunities for them to develop positive mathematical identities, with Aisha doing more mathematical work than in whole-class settings and Jeremiah receiving fewer corrections. But independent work is rarely a focus of mathematics education research. In this study, the children’s freedom from the teachers’ potential censure for their behavior seemed to be particularly important, but opportunities and costs of independent work may look different in other contexts.

Significance

This study contributes to growing understandings of the relationships between mathematical identity and learning. As Langer-Osuna and Esmonde (2017) point out, studies of mathematical identity have typically focused on “micro-identities,” where moment-by-moment interactions are unpacked or on “thickened identities,” where researchers look at the way a more stable identity develops over time. However, even most studies of thickened identities typically take place over a school year or less. By following children through multiple classrooms over 3 years, this study helps to show how identities develop over longer periods of time, demonstrating how thickened identities grow more and more stable. The study also points to the importance of exploring the development of mathematical identity in the early years of schooling, a time when children first learn to be students, both in mathematics and more generally. Ethnographic studies that look at opportunities for disrupting unproductive identities in later years would also be of interest, as would studies that look at how identities thicken and change during points of transition, such as in moves from elementary to middle school.

Broadly, this study suggests that researchers and teacher educators need to attend not just to teachers’ classroom practices with “children,” but to the ways that they engage with particular children who differ from each other in a variety of ways including race, temperament, gender, as well as other identity markers, moving away from reporting the “average experience of the average child,” (Pianta et al., 2008). Studying interactions between temperament, mathematical performances gender, race and participation in varied classroom settings will require diverse methodological tools that open up analyses to data beyond what is said in public spaces in classrooms to include descriptions of body language, gaze, and under-the-breath comments, such as have been used in studies of groupwork (e.g., Esmonde & Langer-Osuna, 2013; M. Wood, 2013). These sorts of analyses may capture more of what mathematical discussions (good or bad) may be doing (or not doing) for the children who sit silently. These kinds of analyses that attend to the body will make it possible for researchers to more fully consider the role of affect in learning, and ways that lessons contribute to the development of long-term identities. A comprehensive picture of learning in mathematics classrooms requires studies that place children, rather than teachers, at the center of the classroom experiences.