Ethnography of Education: Engagement and Authority at a UCLA Math Circle

Written for ANTHRO-152R: Language, Culture and Education at UCLA, Winter 2026.

Introduction

Every Sunday, roughly 25 high school students gather voluntarily in Math Sciences Building 5127 on the UCLA campus to work through challenging mathematical problem packets. There are no grades, no formal disciplinary structures, and no compulsory attendance. The six undergraduate instructors who guide the sessions walk around looking for confused faces rather than enforcing participation or attention. The UCLA Olga Radko Endowed Math Circle (ORMC) Advanced 2 class operates this way, and it raises immediate questions about how engagement and authority work when the usual structures of schooling are not there.

This paper reports on an ethnographic study of the ORMC Advanced 2 class conducted over two observation sessions in the Winter 2026 quarter. Drawing on participant observation and informal conversations with instructors and students, this study examines how engagement is communicated and interpreted in this setting. Specifically, this paper is guided by two research questions: How do students signal when they need help versus when they want to be left alone in this informal mathematics learning environment? And how do instructors interpret and respond to student engagement cues in a non-compulsory educational setting?

To make sense of what was observed, this study draws on frameworks from linguistic anthropology and educational psychology, including research on epistemic authority and role negotiation, autonomy-supportive pedagogy, and Goffman's micro-sociological tools for analyzing interaction. The goal is a close reading of this particular site and what it reveals about how students and instructors navigate engagement when participation is entirely voluntary. Preliminary observations suggest that students rely heavily on subtle embodied cues to regulate instructor involvement, while instructors vary in how they interpret and respond to these signals.

Literature Review

Classrooms are social spaces where teachers and students constantly negotiate power, knowledge, and choices. In informal learning environments like math circles, the usual boundaries between teachers and students often blur. This blurring makes it important to look closely at how social and intellectual roles form. This literature review examines how students and instructors negotiate autonomy and authority through everyday classroom interaction.

Epistemic Authority and the Co-Construction of Roles

Authority in the classroom is not just a fixed rule. Instead, it is something people create together based on who holds and controls knowledge. Friedman (2017) studies how people negotiate "epistemic authority," which is the right to claim knowledge. While Friedman focuses on the differences between native and nonnative speakers, the basic ideas apply to any classroom. Friedman shows that teachers usually assign expert status, but students can accept or resist these roles based on their own experiences. This means authority is built together. Teachers might accidentally reinforce strict power dynamics, but students can claim their own authority by sharing personal experiences or changing the topic of conversation.

Dyer, Machaka, Parr, and Krist (2021) apply this idea to math classes by looking at "joint exploration." They define joint exploration as collaborative problem solving where everyone works together to learn something new. Dyer and colleagues argue that traditional power imbalances between teachers and students limit this type of learning. To achieve real mathematical inquiry, instructors must intentionally step back from being the main authority figure. When teachers change their role by using subtle speech cues or physically stepping away, they share intellectual authority. This shift encourages students to rely on themselves, their peers, or outside tools for answers. As a result, students can keep exploring problems without needing constant teacher approval. Together, Friedman (2017) and Dyer et al. (2021) show that authority is negotiated. Specific shifts in interaction are required to move from a traditional teaching model to collaborative learning.

Autonomy-Supportive vs. Controlling Pedagogy

The way people negotiate authority is closely tied to how much freedom students have. Based on Self-Determination Theory, Reeve and Cheon (2021) explain autonomy-supportive teaching and compare it to controlling teaching styles. Autonomy support starts with focusing on the student and using an understanding tone. Teachers show this by considering the students' perspective, encouraging personal interests, explaining why tasks matter, acknowledging negative feelings, and using inviting language.

In contrast, controlling teaching uses behavioral and mental pressure to force students to comply. This approach restricts student freedom. Reeve and Cheon show that teachers can learn and adopt autonomy-supportive behaviors. When teachers use these methods, the classroom climate changes from strict and conflict heavy to supportive and equal. For research purposes, Reeve and Cheon's framework offers a clear way to categorize how instructors respond to students. If a student seems bored or frustrated, a researcher can label the instructor's response as either autonomy-supportive (like having a conversation or relating to the student) or controlling (like raising their voice or forcing the student to finish the task). This framework directly helps measure the autonomy aspect of the research project.

While prior research has examined authority and autonomy in classroom settings, less attention has been paid to how these dynamics are negotiated through moment-to-moment interaction in informal, non-compulsory environments such as math circles.

Conceptual / Theoretical Framework

Micro-Sociological Mechanics: Framing, Footing, and Alignment

To analyze the verbal and nonverbal cues through which autonomy and authority are negotiated, this study draws on micro-level sociological frameworks. Erving Goffman's (1979) concept of "footing" is highly important for breaking down how conversations work. Goffman defines footing as the stance or posture a person takes during an interaction. Changes in footing represent shifts in the situation. These shifts are marked by cues like changing speech styles, pitch, rhythm, volume, and physical distance.

Goffman's framework breaks down the traditional model of a speaker and a listener. It separates official participants from bystanders. It also categorizes side conversations, such as whispering among active participants or chatting between participants and bystanders. Furthermore, Goffman breaks the speaker role into three parts: the animator, the author, and the principal. This breakdown allows researchers to analyze how instructors and students adopt different roles during a conversation, such as a teacher speaking as a strict rule enforcer versus speaking as a friend.

Wine (2008) provides a helpful summary of Goffman's ideas about framing, footing, and alignment. Wine explains that frames act as mental boundaries that organize interactions, set expectations, and shape how people interpret situations. Shifts in footing act like gear shifts that change the task, tone, and power dynamics between people. Most importantly, Wine expands on the idea of "alignment." She distinguishes between true solidarity, which is genuine emotional or intellectual agreement, and pseudo-accord, which is fake agreement used to manage impressions, avoid conflict, or handle power imbalances.

For this research project, Goffman (1979) and Wine (2008) provide the exact tools needed to analyze field data. If an instructor shifts from a loud, authoritative voice addressing the whole room to a quiet, casual voice while kneeling next to a student, this is a distinct shift in footing that changes the power dynamic. Similarly, when students engage in off-topic joking, researchers can use Wine's concept of alignment to analyze the situation. They can look at whether instructors join the joking to build true solidarity, or if students use the joking as a fake agreement to avoid doing their math work.

Research Site

The research site for this study is the UCLA Olga Radko Endowed Math Circle (ORMC) Advanced 2 class, an informal mathematics enrichment program designed for high school students. The class meets weekly on Sundays in Mathematics Sciences Building 5127 on the UCLA campus and is led by undergraduate instructors. Each session lasts two hours and centers around a problem packet developed by the instructional team.

Attendance typically ranges from approximately 20 to 25 students, although variation occurs across sessions. Students work through the problem sets either independently or in self-selected groups, often consisting of two to four individuals. The instructional team, composed of undergraduate students, circulates throughout the classroom, offering assistance and responding to student questions rather than directing the session from the front of the room.

The structure of the ORMC Advanced 2 class differs from traditional classroom environments in several key ways. The program operates without formal grading structures or compulsory participation, and instructors do not enforce attention or productivity. Students are not required to complete the problem sets or remain continuously engaged. Instead, interaction is largely initiated by students or guided by instructors responding to visible cues such as confusion, hesitation, or requests for help.

This organizational structure creates a learning environment in which participation, engagement, and authority are not formally imposed but are negotiated through ongoing interaction. As a result, the ORMC Advanced 2 class provides a useful site for examining how students and instructors manage autonomy and engagement in an informal, non-compulsory educational setting.

Data Collection Methods

This study is based on two participant observation sessions conducted on January 18 and February 1, 2026, each lasting approximately two hours. Both sessions took place in the ORMC Advanced 2 classroom in Mathematics Sciences Building 5127 at UCLA. During the first session, approximately 25 students and six undergraduate instructors were present, while the second session had a smaller attendance ranging from 11 to 18 students and five instructors.

I conducted participant observation by sitting within the classroom and observing interactions between students and instructors as they worked through weekly problem packets. While my primary role was observational, I was occasionally drawn into informal interactions with participants, including answering questions about my presence and engaging in brief conversations. This positioned me as a semi-participant observer rather than a fully detached observer.

Fieldnotes were recorded in real time and expanded immediately after each session. The notes focused on both verbal and nonverbal aspects of interaction, including speech patterns, tone, posture, gaze, spatial positioning, and movement within the classroom. Particular attention was given to moments where students signaled confusion, disengagement, or independence, as well as how instructors responded to these cues.

Following Goffman's (1989) approach to ethnographic observation, emphasis was placed on capturing detailed, moment-to-moment interactions rather than relying on general impressions. This included documenting specific exchanges, shifts in voice and stance, and patterns of instructor movement throughout the room. These detailed observations allow for a micro-level analysis of how engagement and authority are negotiated in real time.

All participant names used in this paper are pseudonyms.

Findings

The following findings are drawn from two observation sessions conducted at the ORMC Advanced 2 class on January 18 and February 1, 2026. The data are organized around two interconnected themes that correspond to the study's research questions. The first concerns how students signal their engagement status, whether they want help, want to be left alone, or are somewhere in between. The second concerns how instructors read and respond to those signals, and how individual instructors vary in their approach. These two themes are not independent: student signals shape instructor behavior, and instructor behavior in turn shapes how students signal. Together they paint a picture of a classroom where engagement is constantly being negotiated rather than assumed.

Student Signaling Behaviors

One of the most consistent patterns across both observation sessions was how students used their bodies to regulate instructor involvement. The most direct signal was the raised hand, which functioned as an unambiguous request for help. When one student raised his hand during the February 1 session, Adam approached immediately and sat down next to him without hesitation. Similarly, a student who wanted to show Chris his completed work presented his paper directly, saying "Look Chris I did it," making his desire for acknowledgment explicit. These direct signals appeared less common than subtler embodied cues, however. More often, students communicated their engagement status through these less explicit signals.

Headphones were one of the most visible of these cues. Several students across both sessions wore AirPods, sometimes for the entire duration of the class. Instructors generally did not approach students who had both AirPods in, seemingly reading this as a cue that the student did not want to be interrupted, whether because they were focused or disengaged. The ambiguity did not seem to matter much in practice; in both cases the effect was the same, which was that the student was left alone. Body orientation worked similarly. Students who sat facing their problem packet, leaning forward or writing, tended to attract instructor attention when they looked stuck. Students who leaned back, had their phones out, or turned away from their packet were less likely to be approached.

Not all signals were this legible, however. Maya, a recurring presence in both sessions, occupied a space that was difficult to categorize. She spent considerable time socializing, on her phone, and chatting with instructors about topics unrelated to mathematics, yet she also periodically locked in and worked through problems. Her engagement was genuinely variable rather than consistently disengaged, and the instructors seemed to recognize this. Adam commented at one point that "Maya is distracting everyone," but this was said with a degree of amusement rather than frustration, suggesting that her behavior was tolerated as part of the social fabric of the class rather than treated as a problem to be corrected. This tolerance itself reflects something about the ORMC's informal structure: the threshold for what counts as acceptable disengagement is considerably higher than it would be in a traditional classroom.

Social movement within the classroom was another form of signaling worth noting. Students would shift seats, join or leave groups, and reposition themselves throughout the session without asking permission. When Chris vacated his chair during the February 1 session, Maya immediately took it and moved to sit with a different group. This kind of spatial self-organization was unremarkable within the setting, but it communicated meaningful information about where students wanted to be and who they wanted to work with, without any of it needing to be stated explicitly.

Instructor Response Patterns

While students varied in how they signaled engagement, instructors varied considerably in how they read and responded to those signals. Across both sessions, four instructors were observed closely enough to identify distinct patterns: Adam, Ben, Chris, and Leo. Rather than representing a simple spectrum from effective to ineffective, these patterns reflect different implicit orientations toward the balance between autonomy and intervention.

Adam was the most structurally authoritative instructor in the room. He was the one who took attendance, announced the break, and used a loud voice that carried across the whole classroom. His physical presence was assertive; he gestured frequently with his hands, used his fountain pen to write directly on students' problem packets, and was one of the few instructors who spent extended time with individual students, sometimes remaining with one student for the entirety of the second half of class. At the same time, Adam's authority did not feel heavy-handed within the context of the ORMC. When Maya deflected his comment about her distracting others by saying "he hates me, you should write that down," Adam did not push back. His response was more characteristic of someone managing a social dynamic than enforcing a rule.

Ben occupied a middle position. He sat with students for long stretches, leaning forward and engaging directly with the mathematical content, and was observed on multiple occasions taking a student's packet to inspect it after asking "Can I see?" He was occasionally more assertive, asking Maya in a pointed tone whether she was going to finish the packet that day, but he accepted her "maybe" without further pressure. Ben also showed an awareness of the research process itself, joking to students that they were "participating in an anthropological study," and at one point walking behind the observer to read the field notes. His comment that the class was "particularly distracted" that day suggested he was monitoring engagement at a room-wide level, not just responding to individual students.

Chris presented the most peer-like interactional style of the instructors observed. His voice was noticeably quieter than Adam's and Ben's, and he engaged students in conversations about anime, music, sports, and personal topics with an ease that made the boundary between instructor and student feel thin. When helping students with problems he tended to kneel beside them rather than sit, alternating between standing and kneeling in a way that kept the interaction feeling informal. His response to a student's question about his middle name, where he became visibly uncomfortable when it was compared to a drug, was one of the few moments where Chris signaled clear social distance, suggesting that even his relaxed style had limits.

Leo stood out as the instructor who engaged least with students across both sessions. He was observed pacing in the center of the room without approaching anyone for extended periods, and when he did interact with students, he remained standing and spoke quietly. He stuttered slightly at the beginning of one exchange when a student asked him a question, suggesting some discomfort with the role. Unlike the other instructors, Leo did not appear to initiate contact with students and seemed to wait to be approached rather than actively scanning the room for confused faces. Whether this reflected a different pedagogical orientation or simply less confidence in the role was not possible to determine from observation alone.

Across all four instructors, one pattern was consistent across the observed sessions: none of them made sustained attempts to redirect students who were visibly disengaged. Students on their phones, playing chess online, or engaged in extended off-topic conversations were generally left alone unless they signaled a need for help. This tolerance for off-task behavior appears to be a meaningful feature of the ORMC's interactional culture, and it shaped the overall dynamic of both sessions.

Taken together, these two patterns reveal a classroom where engagement is managed through a largely implicit set of signals and responses rather than explicit rules or expectations. Students have developed ways of communicating their needs and preferences without stating them directly, and instructors have developed, with varying degrees of awareness and skill, ways of reading and responding to those cues. The result is a social order that looks informal on the surface but is in fact governed by a quietly negotiated set of interactional norms. What counts as engagement, when it is appropriate to approach a student, and how much disengagement is tolerable are all questions that get answered not through policy but through the accumulated weight of small interactional choices made across each session.

Discussion

The findings from this study illuminate how engagement and authority are negotiated in a learning environment that deliberately removes the institutional scaffolding most classrooms rely on. Drawing on the theoretical frameworks outlined earlier, this section reflects on what the observed patterns reveal about the interactional dynamics of the ORMC Advanced 2 class.

Goffman's concept of footing proved particularly useful for making sense of the moment-to-moment shifts observed across both sessions. The contrast between Adam announcing the end of break in a loud voice carrying across the whole room and Chris kneeling quietly next to a student to discuss a problem represents more than a difference in personality. These are distinct shifts in footing that signal different relationships to authority. Adam's whole-room address positions him as the principal, the person whose words carry institutional weight. Chris's kneeling posture and quiet voice position him closer to a peer, an animator of ideas rather than an enforcer of expectations. Wine's distinction between genuine solidarity and pseudo-accord also maps onto the data in interesting ways. When Chris discussed anime and sports with students, this appeared to reflect genuine rapport rather than a strategy for managing behavior. By contrast, the general tolerance for disengagement across all instructors could be read as a form of pseudo-accord, a tacit mutual agreement to avoid the discomfort of enforcing engagement in a setting where enforcement has no real teeth.

Friedman's work on epistemic authority helps explain why the instructor-student boundary in the ORMC feels as permeable as it does. The instructors are UCLA undergraduates rather than credentialed teachers, and in some cases their mathematical authority appeared less institutionally secure than it would in a traditional classroom. This means that authority cannot be assumed on the basis of institutional role alone and must be established through interaction. The moment where a student told Ben, "You already said that," is a small but telling example of a student declining to ratify the instructor's epistemic authority, something that would be less likely in a traditional classroom with higher social stakes. Dyer et al.'s concept of joint exploration also resonates here. Ben's practice of taking a student's packet and inspecting it together, or Ryan and Ben working through a problem involving the cost of a goat with a group of students, are examples of instructors genuinely sharing intellectual authority rather than simply delivering answers.

Reeve and Cheon's framework for autonomy-supportive versus controlling pedagogy helps organize the variation observed across instructors. Adam's style sat closest to the controlling end of the spectrum, not in a heavy-handed way, but in the sense that he was the most likely to use assertive language and whole-room directives. Chris sat at the autonomy-supportive end, allowing students to set the pace and tone of interactions almost entirely. Ben occupied a middle position, shifting between the two depending on the situation. Leo is harder to categorize through this framework because his limited engagement with students meant there was little data on how he responded when students needed help. His behavior might reflect an autonomy-supportive orientation taken to an extreme, or it might simply reflect uncertainty about his role. The data do not allow for a clear determination.

It is worth acknowledging the limitations of this study directly. Two observation sessions provide a narrow window into a setting that presumably varies considerably across weeks and across the academic year. Ben himself noted that the February 1 session was "particularly distracted," suggesting that what was observed may not be representative of the class at its most engaged. The study also relies entirely on observation, which means that the motivations and interpretations of participants remain largely opaque. Why Leo engaged so little with students, whether students who appeared disengaged were processing ideas internally, and what instructors understood themselves to be doing when they tolerated off-task behavior are all questions that would require interviews to answer properly. These limitations do not undermine the findings but they do suggest that the patterns identified here should be treated as starting points for further inquiry rather than definitive conclusions.

Conclusion

This study set out to examine how engagement is communicated and interpreted in a non-compulsory learning environment, using the ORMC Advanced 2 class as a research site. The two observation sessions revealed a setting that is more socially complex than its informal surface suggests. Students have developed a repertoire of subtle signals, embodied, spatial, and social, through which they communicate their needs and preferences without stating them directly. Instructors, in turn, have developed varying approaches to reading and responding to those signals, shaped by their individual styles, their comfort in the role, and the implicit norms of the setting itself.

What the ORMC case illustrates is that the absence of formal structure does not mean the absence of structure altogether. The rules governing who approaches whom, when it is acceptable to disengage, and how much social conversation is appropriate before redirecting to mathematics are all very much present. They are just carried in behavior rather than policy. This is consistent with Goffman's understanding of social interaction as governed by implicit frames and footings that participants orient to continuously, even when they are not consciously aware of doing so.

The study has clear limitations. Two sessions is a small sample, and the reliance on observation alone leaves important questions unanswered. A fuller account of the ORMC's interactional culture would require more sessions, interviews with both students and instructors, and attention to how dynamics shift across different problem packets and different points in the academic year. Within those constraints, however, the findings suggest that informal learning environments like the ORMC are worth taking seriously as sites where the negotiation of authority and engagement plays out in particularly visible ways, precisely because the usual institutional mechanisms for managing those things are not there to obscure it.

References

Dyer, E. B., Machaka, N., Parr, E. D., & Krist, C. (2021). Understanding joint exploration: The epistemic positioning in collaborative activity in a secondary mathematics classroom. In Proceedings of the 43rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 546-553).

Friedman, D. A. (2017). Negotiating epistemic authority and co-constructing difference: Socializing "nonnative speaker" teachers in a U.S. graduate program in TESOL. Linguistics and Education, 42, 14-23.

Goffman, E. (1979). Footing. Semiotica, 25(1-2), 1-29.

Goffman, E. (1989). On fieldwork. Journal of Contemporary Ethnography, 18(2), 123-132.

Reeve, J., & Cheon, S. H. (2021). Autonomy-supportive teaching: Its malleability, benefits, and potential to improve educational practice. Educational Psychologist, 56(1), 54-77.

Wine, L. (2008). Toward a deeper understanding of framing, footing, and alignment. Working Papers in TESOL & Applied Linguistics, 8(2), 1-21.