Building Metaphors that Describe Mathematics Identity and Experiences Among Mathematics Graduate Teaching Assistants

Introduction

Recent work in higher education in the United States (U.S.) has focused on closing the equity or opportunity gaps that affect the outcomes of postsecondary education for underserved populations (e.g., Ehrmann, 2021). Inequities that are embedded in educational systems are reflective of generalized bias, which is often invisible to those who are afforded privilege in higher education. Such an environment has allowed systemic issues and structural racism to flourish even in locations where it is believed that access is available to anyone who puts forth the effort. These environmental attributes are important to consider because, as Tatum (2019) has suggested, identity, or the “Who am I?” question, is informed by context. We know who we are, in part, by what others who surround us say about us through their words and actions. Put simply, if others do not believe in us or in people like us, we may have a hard time believing in ourselves. In contrast, when others believe in us or we see ourselves mirrored in others who are succeeding, we are more likely to believe we can undertake similar challenges. We undergo, usually unconsciously, identity development that is racialized due to identity markers (Gee, 2000). We know that whether we have been surrounded by people like ourselves or been a minority in the community in which we have lived, we are affected by those circumstances.

Our identity, Tatum (2019) reminds us, is multidimensional and includes features besides skin color, such as gender, sexual orientation, socio-economic status, religion, language, age, and degree of able-bodied-ness, to name only a few. Hurtado (2019) adds that the racialization process described by Tatum is imposed and reinforced structurally by institutions, law, and government and then is enacted relationally through social interactions. Harro (2000) posits that we are all part of a socialization cycle that we are born into, the mechanics already in place. However, as Harro also argues, we are not inevitable victims nor inevitable perpetrators of this cycle. We can become active participants in changing it, influencing not only our own outcomes but the outcomes of others.

In this paper, we share data from a professional development activity focusing on race. We worked with mathematics Graduate Teaching Assistants (GTAs) who engaged in reflective writing that responded to short biographies of mathematicians of color. GTAs’ reflections, as a result, focused on their own racial identities and lived experiences. This paper aims to add to the growing body of literature that focuses on student experiences. In particular, we see our work as contributing to the knowledge of GTA identity—how they see themselves as both educators and mathematicians. We see this project as a way to better understand GTA identity and see GTA professional development as a mechanism for GTA reflection on identity for both themselves and the students they teach.

Our hope is that the entry-level college teacher, today’s GTA, will one day be part of a new professoriate that will guide us toward improved educational experiences. Our exploratory study, which emerged from a pedagogical experiment, suggests that GTAs, who are both students and emerging educators, may be positively affected by the reading of biographies of successful mathematicians from underrepresented populations, especially when accompanied by self-reflexive writing. Such biographies can provide a new lens through which they can view the field of mathematics.

GTA Identity

Adiredja and Andrews-Larson (2017) argue that how students are positioned in their academic environments can impact their learning in significant ways. Students are continually grappling with their identities and classroom roles or belonging as they engage in learning experiences (Adiredja & Andrews-Larson, 2017; Boaler, 2002; Brown, 2018). It is also important to acknowledge that some classroom strategies may reinforce dominant narratives about success in a given field, such as mathematics. It is therefore important to consider the narratives and perspectives that are being centered and celebrated in course lessons. Such a consideration aligns with a tenet of critical race theory from Ladson-Billings and Tate (1995) regarding the use of first-person accounts and counter-storytelling. Ladson-Billings and Tate suggest that the centering of non-dominant personal experiences is necessary for students. This centering enables students to expand what is seen as typical and to include those who have been systematically excluded. Each person has multiple identities and they are all related to their performances in society. These identities include aspects of one’s race, gender, ethnicity, religion, moral values, discourse choices, and many other attributes. Context can range from identity in a particular working group to who someone is/becomes at family gatherings. For our purposes, we note that GTAs navigate multiple identities throughout their week, including institutional identities which Gee (2000) would describe as their role(s) or position(s) in the greater institution they are working within. GTAs fulfill the institutional roles of both student and teacher and are thus navigating their core or central identities within multiple layers of positionality.

Concerns around GTAs and identity have tended to focus on the development of teacher identity. For instance, Hesse (1993) suggested that GTAs may be in an ideal position to understand undergraduate needs due to their dual roles as both teacher and student and their rich potential for empathic understanding of the role of the learner since they share that identity with the undergraduate, albeit at a different level. Hesse also points out, however, that the graduate student must develop theories to explain their students’ practices which “invites dissonance and thus always has a cost. It denies smugly rejecting what doesn’t neatly fit our worldviews” (p. 231). Grouling’s (2015) findings, however, suggest that such dissonance sometimes exceeds the graduate student’s capability. Grouling’s research subject, “Blair,” for instance, rejected notions that undergraduates were smart and capable, stating, “I’ve been looking down on freshmen since my sophomore year” (Moving On: The Reconciliation of Student and Teacher Identities section). Grouling argues that Blaire’s inability to step out of a perception of self as a superior intellect compared to those slightly behind interfered with her ability to effectively understand and teach undergraduate students. In turn, Turnquist (2019) found that among the challenges faced by GTAs in their professional development as teachers was the inadequacy of role models among graduate teaching supervisors, leading Turnquist to argue for more strenuous review and evaluation of those supervising graduate students. These findings, when taken together, suggest that the problematic formulations of one generation are too often passed along to the next. This notion aligns with Lortie’s (1975) theorization of the apprenticeship of observation which holds that teaching is resistant to change precisely because old experience carries more valence than new learning, a point also argued by W. Bishop (1990) even in contexts of strong professional development of GTAs.

Gallagher’s work (2016) goes some distance toward explaining the difficulty of developing teacher identity in the higher education setting. Gallagher explains the challenges within the literature of communities of practices, addressing specifically the domain of mathematics education. Gallagher argues that graduate students in mathematics mainly identify as mathematicians as opposed to identifying as teachers because the context of graduate training tends to reinforce identities that revolve around research productivity and new knowledge creation. As a result, GTAs tend to see themselves, or identify, just as their own professors do, first as mathematicians. This phenomenon has implications for GTAs’ emerging effectiveness as teachers, even as Gallagher concedes the necessity of differing identities arising among those learning to teach in higher education versus those learning for K12 contexts. Nevertheless, Gallagher’s findings have implications at a time when STEM education of the populace is of increasing importance, given that most undergraduates receive their instruction in mathematics from graduate students who do not principally identify as teachers. Gallagher’s point is that since strong performance in STEM courses tends to lead to STEM majors and since GTAs carry a heavy responsibility for undergraduate teaching, especially at the introductory level, GTA identity may be substantially tied to the success of the most vulnerable college student populations.

Gallagher’s (2016) findings are particularly concerning given what McNeill et al. (2022) have found with regard to discourses of color- and gender-blindness in the teaching of mathematics. These discourses, they argue, embrace a “color-evasive, gender-neutral” understanding of the mathematics classroom, holding firm to the idea that every student has the same shot at success. Leyva, McNeill et al. (2021) further found that the logics of classrooms, including faculty positioning as ultimate authority and mathematics classrooms as sites of “weeding out” or gatekeeping, contribute alongside other enculturated structures of bias to inhibit success among underserved students (p. 800). Such findings reinforce what Leyva (2021) describes as “P–16 mathematics education as a white, patriarchal space” requiring disruption and intervention to encourage within-group solidarity (p. 118).

For some time GTA teaching identity has been discussed as a question of the priorities of the GTA, supervising faculty, and the discipline. In recent years, renewed interest in GTA identity has emerged given the central role that GTAs play in the delivery of undergraduate education and given increased recognition of structural bias in STEM classrooms. The COVID-19 pandemic was a large structural change that not only brought to light existing biases, it also inherently impacted how education was taking place and therefore, how GTAs were participating in the process. Funk and colleagues (2021) discuss how COVID-19 may have changed GTAs’ identification as teachers with most GTAs no longer identifying as teachers since their transition to asynchronous or online instruction. In their case study Kress (2020) found that GTAs’ identity within their role includes not only their sense as a teacher and student but also their own relationship with mathematics and how they may—or may not—align with the field. Bjorkman (2024) explored this development in math tutors, going beyond GTAs, and found that serving as a role model and source of support for other students provided both social and mathematical identity development. Tutors reported feeling a greater sense of belonging in the field of math as a result of their identity shifts and mentioned feeling more adequate in their math knowledge than before tutoring. GTA identity is an area of study in higher education research of increasing focus, and we call for more attention to this important contributing factor to student success, especially among underserved student populations.

In this paper we begin to explore GTAs’ perspectives on other mathematicians and how they connect their own identity and experiences to those of others. Style’s (1996/1988) work on curriculum provides us with a framework that can be modified to characterize GTAs’ experiences.

Building a House: Windows, Mirrors, and Doors

Style (1996/1988) introduced the concept of windows and mirrors in the education of learners who have been historically underrepresented in certain domains, including classroom spaces. R. S. Bishop popularized a similar notion in 1990 but focused on children’s literature—positioning books as a mechanism for expanding children’s perspectives on identity and differences. In our study, we leverage Style’s work, which focused specifically on the need for curriculum to function as both a window and a mirror, where windows allow for one “to see the realities of others” and mirrors provide opportunities for one to “see her/his own reality reflected” (p. 1). Windows in students’ educational experiences allow students to see possibilities outside of their own experiences, and mirrors enable students to see a reflection of themselves and experiences as well as their talents reflected back. These structural metaphors resonate with us. For some time, scholars have noted that the way traditional mathematics is taught is largely built in a context of whiteness (Battey & Leyva, 2016; Joseph, 2021). From Styles’ theorization, it would seem that an absence of mirrors reflecting women and students of color along with windows showing only a view of others who are unlike them may contribute to negative outcomes for underserved students of mathematics (Leyva, McNeill et al., 2021).

To elaborate, women and students of color are underrepresented in STEM (Fry et al., 2021). Their mathematics experiences are often viewed through windows of white others, with few opportunities for mirrors (Leyva, Quea et al., 2021). As Gutiérrez (2018) put it, students “should see aspects of themselves reflected back (mirror) as well as obtain views of new worlds outside of their own (windows). Unfortunately, for many students, mathematics classrooms are experienced almost exclusively as windows” (p. 1). Abo-zena et al. (2019) expand on the windows and mirrors metaphor by including doors. Doors are a way to foster engagement between diverse individuals and provide opportunities for sharing of ideas and perspectives among those with different backgrounds and lived experiences. Abo-zena et al. (p. 175) state the importance of each of these metaphorical views of the self, including mirrors so that

diverse students can see themselves and their experiences reflected in developmental scholarship, windows for the field to see authentic representations of communities and experiences that are often marginalized, and doors that enable the co-construction of knowledge and engagement in research and practice.

In contrast to mirrors and windows, doors are not specific to seeing oneself in someone else or seeing a new reality, but instead evoke the desire for or recognition of a reciprocal relationship. This desire for reciprocation demonstrates another facet of how identity is directly influenced by context and perceptions of others (Tatum, 2019). Taken together, mirrors, windows, and doors suggest structural or building components as metaphors for how individuals may visualize self in relation to others.

Positionality

We come to this analysis of diversity, equity, and inclusion in the mathematics classroom and in the professional development of graduate teaching assistants as three cis-gendered white women working at state institutions in the western part of the United States. Author 1 is a tenured associate professor of Mathematics Education, specializing in the scholarship on the teaching of mathematics and the professional development of instructors at the postsecondary level. In addition, Author 1 designs and facilitates professional development focused on active learning, equity, and inclusivity. Author 2 is a doctoral candidate in Mathematics and Science Education, specializing in critical and justice-centered perspectives in STEM education course materials. Author 2 has served as a GTA in several undergraduate mathematics courses during her graduate studies. Author 3 is a tenured professor of English, specializing in writing, rhetoric, and social change, who serves as director of her university’s teaching and learning center. Author 1 taught the professional development course that participants took.

The three of us work at the intersection of pedagogy, faculty development, and antiracist commitment in the college and university setting. The objective of this study is to address the equity and opportunity gap that is so often associated with college-level mathematics education by improving the preparation of graduate teachers.

Methods

For this project we did not set out to frame the assignment around one or more research questions. Rather, we examined online discussion postings post hoc to see what interesting stories they could tell. Our efforts began with pedagogical necessity. We sought to implement a GTA professional development activity that would support GTAs as both students and teachers of mathematics. We landed on the idea of GTAs reading biographies of underrepresented mathematicians and then writing reflectively and critically on the value of the biographies to their emerging sense of self as mathematicians and teachers of mathematics. Given that we did not start with research questions, our post-hoc examination of GTA-generated discussion posts attempted to understand the value of positive models for students. Using the concept of windows, mirrors, and doors and analyzing GTA discussion posts with these concepts in mind, we attempted to understand perspectives and experiences of the GTAs with whom we worked as evoked by the reading of mathematicians’ biographies.

The reason for giving this assignment was to provide GTAs examples of resilience to validate the challenges and lived experiences that GTAs experience in graduate school. While it is widely understood that GTAs grapple with the complexities of dual roles (teacher and student), with advancing their education by undertaking tasks such as identifying an advisor, and undertaking the challenge of a sustained, graduate-level research project, our findings suggest that GTAs also sometimes struggle to see themselves as mathematicians. Self-doubt is amplified by prolonged and pervasive experience with bias.

Data Collection and Analysis

Each of the seven GTAs made an original post, and then subsequently responded to one or more posts from other GTAs. As can be seen in Table 1.

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Table 1.

Discussion board posts and responses layout by GTAs and instructor.

Initial post by:	Thread 1: Ricardo	Thread 2: José	Thread 3: Silvia	Thread 4: Mai	Thread 5: Kevin	Thread 6: Elena	Thread 7: Maya
Replies:	José	Silvia	Ricardo	Instructor	Elena	No responses	No responses
		Ricardo	Mai
			Maya

As Table 1 shows, each GTA was associated with a thread and all GTAs except Maya and Elena had at least one response to their initial post. We note that Maya and Elena posted their reflections just before the due date, which may explain why they received no responses from their posts.

Two of the researchers completed initial codings of the GTAs’ responses independently using inductive open coding (Miles & Huberman, 1994; Strauss & Corbin, 1994). This initial step allowed the themes to emerge organically before the research team reviewed the themes as a group. The full team met to discuss the themes that were arising in the data. The use of thematic analysis allowed the researchers to then identify common patterns emerging in the excerpts (Clarke & Braun, 2014). Among the patterns, it was noted that each GTA responded to the prompt uniquely but there were several themes—belonging, inequity/equity, identity, building community, shared experiences, inspiration, realization, imposter syndrome, and representation—that appeared across multiple individuals. Once patterns were identified and agreed upon by the researchers (all authors on this paper), final codes were established. All researchers coded all data separately and then resolved all discrepancies through discussion.

It should be stated that from the start we sought to better understand the perspectives of mathematics GTAs by examining their reactions to stories from Living Proof. We did not seek to make generalizations about their promise as graduate students, as mathematicians, or as future teachers. Working from the data inductively towards conclusions, subsequent to initial coding, we sought literature that would help us to understand and frame the themes we were seeing in the responses. We consider this post-hoc identification of pertinent literature and framing to be part of our method. The work of Style (1996/1988), Gutiérrez (2018), and Abo-Zena et al. (2019) proved useful as a framework for connecting the themes we were seeing. This literature uses physical features of buildings, or what we called “house metaphors,” such as mirror, window, and door, to describe the ways that GTAs encounter role models that are more or less helpful to their visions of themselves as future teachers of mathematics. Aligning with the house metaphor, we added the code locks to capture statements that highlighted obstacles, challenges, or adversity that the GTAs described.

This framework was applied twice—once to solidify code definitions and a second time to reexamine uncoded excerpts. The final set of codes and their meanings is an agreed upon synthesis of each coder’s individual codings (Table 2).

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Table 2.

Final codes applied and corresponding examples.

Code	Application of code to excerpts	Example
Mirror	Recognition of or identifying with another’s reality or experience	“I was looking at all the successful faces in this chapter trying to find myself. I was looking for someone who shared similar experiences of isolation and doubt in math as a Latina”
Window	Seeing another’s reality or experience. Relating one’s own experience to another’s different reality	“I found it surprising that Dr. Harris still continues to experience harassment about being a woman of color in math as a professor and as someone who has earned a PhD in mathematics”
Door	Includes a call to action for change or an opportunity to co-construct or share knowledge or experiences	“I think we as women of color need to embrace the responsibility of being more vocal so future generations don’t feel like imposters”
Lock	Recognizes a need in the field or context that is related to social justice or equity. This does not include a call to action	“Unfortunately, the STEM industry is still ran [sic] by mostly men and it is sad to see that when you look around your classes, there are very few women”

This coding framework allowed for the distinction of identity of self (Mirror) and identity of others (Window). For example, the following two excerpts from Silvia’s and José’s initial posts illustrate the Mirror and Window codes, respectively.

Excerpt 1 (Silvia; Mirror): I was looking at all the successful faces in this chapter trying to find myself. I was looking for someone who shared similar experiences of isolation and doubt in math as a Latina...I felt heard and my feelings were validated. It’s as if someone has understood my academic journey so well and looked at the positive that has come from it.

Excerpt 2 (José; Window): I found it to be very inspiring to see not just a woman in mathematics, but a mother raising a daughter while trying to earn a PhD. And I was pleasantly surprised to find out that she was able to find a program that was specifically designed to help bring diversity to the graduate program she was in.

Silvia was looking for a mathematician who had experiences that aligned with their own, and José was inspired by the experience of someone else, which was different than their own. José goes on in their post to state that they “connected with the part of her story when she spoke about her parent’s lack of education.” This aspect of José’s post was coded as a Mirror as it was an aspect with which they could personally identify.

Some definition refinement after selecting the four codes included the clarification that a Door can involve an existing two-way relationship between individuals or groups or illustrate the need for that relationship. This distinction can be seen in the two excerpts below (underlined). Excerpt 1, written by Mai, takes an existing relationship between GTAs and encourages ongoing support for each other through that relationship. Excerpt 2 from Silvia references the need for a Door to raise awareness of how parents’ educational experiences influence their children’s experiences.

Excerpt 1 (Mai; Door): I am so impressed with your academic journey and inspired and encouraged for how hard we are all working towards our dreams. Let’s help each other on the path.

Excerpt 2 (Silvia; Door): I also connected to Alejandra's experience feeling as though she needs to take full advantage of higher education when her parents lacked those opportunities. It should be talked about more and those conversations start among us as graduate students.

Locks were originally coded as challenges or obstacles. Upon further examination, the specific challenges and obstacles highlighted in excerpts were related to social justice, equity, and/or inclusion. Thus, we found the code Lock to better capture this phenomenon and support the house metaphor. In the exchange below between Kevin and Elena we can see the Locks (underlined) reflecting the issues of representation in STEM.

Kevin’s initial post (Lock): I chose John Urschell, because he is a black mathematician, and we have similar backgrounds. When I was reading his story, something I found surprising was that he teaches at MIT in Massachusetts. To be a black and a professor at that school, it's impressive, because I can only imagine how very few black instructors they had in the past. Moreover, there aren't many black instructors and mathematicians in the math universe. I wish the [sic] wasn't the case, but that's the reality. I want to emulate what Dr. Urschell has done, and be the best mathematician/professor I can be, so one day I can aspire [sic] other black students and minorities to maximize their mathematical abilities, and also help them understand there are other jobs out there besides being athletes or musicians.

Elena response (Lock): I love this post! Like you, I also want to inspire women and minorities to try their hand at the STEM field. Unfortunately, the STEM industry is still ran by mostly men and it is sad to see that when you look around your classes, there are very few women. I want my students to see themselves in me, and if I could finish it, they can to.

The four codes (Mirror, Window, Door, Lock) were applied to the data through multiple rounds of code refinement and inter-coder discussion. In the next section we discuss these results and demonstrate patterns we see in the responses.

Results

Overall, the Mirror code was the most common (Figure 1). The Mirror code occurred in all seven discussion board threads, many of which contain posts by more than one GTA.OPEN IN VIEWER

Frequencies were counted by the occurrence of an idea per respondent. The following excerpt, for example, contains two instances of a Mirror (underlined), but both are referring to the same idea where the GTA has identified parallels in their own mathematical experience. Thus, the Mirror code frequency in the following post from Elena is one.

I chose to read Jennifer Quinn's story, An Accidental Mathematician. The reason why I chose this is because I could relate to her title. In the beginning, I did not love Math either and actually was not very good at it. In time, after a lot of hard work, I saw that it was tolerable and it became a subject I saw massive improvement. After majoring in it, I began to truly see the beauty of it and how we can use math to make sense of the world. I began to see why it was so important to study it, not just because someone told me to. Jennifer said that she felt like she was the “dumbest of the smart kids” and sometimes, you have to “fake it until you make it.” This was surprising as I never thought anyone could sum it up so well. She knew exactly how I felt at times. Unfortunately, I still do feel that way but it is reassuring that I am not the only one who feels this way.

The above quote demonstrates how reading the story was “reassuring” as it positively comforted this person with regard to their own fears and insecurities.

The following exchange between Silvia and Ricardo highlights their similar feelings of isolation, doubt, and imposter syndrome. Because these ideas are echoed by each of their experiences, the Mirror code frequency is two, once for Silvia and once for Ricardo. Although there are two sections of Silvia’s post that are underlined, the same idea is being discussed and connected to the autobiography that Silvia read (Dr Pamela Harris).

Silvia’s initial post:

Silvia mentioned how reading the autobiography was a “breath of fresh air” and they were able to feel “validated” by learning about the mathematician’s experiences. This sense of shared experience was then reinforced by Ricardo sharing in feelings of insecurity. The following example illustrates two applications of a Mirror code in a statement from Ricardo:

There were a few parallels I saw between Tim's story and my own. He says “I thought of math as simply something I was good at.” I can say with confidence that I have had this exact same thought before. I always excelled in mathematics as a kid, but I thought nothing of it. In fact, until 10th grade, I was pretty certain that I was going to join the workforce once I graduated high school. Funny enough, another instance where Tim's story and mine were similar is that the first instance where he found joy in mathematics was 11th grade.

The first Mirror reflects the parallel between Ricardo’s and the autobiographer’s (Tim’s) abilities to do mathematics, and the second Mirror reflects the joy in mathematics that they both experienced during their high school years. In response to Ricardo, José responded: “I think it’s amazing you were able to connect with Tim’s story in the way that you did. I think being able to find joy in mathematics at a young age is very important and should be encouraged more often.” Here, José found it “amazing” to see these connections between different life experiences in mathematics. Further, the underlined text represents a Door, as it highlights a call to action.

Every Lock code was linked to a Mirror code. Five of the six Locks occurred where GTAs were looking for a connection to autobiographers through personal challenges. In these cases, they saw the obstacles that they themselves had faced in their mathematics career reflected back by successful mathematicians who looked like them (e.g., Silvia’s initial post). One of the six incidents of Lock occurred when Ricardo related to parents’ lack of education discussed by José:

I didn't read Alejandra Alvarado's autobiography, but based on your post, I have some idea of the themes and struggles Alejandra faced. I agree that the lack of education of parents is something that is not pointed out enough. I am a first-generation college student, and I believe I can bring a higher standard of living to my family (not just on the financial side).

Often Locks were followed up by Doors in discussion threads. When a challenge or obstacle was brought up, it was not uncommon for a call to action to be offered.

As might be expected, Doors were never stand-alone statements in the GTA responses. Doors linked or connected with other statements that had other codes. In other words, Doors—the desire to co-construct knowledge or a call to action—came up when a GTA was talking about a Mirror, Window, and/or Lock. For example, when José talked about connecting with her autiobiographer’s (Alejandra) story, Silvia shared a similar connection, stating “I also connected to Alejandra’s experience feeling as though she needs to take full advantage of higher education when her parents lacked those opportunities. It should be talked about more and those conversations start among us as graduate students.” Here Silvia saw aspects of themself (i.e., Mirror) in Alejandra’s story and used that as a springboard to call for action, increasing conversations about being the first in one’s family with certain educational opportunities. Overall, Doors often highlighted the importance of (1) diversifying STEM, (2) students seeing instructors from diverse backgrounds, and (3) having conversations to raise awareness about these issues.

An interesting aspect of the discussion board was that there were statements by GTAs that carried meaning that did not fall into the Window, Mirror, Door, and Lock codes. These were generally self-reflective statements, which is not surprising, given that the assignment itself called for self-reflection. Such “outlier” statements included discussion about finding inspiration (in others or self), loving mathematics, and recognizing the impact one’s story can have on others. While similar to Doors, these statements were different from Doors as there was not a call for action or a recommendation for change. Ricardo, for example, noted “I am a first-generation college student, and I believe I can bring a higher standard of living to my family (not just on the financial side).” This was a personal statement and indicated a goal that Ricardo had for themselves. Self-reflective statements that did not fall into our coding paradigm were found in all threads and added depth to the “house metaphors” of Mirror, Window, Door, and Lock, causing us to wonder if they serve as sufficient analytic tools.

Discussion

We were not surprised that the Mirror code occurred more often in responses than the Window code, as mirrors are seeing one’s own reality reflected in another’s experience and windows are seeing a new reality that is not similar to one’s own. The autobiographies in Living Proof were stories that documented challenges scholars encountered (and continue to encounter) while studying mathematics. Graduate school is often a difficult time for GTAs, and the autobiographies they read may have provided validation for their own lived experiences. When selecting a story, GTAs seem to have been drawn to those that were familiar as they often selected scholars of color who may help them to understand and identify overlaps with their mathematical journeys. Interestingly, GTAs expressed surprise that successful mathematicians had lived experiences that were similar to those of the GTAs, including imposter syndrome.

Relatedly, most of the obstacles highlighted by the GTAs were those related to lack of representation in mathematics with regard to gender and race. In response, GTAs advocated for change, calling to support each other, advocating for broader representation of people of color in mathematics, and increasing awareness about lack of representation. Because of this, they, as future mathematicians of color, seem to have recognized the importance that their own representations can have on students with whom they interact.

The application of building/structural metaphors to the data collected from the reading responses of GTAs has provided support for the idea that students benefit from visions and versions of futures that lie outside the graduate student’s direct experience. For us as researchers and educators, the use of metaphor here has helped deepen our understanding of these GTAs’ lived experiences and identity. However, we recognize that this activity has only captured a small piece of their lives. Our sense is that the Mirrors, Windows, Doors, and Locks metaphors of this study are helpful in offering insight into the experience of the GTA as an emerging mathematician. Through the careful deployment of metaphorical analysis, teachers like ourselves can come to new understandings of the self-perceptions of the GTA.

The context of reading and writing about mathematicians of color may be an important component of discovery and application to self (i.e., self-reflexivity). Such reading and writing provides an enabling context where reading and then writing about others’ stories is important to the visioning of one’s own future, perhaps particularly for budding mathematicians of color and mathematics teachers of color. In particular, centering voices and personal experiences of those who have been historically left out of mathematics spaces can help to improve mathematics as a field and push it to allow room for wider inclusion.

Limitations

We note that our study has multiple limitations that impact generalizability. First, this study has a small sample size of seven mathematics GTAs about whom we did not have detailed information with regard to their racial, ethnic, and gender identities beyond what they declared in classroom conversation or in the discussion board threads. The data also reflects their responses to a single set of prompts and does not include in-depth details about their own lived experiences in relation to their reflections to the prompts. These responses were analyzed and coded under the lens of three white woman-identifying researchers (two tenured faculty members and one graduate student), and the biases and privileges that impact those perceptions regardless of the care taken by the researchers must be acknowledged. In addition, the study was conducted remotely, and instruction took place via Zoom, as it occurred during the COVID-19 initial outbreak in 2020. Also, a word may be in order here about the use of the “house” metaphor which allowed us to develop shared meaning from the data. We found that there were elements of the data that could not be captured with the house metaphor so while the tool was generative, it was also limited. There were important reflective statements, for example, that did not fit into the house metaphor.

Implications

Our experience suggests several ways that metaphors might be used and analysis explored further through consideration of: (1) GTAs as teachers and students of mathematics, and (2) our role as PD providers. Exploring these perspectives may provide insight into the potential benefits of using metaphors to better understand student experiences, barriers in mathematics, and mechanisms for preparing GTAs for their teaching.

Conclusion

Utilizing activities centered around student identity in a classroom can be a meaningful experience for all involved, but it is important to implore a strong sense of mindfulness from the instructor on how the activity may impact students. A reason the discussion board activity led to rich data is because it provides openings for GTAs to reflect on their own experiences and identities within mathematics or school. While this is designed to provide learners with mirrors and windows into other mathematicians’ lives, it has the potential to elicit reflections on challenging experiences on their math journey. This is not something to avoid but should be considered when using a similar assignment with other students. The discussion board format allowed students to spend time selecting a biography, responding with any reaction they had, and commenting on any of the other posted responses. Instructors should prioritize similar methods of interacting with this content to allow students to process and reflect in ways that make sense for them without the peer and time pressures of a traditional in-person classroom task.

While this study was not driven by a predetermined research question, the responses from the GTAs on the reflective writing assignment were more complex than expected, which prompted further investigation. In particular, we wondered: How do biographies of mathematicians of color prompt GTAs to reflect upon their own lived experiences in mathematics? The complexity of GTA responses and discussion board posts in response to reading these mathematicians’ biographies showed us that responses were not just commentary on mathematicians’ biographies. Rather, GTA responses connected their own lived experiences to that of the mathematicians of whom they read. In addition, when GTAs’ experiences resonated with challenges that mathematicians also experienced, GTAs were prompted to think about making change. A dawning awareness allowed for GTAs to link their own student experiences as both undergraduates and graduate students to their dual role as teachers. We are reminded again that GTAs’ dual roles as students and teachers position them uniquely to manifest their rising awareness of the complexities and opportunities associated with learning and teaching (Hesse, 1993), perhaps particularly in mathematics.