Chapter 6: “I Have Gotten Braver”: Growing and Sustaining Critical Mathematics Pedagogies Through a Teacher Community of Praxis
by Emma Gargroetzi, Izzy Hendry, Angela Jeffreys, Andrew Patel, Gina Wei & Critical Mathematics Teachers Collaborative - 2021
Background: Mathematics education is not often identified as the locus of radical social change work, with these topics assumed instead as fodder for social studies or language arts lessons. As such, teachers of mathematics can struggle to find avenues for their commitments to social and educational justice in their mathematics teaching spaces.
Purpose: This study examined the practice and experiences of 10 math educators participating in a voluntary teacher learning community focused on critical pedagogies and math. The purpose was to identify the core learnings and challenges made possible through this learning community.
Setting and Participants: The Critical Mathematics Teacher Collaborative (CMTC) consists primarily of preservice and early career K–12 teachers, all of whom teach math and seek to develop their own math teaching practices through frameworks of critical pedagogy and social justice. An informal, nonhierarchical learning community, CMTC uses a cycle of critical reflection and action: We read and discuss theory to inform visions for critical mathematics teaching, and we workshop participant-designed lesson plans to support moving from vision to action.
Research Design: The study was collaboratively designed among members of the group as a self-study. Bidirectional interviews were conducted among 10 participants, transcribed, and analyzed.
Conclusions: Four core insights were identified. Participating teachers (1) desired to be able to engage in critical work in mathematics teaching spaces, (2) were nurtured by accountability to a community that supports putting ideals into practice and continuing to examine these ideals, (3) benefitted from praxis—having a space to connect theory to the practices of daily classroom teaching, and (4) provided ally-ship to each other in the face of challenges to teaching for social justice, shaping not only their mathematics teaching but also teaching in spaces beyond mathematics. Examples of critical mathematics pedagogies in action in the classrooms of participating teachers are included, as well as appendices with readings and a sample agenda for use by teachers wishing to model a learning community of their own after this one.
Many teachers enter the profession motivated by the idea that as educators, they can empower children to change their circumstances and to change the world. Others become teachers for less explicitly emancipatory reasons but are motivated to explore questions of power, privilege, and educational justice as they come to know their students who, along with their families and communities, have rich histories of resisting oppressions often perpetuated through schooling. Mathematics education is not often identified as the locus of radical social change work, with these topics assumed instead as fodder for social studies or language arts lessons. As such, teachers of mathematics can struggle to find avenues for their commitments to social and educational justice in their mathematics teaching spaces.
The Critical Mathematics Teacher Collaborative (CMTC) is a teacher-initiated informal learning community focused on bringing critical pedagogy to life in elementary and secondary mathematics classrooms, formed in response to the desire to do so and the belief that it is possible. We believe that mathematics education is an appropriate and necessary space for this work. We engage in critical pedagogical practices together, modeling our group after Freirian culture circles (Freire, 1970, 1971), to learn in an informal and antihierarchical community of praxis. Through cycles of critical reflection and action, we explore and move toward enactments of critical pedagogies in our own mathematics teaching. We read and discuss theory to inform visions for critical mathematics teaching, and we workshop participant-designed lesson plans to support moving from vision to action. This chapter shares findings from a self-study conducted by CMTC participants, shedding light on how CMTC has functioned as a community of praxis to support the growth and sustenance of critical orientations and pedagogies in our mathematics teaching and beyond.
CRITICAL MATHEMATICS PEDAGOGIES
Educators who draw on critical pedagogical traditions highlight the dimensions of power and identity throughout their work, and they work with young people to develop the tools to identify, analyze, and respond to problems in their own lives and in the broader world. Critical mathematics pedagogies therefore not only involve the teaching and facilitation of conceptual, inquiry-based mathematics, but also take on issues of power in the world and in the worlds of the participating children (Frankenstein, 1983; Tutak et al., 2011).
Concerns about mathematics education quality in the United States tend to be oriented toward access to participation and success in a fast-paced, rapidly technologizing capitalist economy (National Commission on Excellence in Education, 1983; National Council of Teachers of Mathematics, 2014). Widespread mathematical literacy is communicated as a vital public good to the extent that it supports the economic competitiveness of the United States. In contrast, critical orientations to the purposes of mathematics education support the importance of mathematical literacy to democratic participation (Moses & Cobb, 2002) and citizenship (Tate, 1995), and promote the power of mathematics education as a way not just to play the game of participation in our current world, but also to develop tools to change the game itself (Gutiérrez, 2009).
We use the phrase critical mathematics pedagogies to reference the multiplicity of definitions and priorities that inform mathematics educators with critical orientations. This umbrella term includes, but is not limited to, critical mathematics education (Avcı, 2018; Frankenstein, 1993; Skovsmose, 1994), math for social justice (Gutstein, 2006; Gutstein & Peterson, 2005; Kokka, 2015), culturally relevant mathematics (Leonard, 2008; Tate, 1995), and culturally responsive mathematics (Aguirre & del Rosario Zavala, 2013; Gay, 2009). What all these pedagogies share is an understanding that mathematics education and mathematics itself are human endeavors that are neither value-free nor neutral.
As such, critical mathematics pedagogies entail examining both the role of social, political, and cultural norms in organizing who has access to mathematics and whose ideas are recognized as mathematical in classrooms and throughout history, and the role of mathematics in organizing and sustaining existing power dynamics or providing tools for disrupting and reorganizing these relations of power. Finally, central to each of these critical pedagogies is an emphasis on praxis (Freire, 1970)the coming together of theory and analysis with practice and action.
Tate (1994) provides an example of critical pedagogy in a mathematics classroom: A teacher initiated a unit by eliciting student concerns based in their daily lives. The class eventually decided to explore the problem of students feeling uncomfortable when they encountered inebriated or unconscious people on their daily walks to school. Through a mapping activity, students discovered an overrepresentation of liquor stores in their neighborhood, which they traced to commercial zoning differences. The students then learned how to use infographics to present their data in compelling ways and successfully petitioned their local government to change the inequitable zoning practices. This example captures many of the core tenets of critical mathematics pedagogies: Beginning with concerns generated by the children themselves, students use mathematics to make sense of this world, and then they use mathematics to communicate about the world to others in a way that puts their knowledge into action and compels change.
THE CRITICAL MATHEMATICS TEACHER COLLABORATIVE
The Critical Mathematics Teacher Collaborative (CMTC) is a teacher-initiated informal learning community that brings together preservice and early-career teachers to discuss readings and workshop lesson plan ideas that attempt to bring critical pedagogy to life in the mathematics classroom. During summer of 2016, Emma was an instructor in a required course focused on issues of equity in schools. The course was taken by elementary and secondary teacher candidates from all content areas. When the course ended, one preservice math teacher asked, But what does this all really mean for my math classroom? Others echoed a desire for continued conversation around how to enact humanizing, culturally relevant, and other critical pedagogical approaches in their mathematics teaching. The group formed in response to this desire and began to meet monthly. Four years later, the group includes preservice and early-career teachers in both elementary and secondary teaching contexts, as well as doctoral students studying mathematics education. Monthly meetings usually bring together between five and 15 people. Some people attend only one or two meetings, while others sustain participation through their university studies and into their teaching careers.1
Although most teacher education programs have at least one course that introduces preservice teachers to questions of equity, such as the one for which Emma was an instructor during summer 2016, this is insufficient for developing and sustaining a teaching practice that fundamentally challenges daily practices of schooling present in most schools across the United States. Even in schools or preservice programs organized around teaching as a social justice practice, critical mathematics pedagogy is not the norm. However, standards for mathematics teacher preparation include an awareness of the history of power and privilege in mathematics (see Indicator C.4.4, Association of Mathematics Teacher Educators, 2017). Furthermore, as evidenced in the formation of our group, many teachers desire opportunities to explore theory and enactment of critical pedagogies in mathematics. Teachers leave the profession at alarming rates, especially in their early yearsparticularly when they find themselves unable to actualize a vision for education that enfranchises and empowers children (Glazer, 2018). But, to actualize such a vision demands a concerted effort, over time, with a strong community of support.
CHOOSING TO LEARN IN A COMMUNITY OF PRAXIS
Teacher professional learning communities (PLCs), a common school-based professional learning structure, is one model meant to support school improvement through teacher learning. The structure brings together Deweyan notions that learning happens through inquiry based in problems of practice, and builds on an ethic of care and mutual support central to the notion of community (Stoll et al., 2006). Research also shows that teachers find professional learning opportunities most meaningful when they choose them for themselves (Boston Consulting Group, 2015). Long before the institutionalization of PLCs, Freirian culture circles were being used in Brazil as spaces for learning in community and drawing on lived experience to explore political consciousness. In a message to facilitators of culture circles, Freire (1971) wrote,
A culture circle is not a school, in the traditional sense. In most schools, the teacher, convinced of his wisdom, which he considers absolute, gives classes to pupils, passive and docile, whose ignorance he also considers absolute. A culture circle is a live and creative dialogue, in which everyone knows some things and does not know others, in which all seek, together, to know more. (p. 61)
Although it exists in a vastly different context than Freires 1970s Brazil, CMTC draws on the culture circle model to provide a space where novice and early-career teachers can come together in creative dialogue, asserting that everyone knows some things and does not know others, and that together we can learn more. Specifically, we use this model, a critical pedagogical approach, as appropriate for exploring critical pedagogies and what they may mean for the teaching and learning of mathematicsthe thing that we collectively want to learn more about.
Over the past four years, we have three times kicked off the year with a discussion of the Tate (1994) article with the liquor store example. We return to this example as a touch point for the multiple ways it can inform our visions of critical mathematic pedagogies. Still, we understand that every enactment of critical pedagogy will be different. CMTC participants arrive with different priorities and experiences that shape our interest in, and goals for, learning about critical mathematics pedagogies. Some of us are oriented to issues of racial justice, while others are initially motivated by questions of gender equity or class. By bringing together our diverse priorities and experiences, we each learn from each other. Aligned with Freires insistence that learning happens when it is built from lived problematics in the lives of learners themselves, we leave our definition of critical mathematics pedagogy open, such that the group may pursue the problematics brought to it by those participating.
As such, rather than a strict definition, we set out two guiding principles for our work:
We are all learners as well as people with expertise to share. No single person is the expert. We all have expertise from our own lived contexts, and that expertise is expected, respected, and inspected together. We recognize and embrace this antihierarchical approach as distinct from most models of teacher education and professional development.
We learn together through praxisthe cycle of critical reflection and action (Freire, 1970). We must engage in praxis together if we hope to engage it with our students. Through praxis, we are able and expected to read and unpack theory from different scholars and traditions, apply it to our own contexts, reflect, and repeat, all with the support and encouragement of our community.
Our model is designed to bring together theory and practical implications in every meeting. More than just a book group or a lesson study group, we examine and develop practices of critical mathematics pedagogy in our meetings by combining reading discussion and lesson workshopping. (See Appendix A for an elaborated version of our meeting model accompanied by considerations for starting your own critical mathematics teacher collaborative.)
Rather than asserting one shared definition of critical mathematics pedagogy, we read widely and discuss a range of theory and approaches. Some texts have included, Framing Equity: Helping Students Play the Game and Change the Game by Rochelle Gutierrez, chapters from Reading and Writing the World With Mathematics by Rico Gutstein, and A Talk to Teachers by James Baldwin. Readings have focused on critical pedagogy, culturally relevant, responsive, and sustaining pedagogies, math for social justice, racial justice, gender equity, the involvement of families and communities in classrooms, and questions of equity in mathematics more broadly. We intentionally rotate the role of reading discussion facilitator, taking turns coming up with questions and formats for conversation and actively facilitating during the meeting. (Find a living list of our compiled readings in Appendix B.)
The second half of each meeting is dedicated to workshopping. One or two group members share an idea for bringing critical mathematics to life in their classrooms. This idea can be in any stage of development, from a fully articulated lesson plan with activities, to a nascent inspiration. Group members engage with the work that is shared and provide feedback in the form of questions, suggestions, and encouragement.
Even with the limited documentation of critical pedagogies in mathematics, we know that we are not the first to take this on, and we celebrate that we are not alone. Others have written about their journeys to becoming critical mathematics pedagogues, and the challenges and successes they have faced. There are the obvious challenges of institutional constraints: time, prescribed curricula and state and district assessment demands. As Kokka (2015) and Rubel (2017) noted, even in spaces where teachers desire to teach with critical orientations, there is often a dilemma of existing biases that points to the importance of interrogating and nurturing sociopolitical consciousness. Stinson et al. (2007) who researched their own experiences as teacher and students in a university course on critical theory and teaching for social justice in mathematics, found not only that those in the class were able to develop new language with which to think and talk critically about mathematics teaching, but also that the legitimization of a desire to teach math for social justice was centrally important. In undertaking a self-study within CMTC, we hoped to learn more systematically what being a part of the group meant to us and our developing practices, as well as what might be particular to our model for learning together that mirrors tenets of critical pedagogy that we work to develop in our teaching.
SELF-STUDY AND CO-CONSTRUCTING MEANING
Early in 2018, when the group was in its second year, we began a process of self-study. That fall, we presented the model of our group at a conference, Creating Balance in an Unjust World, held in San Francisco, California. The positive response from participants in our session was overwhelming, far greater than we had expected. Given the value we experienced participating in CMTC ourselves, and the interest expressed by others, we decided to embark on a more systematic process to understand and document what we were learning together and the impact of participation in our lives.
To begin this process, we wrote a set of interview questions and then organized ourselves into pairs or trios to conduct bidirectional interviews. Ten people participated in these interviews, and two more responded to the questions in written form. Each participant in the bidirectional interviews took turns asking and responding to each of the questions. Participant roles and identities (teaching role, CMTC participation, and racial/ethnic and gender identity) are embedded after their names the first time each person is quoted in order to connect voice and positionality more directly.
We asked each other these questions:
Why did you join this group? What keeps you coming? What do you get here that you dont find elsewhere?
How has this group impacted your ideas about what it means to teach math or your teaching practice itself?
Can you describe a lesson, unit, or task you have implemented (or hope to) based on your work with the group?
What challenges have you faced or do you anticipate facing in terms of implementing critical mathematics pedagogy in your classroom?
All interviews were transcribed. Multiple members of the group read through the interview transcripts and annotated them based on recurring themes. Through reviewing each others written notes and follow-up conversations, we developed these themes into the four core insights presented next.
In presenting these insights, we use quotations directly from the interviews to illuminate each insight. We recognize that the use of interview transcripts as a form of text is nontraditional for scholarly writing. However, we made this choice with intention, in alignment with our commitments to antihierarchical participation of teachers and researchers and a Freirian orientation to valuing local experience alongside existing published texts, as we do in our meetings. In the case of a self-study and the work of coauthoring a chapter, while Emmas professional commitments include traditional scholarly research and writing, the professional commitments of the participating teachers entail long hours with children, preceded and followed by more hours in preparation for and reflection on further facilitating the education of those children. The incredible labor of teaching simply does not leave much time for traditional notions of academic writing. This means that while there is much research on teachers, teacher voices are largely absent in the space of education scholarship (Cochran-Smith & Lytle, 1990). To provide space for the voices (in this case, literally) of teachers themselves in this piece, we use portions of transcript from audio records as text. We draw our inspiration for this format from the published book We Make the Road by Walking (Horton & Freire, 1990), which turned a recorded conversation between Paolo Freire and Myles Horton into bound and published text.
We collectively made decisions about the transcript portions to include based on the core insights we identified, with the intent of illuminating each insight through a diverse and multivocal lens. Hence, we chose quotes to avoid repeating the same idea (even though many ideas were repeated across interviews), instead providing a range of perspectives from the group. The quotes we share represent a diversity of perspectives and experiences, with no single voice intended to speak for the collective.
Reflections from practice were written, collected, and collectively reviewed two years after the bidirectional interviews. We collectively selected the reflections to include, again opting to represent a range of CMTC teacher experiences with critical approaches to mathematics, evenand especially whenthey might diverge in the ways that critical approaches to mathematics were understood or enacted. We do not hold ourselves to a single convergent definition or conceptualization of what counts as critical mathematics pedagogy, as long as it addresses questions of power in the lives of teachers and students.
We assert that we, and likely other teachers of elementary and secondary mathematics in particular, need not only communities of practice, but also communities of praxisnonhierarchical learning communities in which they can participate in the cycle of reflection and action, develop powerful visions for critical mathematics teaching, and hold each other accountable for the work it takes to make these visions real. To pursue critical mathematics, teachers need both a vision of what it might be and a community with whom to grow and sustain that vision through action, even as our visions and enactments may differ.
We identified four core insights: (1) We desire to be able to engage in critical work in our mathematics teaching spaces, (2) we are nurtured by a community that holds us accountable for putting our ideals into practice and that challenges us to continue examining these ideals, (3) we benefit from praxishaving a space where we can connect theory to the practices of teaching in our own classrooms the next day, and (4) the ally-ship we provide each other in the face of challenges to teaching for social justice shapes not only our mathematics teaching but also our teaching practices in spaces beyond mathematics.
INSIGHT #1: WE DESIRE TO BE ABLE TO ENGAGE IN CRITICAL WORK IN OUR MATHEMATICS TEACHING SPACES.
Many of us came to teaching for its liberatory potential, yet teaching for social justice in formal institutional spaces is a constant upstream battle, and we have few models when it comes to mathematics. We are passionate about pursuing our commitments and further developing them, pushing ourselves to be more critical of and more informed about the multiple perspectives and experiences from both scholars and local communities. As preservice and early career teachers, we are hungry for spaces to explore and nurture critical orientations. What has been provided through our formal teacher education programs and professional development opportunities at the school and district levels is insufficient. We needed more, we found each other, and we created CMTC together to partially fulfill this desire.
Gina WeiHigh school math teacher, cofounder and member since 2016, Chinese cis woman: Our schooling and equity class was great for having conversations about the role of school and entire landscape of inequalities, and how school fit into it, contributes to it, disrupts it and how the outside world seeps into school. Those are the big questions that I came to teaching to keep pushing on. I really wanted to reflect more on the intersection of social justice and math, because I feel like math is the hardest subject to teach critically. Ive only ever experienced [social justice teaching] in kind of niche classes, not in traditional main subjects. I was really hungry for more. [Also,] I just really loved the dynamic of the group, that everyone had something to contribute and that everyones experiences were honored. We get to talk about what is happening in our classrooms, or the baby ideas we have, bring them to the group and have them come to life. I was really searching for more of those conversations, more of those people I could connect with who shared that same interest in critical math pedagogy.
Marcos MuñozSecondary mathematics teacher, participated 20172018, White Latino cis male: Education is my avenue for social justice. I always believed math education in particular had a very important space in terms of social justice education. I always held onto that, but I never could see why exactly. I thought being here at [the University], a lot of great ideas are out there that Id love to hear. That was definitely the reason I came. The people who show up to the group, the ideas they have, the experiences they share, and the readings too. The readings were a huge boost. I remember trying on my own to find a lot of these kind of articles and authors. I ended up with a very short list and couldnt find a way out of it. Having a steady stream of things to read or to listen to [and] people to talk with was super beneficial.
Izzy HendryElementary teacher, cofounder and member since 2016, White cis female: I started going to this group because I felt like there wasnt a lot of discussion about how to address social justice every single day as a core part of my curriculum. I keep coming because Ive learned new things every time. Im constantly pushing myself as a teacher and questioning how I could be doing better in my practice, how I could be taking the next step. This group is a really good way to step back and look at the bigger picture of why Im teaching. Ultimately, why Im teaching is to help kids make sense of the world around them and discover their own power and agency in that world. A lot of times, you lose focus on that in the daily minutiae.
Andrew PatelElementary teacher, member since 2018, mixed race, nonbinary: One of the big reasons why I teach is because the work of teaching is the work of justice. Youre working with children. You have an impact on their well-being and their lives. Often injustices play out in schools. As a teacher, you have a role in functioning to fix those injustices, and you have tools to change that while interfacing with the most important part of schools, the children. I like being in conversations around justice, around equity, and conversations around racial, socioeconomic injustice, and what we do in our classrooms to fix that. I see it as [an] avenue to get better, to talk to people about how to improve practice, how to use tools better to fix injustices that happen in schools. I get a little tired sometimes of bringing up issues of justice and equity, and just having [people] be like, Well talk about this later. [Hearing] that a few times, now I dont bring it up anymore, which sucks. I just dont have the energy to be told well do it later. Its nice to be in a small group where if you bring up an issue of equity or justice you can hash it out. [Also, getting to talk to] people who are actually going to go to the classroom the next day and be the lead teacher and have full responsibility, and be[ing] able to picture myself in that role, and see some of these things that were talking about being played out.
INSIGHT #2: WE ARE NURTURED BY A COMMUNITY THAT HOLDS US ACCOUNTABLE FOR PUTTING OUR IDEALS INTO PRACTICE AND CHALLENGES US TO CONTINUE EXAMINING THESE IDEALS
CMTC provides us a community of critical friends and colleagues that functions to hold us accountable to our own values and pushes us to actually put them into practice. We often need this extra push in the face of the everyday pressures of teaching, grading, planning, repeat. We thrive on continuing to challenge ourselves to become more critically informed. We gain inspiration from authors who are scholars, activists, and teachers, as well as from each other.
Angela JeffreysElementary teacher, member since 2016, White cis female: [This year] I am coming about 70 miles, which can be two and a half hours sometimes. The first six months of teaching youre drowning, but since I had already committed to do the [Creating Balance in an Unjust World] conference, it just helped remind me that there is more than the day-to-day of teaching. There is a higher purpose. At my school, everyones really putting their heart and soul into it, but theres not a lot of social justice aspect. Coming here and doing some readings, it reminds you that theres a depth to teaching, elements of it that youre not going to think about unless youre in that special space. If youre at a [professional development] or a staff meeting, no ones going to be talking about Freire. That happens in this [group]. Its not just the curriculum and management. Theres how to help these kids in a social justice aspect. Teaching is bigger than the curriculum.
Jeff JorgeMiddle school math teacher, participated in 2018: I met this group of educators at the Creating Balance conference. Meeting a bunch of educators committed to teaching social justice through mathematics was very intriguing. I went to a couple of the meetings, and I keep coming because everyone wants to get better at what theyre doing, and they want to integrate social justice more into their pedagogical practices. Especially as early-career teachers, its good to think about this now so we can continue to push on this throughout our careers. Its a great support network. As I continue to push my pedagogy to include more lessons that have to do with social justice, I have a group of other like-minded teachers who are saying, Yeah. Go for it. That sounds like a great idea. I stand behind what youre trying to do. Even if its not the administrators and its not your co-teachers, because your co-teachers are still thinking about the pacing and how well the school does on the test. Its nice to have other teachers who push you to do things, get another perspective, bounce those ideas off of [each other]. It gives you the confidence to try.
Izzy: One thing Ive taken out of this group is that you cant divorce math from our daily reality of power and justice or injustice in the world we live in. Math can actually be a really powerful tool to interrogate that reality, figure out why it works the way it works, and then see what could be done differently. Now, my dream with math is not just theyll be able to balance a checkbook and be responsible, but theyll be able to make real change in the world. Thats how this group has made me understand math. I did not used to see math in this way.
Marcos: You can be doing all the things that we learn [in our teacher education program] growth mindset and all these really progressive mathematical practicesand still be missing half the picture. The idea that youre not touching on the critical axis (Gutierrez, 2009) at all, right? Its left me with the knowledge of what I should be aiming for, and it gives me the tools to think about how to set up my classroom and how to set up the things I want to be teaching and the way I want to frame them. It just opened the window.
INSIGHT #3: WE BENEFIT FROM PRAXISHAVING A SPACE WHERE WE CAN CONNECT THEORY TO THE PRACTICES OF TEACHING IN OUR OWN CLASSROOMS THE NEXT DAY
The bridge from theory to practice is a crucial piece of what CMTC provides. Through mutual support and engagement, we are able to deeply engage with complex theoretical readings and then push each other to extend the implications in ways that we can bring into our classrooms tomorrow. The workshopping of lesson and unit plan ideas means we get our colleagues input to refine our own work, and we benefit from seeing real examples from others. This praxisthe practice of moving back and forth between theory we read and lived implications in our own classroomsis how we bring our visions to life.
RebekahElementary teacher, participated 20172018: The part that keeps me coming is the community of diverse perspectives on teaching math and the workshopping of lessons. I really enjoy hearing the different perspectives and just nerding out on one lesson really deeply. Its really cool to see how [a] one-hour lesson, or a weeklong unit, can be so deeply picked apart and enhanced by multiple perspectives. All of our opinions, and then to think about that on a grand scale, like, oh man, this can always be done! I find that really inspiring. That community aspect, of having a casual setting, it doesnt have to be someone observing your lesson, and it doesnt have to be a perfect lesson coming to workshop. We can just discuss this topic that were really passionate about in a way that builds all of our knowledge.
Marcos: There was definitely a lot of in-the-moment sparks of inspiration during the workshopping times. Its such an exciting feeling to have all these ideas floating around and then your mind sort of runs off with something. The one that really hung around in my head is when we were talking about a line of best fit, thinking about how you compute the line of best fit, and how you compute the [residuals]. The conversation turned to the idea that, well, theres actually lots of ways to compute the [residual] and each [one] will get you a different line of best fit. Ultimately this line of best fit is very subjective, but its culturally normed. We have this [one] way to do it and this is how we define best. A lot of the models being used to predict a lot of things are using this formula. [I realized] the possibility of opening that discussion up to students when you introduce this lesson. Instead of focusing on, Heres a way to compute it, lets compute it and practice, having them think about, what are ways you can compute? Lets compare those ways and think about why we are using this one instead of [a different] one.
TrishaElementary teacher, participated 20162018, Asian cis female: I worked with Izzy to create a unit on the gender pay gap in the workplace and then implemented it in the classroom. My students were really fascinated by this topic, so it made sense that we were going into it with a deeper lens. The group provided lots of feedback to the initial version of this lesson; [Izzy and I] tweaked it when we [each] taught it and provided each other feedback on what we would do next time.
INSIGHT #4: THE ALLY-SHIP WE PROVIDE EACH OTHER IN THE FACE OF CHALLENGES TO TEACHING FOR SOCIAL JUSTICE SHAPES NOT ONLY OUR MATHEMATICS TEACHING BUT ALSO OUR TEACHING PRACTICES IN SPACES BEYOND MATHEMATICS
Teaching for social justice and enacting critical pedagogy in traditional schooling spaces can be an upstream battle for numerous reasonsexplicit rejection, time constraints, simple lack of models or examples of how its done. Furthermore, the direction forward is not always clear. Context shapes the opportunities and the meanings of the approaches we take. Context can be understood as who the teacher is and where they come from, the students past experiences of schooling, and the racial, linguistic, and socioeconomic contexts of students and the school. CMTC is a place where we find support to do the work, both on ourselves and in our teaching practice, that is often not fully supported in our school sites. Even when we do not live our critical visions in our mathematics teaching every day, the courage and insight we gain from each other continue to shape our teaching and how we understand ourselves in spaces beyond mathematics.
Meg RussellElementary teacher, member since 2018, White cis female: [In reference to Rochelle Gutierrezs (2009) dominant and critical axes of equity framework,] I think it comes down to how its easier to have the one [dominant] axis than the other [critical axis]. I feel like Im getting stronger and stronger at doing the dominant axis in a way that is better than my inclinations would be. [But the critical axis] is always separate from math. If we do it, [its] like, morning meeting. Thats what Id say the challenge is, getting at the power and identity axis. . . . I have one kiddo who has, at multiple times, brought up. . . . The first time they were reading books by Jacqueline Woodson, and she goes, Its cool that shes Black. We dont read anything by Black people, or something like that. Then another time she was in a book club reading a book by Andrew Clements, and she goes, You know, I dont think any Andrew Clements books have Black people. Both of those times I was a lot more ready to be like, Youre right. How does that make you feel? Or, What do you think about that? In the past, as much as that kind of thing concerns me, I wouldnt have known what to say. And I still dont know what to say, but I think since coming to this [group] Ive been a lot more willing to just forge ahead and be like, Youre right, what do you think about that? See how far it goes.
Izzy: Ive almost found myself un-teaching [students] stuff that theyve learned about how school is done and who has the knowledge and the power in school. The other challenge is time and pacing. I wish I had so much more time to really get deep into my planning and help with these awesome context-based, locally focused units about power and structures in our society. But I feel pressured to cover a million things. Ive had other people in my school be psyched about the work Im doing, but theres no explicit [support] for it really. Theres no time, theres no energy, theres no collaboration put into it. If you want to do it, cool, but youre on your own.
Trisha: It has deeply broadened my ideas of what can be achieved in math and changed my reason for why I teach math. [Also,] I have gotten braver in teaching more social justice in my classroom. Its now a norm to talk about current events and we educate ourselves on the happenings of the world. Weve talked about things like DACA, gender inequality, transgender rights, and gun rights throughout the year.
Nicole: When I was teaching in [a segregated neighborhood], I dont think I was pushed so much to think about this framework cause it was like, all of my kids are Black or Brown. Theyre all getting free or reduced lunch. Obviously, teach them to be mathematicians and not just [to do] math in a drill and kill way! Teach them to be critical and problem solvers! Thats teaching for social justice. But now Im teaching in this context where my kids are of much more stable socioeconomic status. Way more White students, students with more privilege, and a lot of students who also get math help outside of school. And its kind of like, okay, what does teaching for equity look like in this context? Those are pieces to think about. Its percolating.
Angela: As I told [someone] once, Im not particularly woke, but Im waking. So, some of it I [still] dont feel qualified to talk about. There are certain aspects of White privilege and everything else that come into it that can make it, like, Im not quite sure which direction to go.
REFLECTIONS FROM PRACTICE
We encourage readers to engage with these reflections as providing a wide range of examples of how participation in CMTC has shaped our teaching practices. We hope these reflections provide examples of our practice that will bring to life both how we put our pedagogies into practice and some of the challenges to doing so.
RHYTHM TIMEANDREW PATEL (THIRD GRADE)
For me, a social-justice math classroom centers students who have struggled most in math, and plans whole group instructions and routines around their needs and interests. Teaching math for social justice also means being responsive to students emotional needs in the room prior to math learning.
This year I am using drums and rhythm in my classroom to count and consider numbers. We do a lot of skip counting and pattern making using claves and coffee can drums. But, I did not start this ritual with the intention or goal of creating math learning. Early in the year I observed that after recess time, students often appeared stressed out. I consulted with the school social worker, and we planned a rhythm time for when students returned to class after lunch and recess. The intention was to create routine and ritual that helped students feel comfortable in the classroom.
Only after creating this routine did we add explicitly mathematical aspects to rhythm time. My students and I now use this ritual to make patterns and explore multiplication. Students are intended to learn multiplication in third grade, but we are taking this circuitous route, using the rhythm time routine to support number sense and automaticity. We also continue to use rhythm time for free drumming and music making; this makes the routine more meaningful and joyful for the students.
Giving students ritual and routine creates a sense of safety and connection within our classroom and with numbers. We sincerely enjoy rhythm time. That joy allows us to think more deeply about numbers and create positive relationships with mathematics and with each other. These relationships with each other and with mathematics can be vital tools for organizing and participation in liberatory change.
MODELING LIFE WITH QUADRATICSGINA WEI (GRADES 10, 11, AND 12)
Teaching students for social justice means that the understandings students build in the field of mathematics are inseparable from the understandings they are building in other realms of their lives. As a teacher with three years of experience, I still have a lot of work to do in discovering more about my students and about mathematics in order to make these connections in my classroom.
One moment when I felt this connection happened in my teaching was last year in my Integrated Math 3 class at my previous school in San Jose, California. We were using quadratic equations to model real-life situations. The examples in the curriculum included physical situations, such as the path of a jumping rabbit and the path of a soccer ball. But quadratic relationships are experienced in student lives beyond physical movement. I created a task where students worked in pairs to come up with an abstract situation that could be modeled by a quadratic.
I started by showing students two examples. The first was a graph of a quadratic where -y represented my happiness, and -x represented the number of potato chips eaten. The quadratic crossed through (0,0), peaked at x=40 (I really like potato chips) and crossed through the x-axis again at 80. I told them that quadratics come into play when there can be too much of a good thing, which applies to many situations. I got a few chuckles from the class because of the ridiculousness of my example. Next I introduced the Laffer curve. The Laffer Curve is a theoretical relationship between tax rates and tax revenue. When the tax rate is 0%, no tax revenue is collected. As the tax rate increases from 0%, the tax revenue will increase, because more taxes are being collected. There comes a tipping point, however, when the tax rate is high enough that some people at the margin will work less (if were talking about an income tax), therefore resulting in lower revenues. Then, as the tax rate continues to increase from this tipping point, the revenues will continue decreasing until they reach 0. The tax rate that maximizes tax revenues is the optimal tax rate. Students and I discussed how some politicians argue taxes should be decreased because current tax rates are above the optimum, therefore lowering taxes would actually result in higher revenue, and other politicians argue the opposite.
The Laffer Curve sparked many students interests. One student said he really likes talking about politics, which is something not usually brought up in my math class. I believe another element that made the example exciting to some students was how the mathematics and the arguments behind taxation helped students deepen their understanding of both concepts, supporting them in becoming mathematically knowledgeable participants in the political life of a democracy. They were inextricably linked.
As the students worked, they came up with brilliant situations of their own. What struck me was the everydayness of their scenarios. One student modeled the likelihood of getting sunburnt versus the time of day. Another pair modeled the popularity of Yeezy shoes versus the number produced, which demonstrated the students understanding that producing too much of a rare good can lower its worth. Another pair modeled their enjoyment of a Spiderman game versus the number of days they played. One pair modeled the benefit to students versus the number of hours at school. A different pair modeled demand versus rent prices, demonstrating reasoning about how both low and high rent prices could turn away potential renters.
The power of this lesson was that students were deepening their understanding of both mathematics and their experiences through their work in the classroom. Their understanding of the math was necessary in elegantly explaining and visually expressing the relationship between the two variables they chose. And the math already existed in these situations that they were experiencingthey just needed the opportunity to connect them. At the end of the lesson, I was able to highlight their own brilliance as well. I told them, What youre thinking about are questions that a lot of researchers are trying to answer! You all are thinking of things Ive never thought of. We all need to learn from your great ideas! That was the most exciting part.
THE WAGE GAPIZZY HENDRY (THIRD GRADE)
A few years ago, Trisha Huynh and I wrote a unit on gender and race pay gaps in the workplace. In the lesson, students use graphing and fractions to investigate wage disparities based on race and gender. We were teaching different grade levels (fourth and third, respectively) at different schools, so we tweaked the lessons for our own contexts. The version that follows exists after three years of teaching and revising it with support from each other and CMTC. In CMTC, we often help each other make the math more authentic in a lesson that initially forefronts a social context, or we make the social context more authentic in a lesson that forefronts the math. Each time I teach this lesson, my students get really into the math and the social aspects, making it feel successful from that standpoint.
The Wage Gap unit builds on student learning about fractions and part-whole relationships. Students learn to construct a scatterplot to compare and analyze information, and deepen their understanding of fractions, especially unit fractions such as 1/2. We also build on previous class discussions around current movements such as Black Lives Matter or the Womens March, and the concept of solidarity more broadly.
During the three-day unit, students first explore the concept of comparing wages using fractions, such as discovering that one person makes 8/10 of another persons salary. On the second day, students are provided a table with wage data over time. The table provides the amounts that Latinx and Black men and women, and White women made for every dollar made by a White male for each decade since 1970 (data retrieved from United States Department of Labor, 2017). I modify the data slightly, rounding to increments of 5 and 10 cents so that students can use dimes and nickels to visualize the wages as fractions of 1 dollar. Students create a scatterplot where each demographic group is represented in a different color, and then connect the points to observe salary change over time (see Figure 1).
Figure 1. Template for plotting of comparative wage data by demographic group over time
On the third day, students reflect on their mathematical and social learning. We focus on big questions: Does women (or another group) making less money only affect women? How might it affect their community too? What could change this injustice? What does this have to do with math? We first collect student ideas with a chalk talk (Hammond, 2014), in which students respond to a question in writing on a shared poster, and read and respond in writing to others thoughts. Then we transition into a whole-class discussion. Postunit follow-up has taken different forms in different years depending on current events and student interest. This year, we made infographics to educate our community and posted them outside our classroom. We also drew connections between this unit and the U.S. womens national soccer team fair pay fight and Dr. Martin Luther King Jr.s Poor Peoples March.
When asked what they learned, students discussed both their mathematical and social noticing. Two students discussed the importance of attention to detail: Yo aprendí que no tengo que hacerlo del mismo color y que no tenemos que hacerlo lo mismo que los anteriores [I learned I shouldnt make the lines the same color and or the same as the previous ones]. A classmate responded, Yo estoy de acuerdo contigo porque después si usas el mismo color te confundes [I agree with you because if you use the same color you get confused]. Questions about social injustice were brought to the surface. One student wrote, Una pregunta que aún tengo es porque le pagan a un hombre blanco un $1 dólar si las otras personas tienen el mismo trabajo y los pagan diferente? [A question I still have is why they pay White men $1 if the other people have the same job and they pay them differently?].
What stands out to me in student responses is that my students were thinking about the mathematics and they were thinking about social injustices. They were noticing, for example, the importance of details, such as color coding the graph when we communicate about the social world using math. They were left with many questions about why injustices exist in the world. And rightly so!
Questions remain for me as well. Many of these I have brought to CMTC when I workshopped this lesson in the past: How do I teach graphing in a more constructivist manner? How can I make sure that this series of lessons is presented in a way that empowers students, rather than discourages them? Through learning from my own students, and ongoing support from CMTC, this unit continues to be revised.
WHEN A SCHOOL CULTURE IS ANTITHETICAL TO CRITICAL PEDAGOGYANGELA JEFFREYS (FOURTH GRADE)
Teaching is wonderful. Connecting with kids is easy for me, and extremely rewarding. My struggles with the educational system fall not to teaching but to discipline and what that means in my current school context.
My current school site recently looks like it is right out of the 1980s. Kids are in desks. Students are praised by how well they stay in said desk, or how quiet they are when you are teaching. Sure, we can move the desks into groups or in rings or wavy lines (my latest attempt), but the fact remains that the kids are in desks the absolute majority of the day. The principal is committed to Explicit Direct Instruction (EDI)attempts at inquiry and student-centered teaching must be very quietly pursued. Behavioral infractions as minimal as chattiness lead to punishments like a week of detentions or copying pages out of the dictionaryyes, I too was shocked to find this still goes on in schools!
What I have noticed is that students who have been brought up in this traditional school environment come to expect it. When things have gone south in my classroom this year, almost exclusively because of excess chattiness, I bring my issue to the class to attempt to solve it. The solutions students suggest include various punishments and are absurdweeks of detention and dictionary copying. Ubiquitous classroom management structures in my school include the use of a clip chart where students have a clip or clothespin that gets moved up or down based on their behavior, and ClassDojoa digital system that allows teachers to track individualized points that can be given or taken away by the teacher.
In my third year, I finally trust myself enough to begin to fight these systems. They are deeply antithetical to why I got into education and how I feel students should be treated. It was this year, where I reflected on how much I despise the shame aspect of such individualized methods as the clip chart and the ClassDojo system. These systems that emphasize behavior control of individual students are directly contrary to critical pedagogies that center student agency and collective liberation. So, I started to go in a different direction with those tools. I transitioned the clip chart into a bathroom pass system, and use ClassDojo points as a type of marble jar system so that everyone is earning them collectively and they go towards a class party. It is a point of pride that I have never taken away ClassDojo points this year, nor had anyone clip up or down.
CMTC is a space where I reconnect to teaching and why I got into education in the first place. I want my students voices to be honored and for them to be given opportunities to think about issues that are relevant to their lives. I struggle with simply feeling so alienated by the discipline systems and direct instruction teaching expectations at my school. Teaching a lesson that actually reflects critical pedagogy in this context is still beyond my grasp. For now, I am finding small ways to resist, but teaching with critical pedagogy is the goal that I keep working toward. Being in touch with people from CMTC inspires me to not give up.
DISCUSSION AND CONCLUSIONS
We believe that our model provides potential to support others who desire, like us, to engage in critical work in the mathematics classroom. Our insights suggest that without the nurturing community of praxis to help us develop our visions and ourselves and put those visions into action, the challenges to teaching math for social justice can appear insurmountable. On the other hand, a community of praxis focused on the teaching of math can support the development of critical orientations and practices that extend beyond mathematics.
Clearly, a single learning space, no matter how powerful, cannot independently make possible the actualization of this vision. What it can do is open a window, like it did for Marcus, or make a teacher braver and more willing to take risks, like it has for Trisha and Jeff. It can lead to an exploration of wage disparities by race and gender in a unit on fractions in the third or fourth grade. And it can prepare a White teacher to be able to validate a Black students emerging recognition of the problems of racial nonrepresentation as the child begins to theorize her own world (hooks, 1994). Like Angela suggested when she said, Im not particularly woke, but Im waking, finding and enacting a critical orientation to teaching is an ongoing process. Teachers in this group reflect on learning about critical mathematics, but also about becoming critical educators across their teaching practice. There is no singular pathway to enacting critical mathematics, and for all of us, this continues to be a work in progress. As is clear from the diversity of the reflections from practice, this path will be distinct for each of us, even as we support each other. We do not expect our visions or practices to converge. As divergent as our approaches may end up being, the community of praxis offered through participation in the Critical Mathematics Teacher Collaborative makes possible not only the exploration, but also the enactment of critical mathematics pedagogies for each of us where it has previously been absent.
We would like to acknowledge the many participants in the Critical Mathematics Teacher Collaborative who have sustained us and our community over the years whose voices are not explicitly represented here, as well as those whose are: Trisha Huynh, Jeffrey Jorge, Nicole Mitchell, Marcos Muñoz, Meg Russell, and Rebekah. We are thankful for the Deans Collaborative Learning Fund at the Stanford Graduate School of Education whose funding has made it possible to provide dinner for participants at monthly meetings.
Over the years, many of the teachers participating in this group have identified as White. This phenomenon has come up as a concern and is the topic of ongoing conversation. We have asked each other, what about the group may not serve the needs or desires of some of our colleagues of color? We have wondered together if Emmas leadership as a White cis-woman has specifically attracted other White women seeking a safe space to expand their knowledge and practice in areas of teaching that address power and privilege, and, as such has been less attractive to those with other identities and experiences. With new leadership and new participants, this conversation remains on the table. However, it was not central to the self-study reflected in this chapter. Still, we believe it merits ongoing and particular examination.
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Our model is elaborated below. For each component we include an objective, some details of the activity itself, and relevant considerations for those interested in starting their own group. Times are based on a two hour meeting.
Checking In (~20 min)
Objective: Mark the opening of the meeting and build community through name and experience sharing.
Activity: Go around to share name, grade level, school and answer a check-in question. Check-in questions might include:
Share something that lifted your spirits this week.
How did a student make you excited or curious recently?
Considerations for your group:
Dont rush check-in time. Time to connect is crucial for building trust and entering challenging conversations.
Do introductions at every meeting, even if many people already know each other. This makes being a newcomer easier to navigate.
Reading Discussion (~45 min)
Objective: Think deeply with the support of others about a relevant text chosen and read by group members prior to the meeting
Activity: A previously identified facilitator provides a summary and questions to support discussion. Conversation may begin in pairs, full group, or some combination. Example questions include:
How does the author define the difference between mathematics goals for social justice and social justice goals for mathematics?
How could you see this working in your classroom?
Considerations for your group:
Make sure everyone has access to readings at least one week in advance of the meeting.
Consider length of readings and bandwidth of your group members. Consider alternative media such as podcasts.
Choose a facilitator in advance. Rotate this role. Taking on this role should be voluntary, but those who havent done it can be encouraged to do so.
Workshopping (~45 min)
Objective: Engage with a critical mathematics inquiry-based lesson plan, activity, or classroom design idea. Provide support and input to move from idea to implementation.
Activity: One or two members of the group share a plan or idea that is under development. Participants may engage in the activity or task under construction. Participants ask clarifying questions, and provide additional ideas, insight and feedback.
Considerations for your group:
It can be useful for the workshopper to prepare questions to guide the feedback they desire. For example,
What kinds of challenges do you think 3rd graders will have with this lesson?
What is important for facilitating a careful conversation about gentrification?
Closing (~10 min)
Objective: Make plans for the next meeting while together in order to ensure shared leadership and hold each other accountable to attending.
Activity: Pick a date for the next meeting. Pick a reading (or two). Identify a facilitator for the discussion. Identify who will workshop a lesson at the next meeting. Encourage different people to suggest readings based on their own diverse knowledges and interests. Encourage everyone to workshop at some point, including pre-service teachers who may be hesitant.
Considerations for your group:
Leave time for closing. If you skip this part, it is more likely that the same people will end up choosing readings, facilitating discussion, and workshopping their ideas rather than rotating roles as intended.
Setting a time for the next meeting makes people feel accountable for attending. And, its easier to coordinate in person.
The group reads, listens to and watches texts that are directly related to mathematics education as well as texts that touch on broader issues of educational justice.
Podcast interview with Laurie Rubel (Otten, 2018)
Race, Retrenchment and the Reform of School Mathematics (Tate, 1994)
Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice. Chapters 1 and 2. (Gutstein, 2006)
Framing Equity: Helping Students Play the Game and Change the Game (Gutiérrez, 2009)
Critical Pedagogy for Critical Mathematics Education (Tutak et al., 2011)
Critical mathematics pedagogy (Critical Mathematics Pedagogy, n.d.)
What Do We Mean When We Talk About Social Justice and Math? (Chen, 2017)
Tinkering Spaces: How Equity Means More Than Access (Schwartz, 2016)
Plotting Inequalities, Building Resistance in Teaching for Black Lives (Renner, Brew, & Proctor, 2018)
Cognition and Cultural Pedagogy in Culturally Specific Pedagogy in the Mathematics Classroom: Strategies for Teachers and Students (Leonard, 2008)
The Culture of Exclusion in Mathematics Education and Its Persistence in Equity-Oriented Teaching (Louie, 2017)
Addressing Dilemmas of Social Justice Mathematics Instruction Through Collaboration of Students, Educators, and Researchers (Kokka, 2015)
Rehumanizing Mathematics: A Vision for the Future (Gutiérrez, 2018)
Culturally Relevant Pedagogy in Mathematics (Jones, 2016)
Elementary Mathematics and #BlackLivesMatter (Chao & Marlowe, 2019)
Education and Justice
Theory as Liberatory Practice in Teaching to Transgress (hooks, 1994)
Healing the Hidden Wounds of Racial Trauma (Hardy, 2013)
Pedagogy of the Oppressed, Chapters 1 and 2 (Freire, 1970)
It Takes a Village to Teach First Grade in Black Ants and Buddhists: Thinking Critically and Teaching Differently in the Primary Grades (Cowhey, 2006)
A Talk to Teachers (Baldwin, 2008)