Module VIII: Equity in Classroom

“Our Issues, Our People—Math as Our Weapon”: Critical Mathematics in a Chicago Neighborhood High School

Eric “Rico” Gutstein University of Illinois at Chicago

This article provides an example of, and lessons from, teaching and learning critical mathematics in a Chicago public neighborhood high school with a social justice focus. It is based on a qualitative study of my untracked, 12th-grade mathematics class, a full-year enactment of mathematics for social and racial justice. Students were Black and Latin@ from a low-income, working-class community with a tradition of resis- tance. Any neighborhood student could enroll without selection criteria. The class goal was for students to cocreate a classroom in which they would learn and use college- preparatory, conceptually based mathematics to study and understand social reality to prepare themselves to change it. Through analyzing my practice, I address possibilities and challenges of curriculum development and teaching, examine student learning, and pose questions and directions for further research and practice.

Key words: Critical mathematics; Equity; Freire; Mathematics for social justice; Race; Racism

I wrote the numbers on the board as class wound down. Students silently and soberly stared at 150,000 – 291,000 = 92,000 (an “equation” derived from a recursive function, modeling a mortgage). I talked as I wrote: “You’ve paid two hundred and ninety-one thousand dollars on a one-hundred-and-fifty-thousand-dollar mortgage, and you still owe ninety-two thousand dollars. Check that math out. One hundred and fifty thousand minus two hundred and ninety-one thousand equals ninety-two thousand.” I paused as students looked and mumbled to themselves and neighbors. “Think about that. Hey! You started with a hundred and fifty—you paid two ninety-one—and you still owe ninety-two thousand dollars. What’s going on here?”

Antoine: “They’re taking your money.”1

Daphne: “The bank is taking advantage of you.”

This research was supported in part by a fellowship from the Great Cities Institute, University of Illinois at Chicago. Acknowledgments to student “Crew” members (Vero González, Rut Rodríguez, Amparo Ramos, George Carr, Channing Redditt, Alex Hernández, Amy Maldonado, Nikki Blunt, Darnisha Hill, and Rogelio Rivera); Research Team members (Anita Balasubramanian and Patricia Buenrostro, doctoral students who shared in this analysis and the curriculum framework development); and graduate students who helped develop curriculum frameworks (Joel Amidon, Phillip Caldwell, Paula DeAnda, Naama Lewis, Carlos Lopez Leiva, Craig Willey, and Caro Williams).

Journal for Research in Mathematics Education 2016, Vol. 47, No. 5, 454–504

1 Student names are pseudonyms except when I discuss public settings.

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455Eric “Rico” Gutstein

Mr. Rico:2 “This is legal—this is how banks loan money and make money.” Silence. I paused and repeated slowly. “This is legal—this is how banks loan money and make money.” I paused again.

As class ended, I asked, “What are some questions you could ask here?” Renee said, “Why is it legal?” and Daphne asked, “Why don’t more people look into it so they don’t end up in the same situation?”

So went a more-or-less typical day in a 12th-grade mathematics class I taught with the goal that students learned to read and write the world with mathematics (RWWM). That is, my aim was for students to develop a deep understanding of mathematics through using it to study their social reality and create (and extend) their own analyses of contradictions in society and their lives—to prepare them to change reality as they saw fit. In this article, I describe and analyze occurrences in this classroom during an entire school year to explicate the teaching and learning of mathematics for social justice, or critical mathematics (CM).3

Overall, there is insufficient research on enacting CM in K–12 classrooms, and although theory exists, it has not been well worked out in practice and developed over time. I heed Sleeter and Delgado Bernal’s (2004) call: “Within the United States it [critical pedagogy] has been developed mainly at a theoretical level, often leaving practitioners unclear about what to do”; “there is a need for practical guidance that does not, in the process, sacrifice conceptual grounding” (p. 244). I attempt here to provide both grounding and guidance. My research addressed the following questions:

• What are some complexities in actualizing RWWM practice? • What do students learn through the process?

Conceptual Framework: Reading and Writing the World With Mathematics

For me, RWWM means that students use mathematics to comprehend and change the world—and through the process, deepen their knowledge of both mathematics and their social reality. RWWM encompasses an epistemology of learning by doing, but the doing has the explicitly political intent of transforming society toward equity and justice. Freire (Freire & Macedo, 1987) introduced the idea of reading and writing the world (p. 35) and framed education as emancipatory praxis—the dialec- tical interconnection of action and reflection for the purpose of liberation and full humanization. In the United States, Frankenstein (1987, 1995, 1998) extended Freire’s ideas to mathematics (in adult education), and Skovsmose (1994, 2004, 2005)

2 My classroom name. 3 Throughout this article, I use three terms interchangeably: reading and writing the world with

mathematics, critical mathematics, and teaching and learning mathematics for social justice. In my previous work, I used “reading and writing the world with mathematics” synonymously with “mathematics for social justice,” a practice that I continue here. Although multiple interpretations exist for these terms, U.S. educators mainly refer to “mathematics for social justice,” and those in other countries refer to “critical mathematics,” often with roughly the same meaning.

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developed a CM philosophy in a European context. I draw from both and build on their and others’ contributions.

By reading the world, Freire meant that people, through interacting with reality, develop an increasingly critical sociopolitical consciousness and come to understand the roots of injustice. We read the world as we shape it and are shaped by it, engaging in the praxis that Freire (1970/1998) called our “ontological and historical vocation to be more fully human” (p. 37). For him “the fundamental theme of our epoch . . . [was] that of domination—which implies its opposite, the theme of liberation, as the objective to be achieved” (p. 84). Reading the world, in Freire’s framework, always serves the goal of freedom from oppression.

We read the world while using various literacies (e.g., media, textual, visual) in social practice. With respect to mathematics, my view of reading the world is

to use mathematics to understand relations of power, resource inequities, and disparate opportunities between different social groups and to understand explicit discrimination based on race, class, gender, language, and other differences. Further, it means to dissect and deconstruct media and other forms of representation [and mathematics itself] and to use mathematics to examine these various phenomena both in one’s immediate life and in the broader social world and to identify relationships and make connections between them. (Gutstein, 2003, p. 45)

Merely understanding social reality, however, does not liberate people, though it is both a precondition for, and an effect of, consciously transforming the world. Reading the world needs writing the world, which, for Freire, meant to change reality. In the process, people develop social and individual agency, whether or not they engage with mathematics (Gutstein, 2006a).

Reading and writing the world are interdependent and do not proceed linearly. They dialectically interweave as people participate in daily life and reflect on their actions (Freire’s praxis). People think about their lives and learn while acting and transforming, and they learn to more deeply read their reality through social practice. One’s reflections lead to learning, which influences action; as one acts, one looks back on experiences and learns.

In school, reading and writing the world can connect to academic content. Freire (Freire & Macedo, 1987) called this “reading the word” (i.e., learning to read text)— or for me, reading the mathematical word. Freire (1994) noted that in human evolu- tion and development, reading the world precedes reading the word, but the two are always linked: “Not a reading of the word alone, nor a reading only of the world, but both together, in dialectical solidarity” (p. 105). This relationship was salient in the literacy campaigns in which he participated in Latin America and Africa (Freire, 1978, 1994). He wrote, “From the beginning, we rejected the hypothesis of a purely mechanistic literacy program and considered the problem of teaching adults how to read in relation to the awakening of their consciousness” (Freire, 1973, p. 43). So, reading the world with mathematics has two interrelated processes—reading the world and reading the (mathematical) word. A challenge is to develop and teach curriculum that supports students in thoroughly integrating both.

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With respect to reading the word, RWWM must ensure that students genuinely learn mathematics, especially those who have weak educational backgrounds or are otherwise marginalized. This is not optional—education that one calls liberatory but fails to teach mathematics (or any subject) is a charade. Students need full oppor- tunities to learn mathematics for many reasons—economic survival for themselves, their families, and their communities; future education and meaningful vocational or career plans; reading and writing the world; and the full actualization of their human potential. Fundamentally, students need the right to choose to embrace mathematics as a human cultural production (D’Ambrosio, 1985).

In summary, there are two sets of dialectical relationships associated with RWWM (Figure 1). One is reading the world and the word, and the other is reading and writing the world. In relation to mathematics, reading the world (in general) and reading the mathematical word merge into reading the world with mathematics— and reading the world with mathematics stands in dialectical relation to writing the world with mathematics. In essence, my theoretical framing of RWWM is that students develop deeper sociopolitical awareness through learning and using math- ematics to study reality, which prepares them to shape society by using mathematics, at the moment and in the future. They come to view mathematics as useful in this process (a dispositional shift), recognize some of its limitations, and also learn that mathematics is but one way to read and write the world. Through understanding and acting in the world—even if their actions are limited by being in school—they also transform themselves.

Figure 1. Reading the world (in general) and reading the mathematical word dialectically interrelate and together constitute reading the world with mathematics; this then dialectically interrelates with writing the world with mathematics.

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Three Types of Knowledge: Community, Critical, and Classical In earlier work, I addressed several aspects of RWWM pedagogy (and curriculum

development),4 including what I called creating a pedagogy of questioning, normal- izing politically taboo topics, and developing political relationships with students (Gutstein 2006a, 2007b, 2007c, 2008). Also, I described RWWM as involving three types of fluidly interconnected knowledge, which I refer to as the 3 Cs: (a) commu- nity knowledge (or popular knowledge; Freire, 1994), knowledge of one’s reality, experiences, culture, language, neighborhood, and informal ways of relating to and interpreting the world; (b) classical (mathematical) knowledge, academic mathe- matics; and (c) critical knowledge, reading the world, in this case, with mathematics (Gutstein, 2006a, 2007b). These types of knowledge mutually interact, evolve, overlap, and are never entirely distinct from each other.

Freire (1970/1998) believed that community knowledge—learners’ knowledge of their own reality—was the starting, but not the ending point, for liberatory educa- tion. He wrote, “The point of departure [for emancipatory education] must always be with men and women in the ‘here and now,’ which constitutes the situation within which they are submerged, from which they emerge, and in which they intervene” (p. 66). The people’s “here and now” is expressed in their generative themes—the dialectical relationship between key social contradictions in their lives and how they perceive and interact with them—a central component of their community knowl- edge (pp. 77–105). For Freire, beginning an educational program of reading the world with people’s generative themes was always to support them in deepening their critical knowledge of that reality. Such an education “involves a constant unveiling of reality [reading the world] . . . . [that] strives for the emergence of consciousness and critical intervention in reality [writing the world]” (p. 62). Given the role of textual literacies in developing sociopolitical consciousness, people also need to develop various types of classical knowledge.5

Connecting the three types of knowledge by starting from, and building upon, community knowledge then interweaving and developing critical and classical knowl- edge with the goal of transforming reality6 is part of the theory of RWWM pedagogy (see Figure 2). This was a goal in my class, but it is difficult to actualize. I raised this complexity in Gutstein (2006a): “The key question is: how does one connect and synthesize all three knowledge bases, while fully honoring and respecting each, to develop liberatory mathematics education [i.e., prepare students to write the world] in urban schools given the current high-stakes accountability regimes and larger political climate?” (p. 207). Essentially, my purpose in this article is to address this question.

4 Although I mainly discuss RWWM pedagogy here, it is linked closely to curriculum development, as I describe below.

5 Of course, in societies with different forms of literacies (e.g., nontextual ones), classical knowledge could take different forms.

6 People (including youth) already have their own analyses, critiques, and theories about reality— some of which are quite critical. Even if students have little in-school experience reading the world with mathematics, they may have done so within other disciplines or in out-of-school contexts, including using mathematics (e.g., to understand their finances).

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From Theory to Practice Freire’s work lays the foundation for RWWM, but there also exists a diverse body

of related literature (Powell, 2012) that is theoretical (Rands, 2013; Skovsmose, 1994), methodological and paradigmatic (Valero & Zevenbergen, 2004), philo- sophical (Ernest, 1991, 2002; Skovsmose, 1994), and empirical (e.g., Bartell, 2013; Foote, 2010; Gonzalez, 2008, 2009; Jacobsen, Mistele, & Sriraman, 2012). I briefly outline several studies to give readers a feel for the developing breadth of issues that researchers are addressing and what we have learned as we deepen our under- standing and practice of CM.

Several studies are of CM projects outside regular K–12 classrooms. These include Terry’s (2009) dissertation on a summer program with Black male high school students, which merged CM, participatory action research, critical race theory, and what Terry called critical Blackness; Varley Gutiérrez’s (2009) dissertation on an after-school program with fifth-grade girls integrating justice concerns and math- ematics; Turner, Varley Gutiérrez, Simic-Muller, and Díez-Palomar’s (2009) study of an after-school mathematics club investigating issues centered on students’ experiences (third through sixth grades); Brantlinger’s (2007) dissertation that focused mainly on a 9-week night-school recovery class with CM applications for high school students who failed geometry; and Lam’s (2012) study of an after-school social justice club with high school females that integrated mathematics.

Numerous studies also exist of in-school settings into which teachers integrated CM. These include Turner’s (2003) dissertation on a sixth-grade classroom in which she collaborated with the teacher to integrate social justice mathematics projects; R. J. Gutiérrez’s (2013) dissertation on a 12th-grade classroom in which he collaborated

Figure 2. Connecting the 3 Cs—Building on community knowledge (generative themes) to develop critical and classical knowledge.

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with the classroom teacher to intersperse CM; Gregson’s (2012) dissertation on a middle school mathematics teacher who integrated CM contexts; Amidon’s (2011) dissertation on a ninth-grade high school class in which he taught one semester with social justice mathematics ideas; Brelias’s (2009) dissertation in which she investi- gated 11th-grade and 12th-grade students’ views of mathematics as a means of studying reality; Tanko’s dissertation (2012) of an all-female mathematics college class in the United Arab Emirates that he taught using social justice pedagogy; and Balasubramanian’s (2012) and Buenrostro’s (2016) dissertations about my 12th-grade classroom. Beyond dissertations, there is other research on my 12th-grade classroom (Gutstein, 2012a, 2013b), my middle school studies (Gutstein, 2003, 2006a, 2006b, 2007a, 2007c, 2009b, 2013a), and Peterson’s (2003, 2013; Gutstein & Peterson, 2013) writings about his fifth-grade classroom.

All these studies together suggest several points. First, teachers can enact CM in various settings, from traditional public schools to college and out-of-school contexts. Studies ranged from working-class and low-income settings to middle- or upper-middle-class settings, from all African American and all Latin@7 settings to racially diverse ones, from third grade to university classrooms, and in both single- and mixed-gender situations. The research documents that students learned both mathematics and about their sociopolitical reality. Second, the literature examined conditions that supported, as well as challenged, students in reading the world with mathematics. Facilitating conditions included teachers’ knowledge of sociopolitical contexts and students’ communities, students’ and teachers’ cocreation of class- rooms supporting the investigation of generative themes, and relationships between students and teachers that involved shared political analyses and solidarities (Balasubramanian, 2012; R. J. Gutiérrez, 2013; Gutstein, 2006a; 2012b; Terry, 2009; Turner, 2003; Varley Gutiérrez, 2009). Challenges included preparing and supporting teachers to teach mathematics for social justice given their existing dispositions, knowledge, and experience (Bartell, 2013; Foote, 2010; González, 2009; Jacobsen et al., 2012); dealing with curricular mandates and practical or structural issues (e.g., test preparation, teachers’ conditions of work; Gregson 2012; Gutstein, 2006a); navigating complex interconnections of creating and teaching interdisciplinary curriculum that pays sufficient attention to the multiple types of knowledge involved (Balasubramanian, 2012; Brantlinger, 2007; Gregson, 2012; Gutstein, 2006a; Turner, 2003); and working with students’ resistance to nontradi- tional curriculum and pedagogies (R. J. Gutiérrez, 2013). Third, CM researchers have explored aspects of identity, including students’ social class (Brelias, 2009; Lam, 2012; Wonnacott, 2011), race (Gutstein, 2003, 2006a, 2013a; Terry, 2009), and gender (Tanko, 2012; Terry, 2009; Varley Gutiérrez, 2009).

Though a theory of RWWM pedagogy exists, as outlined above, the research has at least two limitations with respect to practice. First, most CM studies examine teachers integrating limited-duration social justice projects into their mathematics

7 I use Latin@ as a gender-neutral pronoun to refer to both males and females (i.e., Latinas and Latinos). I use Black and African American interchangeably.

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classrooms—with or without researcher support—or in out-of-school settings. I know of no studies (other than the present research) of a full-year enactment of developing and teaching CM in regular K–16 classrooms in which the entire focus was for students to learn and use mathematics to understand and change their world.

Second, in most studies, teachers chose the contexts that students investigated based on the teachers’ understanding of the world—the situation in my earlier middle school teaching— rather than on students’ community knowledge (Freire’s starting point for liberatory education). Exceptions besides the present research include Turner’s (2003) study, in which she and the teacher created three short-term units (from 2 to 5 weeks) based on student-generated topics for them to learn particular content and generate their own data; Terry’s (2009) summer-school project with students who chose sociopolitical contexts from which he developed curric- ulum; and studies of after-school clubs in which students selected contexts (Turner, Varley Gutiérrez, Simic-Muller, & Díez-Palomar, 2009; Varley-Gutiérrez, 2009).

In short, what we lack is research on a full-year example of teachers attempting to connect the 3 Cs in an RWWM context, especially one of full-time classroom teachers. In this article, I address connecting the 3 Cs starting from community knowledge (generative themes) and simultaneously developing classical knowledge (reading the mathematical word) and critical knowledge (reading the world) to support students in reading and writing the world with mathematics. Though a limitation of this study is that I taught only one class during the year—a constraint that future research needs to address—I describe my attempt to build on and extend existing literature and theory and my previous middle school work by going from interjected CM projects (15–20% of the time) to a year-long curriculum entirely focused on RWWM, and from contexts that I chose to ones that students selected through articulating their own generative themes.

A Caveat on What Is Missing—Antiracist Critical Mathematics Critical mathematics has not dealt much with race, racialization, and racism (or

much with other injustices, like sexism, ageism, or heterosexism; see Rands, 2013). Though I ground my work in Freire, he provided insufficient guidance on these questions (Ellsworth, 1989; Ladson-Billings, 1997; Murrell, 1997). Freire did not understand “that racial epistemologies, and more specifically racist epistemologies, have an ontological content that compels one whom is ‘black’ to constantly ask himself or herself not ‘who am I?’ but ‘what am I?’” (Haymes, 2002, p. 157). Thus, I turned to liberatory Black philosophies and pedagogies for deeper understanding of antiracist education (e.g., Anderson, 1988; hooks, 1994; Perry, 2003; Woodson, 1933/1990). Furthermore, CM has not broached the interrelationships of race and class and the specific historical and present intersections of racism and capitalism (e.g., Marable, 1983) or how these are mutually constitutive and the implications for mathematics education. Although I cannot address this in depth in this article (see Gutstein, 2016), I do discuss ways in which I did and did not sufficiently address antiracist mathematics as a way to problematize my work and point out future directions.

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Method This study incorporated aspects of practitioner research (Anderson, Herr, &

Nihlen, 1994; Cochran-Smith & Lytle, 1993), ethnography (Hammersley & Atkinson, 1983), and critical, collaborative participatory action research (Carr & Kemmis, 1986) as I analyzed my classroom and teaching to better understand how to enact RWWM pedagogy. The work that I report here involved

• collaborating with students to uncover their generative themes; • creating mathematics curricula based on these themes, so that students learned

conceptually based mathematics and simultaneously deepened their under- standing of their social reality—to prepare them to shape the world;

• teaching, and necessarily modifying, the above curriculum in a classroom community, cocreated with students, that insistently focused on integrating mathematics and sociopolitical contexts;

• doing this work in an untracked,8 neighborhood, Chicago public high school in a low-income, working-class Latin@ and Black community while supporting students’ racial, social, and cultural identity formation; and

• studying my practice and developing and extending theory with students as active participants and as informal coresearchers. This work occurred within one particular setting, and like all contexts, mine had its unique dynamics. What is feasible here needs to be reinvented there (Freire & Faundez, 1989; Freire & Macedo, 1987). I say more about this reinvention below.

Context and Setting The site for this study was the Social Justice High School (commonly known as

Sojo), which is part of the Little Village Lawndale High School Campus in Chicago, Illinois. I identify the school because its history is central to the story. Sojo is located in Lawndale, a Latin@ and African American neighborhood on Chicago’s West Side (a “Black and Brown” community, as many residents call it). North Lawndale is an overwhelmingly Black and low-income neighborhood. Across a railroad viaduct lies South Lawndale (Little Village), a dense, mainly working-class, low-income Mexican immigrant neighborhood,9 part of the largest contiguous Mexican commu- nity in the United States outside of Los Angeles. Lawndale has suffered disinvest- ment and neglect for years but has a strong history of activism and efforts toward community betterment. White flight occurred in the 1960s and 1970s, and Lawndale became overwhelmingly populated by people of color, many working in nearby factories. Deindustrialization seriously injured Lawndale: Chicago lost 330,000 manufacturing jobs from 1967–1990 (Betancur & Gills, 2000). Concurrently, the city cut services, property tax revenue fell, and the area suffered. Nevertheless,

8 Tracking or streaming refers to students being placed into particular education pathways that are narrowly focused and are difficult to transcend (e.g., vocational, academic, “gifted”).

9 In Little Village (and Chicago), people often use “Mexican” to refer to transnational Mexican or Mexican American communities.

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individuals, community and civic organizations, and churches throughout the neigh- borhood have worked hard to improve conditions, secure more city support, and develop cultural programs and social services.

For years, Little Village residents organized for a new high school, eventually securing a promise from the school district in 1997. However, in 2001, after Chicago Public Schools (CPS) administrators had claimed that they ran out of money but built two selective-enrollment high schools in Whiter, wealthier neighborhoods, 14 resident Latin@ activists went on a 19-day hunger strike (Russo, 2003; Stovall, 2005). CPS capitulated and built a new campus with four small high schools, each with a community-determined theme. One theme was social justice, ensuring that the values of the hunger strike, including solidarity, transparency, and self-determi- nation, stayed prominent—and Sojo was born. A colorful mural of hunger strikers and community and national civil rights leaders graces Sojo’s walls, bearing the school’s slogan, “Born out of struggle,” to keep the memory and spirit of collective sacrifice alive. Although young and trying to survive despite the dismantling of urban public education (Lipman, 2011b) and the pain and challenges of poverty and racism facing Lawndale, Sojo has been able to build on community strength and create a lively space for learning with a spirit of family and collectivity.

Each of the four schools on campus has roughly 375 students, originally 70% Latin@, mainly from Little Village, and 30% Black, from North Lawndale.10 Neighborhood public schools have no selection criteria and accept all students in the community. Like many such CPS schools, Sojo has overwhelmingly low-income (approximately 98%) students of color (100%). Students’ ACT scores averaged 16.0 from 2008 through 2012, and although Sojo’s academic performance has fluctuated, a relatively high percentage of students graduate (approximately 70%) for a neigh- borhood Chicago high school.

I was part of the design team that created the plan for and founded Sojo. From the time it opened in the fall of 2005 until 2011, I worked closely with the teachers and students at Sojo in creating and coteaching mathematics for social justice (Gutstein, 2009a; Gutstein & Sia, 2007). Throughout Sojo’s first 3 years, I worked with the entering class (initially 98 students) as they passed from 9th to 11th grade. During that time, the mathematics teachers and I developed and cotaught several 1-week to 2-week social justice projects. These included mathematizing who was left behind in New Orleans (in terms of class and race) during Hurricane Katrina’s evacuation, examining racial profiling, and investigating the probability of racially dispropor- tionate jury compositions (the trial of the Jena 6; see Gutstein & Sia, 2007). But Sojo’s teachers primarily taught mathematics using the Interactive Mathematics Program (IMP). During Sojo’s third year, the faculty and I agreed to offer a 12th-grade math- ematics class in 2008–2009 entirely focused on social justice contexts that I, as an experienced CM teacher, would develop and teach.11 It was to be a learning year

10 This balance has shifted over time. As of 2015, Latin@s comprised at least 80% of the student body.

11 The class was colisted with my university as an education elective, so students received both high school and college credit.

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during which Sojo’s three mathematics teachers would rotate regularly into my class and participate or coteach, and we would collectively study the process. Unfortunately, we all taught simultaneously, so the three teachers were unable to attend my class. This class was the setting for the research that I report in this article.

Data Sources Data for this study were collected by a research team that included myself (a

university-based mathematics educator) and two doctoral students who were participant observers in the classroom for most of the school year: Anita Balasubramanian and Patricia Buenrostro. Both students’ dissertation studies (Balasubramanian, 2012; Buenrostro, 2016) were based on data from this class. We collaborated closely throughout the study, sharing the work of collecting and analyzing data.12

Data sources from which we drew included open-ended student surveys (one at each semester’s end); five focus-group and 14 individual audio-recorded interviews of students; video recordings of students in two public community presentations and other student presentations;13 my reflective teacher journals (103 entries); field notes written by Balasubramanian and Buenrostro (74 entries); audio recordings of 35 classes; video recordings of 41 classes;14 and electronic communications between students and me, during and after the class ended. Our reflective journals and field notes included reflections on classroom culture and discussions, students’ mathe- matics work and dispositions related to their analyses of sociopolitical reality, and interactions among class participants (including adults). I collected classwork, homework, tests, projects, ACT scores, 17 sets of student journals (mostly weekend homework assignments), an op-ed piece students collectively wrote that was published online in the Huffington Post (Koehler, 2008), the 81-slide PowerPoint presentation that students created for their end-of-year community presentations, and other documents.

Data Analysis We analyzed data both during and after the school year. During the year, analysis

was generally informal and was used primarily to improve students’ learning, although occasionally our research team discussed student work or audio or video recordings and even debriefed with students a couple of times. The purpose of these sessions was to get a better sense of major issues and trends within the data and to assess the overall progress and process of the class. On a regular basis, I debriefed with one or both coresearchers for approximately 15 minutes after class. I generally wrote my teacher journals after teaching and reviewed them regularly. I occasionally

12 When appropriate, I use “we” to refer to the research team or the class as a whole. 13 Seven students in my class comprised an out-of-school mathematics for social justice advocacy

group called the Crew. The Crew presented at 17 local, regional, and national education research and practitioner conferences over 3 years; for example, six students presented at the 2009 annual meeting of the American Educational Research Association (Gutstein et al., 2009).

14 Audio and video recordings were not necessarily of the whole class period.

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discussed class with the students after school or via phone, email, and text. I regularly modified my instruction and plans for the class based on our analyses (see Gutstein, 2012a).

More systematic analysis continued after the school year. We started by analyzing field notes and proceeded to look at video or audio recordings, student work, and other data. We used open and focused coding (Emerson, Fretz, & Shaw, 1995) and iteratively looked for patterns and relationships that emerged and guided further analysis, and we either audio-recorded or took notes on data-analysis sessions for further reflection. We coded and analyzed data by hand as well as using qualitative data analysis software (Atlas-ti), and we relied on multiple data sources for richer interpretations. We also chose 3 weeks (one from each of the three most extensive curricular units in the year—elections, displacement, and HIV-AIDS), selectively transcribed portions of audio or video recordings, and discussed our analyses collec- tively.15 Overall, we were interested in the meanings that students made of their class experiences and especially how they integrated mathematics (classical knowledge) and sociopolitical understanding (critical knowledge) of the contexts that they studied (which they had chosen). We paid specific attention to how students and teacher cocreated a culture that supported collaboration, students’ own voices and questions, and serious disciplined study (Balasubramanian, 2012)—even when students found class especially challenging (Buenrostro, 2016). Throughout our analyses, we looked for evidence of the ways that I facilitated or hindered students’ capacity to read and write the world with mathematics (see Balasubramanian, 2012).

I took the following as evidence of students’ learning. First, I looked for students’ assertions of understanding both mathematical and sociopolitical ideas and the extent to which these were integrated. Second, I examined students’ mathematical work and explanations for evidence of the National Council of Teachers of Mathematics (2000) process standards and the National Research Council’s (Kilpatrick, Swafford, & Findell, 2001) strands of mathematical proficiency.16 Third, I looked for students’ participation, perseverance, and engagement (individual and collaborative) in the classroom community (Lave & Wenger, 1991). Fourth, I consid- ered grades and successful (or unsuccessful) completion of assignments, projects, and other assessments (e.g., answers, work shown, reasoning). Fifth, I looked at students’ sense making of mathematical representations in relation to sociopolitical reality, as evidenced in their work and oral or written words, and their fluent use of tools with conceptual understanding. Finally, I searched for examples of students’ sense of social agency (Gutstein, 2006a), or writing the world (with mathematics),

15 Our selection criteria included whether the portions illustrated aspects of RWWM pedagogy (e.g., connecting the 3 Cs), highlighted tensions or constraints, or were otherwise notable in various ways.

16 The National Council of Teachers of Mathematics (2000) process standards are communication, connections, problem solving, reasoning and proof, and representation. The National Research Council’s (Kilpatrick, Swafford, & Findell, 2001) strands of mathematical proficiency are adaptive reasoning, conceptual understanding, procedural fluency, productive dispositions, and strategic competence.

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as evidenced by their actions and expressed orientations to share with and teach others about what they learned. I always look to multiple data sources for more complicated interpretations.

Researcher Positionality I try to be aware of power relations. I am a White male professional, and I was in

my late 50s when I taught the class. My Latin@ and Black students were between 17 and 19, all from low-income or working-class families. There were social distances and power differentials—race, class, status, and more—between the students and me, though I had some life experiences like theirs and grew up in a New York neigh- borhood similar to Lawndale. Buenrostro and Balasubramanian, as doctoral students, also had social distances from students, though both are women of color— Balasubramanian is Indian and came to the United States for graduate school, and Buenrostro, a Latina, had lived in Lawndale for years (and still does).17 Power rela- tions also existed because I was their dissertation advisor. It was our practice, however, to make visible and problematize differences of which we were aware.

I know that one cannot speak for another (especially when crossing boundaries of culture, race, gender, class, age, and language), and I accept Delpit’s (1988) epis- temological stance that “we must keep the perspective that people are experts on their own lives. . . . they can be the only authentic chroniclers of their own experi- ence” (p. 297). Thus, I valued and treated students’ self-reflections and observations as meaningful data that shaped my analysis and development of theory. If students said that they experienced something in a particular way, I acknowledged and looked at it in relationship to other phenomena. I let students speak for themselves as much as possible in this article, understanding that my interpretations do not necessarily capture students’ full meanings (Weis & Fine, 1996).

My perspective on how to theorize and for what purpose builds on Freire’s (1994) description of meetings with groups of workers involved in organizing and political struggles. Their approach in analyzing their experiences was to “sneak up on the theory that was imbedded in their practice” (p. 126). Similar to action researchers, their “interest always centered on a more critical understanding of practice in order to improve future practice” (p. 126). This aptly describes my approach. As I attempted to teach in ways that encouraged students to read and write the world with mathematics, our research team reflected on, interpreted, and analyzed those expe- riences to support students’ learning. This is similar to Freire’s (Freire & Macedo, 1987) work on literacy campaigns in São Tomé and Príncipe (former Portuguese colonies in Africa) for which he developed Practice to Learn, an exercise workbook for beginning readers. A line in the workbook states, “To practice always to learn and to learn in order to practice better” (p. 71). This captures the sense of this dialec- tical process, a never-ending praxis cycle of practice informing theory informing practice. . . .

17 Buenrostro is the parent of a recent Sojo graduate and was one of the activists who participated in the 2001 hunger strike.

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Developing the Class—Creating Curriculum Based Upon Student-Articulated Generative Themes

My class had 21 students—15 females and six males, and 15 (non-Black) Latin@s (Mexican except for three, who were Mexican and other Latin@) and six African Americans. All Latin@s spoke Spanish to varying degrees, and it was the first language of most because they were all from immigrant families. All students were from working-class, low-income Lawndale families. As Sojo seniors, they were college-oriented and relatively successful in school compared with roughly half of CPS neighborhood high school students who do not graduate (Swanson, 2008). Almost all attended neighborhood CPS elementary schools, and their mean ACT score was 16.8.

At the time of this study, Sojo students took 4 years of mathematics and selected senior classes during their junior year. In spring 2008, senior mathematics options were precalculus, IMP IV, or my class. That spring, I met twice with the students who chose my class to discuss and select the contexts that they would investigate. This was to “reinvent” Freire’s work of uncovering learners’ generative themes (Freire & Faundez, 1989) and then to create and teach a curriculum based on them— in other words, to start from community knowledge and connect the 3 Cs as part of enacting RWWM pedagogy. Freire (1993) proposed that through this, students would more fully analyze their social reality and learn the academic disciplines that they used to study the world (O’Cadiz, Wong, & Torres, 1998). The five units selected for study are presented in Table 1. Students suggested HIV-AIDS and criminalization because of their experiences, knowledge, and interest. Displacement was an issue often discussed at Sojo. Although I proposed elections and sexism, the students had substantial dialogue about all of the unit topics and reached a collective decision about what they would study (see Table 1). They also determined the pacing and modification of units during the year. For example, when we ran short of time in March 2009, they decided to incorporate sexism (the last unit) into the third and fourth units.

However, selecting contexts does not determine the content that one uses to study them. Chicago 12th graders take few high-stakes tests, and Sojo’s principal curric- ulum requirement was to mathematically prepare students for further study, college, life, and meaningful work. Consistent with my understanding of RWWM and my previous work, I decided that the over-arching content focus would be for students to mathematically model their social reality (Mukhopadhyay & Greer, 2001). The curriculum that I developed included an eclectic blend of algebraic reasoning and precalculus; discrete mathematics; probability, statistics, and data analysis; and number (especially proportional and quantitative reasoning).18

I worked with several graduate students in spring 2008 to develop curriculum frameworks for two units (elections and HIV-AIDS). A key question driving the development was, How do social scientists use mathematics to study these issues? We learned that epidemiologists modeled HIV-AIDS with differential equations and

18 See Appendix A for the mathematics content in each unit.

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that Freeman (Freeman & Bleifuss, 2006) investigated fraud in the 2004 presidential election with binomial and normal probability distributions; these set outlines for students’ explorations. However, even with partial frameworks, I created most of the curriculum materials (for all units) while teaching, borrowing from and adapting IMP and Modeling Change With Discrete Dynamical Systems (Garfunkel, Godbold, & Pollack, 2000) from Mathematics: Modeling Our World: Pre-Calculus.

The order of the units was intentional and codecided with students. Starting the year with the elections unit made sense because school started in September and the U.S. presidential election was in early November 2008. At least a third of the students were already 18 (voting age in the United States), and several students had already been, or planned to be, poll watchers to help certify honest elections. Barak Obama was from Chicago and had the real potential to be the first Black president in the country. On Chicago’s West Side, people were tremendously excited and supportive.19 For the students, the idea that someone could potentially steal the presidential election from Obama—possibly as had been done with John Kerry in 2004—was untenable and violated the generative theme of enfranchisement. The unit on displacement followed because it had the most potential to unveil the complicated sociopolitical realities impacting Lawndale in ways that were similar but distinct in the commu- nity’s two parts (which I explain in more detail below). The HIV-AIDS unit came after displacement because students used similar but increasingly complex mathe- matics in the two units and they needed to work through the displacement unit first. Finally, we chose the criminalization and sexism units to finish out the year, with criminalization first because students explicitly named that theme.

In this article, I focus on the displacement unit because I have written elsewhere about the HIV-AIDS unit (Gutstein, 2012a) and Balasubramanian (2012) focused on both the HIV-AIDS and elections units in her dissertation. Although every unit had its particularities and lessons,20 the common goal across all units was that I wanted students to read the world with mathematics.

Table 1 RWWM Units in Chronological Order 1. Elections (11 weeks; Early September–Mid November 2008). Statistical

anomalies in the 2004 presidential election (Freeman & Bleifuss, 2006); implications for the then-upcoming 2008 Obama–McCain presidential race.

2. Displacement (13 weeks; Mid November 2008–Early March 2009) 3. HIV-AIDS in Lawndale (7 weeks; Early March–Late April) 4. Criminalization of youth and people of color; Community presentations

(7–8 weeks; Late April–Mid June) 5. Sexism (integrated into HIV-AIDS and Criminalization, by student decision)

19 One student, Jenny, served as a poll watcher in her North Lawndale voting precinct. The day after the election, she brought to school a copy of the official voting record for her precinct—292 to 0, in Obama’s favor.

469Eric “Rico” Gutstein

Interweaving Learning Mathematics and Reality—Doing the Dance Our research team coined the term doing the dance to represent enacting RWWM

in the classroom (Balasubramanian, 2012; Gutstein, 2012a). This metaphor captures the ways humans respond to and interact with each other in unscripted ways. It signifies the braided interconnections that people make between mathematics and sociopolitical reality and the ways in which a CM teacher might support students to connect the 3 Cs. The dance of RWWM is not simply a linear movement from community knowledge to a relational understanding of critical and classical knowl- edge but rather a back-and-forth, dialectically linked interweaving between the multiple types of knowledge. Students do not leave behind their community knowl- edge as they learn the mathematics of displacement and understand better how that very mathematics can help move them out of their community. Rather, as they connect a mathematical and sociopolitical understanding of their displacement, their awareness of their lived reality grows as well. This process, in a high school math- ematics class, as actualized by teacher and students collaborating together, is what we mean by the dance of RWWM. In this section, I describe the displacement unit and analyze two examples to illustrate the dance. I briefly comment on each example and then more fully discuss them and how the dance exemplifies RWWM theory and practice.

Displacement Unit

We are reading the world with mathematics by understanding different mortgage loans and how they affect our community. We are learning about the grass root of the problems in our community. (Mónica, Journal 12, February 24, 2009)

Displacement was a generative theme—students lived with it and saw it around them in multiple ways. I started the unit by telling the story (with family permission) of how the grandmother of a student in class (Carmen) lost her home to a subprime21 home-equity loan, despite having paid off her mortgage years before. Other students’ families had lost homes or were struggling to keep them. Boarded up homes were everywhere in the community, and we examined data and graphs showing that foreclosures tripled in both North Lawndale and Little Village from 2005 to 2008 (Figure 3).

Displacement occurred in similar and different ways in the two Lawndale commu- nities. Both experienced foreclosures, but North Lawndale was being gentrified, while Little Village was not.22 However, Little Village’s nemesis was the ubiquitous

20 For example, the elections unit acculturated students into RWWM, involved them in cocreating a collective classroom space, and challenged their tendencies to conflate nonrandomness with malevolent actions, whereas the HIV-AIDS unit laid bare my preoccupation with challenging mathematics at the expense of student sense making—see Balasubramanian (2012) and Gutstein (2012a).

21 Subprime loans are typically high-cost loans with initially attractive (low) but changeable rates.

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threat of displacement not from the community but from the country—deporta- tion—because thousands of undocumented migrants lived there.

The driving question of the whole displacement unit was this: Whose community is this? The first part was on gentrification, subprime mortgages, and affordability, and the second part was about immigration and deportation. Initially, students examined median income, housing prices, and trends (learning about regression) and then learned to use discrete dynamical systems (DDSs)—discrete versions of differential equations—to model mortgages. DDSs are complicated models involving precalculus and discrete mathematics concepts (e.g., recursion, shape of curves, rate of change, equilibrium values, limits). Students developed mathematical models to better understand neighborhood displacement, starting with simple linear models and proceeding to more complicated ones (DDSs) while also examining the limitations associated with modeling. Students became proficient at interpreting and graphing fixed-rate and subprime loans (adjustable, pay-option, and balloon). They learned about and calculated the interrelationships of payments, interest rates, terms, and balances on different mortgages and determined “affordability” for both North Lawndale and Little Village. They studied the concentration of subprime mortgages among different demographic groups and learned how people of color were dispro- portionality impacted, an example of how racism and capitalism interrelate. DDSs gave students the means to mathematically unpack mortgages and to understand the subprime mortgages devastating their community. They used easier-to-grasp DDSs (with two equations) during this unit and then progressed to working with qualita- tively more complicated ones (with four equations) in the subsequent unit on HIV-AIDS (see Appendix B).

Figure 3. North Lawndale and Little Village foreclosures 2000–2008.

22 This was at least partially because North Lawndale had better housing stock, more vacant land, and better transportation to downtown.

471Eric “Rico” Gutstein

In the second part of the unit, students studied, analyzed, and predicted immigration rates and trends using linear and quadratic regression. They graphed and analyzed the concentration of U.S. government subsidies to agribusiness (Fanjul & Fraser, 2003), which contributed to displacing Mexican farmers from their land to maquiladoras along the United States and Mexican border (Bacon, 2008; Bigelow, 2006) and even- tually to Little Village—where they experienced further displacement threats through deportation.23 Students looked at the loss of manufacturing jobs in Lawndale and the United States due to the North American Free Trade Agreement and examined how government policies could simultaneously push people out of Lawndale while creating conditions for them to immigrate to the community (Bacon, 2008).

An explicit goal of the unit was for students to develop a more complicated political and mathematical understanding of how displacement affected the Lawndale communities. Students uncovered that larger economic interests, like real estate developers and investment firms, stood to gain and that, overall, similarities outweighed dissimilarities of race and culture between North Lawndale and Little Village—two communities of color being displaced for profit. Racism and capi- talism intersected throughout the unit, although I did not make this sufficiently explicit (as I explain below).

Examples of the Dance To start the unit, students interviewed family members to answer the question,

How did I get here [meaning to Lawndale]? North Lawndale (and Chicago’s Black community overall) has mostly Mississippi and Arkansas roots (Wilkerson, 2010), and all students (and the research team) had a migrant story, which several shared in class, establishing parallels across neighborhoods. We watched short video clips of North Lawndale residents discussing the pros and cons of gentrification and clips of other communities’ gentrification stories. We examined North Lawndale and Little Village demographics and housing prices over time.

I chose the following examples because the first gives a sense of the ebb and flow over three class periods (skipping 1 day) and the second is a tightly connected RWWM instance that occurred within a single class period.

Example 1A. On November 25, the class period started with a discussion of a miniproject, which was due soon:

I then talked about mathematical models, but that quickly led me to briefly discuss casino capitalism and related manifestations. I told students about the [Pentagon’s proposed] “trading on death” market,24 and Daphne couldn’t believe it. . . . One senator said that

23 Maquiladoras are free-trade-zone factories along the border of the United States and Mexico. However, many jobs there were lost, especially in textiles, as companies shifted production to lower cost countries like Bangladesh, Haiti, and China. Some of these displaced workers went north to the United States, winding up in communities like Little Village.

24 The Policy Analysis Market was a futures exchange that the U.S. Department of Defense proposed in 2003 in which traders could bet—and make money—“on forecasting terrorist attacks, assassinations and coups” (Hulse, 2003, para. 1). The Pentagon rapidly withdrew the exchange due to widespread opposition.

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the idea was so preposterous that he had trouble convincing people that it wasn’t a hoax. I shared M.’s story about borrowing $130,000 to buy her Little Village house and ending up owing over $300,000, and I explained that this was how banks made money. People had questions, which we discussed. It was pretty interesting.

We then segued into the homework, to extend the graph of North Lawndale median house prices to the present (Figure 4). Students had not yet learned regression. Instead, they presented and we discussed their informal solutions. Vanessa found the difference between 2006 and 2001, about $127,000, and estimated that it was about 2.5 years between the last data point (first quarter 2006) and the present (fourth quarter 2008). She divided 127,000 by 2, added that to the last data point, and proclaimed, “somewhere close to $350,000.” Next, Roxanne added and subtracted the amounts between each period (paying attention to positives and negatives, because, as Antoinette pointed out, prices did not always rise). She divided the total by 19 and extended it 11 more periods for her answer. Next, Gregory did some research and created a graph with his estima- tions. Mónica looked at the last year and thought that it was too different from the other years, so she found the difference from the last data point and 2 years prior (about 80k). She divided that by 2 for a year (40k), and added 2 years and 3 quarters (110k) to the last data point. Finally, Carlton just estimated that prices increased, on average, by 5% a year. He had no real rationale for the 5% a year, but since he did it incrementally, it was a nice lead-in to compounded interest.

So we have 5 ways of doing it. Renee asked how I would have done it, and I told her that I would use a “line of best fit.” Carlton liked that. I said something about having calculator tools, and he said, “Why didn’t you just tell us before?” As usual! Tomorrow, I will go into linear regression. Nice class! (Data sources: teacher journal, student work, 11/25/08)

Discussion. The flow of this class period moved from principles of mathematical modeling to examining how “casino capitalism” speculates to maximize profit, how

Figure 4. North Lawndale housing prices by quarter, 2001–2006.

473Eric “Rico” Gutstein

banks make money, and how one could borrow $130,000 but end up owing $300,000—integrating mathematics and reality. Students then investigated a specific model related to their lives—a graph of Lawndale housing prices—and extended it to the present. The students’ solution inventions (evidencing conceptual understanding and mathematical learning), multiple perspectives, and classroom discourse are suggestive of the pedagogical practices advocated in the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), which I refer to as reform mathematics. Students explored mathematical models of their reality and moved towards learning DDSs (reading the word) and beginning to read the world.

Example 1B. In class the previous day, on December 1 (when we returned from the Thanksgiving holiday), students had learned how to use linear regression to find the best-fit line. I started December 2 by collecting a challenging RWWM weekend assignment about income inequality (Appendix C). I told students how much I enjoyed teaching them but also that I was concerned that I was doing them a disser- vice by not pushing hard enough academically. I also connected solving mathematics and social problems:

If nothing else, a lot of this is making meaning out of the world, making meaning out of the mathematics, and bringing them together. And it takes work! You’re capable of doing this work! You’ve learned in school that if a math problem can’t be solved in 30 seconds, then it can’t be solved. That’s bull. People have worked on math problems for years. And how about social problems? Let’s stop displacement in North Lawndale and Little Village—everybody knows that’s not a 30-second problem, right? Mathematics is not a 30-second problem either. (Data source: audio, 12/2/08)

The class then segued to homework (Figure 5). A few students answered Question 1; most struggled on Question 2. I saw that few could explain what the y-intercept meant, why it was negative, or precisely what slope meant, and students appeared to have rote understanding, as this brief interchange suggests:

Mr. Rico: How do you know b is the y-intercept—how do you analyze it, justify, explain it?

Gregory: ’Cuz that’s the way we were taught. Mr. Rico: Yeah, the way you were taught, but there’s a meaning behind it, there’s

a meaning behind everything. (Data source: audio, 12/2/08)

I then drew a coordinate graph on the board and asked, “Where does the [best-fit] line go?” Students’ answers included “the middle,” “the third one,” “zero zero,” “lower right,” “the negative,” “upper right,” “from the upper right to the lower bottom,” and “no, it’s on the upper right because it rises all the way to 2000.” I placed the line on the graph and asked, “Now why did I put it there and is it right or wrong?” Some students said that it was wrong, but not all agreed. They continued to discuss and struggle. I wrote y = mx + b and asked its meaning, engendering discussion

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about intercepts, slope, and zeroes. Calvin asked, “But why is it that letter, why can’t it be, like, the x-intercept?”

For the rest of the period, students and I discussed various mathematical ideas and representations, created in-out tables (functions), graphed points, debated if all the points made a line, argued whether the line was curved or straight, linked algebra to geometry (similarity and right triangles), and deconstructed slope. Near the end of class, Gregory returned to the relationship between mathematics and the under- lying context, saying, “I still don’t get what we’re doing with this situation [of reality].” Ann added, “Yeah, what are you trying to show us?” I responded, “Hang on, stay with it. We’re going to understand what slope is and what a line is.” A short time later, I continued,

We’re trying to figure out what that 21,000 [see Figure 5] means. And to understand that, we’re backing up to understand what a line is, what the slope is, what the intercept is, how we understand the meaning, the equation, of a line. We’re looking at a graph, an in-out table, and the equation. All three of them, making meaning out of that, so we can understand what the 21,000 is. That’s where we’re going on this. To connect it for us. (Data source: audio, 12/2/08)

My last words that period were:

You have to work this out in your mind to understand what a line is, what slope is, what an intercept is, and how to figure out and take meaning from this for the situation in North Lawndale. You gotta understand the mathematics to understand the political situation. (Data source: audio, 12/2/08)

Discussion. In many ways, this was a typical class period. We started with a reading-the-world-with-mathematics assignment about economic injustice. I did a public reflection or self-criticism grounded in my self-questioning and the need to adequately prepare students for the future, and I linked the complexity and necessary persistence of solving real-world and mathematics problems. When I assessed that

Figure 5. Homework questions discussed on December 2, 2008.

1. The equation of the line of best fit is y = 21711.28x – 43331109.77. What does the slope mean in this situation?

2. What is the y-intercept of this line, and what does it mean in this situation? Why is it negative?

3. If you set your window values correctly, you can predict relatively far into the future, say to 2020. When you do this, you are creating a mathematical model of this situation (house prices in North Lawndale). What assumptions is your model making? Do you agree with those assumptions? Why or why not?

475Eric “Rico” Gutstein

students were not fully grasping the political implications because they struggled mathematically—that is, their critical comprehension was impeded by their weak classical knowledge—I had them explore more abstract mathematics to explicitly link the meanings in both the mathematical and sociopolitical realms, which is a theme of the dance. This challenged them. Although student participation was good overall throughout the year, every student can be heard on this day’s recording in more and less substantive ways, even Miriam, who voluntarily went to the board to verbalize her thinking and ask questions, which was unusual for her.

Example 1C. I began class with a 6-minute discussion about the history of December 4 on Chicago’s West Side (Lawndale’s community). On December 4, 1969, less than 5 miles from Sojo, police raided the Chicago Black Panther Party office and murdered its leader, Fred Hampton, in his bed and killed another Party member, Mark Clark. I told students that Hampton organized nearby and that a federal civil rights suit eventually found the city guilty of violations and forced Chicago to pay almost two million dollars to the victims’ families (People’s Law Office, n.d.). Students asked questions including whether police were charged with crimes and, if not, why not. We then returned to investigating lines (again, the best- fit line for North Lawndale housing prices). I drew four lines through labeled points on a coordinate graph on the board and gave students 10 minutes to collaborate, find the equations, and explain their reasoning.

This excerpt from my journal describes the rest of the class after the 10 minutes for collaboration. I reviewed the video of the class period to corroborate this descrip- tion of events.

I dragged people to the board [almost all stood for the last 20 minutes]. We had 19 students, so it was crowded, but we managed. I started w/ the line y = –4. A few people understood that all points on the line had y-values of –4 (there were two points on the line with y-coordinates of –4). But it took a while to establish that all points on a hori- zontal line had the same y value. We worked for a while, then switched to the vertical line. Miriam and Ellen, again, debating whether the equation was y = –7 or x = –7. Good conversation. We then worked on the “diagonal” line [y = x], as someone called it, and Vanessa said “well, y equals x,” but she (and others) did not understand that what she said was the equation.

I finished up with HW to find the equation for a line with two points, (6,4) and (12,2). It will be interesting to see what students do.

The larger direction of this “digression” is this—the regression line is y = 21,711.28x – 43,331,109.77. So if you ask, which I did, “What’s the meaning of the y-intercept?” then you’re confronting that the y-intercept means that in the year of Christ’s birth, the median house price in North Lawndale was about negative $43 million. This shows that a regression line (an instantiation of mathematical modeling) can be meaningless unless in context. The point is that modeling has to be critiqued, questioned, challenged, so we don’t end up bowing before the math gods. This is a good example. But unless one understands what a line represents and what the y-intercept is all about, it doesn’t follow. That means (in my logic), that in this particular example, these ideas are connected— understanding what a line is can lead to understand the issues and problems of

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mathematical modeling. So that motivates my spending the time on this. Not to mention that students’ grasp of these ideas is weak. (Data sources: video, teacher journal, 12/4/08)

Three-day summary. These 3 days show the dance—back-and-forth, connected movement between real-world contexts with mathematics, decontextualized math- ematics, and real-world contexts without mathematics. The principal emphasis is the relationship of sociopolitical context—North Lawndale housing prices and the omnipresent threat that students may not be able to live there—and mathematics content—understanding what the symbols, relationships, equations, and graphs mean and how they relate, not only to each other (geometry–algebra, tables–graphs– equations) but also to the reality that they represent; this is the essence of RWWM and illustrates the connection of the 3 Cs. A reasonable question to ask is: Why would 19 teenagers stand for 20 minutes trying to understand the meaning of a line? I assert that students’ desire to comprehend what was happening to their community contrib- uted to their perseverance in learning mathematics.

Example 2. This example, from January 6, 2009, is the class period from which I pulled the quote used at the beginning of this article. Students were beginning to learn DDSs. The previous day, I assigned the homework task presented in Figure 6.

For the first 25 minutes, Marisol, Vanessa, and Minerva each presented and led discussion as students collectively answered Questions 1–3a while I sat in back and said little. But no one answered Question 3b. I then worked with students to help them realize that they needed to subtract a payment in the difference equation and that 6% was the annual, not monthly, interest. Together, students produced the DDS for Little Village:

Starting equation: u(1) = 150,000 /* initial mortgage value Difference equation: u(n) = u(n–1) + .005u(n–1) – 808 /* amount owed at start of month n

Daphne was confused, so Ann used our overhead projector with the graphing calculator to explain that after 1 month, the family would owe $149,942.00 (Figure 7). Vanessa did not understand the full equation, so students collectively discussed this, questioning and answering each other. I pushed Vanessa to explain every number on the screen (Figure 7). The following interchange then took place.

Mr. Rico: How much less than your initial payment is what you owe now? Antoine: 58 dollars. Mr. Rico: How much did you pay? Gema: 808 dollars. Mr. Rico: 808. Out of the first payment of $808, $750 goes to interest to the bank.

Only $58 reduces your loan balance. Understand how capitalism works, how banking works. Almost 15 times as much money goes to

477Eric “Rico” Gutstein

the interest as goes to the principal—the money you owe. That is profit for the bank. Yes, they have to pay their employees, but there is still a huge profit. That’s how they make money. (Data source: video, 1/6/09)

I asked Ann to iterate monthly so that students could watch the balance change (149,883.71 after 2 months). I refocused students on the question (Can we afford to

Figure 6. Homework discussed on January 6, 2009.

1. Suppose you deposit $500 in a savings account that pays 3% interest a year, but pays you interest each month. Create a dynamical system (write the equations for the situation) and figure out how much you will have after 1 year.

2. Now assume you have a $500 deposit that pays 3% per year, but you also withdraw $25 a month. Create the dynamical system—how long before you run out of money?

3. Currently, mortgages average 6% a year for a “fixed-rate, 30-year, conventional mortgage.” This means that the interest rate does not change and you pay (monthly) for 30 years unless you sell the house, default, or re-finance the mortgage. According to the U.S. census, North Lawndale median family income in 2000 was $20,253, and Little Village’s was $32,317. According to the Federal department of Housing and Urban Development (HUD), “The generally accepted definition of affordability [without “hardship”] is for a household to pay no more than 30 percent of its annual income on housing.”

a) What is the maximum monthly amount a North Lawndale family at the median income should pay for housing? A Little Village family?

b) If a family in your community (either North or South Lawndale) with the 2000 median income acquires a 30-year, fixed-rate $150,000 mortgage at 6% interest:

1) Create a dynamical system with u(1) being $150,000 and interest at 6% a year.

2) Figure out whether that family can afford the mortgage “without hardship.” If they can, figure out how big of a mortgage they can afford; if they cannot, explain why not.

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stay in this neighborhood?) and had Ann (with help) quickly press the calculator button once for each of 360 months to find the balance after 30 years, $91,738.13.25 I had students calculate how much the family paid over 30 years (approximately $291,000) and wrote the equation on the board, 150,000 – 291,000 = 92,000, and then we had the discussion that appears at the beginning of this article.

Discussion. On this day, students explored mortgage affordability for Lawndale families while posing questions and encountering complex mathematical and real-world ideas. Students led the initial 25-minute discussion, mainly talking among themselves to answer problems that supported their learning of DDSs and led toward the question of who could afford to live in Lawndale. They learned and used mathematics to collectively begin to understand how banks and mortgages work, mathematically and politically. Students’ developing grasp of DDSs informed them about, and further uncovered, their sociopolitical reality. Their question (Can we stay here?) demanded an answer, and mathematics was one way to address it. The more students grasped the mathematics, the better they realized, overall, what was happening to their community; the better they comprehended their reality using mathematics, the more mathematics made sense as an explana- tory and revealing phenomenon. This shows the close interweaving of studying intellectually challenging mathematics and unveiling social forces affecting students’ lives. The example illustrates how RWWM pedagogy intends to start from students’ generative theme of displacement to develop both critical and clas- sical knowledge in a connected way.

Doing the Dance The examples presented above illustrate the dance of RWWM at Sojo, including

some of the challenges and the inherent messiness. The types of interactions ranged from student led (the first half of January 6), to teacher talk (my initial words to students December 2 and December 4), to extended interchanges among class participants and me (the second half of December 4 and January 6). We view

Figure 7. Ann’s work on the graphing calculator, projected for the class, to show the balance due on a mortgage after 1 month.

25 My reason was to motivate a more effi cient, technological means and have students see the balance change over 30 years.

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the dance as moving between mathematical and sociopolitical poles, always with an eye on circling back to the center where interrelationships are clearer and the 3 Cs are more tightly connected. The class may have swung briefly towards doing abstract mathematics or, conversely, discussing context, but the intention was always to dialectically read and write the world with mathematics. When students explored mathematical concepts, even if decontextualized, content was tied to underlying sociopolitical realities. As illustrated in Example 1, students made sense of a line and its slope, equation, and intercept to “understand the mathe- matics to understand the political situation,” as I told them on December 2. Example 2 illustrates the tight integration of using and learning mathematics to more fully understand Lawndale displacement. This teaching and learning inte- grated the development of classical and critical knowledge and was constantly linked to a particular sociopolitical context, for a specific political purpose, driven by a deeply felt, nontangential question emerging from students’ knowledge of their community context: Will we be able to continue to live in our own neighborhood?

Doing the dance may involve the students and teacher discussing ostensibly nonmathematical phenomena related to students’ lives, which helps explain why the dance is more that just connecting the 3 Cs. These issues may connect math- ematically to students’ generative themes, such as how Carmen’s grandmother lost her home. However, these apparent digressions also have another purpose—to help “establish the intersubjectivity needed between learners and teachers in a Freirean [mathematical] literacy learning framework” (Murrell, 1997, p. 37). Murrell was discussing White teachers and Black students, but I extend this to White teachers (like myself) and students of color more generally. I argue that discussions about Fred Hampton’s assassination, for example, unconnected to reform mathematics—some might say inappropriate—have a purpose in teaching critical mathematics. Such discussions can reduce the challenge of “educating other people’s children” (Delpit, 1988) and build the intersubjectivity needed to explore painful issues (e.g., displacement, HIV-AIDS, criminalization) in ways that foster learning—and resistance, strong identities, collective solidarities, and hope. I am not suggesting that these Black and Brown youth stood for half a period and struggled together over the meaning of a line because their teacher started class by commemorating Fred Hampton, but I consider discussions that normalize politically taboo topics and develop political relationships as core components of RWWM pedagogy (Gutstein, 2006a, 2008). They are part of what Murrell (1997) called “the process of recovering narratives and motifs that are the ‘dangerous memories’ to racial oppression, cultural hegemony, and socioeconomic domina- tion” (p. 37)—memories, such as what Fred Hampton stood (and died) for and his meaning and connection to Lawndale youth.

In short, the RWWM dance involves substantial challenges—connecting the 3 Cs (e.g., helping students address their own questions, such as Whose community is this? Why our neighborhoods?), teaching across differences of race and other aspects of identity, and ameliorating the weak academic preparation (mis-education;

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Woodson, 1933/1990) of many students of color in underresourced, disinvested-in (Chicago) public schools. I turn now to what students learned.

Student Learning—Mathematics and Sociopolitical Reality In examining student learning, I am most interested in the interconnections of the

mathematical and sociopolitical realms given the dialectical nature of RWWM. This complicates the analysis and makes it difficult to unravel and explicate. My experi- ence is that attempting to separately assess learning mathematics and learning about social reality is impractical and does not adequately portray the spirit of RWWM (Gutstein, 2006a). If students learned mathematics and learned about their world separately, I would not be satisfied because a fundamental purpose of RWWM is for students to understand and transform the world through and with mathematics so that they come to value (mathematical) knowledge in shaping reality. Therefore, I looked for data showing the interrelationship of the two spheres. As I mentioned earlier, I accept that people are experts on their own experiences and view learners’ self-reflections and analyses as valid and important in understanding what they learned. I also acknowledge that a potential limitation of this study is that I have no so-called objective assessments of student learning. What I do have are assessments from the displacement unit, which include homework, an exam (Appendix D), a group project in which students created and analyzed mortgage scenarios for three subprime loan types, and a final essay reflection. In this section, I present evidence on learning mathematics and sociopolitical context, and I ask readers to reflect on students’ engagement and participation (including the difficulties and confusion they experienced), illustrated in the examples presented above.

Ann, a student who generally grasped mathematics, was one of many classroom leaders. She consistently pushed for conceptual clarity, was sometimes frustrated when things did not gel, and was unwilling to let me leave issues unconnected (Example 1B). In her displacement unit essay, she wrote:

In the displacement unit we learned how to calculate the amount of interest per both “n” months and “n” years, the total amount of interest after “n” years, the interest rate, the amount of years it would take a family with a starting value of “U(1)” to pay off their loan, and we also learned how to figure out whether or not a family would have been able to pay off their loan (of however much) with interest (of however much). We were also able to create a dynamical systems equation to help us understand the mathematics of three [types of] subprime mortgages. This helped us to understand how each part of our equation: u(n) = u(n – 1)+ ?u(n – 1) – ? affects the other and how the principal and interest affect all subprime mortgages.

From this project I learned not only how but why so many people’s houses are fore- closed. In understanding the mathematics of subprime mortgages I now understand how and why it is important to be educated about mortgage plans. The mathematics we did helped me realize how few families, in both Little Village and North Lawndale, are actually able to pay off their mortgage.

Ann’s claim of understanding each equation component and its interconnections within the DDS is corroborated by other data (e.g., her explanation of the DDS in

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Example 2), and her assertion of learning mathematics to help understand how and why banks foreclosed houses in her neighborhood was a unit goal.

I regularly solicited student feedback on the class. On an end-of-year survey, I asked the following questions:

As you know, the goals for this class were for you (1) to learn mathematics and (2) to learn to read and write the world with mathematics. We tried to do this by using math to study real-world issues related to your lives. Did this class meet its goals? Why or why not?

Ann responded:

NO!!! Just playing. It did meet its goals. I think this because the way we speak and write not only about math but the social justice issues is powerful. We watch what we say and really think about it. We have proven that the class met its goals because we are articu- late in the sense that we can answer questions and REALLY answer them.

These and other data—her participation, reflections, tests, projects, and class work—suggest that she met the class goals. Her response speaks to her sense of the class’s collective capacity to write the world through speaking and writing about mathematics and social justice in “articulate,” “powerful” ways.

Carlton responded to the same question:

This class did meet its goals because I did learn mathematics this year. I learned a lot of USEFUL mathematics as well. The stuff we learned this year is things that we can use anytime and anywhere. We learned how to read and write the world with mathematics because I can now look at numbers and figure out whether or not they are dispropor- tionate. I can discuss with my peers things that are right in society, using math.

In his displacement unit essay, he wrote:

Learning the dynamical system helped me really understand how and why people were losing their homes. It showed how small an income the average black/brown family was making and how since it was a small amount, how hard it was to pay off the mortgage loans. Not only was it hard but the banks were really stealing money from these people because they would end up paying a double amount of money than they took out because of this thief called Interest. The difference equation [DDS] helped in this process, too. It helped me get a broader picture of how families should pay for their homes. These were also the most important things that I learned because they helped me understand what I know now. It helped me to be able to predict whether or not a family would be able to pay off their loan with certain types of mortgages, and this is very important in being able to read the world so we can be able to share with the world.

Carlton provides evidence of his sense of being able to use, share, and discuss important mathematical ideas that he learned related to his family and community.

Every student who completed the survey (one was absent) wrote that the class met its goals. Responses ranged from Ann’s and Carlton’s elaborated ones to Vanessa’s succinct, “Of course it did. We did both in this class, learn math and read our world. I think this was clearly shown in our [end-of-year] presentations to the community.”

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Like Carlton and Vanessa, several others also mentioned discussing issues with peers or presenting to their community. In their reflections on their two community presentations (every student participated, most in both), many wrote about helping their community and working for justice in various ways. Two years after the class, I emailed Ann and asked why students titled their presentations “Our Issues, Our People: Math As Our Weapon.” She responded:

We chose to name it this way because of the way our curriculum was chosen [by students themselves]. We were able to learn and teach math that we felt actually represented not only ourselves but the people in our communities. Our way of fighting back against those who try to oppress us and against ignorance was by using the one thing we had at our disposal, math.

Despite students’ overall positive assessments, some struggled. Renee’s out-of- school life often took priority. She was focused on helping her family keep their house, which they were in danger of losing because of financial troubles. She rarely did homework, missed class occasionally, was sometimes distracted (and distracting), and occasionally resisted participating . Her first semester grade was a C, and she barely earned a B for second semester. However, she also spoke her mind in class and contributed much. In her displacement essay, she wrote:

In the displacement project, I learned a lot. At first, I had problems with the math but eventually understood it better. It’s crazy how banks give you this loan with a monthly payment that eventually people don’t really get out of debt. . . . People need to know what happens—why they get into debt, especially what the banks do is legal with our government. People such as my sister lose their homes because they don’t read the papers they are signing when they get a loan for a mortgage. People need to know the difference between the different loans that are out there. The only question that I have after this unit is can what the banks do be made illegal?

Renee discussed multiple issues, but the role of mathematics was unclear although she claimed to understand it. However, in the next paragraph of her essay, she wrote the following:

The most helpful part of this unit was the dynamical systems. As soon as I really learned how to work with the dynamical systems I came home and grabbed my credit card bill and the mortgage and plugged them in the calculator. Paying the minimum balance on my credit card wasn’t enough. I would have to pay double my minimum balance to get out of it in less time. Obviously, what my mother [a low-paid factory worker] is paying isn’t enough to finish paying the house in 30 years. The worst part about this is that what she pays isn’t 30 percent of her income, it’s more.

She used the mathematics that we studied to critically read her own reality, relate it to class, and name what she thought was wrong. Her words show her analyzing the sociopolitical context of her life through a mathematical lens. This epitomizes reading the world with mathematics.

To further consider Renee’s growth, I examined a (transcribed) classroom video of her analyzing an adjustable-rate mortgage scenario that she created for her displacement project. For 11 minutes, Renee sat in front of class using the overhead

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calculator projector as she explained. She told us that the initial mortgage was 234,000 with an annual interest rate of 5.42% and a lifetime cap26 of 4%. In her scenario, the rate jumped to 7.42% after 3 years (she showed the balance due of $223,825.67) and jumped to 9.42% 10 years later for the remaining balance. Along the way, she graphed the equations, made and corrected mistakes, forgot what she had done and had to consult her notes, was helped by classmates to fix an error, adjusted window variables on the calculator, debugged two errors in her calculations, analyzed why the graph looked skewed, and kept up a running commentary while rapidly pressing keystrokes and fluently manipulating the calculator in meaningful ways. Despite moments of uncertainty in her presentation, she demonstrated her evolving understanding of the representations, their relationship to reality, and her grasp of the issues, language, and technology. Renee was learning and using math- ematics to understand her social reality and turned in a detailed three-page, single- spaced analysis with her computations (mostly correct), reasoning, and conclusions; this included how much the family paid in interest and principal under her subprime scenario and how much more interest they paid than if they had a fixed-rate, stan- dard-term loan (over $70,000).

It took time and effort for students to develop the fluency to generate scenarios of different subprime loans and to create, graph, analyze, and understand DDSs (or apply them to their lives, as Renee did). As expected, students varied in their concep- tual understanding and other aspects of mathematical proficiency, as can be seen in the examples detailed above (e.g., their struggles to fully comprehend the meaning and representation of a straight line).

There were also issues of mathematics dispositions. Jenny entered class claiming to hate mathematics and, at year’s end, said she still hated it, for which I still feel responsible. Ellen and Miriam described their elementary school mathematics experiences as terrible. Ellen wrote “math = terror,” and after the year, emailed me to say, “I hated math before the 12th grade. I felt that when I reached my first year of high school I was not prepared at all. I was not taught many things in my grammar school.” These three students struggled to understand many mathematical ideas all year and did not grasp them to the extent that I wanted. However, they participated regularly, worked hard, and came to see themselves as mathematical learners. Ellen wrote on the year-end survey, “Some of us entered this room without having a lot of prior knowledge in mathematics. We now know how to read and write the world with mathematics!” And Miriam wrote:

I think that our math class did meet our goals and more. I’m not only talking about me but also my peers. We as a class, we learn a lot of math and hard math, that sometimes we just wanted to drop the class. We also learned how to read and write the world with math. To be honest with you, now every time I read or hear about an issue I always think, “Can we use math to solve it.” 

26 A lifetime cap on a loan means that interest rates cannot increase or decrease more than the size of the cap.

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All 21 students participated, persevered in, and passed the class (two with D’s and five with A’s), despite several saying that it was the hardest class that they took (Buenrostro, 2016). Even Jenny, who said, “I still hate math,” when Vanessa (another student) video-interviewed her at year’s end, continued with:

But I think that it [the class] has taught me to care about things and view things in a wider selection, instead of always being close-minded about things. Because before, I never really cared about the world and never gave it a second thought. But now, it makes me want to do something.

The final question that Vanessa asked was, “What do you think is the most impor- tant thing that you learned in this class so far?” Jenny responded,

Phew, that’s a hard question. I think that the most important thing that I learned in this class was that you can’t always believe what you see. You always have to question and research things for yourself to get an answer.

Thus, I argue that although her mathematics learning may have lagged compared with some others—which I ascribe in part to my own pedagogical limitations—she took much from the class and, as of summer 2016, earned her master’s degree and certification as a high school English teacher and was hired to teach English in a neighborhood Chicago public high school in her Lawndale community. Her partic- ipation was noticeably greater when we discussed social issues (e.g., HIV-AIDS), but we have video of her leading class, describing her mathematical thinking, and persevering in raising her own questions to clarify her understanding.

A central political goal of the displacement unit was for students to study and begin to understand the social and economic forces that drove displacement differ- entially and similarly across their communities while coming to realize that, despite appearances, commonalities likely outstripped differences. This was to support students in building unity in Lawndale in the face of social forces that often posi- tioned them against each other. That students moved toward understanding these commonalities was shown in several ways. For example, Renee wrote:

The unit made many relations between black and brown communities. There are so many misconceptions about black people as well as brown. . . . One in the black commu- nity might say that Mexican people don’t belong in this country because we’re illegal aliens. Some Mexican people will say that all black people have a Link card [receive government food aid] and spend all their money on clothes, etc. What people don’t understand is that we both have the same struggles. They might seem different because of the color of our skin but deep down inside our parents struggle to get by with sick- nesses, drug addictions, or unemployment. People are dealing with foreclosures and then become homeless. . . . When we did the 30 percent of the median household income for both communities we figured that we can’t afford the house that we are living in. Our family members kill themselves in factories trying to make ends meet. This unit taught us that we have the same struggle. People always ask what similarities do we all have and this unit tells us why we are the same.

Others argued similarly. Rut, a non-Black, Latina student explaining gentrifica- tion at a teachers’ conference, said, “my people, Black and Brown, are being pushed

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out.” Mónica, now a middle school mathematics teacher in Little Village, her home community, wrote on her essay:

I want the people in my community to know that we are really similar with these situa- tions. That there is more that makes us similar, less that makes us different. If we want to fight the bigger people out there, the best way is to unite. Fighting each other is not going to take us anywhere.

At the end of the year, students created an 81-slide PowerPoint presentation to share what they had learned with their communities. Erika made the last slide, which is shown in Figure 8. I do not claim that every student reached this level of political clarity. But many did, and all were part of and helped create a classroom community that normalized discussions about these issues.

With respect to the year’s trajectory, students entered the class with no knowledge of significant mathematical content areas that we covered (binomial and normal probability distributions in the elections unit as well as discrete mathematics and linear and quadratic regression in the displacement and HIV-AIDS units) and under- developed knowledge of others (proportionality as well as fundamental algebraic concepts and reasoning). They knew nothing of statistical anomalies in the 2004 U.S. presidential election or the possibility of disenfranchisement, nothing about how mortgages functioned or how governmental agriculture subsidies contributed to the displacement that Lawndale experienced, nothing about how mathematics could help understand disease transmission, and very little about racial dispropor- tionality of criminalization or HIV-AIDS prevalence. The data over the year—their year-end exam, their HIV-AIDS unit exam (7 weeks before the end of school), their self-reports of what they learned (at the end of both semesters), and their community presentations together provide evidence that all students learned about these topics, both mathematical (classical knowledge) and sociopolitical (critical knowledge) through using and learning mathematics to study their generative themes. They did so in multiple ways, with different challenges, and to varying degrees, even the students who experienced the most struggles. The self-assessments for each student mirrored these findings as well.

Figure 8. The last slide of the students’ presentation at the end of the year.

Why Should We Care [about displacement]?

• Both communities face the same problems but different situations.

• There are many lies and stereotypes about both Mexicans and African Americans.

• “Mexicans steal the jobs of U.S. citizens.”

• “African Americans are lazy.”

• Don’t let them pit us against each other!

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Discussion From a Freirean (1970/1998, 1993) theoretical perspective, students’ community

knowledge (e.g., generative themes) is the starting point for liberatory education that supports them in developing interconnected critical and classical knowledge. This article gives a concrete example of that theory in practice in an untracked, neighborhood, Chicago public high school with low-income Latin@ and African American students. It also provides evidence of students’ learning and challenges. In my effort to enact a Freirean education, I have extended the research literature in two ways: (a) the whole school year focused on CM (rather than interjected, shorter projects, units, or activities), and (b) students (not the teacher) determined the contexts to investigate. To summarize, the dialectics of reading the world and word materialized in the dance, as students learned challenging mathematics— read the word (developed classical knowledge)—through using that mathematics to more deeply understand the forces devastating their community—read the world (developed critical knowledge). That the generative themes the class studied came directly from students is what it means for the dance to begin with and build upon their community knowledge, interconnecting the 3 Cs. And students’ commitment and actions to use mathematics to spread and teach what they learned to their families, friends, neighborhood, and the larger public (community presen- tations, Huffington Post op-ed piece, and other public presentations) are part of writing the world with mathematics. Despite students’ and my stumbles and difficulties, together we actualized the precepts of RWWM—students grew in their capacity to do this work, and I believe that I also grew in my capacity to teach this way. Below, I summarize principles and lessons learned and discuss ways to address some of the difficulties in enacting RWWM.

In earlier work (Gutstein, 2006a), I discussed in detail several RWWM dilemmas, including creating space for students to develop their own voices while simultaneously maintaining my own voice and not silencing theirs (Freire, 1994), supporting students to overcome feeling powerless to change formidable obstacles (like gentrification), developing students’ sense of social agency, and connecting critical and classical knowledge (without building on students’ community knowl- edge). Here, I discuss the challenge of teaching a full-year RWWM program and using generative themes. However, I do not include developing curricula or uncov- ering themes, which are substantial issues themselves.

I also do not discuss some structural challenges that full-time teachers in public schools experience in trying to enact RWWM. I had the luxury of teaching only one section of critical mathematics and did not have 100–150 high school students. No one monitored my teaching to see if I was on page 235 on Day 47 or checked to see if my lessons “covered” various standards. And, unlike when I taught middle school, I did not have to prepare students for high-stakes exams. One could ques- tion how relevant my story is to full-time teachers. My response has consistently been that we can learn from others’ experiences even with markedly different conditions than our own, provided that we understand the others’ context, theory, and practice. Some of my own mentors in reform mathematics said this well in

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their book on Cognitively Guided Instruction (Carpenter, Fennema, Franke, Levi, & Empson, 2015):

Our goal is not to provide models of classroom instruction to serve as a template for you to apply to your own classroom. Rather, we present specific cases that embody first principles of successful CGI classrooms. Our goal is for teachers individually and in collaboration with other teachers to make sense of the principles in relation to their own classes and teaching styles. (p. xix)

Thus, I cannot answer how to do this work under other conditions. Instead, this remains a challenge for other teachers, teacher educators, and researchers working together.

Generative Themes, Full-Year RWWM, and Reform Mathematics Through reflection and studying my own practice over the years, I have come

to understand several principles about teaching from generative themes (e.g., Gutstein, 2012a). These are not empirical findings of this study but rather sugges- tions about ways that teachers can use themes and enact RWWM in classrooms.

First, themes prescribe neither curriculum nor pedagogy. Uncovering and begin- ning to understand students’ themes is an initial step in supporting them to read and write the world (Gandin, 2002; O’Cadiz et al., 1998). This would be followed by developing curriculum based on the themes, which is interwoven with peda- gogical complexities. Teachers need to flexibly modify curriculum based on their gradually increasing understanding of students’ lives and perceptions and on students’ responses to the curriculum—teachers can learn about these in dialogue with students while teaching from themes.

Second, using generative themes will not transform the schooling experiences of students from communities facing inadequate educational resources, inequi- table education, and systemic disinvestment. The sociopolitical realities outside of education, such as racism, poverty, and other marginalization, have tremendous impact on learning conditions, as is well documented. Moreover, on a personal level, students do not easily leave the harshness of life in communities like Lawndale at the classroom door. No matter the depth of their interest and commit- ment, realities outside the classroom sometimes intrude inside and interfere with students’ attention and focus. For both reasons, one should not presume that CM is a silver bullet.

Third, teaching from generative themes creates conditions for classroom democ- racy but also demands them. Teachers have to initiate this process so that students eventually cocreate a space in which RWWM thrives. It is difficult to “renegotiate coercive relations of power” (Cummins, 1989) imposed by educational systems that teach and orient adults to “manage” and “discipline” young people. Students are also taught, and learn, passivity and expectations of being spoon fed, but they also learn how to circumvent and rock the established order. When teachers enact CM based on generative themes, power relationships shift, and teachers and students need to learn—together—how to navigate change.

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Last, teaching from themes is not a motivational gimmick to get students to learn mathematics. The purpose is explicit—to prepare students to transform an unjust reality, with and without mathematics, as they see fit.

There are also two concrete ways that teachers and teacher educators can draw on reform mathematics principles in using generative themes and teaching a full- year RWWM program—mathematizing students’ social reality and building on students’ sociopolitical thinking. In my book about my middle school teaching, I argued that under specific conditions, a strong reform curriculum (such as Mathematics in Context, which I used 75–80% of the time) supported RWWM (the balance of my curriculum) by positioning students as people who arbitrate knowl- edge, invent solutions, create meaning, and make arguments. I claimed that students learned conceptual mathematics primarily through the reform curriculum, which gave me pedagogical space to use CM with less concern for RWWM being the site of rich mathematical learning (Gutstein, 2006a). However, when teaching a full-year curriculum entirely developed from generative themes, reform mathematics can offer different possibilities.

In such settings, a dilemma emerges because context, not content, generally sets direction. In reform mathematics, settings are mere backdrop; however, in CM, students learn mathematics to study reality, answer their own questions, and (prepare to) write the world. Content is essential, but CM places different demands on teachers. It requires them, in real time, to continually mathematize students’ social reality, creating and modifying curriculum. Reform mathematics, in a sense, has an analogous orientation with respect to students’ natural and physical reality but does not intentionally focus on justice and liberation, even if using social contexts. Perceiving mathematics in situations of injustice is similar, but it requires a significant change in perspective to one that is explicitly political.

Furthermore, reform mathematics attends to students’ thinking, but RWWM demands that teachers build on students’ informal mathematical knowledge and knowledge of sociopolitical reality while helping them connect and synthesize both types of knowledge. Reform mathematics can be useful through its focus on devel- oping teachers’ capacity to follow and draw from students’ ideas, but one must extend it to the sociopolitical realm, which requires a critical lens and disposition— again a shift.

Educators can use these facets of reform mathematics to support RWWM, and future research needs to explore other possibilities and interrelationships. In general, I built on my knowledge of and experience with the former to enact the latter. However, I continue to argue that these (and other) aspects of reform mathematics may be necessary for CM but are insufficient. I never assume that questioning math- ematics will necessarily lead to questioning unequal relations of power in society, sexism, racism, or other injustice. Our schools and society generally teach us not to question authority, what Macedo (1994) called literacy for stupidification. As Freire (1970/1998) wrote, “No oppressive order could permit the oppressed to begin to ques- tion: Why?” (p. 67). From a Freirean perspective, a role of education in an unjust reality is to sharply muffle people’s consciousness and obscure their critical gaze.

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This framing differentiates RWWM from reform mathematics. For example, from the latter perspective, a goal of the December 4 class was for students to learn conceptually based, flexible, and meaningful mathematics. Although CM may place deep value on mathematics as a human cultural production of all peoples (D’Ambrosio, 1985), my understanding of its essence is that “students need to be prepared through their mathematics education to investigate and critique injustice, and to challenge, in words and actions, oppressive structures and acts” (Gutstein, 2006a, p. 4). Reform mathematics can play an important role, but specific knowl- edge, experience, and orientations beyond it are needed to teach RWWM.

Critical Mathematics as Antiracist Pedagogy Building upon reform mathematics can strengthen CM, but its antiracist focus

needs sharpening as well. In our class, race and racism were central. The experien- tial knowledge and consciousness of Sojo’s Black and Brown students about race and racism—often outstripping that of their teachers—were integral parts of our classroom. I tried to build on what students brought into school to support their development, identities, and capacity to read and write the world. In each unit, students examined racism in different ways, with the orientation to understand and transform reality. In the criminalization unit, we studied racially disproportionate prison populations, private prison profits, and exploited Black and Latin@ labor. In the HIV-AIDS unit, students explored data showing both correlations and contradic- tions between race, poverty, and gender with respect to HIV incidence (Balasubramanian 2012; Gutstein, 2012a). We also frequently discussed race in the elections unit.

Within the displacement unit, students examined how predatory lenders targeted Black and Latin@ communities like Lawndale. Students learned that in Chicago, and across the urban United States, displacement mostly affects low-income and working-class neighborhoods of color. An expression encapsulating gentrification that one hears in Chicago and that resonates with many is “the land is valuable, but the people are not.” The facets of displacement that we studied—gentrification, foreclosures, and deportation—directly brought race and capitalism together through the devastation wrought on a community of color for economic gain.

Upon reflection, I did not interrelate racism and capitalism as explicitly as I would have liked or as I understand them theoretically. We studied racism and capitalism— separately. The links were there, but implicitly, and I needed to do more given my perspectives. For example, we did not examine, mathematically or politically, how displacement (like slavery) was not solely an economic or racial project but, rather, it was a clear amalgam of both. If I were to teach the class again, I would develop a curriculum for students to explicitly interrogate these interconnections.

I would also, without essentializing, find ways to integrate Mexican, African American, and African mathematics into RWWM for students to better appreciate that they have “math in their blood” (Ortiz-Franco, 2013) and that mathematics is their heritage. This would further support students’ identity construction (academic, racial, cultural, and ethnic). In retrospect, this too was a limitation of my teaching.

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I advocate that teaching mathematics for social justice is teaching it for racial justice (Gutstein, 2013a; Terry, 2009) and that RWWM should directly contest racism.27 This includes providing students with opportunities to analyze whether and how racism is implicated in social phenomena and to understand different forms of racism (individual, ideological, structural, and institutional). Teachers can consciously integrate this into curriculum and pedagogy in ways both explicit and embedded. The intent is to prepare young people to learn and use mathematics to study racism, think about the roots of social problems, understand relations of power, act out their values with confidence and courage, and use their racialized life expe- riences (no matter their “race”) to challenge racism in ways they choose—both in its independence from and interdependence with capitalism.

To clarify, I am not proposing that only low-income students of color learn CM, although many teachers and scholars of color purposefully do (nonmathematical) critical pedagogy work with these students (e.g., Camangian, 2009, 2011; Cammarota & Romero, 2014; Duncan-Andrade & Morrell, 2008; Noguera, 2007; Perry & Delpit, 1998; Stovall & Morales-Doyle, 2010; Yang, 2006). These authors’ writings and practices suggest that they likely agree with Freire’s (1970/1998) contention:

Who are better prepared than the oppressed to understand the terrible significance of an oppressive society? Who suffer the effects of oppression more than the oppressed? Who can better understand the necessity of liberation? (p. 27)

Neither Freire nor any of these authors suggests that students of privilege have no role to play. Tate (1994) addressed this: “Our highly technological society requires that all students, not just African American students [emphasis added], be prepared to use mathematics to defend their rights” (p. 483). Teaching CM does not just apply to places like Lawndale. It is implausible that more economically and racially privileged students lack their own generative themes (e.g., sexism, homophobia, environmental degradation) from which teachers can support them in learning mathematics for social—and racial—justice.

Developing the Knowledge to Teach RWWM—On Becoming Political Militants28 in Mathematics Education

We are political militants because we are teachers. Our job is not exhausted in the teaching of math, geography, syntax, history. Our job implies that we teach these subjects with sobriety and competence, but it also requires our involvement in and dedication to overcoming social injustice. (Freire, 1998, p. 58)

Teaching RWWM as antiracist mathematics is certainly a political act. Although the mainstream mathematics education community is beginning to accept that mathematics is sociocultural (Lerman, 2000) and sociopolitical (R. Gutiérrez, 2013;

27 I am in no way suggesting that CM should not also counter other forms of injustice, such as sexism.

28 “Freire uses this term [militants] to designate persons actively committed to justice and liberation—political activists” (Freire, 1978, p. 73, translator footnote).

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Valero, 2004), education sociologists (e.g., Apple, 2004; Giroux, 1983) have long held this view. As Freire (Shor & Freire, 1987) wrote, “education is politics” (p. 61). Well before Freire, others understood this. Bell hooks (1994) wrote about her expe- riences in segregated Black schools in the U.S. South:

For black folks, teaching—educating—was fundamentally political because it was rooted in antiracist struggle. Indeed, my all-black grade schools [with all-Black staff] became the location where I experienced learning as revolution. . . . Though they did not define or articulate these practices in theoretical terms, my teachers were enacting a revolutionary pedagogy of resistance that was profoundly anticolonial. . . . My teachers were on a mission. (p. 2)

The teaching that hooks described was “fundamentally political” because of the life conditions and “antiracist struggle” facing Black people in the South. For them, their “mission” was not optional. I contend that, for several reasons, we are in similar circumstances today. Briefly, I believe that our work as teacher educators and researchers is increasingly being shaped by the marketization and commodification of knowledge, higher education, and the education sector as a whole—which is estimated to be worth over $5 trillion globally (GSV Advisors, 2012). The neoliberal privatization of education affects all aspects of our work, from college class size to research priorities to curriculum (Giroux, 2014). Science, Technology, Engineering, and Mathematics (STEM) programs, and mathematics education more generally, increasingly serve U.S. economic competitiveness (Gutstein, 2009c). Many of our students (or readers of this article) are, or will be, K–12 teachers. Conservative state governments; corporatized, so-called education “reformers”; and mayor-controlled school districts all push austerity (refusing progressive funding schemes that redirect resources from the truly wealthy) and attack teachers’ unions, professionalism, standard of living, pensions, conditions of work, autonomy, and jobs themselves (e.g., massive school closings in urban centers; Lipman, 2011a, 2011b). Perhaps most profound, our students (or our students’ students) who live in Lawndale and else- where are also under attack—from police murders of Black people that are now reported almost daily in the nation’s newspapers,29 which have finally been brought to international attention by rebellions in Ferguson, Missouri, and elsewhere and by the Movement for Black Lives, to the displacement that I describe above. As we prepare teachers to teach these young people (or teach them ourselves), we have the responsibility to also be “on a mission.”

This requires knowledge beyond what one needs to teach reform mathematics: pedagogical content knowledge (Shulman, 1986); specialized knowledge for teaching mathematics (Hill, Ball, & Schilling, 2008); and curricular, pedagogical, and content knowledge. All are useful for RWWM but are not enough. Even if teachers and teacher educators mathematize natural and physical reality or social reality in nonpolitical ways or build on students’ mathematical thinking, it is insuf- ficient. RWWM demands that teachers and teacher educators cultivate critical

29 In the United States in 2012, one Black person was murdered every 28 hours by police, security guards, or vigilantes (Malcolm X Grassroots Movement, 2013).

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knowledge, sociopolitical consciousness, and dispositions. If we want students, like Sojo youth, to develop critical perspectives on reality, we have to do the same.

I wanted my students to investigate causes of displacement, its interrelationships with racism and capitalism, and how and why it affected North Lawndale and Little Village in similar and different ways. A transnational investment firm was specu- lating on a large gentrification complex almost in Sojo’s backyard, soliciting inves- tors globally.30 Comprehending how and why it was occurring and its relationship to students’ lives and neighborhoods required me to study and understand the interconnections of local and global phenomena. A student and I researched the project, and he shared his analysis in class and at the community presentations. In the HIV-AIDS unit, the data showed that in 2006, Black women had a dramatically higher rate of contracting HIV than other demographic groups, which I had not known and could not initially explain. This demanded study on my part and caused me to significantly shift how I taught the unit so that no students—Black females or not—demonized themselves or others for Black women’s incidence rate (Balasubramanian, 2012; Gutstein, 2012a).

A central problem in reinventing this work is finding the answer to this question: How does one develop such knowledge and orientations? My experience is not mainly from classrooms or libraries (though both are essential) but from political praxis and involvement in efforts to change reality along with disciplined study. This call to activism does not propose that we grab picket signs and hit the street (although when my university’s faculty unionized in 2014, we did strike for 2 days). Like real mathematics problems, social movements are endowed with multiple entry points. Our work as politicized mathematics educators and teachers can take diverse forms, from supporting our students to read and write the world, to pushing ourselves to do the same, to engaging in fights against education privatization and school closings in solidarity with affected communities, to forcefully speaking and writing as advo- cates for racial and social justice, to collaborating across disciplines in and outside of our universities and schools in conjunction with actors in and outside of education, and much more. This engagement is what Freire (1998) meant when he spoke of teachers as political militants, and this is what I propose as one way to gain the experiences and develop the knowledge and dispositions in order to address some challenges in enacting RWWM beyond the structural and practical ones. As I say above in reference to youth—but it also applies to ourselves—this is an epistemology of learning by doing in which the “doing” is transforming society toward equity and justice.

Final Thoughts This study gives an example of some possibilities and challenges of RWWM in

action in a particular context. But, this road is to be made and remade while walking, as teachers, researchers, and teacher-researchers enact and study the practice, and

30 The developers were unable to get funding, probably due to the 2008 recession.

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develop the field. My goal in writing this is that readers might critique and prob- lematize my framing, practice, and reflections to move RWWM forward.

For me, today more than ever, it is insufficient to just teach and learn mathematics in mathematics class, given where humanity finds itself. I argue that education should, above all, develop full, ethical, loving human beings who care about and act in the world for peace and justice. If that is so, then the question of how and why we use what we learn must be part of everything that we do in education. Everything.

It is ours to determine how we engage in “overcoming social injustice” (Freire, 1998, p. 58), but in the present moment, we have little choice. We live in a time of deep, sustained, global crises—sociopolitical, economic, and ecological. They have generated the stirrings of world-wide movements in response and alternative visions of social systems and ways to build them. This moment offers us opportunities to connect the dynamism outside the classroom—struggles of teacher unions and parents for just and equitable public education, and the work of social movements confronting racism, sexism, poverty, housing, health, the environment—to life inside the classroom (in this case, mathematics). If young people are to be prepared for the challenges of the future, involving them in reading and writing the world today is essential for tomorrow.

I end with the collective hope that Freire (1994) embraced, which he linked to fighting for a better world. He wrote that without hope, this struggle dissipates and withers—but without trying to change reality, hope is meaningless. Dialectically, hope and struggle need each other. This emerges in the writing of Channing in addressing my question: What is RWWM and why do we do it? When Channing, a Black student in my class, referred below to “our people,” I believe that he meant both Black and Brown people—in Lawndale and beyond. He read this aloud at an education research conference:

Reading and writing the world with mathematics is to look at important issues that we are faced with in the world and try to understand them using mathematics. It is to try and have a bigger picture of why and how things may have gone wrong or why there are injustices in certain situations. With mathematics, things can become more clear, especially when dealing with situations that are based on quantity or the amount of something or somebody. Once we understand these things, we take it out to the world. We use our knowledge to let others know what we know.

We do this because too many of the problems in this world are happening with our people. We have looked at so many injustices that are done to our people. We want our people to know what is being done to them and what they can do to avoid these things from happening. We want them to be able to collectively think of some solutions to the issues that they are faced with.

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University, Chicago, IL.

Author Eric “Rico” Gutstein, Department of Curriculum and Instruction, University of Illinois at Chicago,

1040 W. Harrison St., M/C 147, Chicago, IL 60607; [email protected]

Submitted September 12, 2014 Accepted January 30, 2015

500 Critical Mathematics in a Chicago High School

APPENDIX A Mathematics Content of Elections, Displacement, and HIV-AIDS Units

Elections Unit: Mathematically, students engaged with the following ideas, representations, and models in this unit: the meaning of nCr (n choose r); use of tree diagrams and tables to find probability of an event; equivalence of using tree diagrams and combinatorics for finding probability; independent events; multiplication of probabilities for multiple events; equivalence of finding probability of a multiple event (coin toss, picking people, etc.); using combinatorics or by multiplying individual event probabilities; developing the binomial formula when an event has only two possible outcomes (H or T, M or F, Bush or Kerry, etc.); binomial formula for equally vs. unequally likely outcomes; distinction between theoretical and experimental probability distribution by simulation on the calculator; and the ideas of sample/poll, sample variation, population, standard deviation, and normal distributions (including confidence intervals). (Balasubramanian, 2012, p. 63)

Displacement Unit: Students worked on predicting housing prices based on previous years’ data using invented strategies and linear and quadratic regression; comparing price predictions with actual prices; recognizing the limits of mathematical modeling; creating discrete dynamical systems to calculate payment on housing loans and mortgages with predatory loans; calculating the price of “affordable” housing; interpreting graphs and data related to median housing income, median housing values, corn subsidies to farmers, and population change over time; analyzing issues of disproportionality with respect to demo- graphics of subprime borrowers.

HIV-AIDS Unit: Mathematically, students worked on the following: creating a DDS with one and two variables; finding equilibrium values (algebraically and graphically); creating a DDS for disease spread (SI models); simulating disease spread using DDS on calculators; interpretation of graphs, statistics, pie charts, and other visual representation of data; proportionality and disproportionality; and prediction using linear and cubic regression. The sub-contexts used included the following: a farm where trees were being periodically cut and replaced at a certain rate; a rental car system between Chicago-Milwaukee; the human body blood-liver system; and data on HIV-AIDS infection and spread in Chicago and the United States. (Balasubramanian, 2012, p. 58)

501Eric “Rico” Gutstein

APPENDIX B HIV-AIDS Unit Assignment

This assignment required students to derive and program into their calculators the four equations below. (I did not give students the equations.) After the assignment, we analyzed why the graph was shaped as it was. These equations describe much simpler disease transmission dynamics than those of HIV-AIDS. The actual assignment starts below the equations.

u(n) = u(n−1) + .0001v(n−1)u(n−1) [u(n) means number of infected people at time n] u(1) = 1 [we start off with one infected person] v(n) = v(n−1) – .0001u(n−1)v(n−1) [v(n) means number of susceptible people at time n] v(1) = 999 [we start off with 999 susceptible people]

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Mathematics for Social Justice—Sojo, 2008–2009— Mr. Rico, Sheet A7, 3/31/09

Creating a Mathematical Model of HIV-AIDS Two weeks ago, we used cubes in a bag to simulate the transmission of a disease

that had only two populations, susceptibles and infecteds. Our graphs looked something like this:

502 Critical Mathematics in a Chicago High School

We discussed why the graph looked like this and you filled out a little chart with the number of possible “dangerous” interactions (contacts between infected and susceptible). One way a disease transmits is through contact between people. So the more possible contacts, then usually in this type of disease, the more transmis- sion occurs. But we also saw three things:

1. When there are a lot of susceptibles and a few infecteds, not many interactions occur.

2. When there are about an equal number of susceptibles and infecteds, many interactions occur.

3. When there are a lot of infecteds and a few susceptibles, not many interactions occur.

So now let’s try to build a model for this using dynamical systems.

CLASSWORK:

1. Suppose you had 1 infected and 999 susceptible people. Answer a-f below based on these numbers. This is too hard to do with cubes, so we can use our yellows [graphing calculators]. a. How could you find the number of interactions in a day using I and S for

the number of infecteds and susceptibles? (Think about what you did to answer this w/ the chart.)

b. What is the minimum number of interactions in 1 day that you could have? c. What is the maximum number of interactions in 1 day that you

could have? d. Suppose there is a probability of .01% (or .0001) that an interaction occurs

between an infected and susceptible person and the susceptible is infected. What is the maximum number of infections that can occur in any 1 day?

e. A dynamical system would tell us that “the number of infected at the start of day n is equal to the number at the start of day n–1 plus the newly infected that day.” What is that as a difference equation? And what is the starting equation for the system?

f. Create the dynamical system (the difference equation and starting equa- tion) for the number of susceptibles, also.

2. Graph both systems on your yellow. (Think about your window variables!) When does your system hit equilibrium, and what are the equilibrium values?

3. Experiment with different values for the probability, changing it slightly. Record your results for the probability value and number of days until equi- librium. What do you learn? What mathematical generalizations can you make?

503Eric “Rico” Gutstein

APPENDIX C RWWM journal assignment, December 2, 2008

JOURNAL 9

Read the following graph and then answer these questions. What story does the graph tell and what does it mean to you? What are the implications of this graph? What does it suggest to you are the things that you need to do? What are your connections? Questions? Critiques? The challenges you see in this story? Communicate and create! Use mathematics to read the world, your world!

Note. This figure above is from “Income inequality continues staggering 25-year growth trend: Surging wage growth for topmost sliver,” by L. Mishel, 2008, Economic Policy Institute. Copyright 2008 by the Economic Policy Institute. Used with permission.

504 Critical Mathematics in a Chicago High School

APPENDIX D Displacement Unit Exam Item

A problem on the mid-unit exam asked students:

We have been studying the idea of “affordable” housing in North Lawndale and Little Village. One student in our class said that his mother makes $24,500 a year.

a) Using the HUD guideline that a family should pay at most 30% of their income for housing, what is the most his mother should pay for housing a month?

b) His family wants to move and was looking at a house that cost $135,000. They have $5,000 saved for a down payment and would need a mortgage of $130,000. Currently, mortgages are available for as low as 5.5% annually for a 30-year fixed-rate mortgage. (NOTE: You will need to set your mode to 4 decimal places.)

i) Show the dynamical system (the starting equation and the differ- ence equation) that represents this situation.

ii) Can they afford this house with this mortgage? Explain why or why not.

iii) With this income, how large a mortgage, to the nearest dollar, can they afford (with the HUD’s guidelines above)?

iv) How large a monthly payment, to the nearest dollar, does this mortgage require (5.5% interest, 30-year fixed-rate mortgage of $130,000)?

v) Using your answer from iv immediately above, how much annual income would the family need for that monthly payment (using HUD guidelines)?